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constrained submodular maximization via technique nov niv moran november abstract study combinatorial optimization problems submodular objective attracted much attention recent years problems important theory practice objective functions general obtaining improvements many submodular maximization problems boils finding better algorithms optimizing relaxation known multilinear extension work present algorithm optimizing multilinear relaxation whose guarantee improves guarantee best previous algorithm given ene nguyen moreover algorithm based new technique arguably simpler natural problem hand nutshell previous algorithms problem rely symmetry properties natural absence constraint technique avoids need resort properties thus seems better fit constrained problems department statistics operations research school mathematical sciences tel aviv university israel email depart mathematics computer science open university israel email moranfe introduction study combinatorial optimization problems submodular objective attracted much attention recent years problems important theory practice objective functions functions generalize example cuts functions graphs directed graphs mutual information function matroid weighted rank functions specifically theoretical perspective many problems combinatorial optimization fact submodular maximization problems including generalized assignment facility location practical perspective submodular maximization problems found uses social networks vision machine learning many areas reader referred example comprehensive survey bach techniques used approximation algorithms submodular maximization problems usually fall one two main approaches first approach combinatorial nature mostly based local search techniques greedy rules approach used early late maximizing monotone submodular function subject matroid constraint works apply specific types matroids later works used approach handle also problems submodular objective functions different constraints yielding cases optimal algorithms however algorithms based approach tend highly tailored specific structure problem hand making extensions quite difficult second approach used approximation algorithms submodular maximization problems overcomes obstacle approach resembles common paradigm designing approximation algorithms involves two steps first step fractional solution found relaxation problem known multilinear relaxation second step fractional solution rounded obtain integral one incurring bounded loss objective approach used obtain improved approximations many problems various techniques developed rounding fractional solution techniques tend quite flexible usually extend many related problem particular contention resolution schemes framework yields rounding procedure every constraint presented intersection basic constraints knapsack constraints matroid constraints matching constraints given wealth rounding procedures obtaining improvements many important submodular maximization problems maximizing submodular function subject matroid knapsack constraint boils obtaining improved algorithms finding good fractional solution optimizing multilinear relaxation maximizing multilinear relaxation point would like present terms formally submodular function set function obeying sets submodular maximization problem problem finding set maximizing subject constraint formally let set subsets obeying constraint interested following problem max relaxation problem replaces polytope containing characteristic vectors sets addition relaxation must replace function extension function thus relaxation fractional problem following format max defining right extension function relaxation challenge unlike linear case single natural candidate objective turned useful thus used multilinear relaxation known multilinear extension first introduced value extension vector defined expected value random subset containing every element independently probability formally every first algorithm optimizing multilinear relaxation continuous greedy algorithm designed calinescu submodular function algorithm finds vector set maximizing among sets whose characteristic vectors belongs interestingly guarantee continuous greedy optimal monotone functions even simple cardinality constraint optimizing multilinear relaxation necessarily monotone proved challenging task initially several algorithms specific polytopes suggested later improved general algorithms designed work whenever solvable designing algorithms work general setting highly important many natural constraints fall framework moreover restriction algorithms polytopes unavoidable proved algorithm produce vector obeying constant solvable recently best algorithm general setting called measured continuous greedy guaranteed produce vector obeying natural feel guarantee measured continuous greedy fact improved years made people suspect optimal recently evidence conjecture given described algorithm special case cardinality constraint improved approximation guarantee even recently ene nguyen shuttered conjecture completely extending technique used showed one get approximation guarantee every solvable polytope inapproximability side oveis gharan proved algorithm achieve approximation better even matroid polytope partition matroid closing gap best algorithm inapproximability result fundamental problem remains important open problem upper set function monotone every polytope solvable one optimize linear functions polytope implies every vector bounded must belong well contribution main contribution algorithm improved guarantee maximizing multilinear relaxation theorem exists polynomial time algorithm given submodular function solvable polytope finds vector obeying arg max multilinear extension admittedly improvement guarantee obtained algorithm compared guarantee relatively small however technique underlying algorithm different arguably much cleaner technique underlying previous results improving natural guarantee moreover believe technique natural problem hand thus likely yield improvements future rest section explain intuition base belief results based observation guarantee measured continuous greedy improves algorithm manages increase coordinates solution slow rate based observation run instance measured continuous greedy discretized version force raise coordinates slowly extra restriction affect behavior algorithm significantly produces solution improved guarantee otherwise argue point extra restriction affect behavior measured continuous greedy reveals vector contains significant fraction available one use technique unconstrained submodular maximization described higher approximation guarantee extract vector large value guarantees belongs well unfortunately use unconstrained submodular maximization technique approach problematic two reasons first technique based ideas different ideas used analysis measured continuous greedy makes combination two quite involved second abstract level unconstrained submodular maximization technique based symmetry exists absence constraint since submodular whenever properties however symmetry breaks constraint introduced thus unconstrained submodular maximization technique seem good fit constrained problem algorithm replaces symmetry based unconstrained submodular maximization technique local search algorithm specifically first executes local search algorithm output local search algorithm good algorithm simply returns otherwise observe poor value output local search algorithm guarantees also far sense algorithm uses far solution guide instance measured continuous greedy help avoid bad decisions turns analysis measured continuous greedy local search algorithm use similar ideas notions thus two algorithms combine quite cleanly observed section preliminaries analysis uses another useful extension submodular functions given submodular function extension function defined extension many important applications see however paper use context following known result immediate corollary work lemma given multilinear extension extension submodular function holds every vector define additional notation use given set element denote characteristic vectors sets respectively sets respectively given two vectors denote maximum minimum multiplication respectively finally given vector element denote derivative respect point following observation gives simple formula observation holds multilinear function observation let multilinear extension submodular function every rest paper assume without loss generality every element larger given constant first assumption justified observation every element violating assumption safely removed since belong second assumption justified observation possible find set obeying constant time constant another issue needs kept mind representation submodular functions interested algorithms whose time complexity polynomial however representation submodular function might exponential size thus assume representation given part input algorithm standard way bypass difficulty assume algorithm access oracle assume standard value oracle used previous works submodular maximization oracle returns given subset value main algorithm section present algorithm used prove theorem algorithm uses two components first component close variant fractional local search algorithm suggested chekuri following properties formally every element max min lemma follows chekuri exists polynomial time algorithm returns vector high probability every vector proof let max let arbitrary constant larger lemmata chekuri imply high probability fractional local search algorithm suggest terminates polynomial time outputs vector obeying every vector moreover output vector whenever fractional local search algorithm terminates assumption every element implies submodularity every set since maximum values get also using observation plugging get exists algorithm high probability terminates operations polynomial function every vector outputs vector obeying moreover output vector belongs whenever algorithm terminates complete lemma consider procedure executes algorithm operations return output terminates within number operations algorithm fails terminate within number operations happens diminishing probability procedure simply returns always belongs since one observe procedure properties guaranteed lemma second component algorithm new auxiliary algorithm present analyze section auxiliary algorithm main technical contribution paper guarantee given following theorem theorem exists polynomial time algorithm given vector value outputs vector obeying ets main algorithm executes algorithms suggested lemma followed algorithm suggested theorem notice second algorithms two parameters addition parameter set output first algorithm parameter set constant determined later two algorithms terminate algorithm returns output first algorithm probability constant determined later remaining probability returns output second formal description algorithm given algorithm observe lemma theorem imply together algorithm polynomial time algorithm always outputs vector prove theorem remains analyze quality solution produced algorithm clearly always better return better two solution instead randomizing however require algorithm either oracle access estimate values solutions using sampling later done using standard sake simplicity chose easier analyze approach randomizing two solutions algorithm main algorithm execute algorithm suggested lemma let output execute algorithm suggested theorem let output return probability solution solution otherwise lemma parameters set algorithm produces solution whose expected value least proof let event output algorithm suggested lemma satisfies inequality since high probability event enough prove conditioned algorithm produces solution whose expected value least constant rest proof lemma devoted proving last claim throughout everything implicitly conditioned conditioning plug respectively inequality get last inequality follows noticing next let denote expected value conditioned given value inequality guarantees ets recall algorithm returns probability otherwise hence expected value output expectation optimizing constants would like derive inequalities best lower bound get end let two numbers let using inequalities notation lower bounded ets rewritten ets ets ets get lower bound need maximize coefficient keeping coefficients ignored due objective formalized following program max ets ets ets solving program get best solution approximately objective function value corresponding solution least hence managed lower bound thus also expected value output algorithm completes proof lemma aided measured continuous greedy section present algorithm used prove theorem proving theorem directly made involved fact vector might fractional instead prove following simplified version theorem integral values show simplified version implies original one theorem exists polynomial time algorithm given set value outputs vector obeying ets next promised proof theorem implies theorem proof theorem given theorem consider algorithm alg given arguments specified theorem executes algorithm guaranteed theorem value random set distributed like output alg output produced algorithm guaranteed theorem let denote output theorem guarantees every given ets complete proof take expectation two sides last inequality observe rest section give proof theorem proof explains main ideas necessary proving theorem uses simplifications allowing direct oracle access multilinear extension giving algorithm form continuous time algorithm implemented discrete computer known techniques getting rid simplifications see formal proof theorem based techniques given appendix algorithm use proof theorem given algorithm algorithm starts empty solution time grows solution time reaches final solution time way solution grows varies time time range solution grows like measured continuous greedy algorithm hand earlier time range algorithm pretends elements exist giving negative marginal profits grows solution way measured continuous greedy would grown given ground set value time algorithm switches two ways uses grow solution thus notation stands switch algorithm aided measured continuous greedy let foreach let arg let arg increase rate return first note algorithm outputs vector observation proof observe eachr time implies also since therefore convex combination vectors thus belongs following lemma lower bounds increase function lemma every proof chain rule dyu consider first case time period algorithm chooses vector maximizing since value thus plugging observation equality yields last inequality holds submodularity similarity algorithm chooses vector maximizing since get time plugging observation equality yields last inequality holds submodularity lemma every time set holds max max proof first note every time element max follows following reason since always obeys differential inequality using initial condition solution differential inequality get tighter bound note every time algorithm chooses vector maximizing linear function assigns negative weight elements since maximum must every element means whenever moreover plugging improved initial condition differential inequality yields promised tighter bound also range next let extension lemma max max max max inequality follows equality follows since inequality guarantees every thus finally inequality follows since max inequality guarantees every thus also submodularity every set plugging results lemma lower bound given lemma improvement function yields immediately useful lower bound given next corollary every ets ets using last corollary complete proof theorem proof theorem already seen output algorithm remains show ets corollary describes differential inequality given boundary condition solution differential inequality within range plugging last inequality get let right hand side last inequality next solve differential inequality given corollary range boundary condition resulting solution ets ets vets plugging value get ets ets vets ets ets ets inequality follows since submodularity note corollary follows weaker version lemma guarantees proved stronger version lemma useful formal proof theorem given appendix max references ageev sviridenko approximation algorithm uncapacitated facility location problem discrete appl july noga alon joel spencer probabilistic method wiley interscience second edition per austrin siavosh benabbas konstantinos georgiou better balance biased max bisection soda pages francis bach learning submodular functions convex optimization perspective foundations trends machine learning boykov jolly interactive graph cuts optimal boundary region segmentation objects images iccv volume pages niv buchbinder moran feldman joseph seffi naor roy schwartz tight linear time unconstrained submodular maximization focs pages niv buchbinder moran feldman joseph seffi naor roy schwartz submodular maximization cardinality constraints soda pages gruia chandra chekuri martin jan maximizing monotone submodular function subject matroid constraint siam chandra chekuri alina ene approximation algorithms submodular multiway partition focs pages chandra chekuri sanjeev khanna polynomial time approximation scheme multiple knapsack problem siam september chandra chekuri jan rico zenklusen dependent randomized rounding via exchange properties combinatorial structures focs pages chandra chekuri jan rico zenklusen submodular function maximization via multilinear relaxation contention resolution schemes stoc pages chandra chekuri jan rico zenklusen submodular function maximization via multilinear relaxation contention resolution schemes siam reuven cohen liran katzir danny raz efficient approximation generalized assignment problem information processing letters conforti submodular set functions matroids greedy algorithm tight worstcase bounds generalizations radoedmonds theorem disc appl cornuejols fisher nemhauser location bank accounts optimize float analytic study exact approximate algorithms management sciences cornuejols fisher nemhauser uncapacitated location problem annals discrete mathematics alina ene huy nguyen constrained submodular maximization beyond focs uriel feige threshold approximating set cover acm uriel feige michel goemans aproximating value two prover proof systems applications max max dicut istcs pages uriel feige vahab mirrokni jan maximizing submodular functions siam journal computing uriel feige jan approximation algorithms allocation problems improving factor focs pages moran feldman maximization problems submodular objective functions phd thesis technion israel institute technology june moran feldman joseph naor roy schwartz unified continuous greedy algorithm submodular maximization focs pages moran feldman joseph seffi naor roy schwartz justin ward improved approximations systems esa pages fisher nemhauser wolsey analysis approximations maximizing submodular set functions polyhedral combinatorics volume mathematical programming studies pages springer berlin heidelberg lisa fleischer michel goemans vahab mirrokni maxim sviridenko tight approximation algorithms maximum general assignment problems soda pages alan frieze mark jerrum improved approximation algorithms max max bisection ipco pages shayan oveis gharan jan submodular maximization simulated annealing soda pages michel goemans david williamson improved approximation algorithms maximum cut satisfiability problems using semidefinite programming journal acm eran halperin uri zwick combinatorial approximation algorithms maximum directed cut problem soda pages jason hartline vahab mirrokni mukund sundararajan optimal marketing strategies social networks www pages johan optimal inapproximability results acm july hausmann korte algorithms independence systems oper res ser hausmann korte jenkyns worst case analysis greedy type algorithms independence systems math prog study jegelka bilmes submodularity beyond submodular energies coupling edges graph cuts ieee conference computer vision pattern recognition jenkyns efficacy greedy algorithm cong richard karp reducibility among combinatorial problems miller thatcher editors complexity computer computations pages plenum press david kempe jon kleinberg tardos maximizing spread influence social network sigkdd pages subhash khot guy kindler elchanan mossel ryan donnell optimal inapproximability results csps siam april khuller moss naor budgeted maximum coverage problem information processing letters korte hausmann analysis greedy heuristic independence systems annals discrete andreas krause ajitsingh carlos guestrin sensor placements gaussian processes theory efficient algorithms empirical studies mach learn january andreas krause carlos guestrin nonmyopic value information graphical models uai page andreas krause jure leskovec carlos guestrin jeanne vanbriesen christos faloutsos efficient sensor placement optimization securing large water distribution networks journal water resources planning management november ariel kulik hadas shachnai tami tamir approximations monotone nonmonotone submodular maximization knapsack constraints math oper jon lee vahab mirrokni viswanath nagarajan maxim sviridenko maximizing nonmonotone submodular functions matroid knapsack constraints siam journal discrete mathematics jon lee maxim sviridenko jan submodular maximization multiple matroids via generalized exchange properties approx pages hui lin jeff bilmes summarization via budgeted maximization submodular functions north american chapter association computational language technology conference los angeles june hui lin jeff bilmes class submodular functions document summarization hlt pages submodular functions convexity bachem korte editors mathematical programming state art pages springer schrijver ellipsoid method consequences combinatorial optimization combinatoria nemhauser wolsey best algorithms approximating maximum submodular set function mathematics operations research nemhauser wolsey fisher analysis approximations maximizing submodular set functionsi mathematical programming maxim sviridenko note maximizing submodular set function subject knapsack constraint operations research letters luca trevisan gregory sorkin madhu sudan david williamson gadgets approximation linear programming siam april jan symmetry approximability submodular maximization problems siam formal proof theorem section give formal proof theorem proof based ideas used proof theorem section employs also additional known techniques order get rid issues make proof section algorithm use prove theorem given algorithm algorithm discrete variant algorithm reading algorithm important observe choice values guarantees variable takes one values point thus vectors well defined algorithm aided measured continuous greedy initialization let let growing foreach let estimate obtained averaging values independent samples arg let arg let let update return begin analysis algorithm showing remains within cube throughout execution algorithm without observation algorithm welldefined observation every value proof prove observation induction clearly observation holds assume observation holds time every inequality holds since induction hypothesis implies similar argument also implies using last observation possible prove following counterpart observation corollary algorithm always outputs vector proof let set values takes execution algorithm observe implies convex combination vectors vectors belong convex convex combination including must next rewrite output algorithm discussion rightmost hand side inequality vector implies since next step towards showing algorithm proves theorem analyzing approximation ratio start analysis showing high probability estimations made algorithm quite accurate let event every time lemma symmetric version theorem let mutually independent set corollary proof consider calculation given time calculation done averaging value independent samples let denote value definition obtained sample let since distributed like definition guarantees every additionally every since absolute values upper bounded last inequality follows assumption every element thus lemma observe algorithm calculates every combination element time since elements times smaller union bound implies probability least one value upper bounded completes proof corollary next step give lower bound increase function given lower bound given corollary follows next two lemmata statement proof corollary next lemma easier following definition let denote set otherwise lemma given every time proof let calculate weight according weight function first inequality follows definition last follows submodularity recall vector maximizing objective function depends objective function maximized assigns value vector similarly objective function maximized assigns value vector thus definition guarantees cases hence first inequality holds definition second equality holds since lemma rephrased version lemma consider two vectors every corollary given every time proof observe every hence lemma max max consider rightmost hand side last inequality lemma first term side bounded hand second term bounded max since definition assumption every lower bound given last corollary terms make lower bound useful need lower bound term done following two lemma corresponds lemma lemma corresponds lemma every time set holds max max proof lemma goes along lines proof corresponding lemma section except bounds coordinates used proof section replaced slightly weaker bounds given following lemma lemma every time element max proof let observe every time first objective prove induction time every time claim holds next assume claim holds let prove last inequality holds since decreasing function complete proof case choosing clearly observing every time remains prove lemma case note every time algorithm chooses vector maximizing linear function assigns negative weight elements since maximum must element means addition proving lemma time range last inequality also allows choose gives ets combining corollary lemma gives following corollary corollary given every time every time ets max proof every time corollary lemma imply together max observe inequality identical inequality promised time range corollary except extra term right hand side since upper bounded due absolute value extra term completes proof time range consider time range time range corollary lemma imply together ets max observe inequality identical inequality promised time range corollary except extra terms max right hand side corollary follows since absolute value terms upper bounded corollary bounds increase terms thus gives recursive formula used lower bound remaining task solve formula get lower bound let defined follows every time ets max next lemma shows lower bound yields lower bound lemma given every time proof let larger constant among constants hiding behind big notations corollary prove induction clearly holds since assume claim holds let prove two cases consider induction hypothesis corollary imply large enough similarly get ets max ets max ets max ets max remains find expression lower bounds thus also let defined follows ets max ets lemma every time proof proof induction assume lemma holds let prove holds also induction hypothesis first inequality holds since decreasing function increasing function range lemma every time proof proof induction lemma assume lemma holds let prove holds also avoid repeating complex expressions let denote ets max notice independent moreover using notation rewrite ets thus every ets ets definition imply immediately would like prove also ets two cases consider first ets ets max ets ets ets ets ets ets ets ets inequality uses fact hand ets ets ets ets ets using observations induction hypothesis get ets ets last two lemmata give promised lower bound used lower bound approximation ratio algorithm corollary ets proof lemma given lemma ets max ets ets ets ets ets ets ets second inequality holds since submodularity imply combining observations get given ets since always implies law total expectation ets ets ets ets second inequality holds since corollary theorem follows immediately combining corollaries

self organizing maps whose topologies learned adaptive binary search trees using conditional rotations jun astudillo john abstract numerous variants maps soms proposed literature including also possess underlying structure cases structure defined user although concepts growing som updating studied whole issue using adaptive data structure ads enhance properties underlying som unexplored earlier work impose arbitrary topology onto codebooks consequently enforced neighborhood phenomenon bubble activity boa paper consider underlying tree rendered dynamic adaptively transformed present methods som underlying binary search tree bst structure adaptively using conditional rotations conrot rotations nodes tree local done constant time performed decrease weighted path length wpl entire tree introduce pioneering concept referred neural promotion neurons gain prominence neural network significance increases aware research deals issue neural promotion advantages scheme user need aware topological peculiarities stochastic data distribution rather algorithm referred ttosom conditional rotations ttoconrot converges manner neurons ultimately placed input space represent stochastic distribution additionally neighborhood properties neurons suit best bst represents data properties confirmed experimental results variety data sets submit concepts novel pioneering sort keywords adaptive data structures binary search trees maps universidad talca merced chile castudillo author assistant professor department computer science universidad talca work partially supported fondecyt grant chile preliminary version paper presented australasian joint conference artificial intelligence melbourne australia december paper award best paper conference also grateful comments made associate editor anonymous referees input helped improving quality final version paper thank much school computer science carleton university ottawa canada oommen chancellor professor fellow ieee fellow iapr author also adjunct professor university agder grimstad norway work author partially supported nserc natural sciences engineering research council canada introduction paper pioneering attempt merge areas maps soms theory adaptive data structures adss put nutshell describe goal paper follows consider som neurons rather neurons merely possess information feature space also attempt link together means underlying data structure could list list binary search tree bst etc intention neurons governed laws som underlying observe concepts neighborhood bubble activity boa based nearness neurons feature space rather proximity underlying accepted premise intent take entire concept higher level abstraction propose modify adaptively using operations specific far know combination concepts unreported literature proceed place results right perspective probably wise see concept neighborhood defined som literature kohonen book mentions possible distinguish two basic types neighborhood functions first family involves kernel function usually gaussian nature second neighborhood set also known bubble activity boa paper focuses second type neighborhood function even though traditional som dependent neural distance estimate subset neurons incorporated boa always case included literature indeed different strategies described utilize families schemes define boa mainly identify three first type boa uses concept neural distance case traditional som best matching unit bmu identified neural distance calculated traversing underlying structure holds neurons important property neural distance two neurons proportional number connections separating examples strategies use neural distance determine boa growing cell structures gcs growing grid incremental grid growing igg growing som gsom som tssom hierarchical feature map hfm growing hierarchical som ghsom selforganizing tree algorithm sota evolving tree topology oriented som ttosom among others second subset strategies employ scheme determining boa depend connections instead strategies utilize distance feature space cases possible distinguish two types neural networks nns simplest situation occurs boa considers bmu constitutes instance hard competitive learning case tsvq tree map sotm sophisticated computationally expensive scheme involves ranking neurons per respective distances stimulus scenario boa determined selecting subset closest neurons example som variant uses ranking neural gas according authors variants included literature attempt tackle two main goals either try design flexible topology usually useful analyze large datasets reduce task required som namely search bmu input set complex nature paper focus former two mentioned goals words goal enhance capabilities original som algorithm represent underlying data distribution structure accurate manner also intend constraining neurons related based neural indices stochastic distribution also based bst relationship furthermore long term ambition also anticipate methods accelerate task locating nearest neuron phase work present details design implementation adaptive process applied bst integrated som regardless fact numerous variants som devised possess ability modifying underlying topology moreover small subset use tree underlying strategies attempt dynamically modify nodes som example adding nodes single neuron layer however hypothesis also possible attain better understanding unknown data distribution performing structural modifications tree although preserve general topology attempt modify overall configuration altering way nodes interconnected yet continue bst accomplish dynamically adapting edges connect neurons nodes within bst holds whole structure neurons explain later achieved local modifications overall structure constant number steps thus attempt use rotations neighbors feature space improve quality som motivations acquiring information set stimuli unsupervised manner usually demands deduction structure general topology employed artificial neural network ann possessing ability important impact manner absorb display properties input set consider example following user may want devise algorithm capable learning distribution one depicted figure som tries achieve defining underlying topology fit grid within overall shape shown figure duplicated however perspective topology naturally fit distribution thus one experiences deformation original lattice modeling phase opposed figure shows result applying one techniques developed namely ttosom reader observe figure tree seems far superior choice representing particular shape question operation rotation one associated bsts presently explained grid learned som tree learned ttosom figure distribution learned unsupervised learning closer inspection figure depicts complete tree fills triangle formed set stimuli seems uniformly final position nodes tree suggests underlying structure data distribution corresponds triangle additionally root tree placed roughly center mass triangle also interesting note three main branches tree cover areas directed towards vertex triangle respectively fill surrounding space around recursive manner identify behavior course triangle figure serves simple prima facie example demonstrate reader informal manner techniques try learn set stimuli indeed problems techniques employed extract properties samples one argue imposing initial topological configuration accordance founding principles unsupervised learning phenomenon supposed occur without supervision within human brain initial response argue supervision required enhance training phase information provide relates initialization phase indeed line principle little automatically learned data distribution assumptions made next step motivating research endeavor venture world neural topology structure learned training process achieved method propose paper namely ttosom conditional rotations ttoconrot essence dynamically extends properties ttosom accomplish need key concepts completely new field soms namely related adaptive data structure ads indeed demonstrated experiments results already obtained applauded research best knowledge remained unreported literature another reason interested integration deals issue devising efficient methods add neurons tree even though schemes currently proposing mentioned earlier paper reported preliminary results study best paper award international conference paper focus tree adaptation means rotations envision another type dynamism one involves expansion tree structure insertion newly created nodes considers different strategies expand trees inserting nodes single neuron essentially based quantization error measure strategies error measure based hits number times neuron selected bmu strategy chosen adapting tree namely using conditional rotations conrot already utilizes bmu counter distinct previous strategies attempt search node expanded case soms usually level leaves foresee advocate different approach ttoconrot method asymptotically positions frequently accessed nodes close root according property root node split observe follow philosophy one would search node higher measure rather conrot hopefully able migrate candidates closer root course works assumption larger number hits indicates degree granularity particular neuron justifies refinement concept using root tree growing som pioneering far know contributions paper contributions paper summarized follows present integration fields soms ads respectfully submit pioneering neurons som linked together using underlying governed laws ttosom paradigm simultaneously restructuring adaptation provided conrot definition distance neurons based tree structure feature space valid also boa rendering migrations distinct adaptive nature ttoconrot unique adaptation perceived two forms migration codebook vectors feature space consequence som update rule rearrangement neurons within tree result rotations organization paper rest paper organized follows next section surveys relevant involves field soms including instantiations respective field bsts conditional rotations section provide explanation ttoconrot philosophy primary contribution subsequent section shows capabilities approach series experiments finally section concludes paper sake space literature review considerably condensed however given survey paper area soms reported literature currently preparing paper summarizes field literature review som one important families anns used tackle clustering problems well known som typically som trained using supervised learning produce neural representation space whose dimension usually smaller training samples lie neurons attempt preserve topological properties input space som concentrates information contained set input samples belonging ddimensional space say utilizing much smaller set neurons represented vector neurons contains weight vector ird associated vectors synonymously called weights prototypes codebook vectors vector may perceived position neuron feature space training phase values weights adjusted simultaneously represent data distribution structure training step stimulus representative input sample data distribution presented network neurons compete identify winner also known best matching unit bmu identifying bmu subset neurons close considered within bubble activity boa depends parameter specified algorithm namely radius thereafter scheme performs migration codebooks within boa position closer sample examined migration factor update effected depends parameter known learning rate typically expected large initially decreases algorithm proceeds ultimately results migration algorithm describes details som philosophy algorithm parameters scheduled defining sequence corresponds tuple specifies learning rate radius fixed number training steps way parameters decay specified original algorithm alternatives parameters remain fixed decrease linearly exponentially etc algorithm som input input sample set schedule parameters method initialize weights randomly selecting elements repeat obtain sample find winner neuron one similar determine subset neurons close winner migrate closest neuron neighbors towards modify learning factor radius per schedule noticeable changes observed end algorithm although som demonstrated ability solve problems wide spectrum possesses fundamental drawbacks one drawbacks user must specify lattice priori effect must run ann number times obtain suitable configuration handicaps involve size maps lesser number neurons often represent data inaccurately approaches attempt render topology flexible represent complicated data distributions better way make process faster instance speeding task determining bmu vast number domain fields som demonstrated useful compendium articles take advantage properties som surveyed survey papers classify publications related som according year release report includes bibliography published year report includes analogous papers published additional recent references including related work year collected technical report recent literature reports host application domains including medical image processing human eye detection handwriting recognition image segmentation information retrieval object tracking etc soms although important number variants original som presented years focus attention specific family enhancements neurons using tree topology tsvq algorithm som variant whose topology defined priori static training first takes place highest levels tree tsvq incorporates concept frozen node implies node trained certain amount time becomes static algorithm allows subsequent units direct children trained strategy utilizes heuristic search algorithm rapidly identifying bmu starts root recursively traverses path towards leaves unit currently analyzed frozen algorithm identifies child closest stimulus performs recursive call algorithm terminates node currently analyzed frozen node currently trained returned bmu koikkalainen oja paper refine idea tsvq defining tssom inherits properties tsvq redefines search procedure boa case tssom som layers different dimensions arranged pyramidal shape perceived som different degrees granularity differs tsvq sense bmu found direct proximity examined check bmu hand boa differs instead considering bmu direct neighbors pyramid also considered tree algorithm sota dynamically growing som according authors take analogies growing cell structures gcs sota utilizes binary tree underlying structure similarly strategies tssom evolving tree explained considers migration neurons correspond leaf nodes within tree structure boa depends neural tree defined two cases general case occurs parent bmu root situation boa composed bmu sibling parent node otherwise boa constitutes bmu sota triggers growing mechanism utilizes determine node split two new descendants authors presented som called growing hierarchical som ghsom node corresponds independent som expansion structure dual first type adaptation conceived inserting new rows columns som grid currently trained second type implemented adding layers hierarchical structure types dynamism depend verification measures sotm som also inspired adaptive resonance theory art sotm input within threshold distance bmu latter migrated otherwise new neuron added tree thus sotm subset neurons migrated depends distance feature space neural distance som families authors proposed called evolving tree takes advantage procedure adapted one utilized tsvq identify bmu log time set neurons adds neurons dynamically incorporates concept frozen neuron explained node participate training process thus removed boa similar tsvq training phase terminates nodes become frozen topology oriented som ttosom central paper som node possess arbitrary number children furthermore assumed user ability tree whose topological configuration preserved training process ttosom uses particular boa includes nodes leaf ones within certain neural distance radius interesting property displayed strategy ability reproduce results obtained kohonen nodes som arranged linearly list case ttosom able adapt grid object way som algorithm phenomenon possessed prior hierarchical networks reported additionally original topology tree followed overall shape data distribution results reported also depicted motivational section showed also possible obtain symmetric topology codebook vectors recent work authors enhanced ttosom perform classification fashion method presented first learns data distribution unsupervised manner labeled instances become available clusters labeled using evidence according results presented number neurons required accurately predict category som possesses ability learn data distribution utilizing unidimensional topology neighbors defined along grid direction case one encounter unidimensional topology forms peano curve ttosom also possesses interesting property tree topology linear details achieved presented detail including explanation techniques fail achieve task novel data small portion cardinality input set merging ads ttosom adaptive data structures adss bsts one primary goals area ads achieve optimal arrangement elements placed nodes structure number iterations increases reorganization perceived automatic adaptive convergence tends towards optimal configuration minimum average access time cases probable element positioned root head tree rest tree recursively positioned manner solution obtain optimal bst well known access probabilities nodes known priori however research concentrates case access probabilities known priori setting one effective solution due cheetham uses concept conrot reorganizes bst asymptotically produce optimal form additionally unlike algorithms otherwise reported literature move done every data access operation performed overall weighted path length wpl resulting bst decreases bst may used store records whose keys members ordered set records stored way traversal tree yield records ascending order given set access probabilities problem constructing efficient bsts extensively studied optimal algorithm due knuth uses dynamic programming produces optimal bst using time space paper consider scenario access probability vector known priori seek scheme dynamically rearranges asymptotically generates tree minimizes access cost keys primitive tree restructuring operation used bst schemes well known operation rotation describe operation follows suppose exists node bst parent node left child right child function relates node parent exists also let relate node sibling node exists shares parent consider case left child see figure rotation performed node follows becomes right child becomes left child node nodes remain relative positions see figure case node right child treated symmetric manner operation effect raising promoting specified node tree structure preserving lexicographic order elements refer tree reorganizing use operation presented literature among simple exchange rules heuristic time record accessed rotations performed upwards direction becomes review necessary brief detailed version found tree rotation performed contents nodes data values case characters tree rotation performed node figure bst rotation performed root tree hand simple exchange rule rotates accessed element one level towards root sleator tarjan introduced technique also moves accessed record root tree using restructuring operation called splaying actually generalization rotation structure called splay tree shown amortized time complexity log complete set tree operations included insertion deletion access split join literature also records various schemes adaptively restructure tree aid additional memory locations prominent among monotonic tree mehlhorn dynamic version tree structuring method originally suggested knuth spite advantages schemes mentioned drawbacks serious others schemes one major disadvantage splaying rules always move accessed record root tree means arrangement reached single access record disarrange tree along entire access path element moved upwards root opposed schemes rule move accessed element root every time reported practice perform well weakness lies fact considers frequency counts records leads undesirable property single rotation may move subtree relatively large probability weight downwards thus increasing cost tree paper uses particular heuristic namely conditional rotations bst shown reorganize bst asymptotically arrive optimal form optimized version scheme referred algorithm requires maintenance single memory location per record keeps track number accesses subtree rooted record algorithm specifies accessed element rotated towards root tree minimize overall cost entire tree finally unlike algorithms currently literature move done every data access operation performed overall wpl resulting bst decreases essence algorithm attempts minimize wpl incorporating statistical information accesses various nodes subtrees rooted corresponding nodes basic condition rotation node wpl entire tree must decrease result single rotation achieved conditional rotation define concept conditional rotation define total number accesses subtree rooted node one biggest advantages heuristic requires maintenance processing values stored specific node direct neighbors parent children exist algorithm formally given algorithm describes process conditional rotations bst algorithm receives two parameters first corresponds pointer root tree second corresponds key searched assumed present tree node access requested algorithm seeks node root towards leaves algorithm input pointer root binary search tree search key assumed output restructured tree pointer record containing method true else end end return record else else end end end algorithm first task accomplished algorithm updating counter present node along path traversed next step consists determining whether node requested key found occurs quantities defined equations computed determine value quantity referred left child parent right descendant less zero upward rotation performed authors shown single rotation leads decrease overall wpl entire tree occur line algorithm method invoked parameter method pointer node method necessary operations required rotate node upwards means node left child parent equivalent performing right rotation parent analogously right child parent parent instead rotation takes place necessary update corresponding counters fortunately task involve updating rotated node counter parent last part algorithm namely lines deals search key case achieved recursively reader observe tasks invoked algorithm performed constant time worst case recursive call done root leaves leading running complexity height tree ttosom conditional rotations ttoconrot section concentrates details integration fields ads som particular ttosom although merging ads som relevant wide spectrum dss focus scope considering structures specifically shall concentrate integration heuristic ttosom explained preceding sections conceptually distinguish method namely topology oriented som conditional rotations ttoconrot components properties terms components detect five elements first ttoconrot set neurons like methods represents data space condensed manner secondly ttoconrot possesses connection neurons neighbor specific neuron based nearness measure third fourth components involve migration neurons similar reported families soms subset neurons closest winning neuron moved towards sample point using vector quantization rule however unlike reported families soms identity neurons moved based proximity proximity finally ttoconrot possesses mutating operations namely conditional rotations respect properties ttoconrot mention following first adaptive regard migration points secondly also adaptive regard identity neurons moved thirdly distribution neurons feature space mimics distribution sample points finally virtue conditional rotations entire tree optimized regard overall accesses unique phenomenon compared reported family soms far know mentioned introductory section general dynamic adaptation som lattices reported literature considers essentially adding cases deleting however concept modifying underlying structure shape unrecorded hypothesis advantageous means repositioning nodes consequent edges seen one performs rotations bst words place emphasis occurs result restructuring representing som case alluded earlier restructuring process done connections neurons attain asymptotically optimal configuration nodes accessed frequently tend placed close root thus obtain new species soms performing rotations conditionally locally constant number steps primary goal field ads structure elements attain optimal configuration number iterations increases particularly among adss use trees underlying topology common goal minimize overall access cost roughly means one places frequently accessed nodes close root also moves towards although adaptation made som paradigm conrot relevant tree structure thus ttosom implies specific must applied achieve integration two paradigms start defining binary search tree som bstsom special instantiation som uses bst underlying topology adaptive bstsom abstsom refinement bstsom training process employs technique automatically modifies configuration tree goal adaptation facilitate enhance search process assertion must viewed perspective som neurons represent areas higher density queried often every abstsom characterized following properties first adaptive virtue bst representation adaptation done means rotations rather merely deleting adding nodes second neural network corresponds bst goal maintains essential stochastic topological properties som neural distance case ttosom neural distance two neurons depends number unweighted connections separate tree consequently number edges shortest path connects two given nodes explicitly distance two nodes tree defined minimum number edges required one case trees fact single path connecting two nodes implies uniqueness shortest path permits efficient calculation distance node traversal algorithm note however case ttosom since tree static distances priori simplifying computational process situation changes tree dynamically modified shall explain implications tree describes som dynamic first siblings given node may change every time instant secondly parents ancestors node consideration could also change every instant importantly structure tree could change implying nodes neighbors time instant may continue neighbors next indeed extreme case node migrated become root fact parent previous time instant irrelevant next course changes entire landscape rendering resultant som unique distinct example clarify consider figure illustrates computation neural distance various scenarios first figure present scenario node accessed observe distances depicted dotted arrows adjacent numeric index specifying current distance node example prior access nodes distance node even though different levels tree reader aware nodes may also involved calculation case node figures show process node queried turn triggers rotation node upwards observe rotation requires local modifications leaving rest tree untouched sake simplicity explicitness unmodified areas tree represented dashed lines finally figure depicts configuration tree rotation performed time instant distance means increased distance unity moreover although node changed position distance remains unmodified clearly original distances necessarily preserved consequence rotation generally speaking four regions tree remain unchanged namely portion tree parent node rotated portion tree rooted right child node rotated portion tree rooted left child node rotated portion tree rooted sibling node rotated even though four regions remain unmodified neural distance regions affected rotation could lead modification distances nodes another consequence operation worth mentioning following distance two given nodes belong unmodified region tree preserved rotation performed proof assertion obvious inasmuch fact remains every path nodes unmodified remains property interesting potential accelerate computation respective neural distances figure example neural distance rotation figure nodes equidistant even though different levels tree figures show process rotating node upwards finally figure depicts state tree rotation equidistant distance increased unity hand although changed position distance remains bubble activity concept closely related neural distance one referred bubble activity boa subset nodes within distance away node currently examined nodes essence migrated toward signal presented network concept valid nns particular ttosom shall consider bubble modified context rotations concept bubble involves consideration quantity radius establishes big boa therefore direct impact number nodes considered boa formally defined node currently examined arbitrary node tree whose nodes note generalizes special case tree simple directed path clarify bubble changes context rotations first describe context tree static presented function ttosom calculate neighborhood see algorithm specifies steps involved calculation subset neurons part neighborhood bmu computation involves collection parameters including current subset neurons proximity neuron examined bmu current radius neighborhood function invoked first time set contains bmu marked current node algorithm ttosom calculate neighborhood input set nodes bubble activity identified far node bubble activity calculated iii current radius bubble activity output set nodes bubble activity method return else child child child ttosom calculate neighborhood child end end parent parent null parent parent ttosom calculate neighborhood parent end end end algorithm recursive call end storing entire set units within radius bmu tree recursively traversed direct topological neighbors current node direction direct parent children every time new neuron identified part neighborhood added recursive call made radius decremented one marking recently added neuron current node question whether neuron part current bubble depends number connections separate nodes rather distance separate networks solution space instance euclidean distance figure depicts boa differs one defined ttosom result applying rotation figure shows boa around node using configuration tree figure rotation takes place boa involves nodes nodes contained bubble subsequently considering radius equal resulting boa contains nodes finally case leads boa includes whole set nodes observe case presented figure corresponds boa around rotation upwards effected configuration tree used figure case radius unity nodes nodes within bubble different corresponding bubble rotation invoked similarly obtain set different analogous case case note coincidentally case radius equal bubbles identical rotation invoke nodes trivially boa invokes entire tree fact ensure algorithm reaches base case figure boa associated ttosom rotation invoked node explained equation describes criteria boa calculated static tree happens result conditional rotations tree dynamically adapted entire phenomenon consequently boa around particular node becomes function time reflect fact equation reformulated discrete time index algorithm obtain boa specific node setting identical algorithm except input tree dynamically changes even though formal notation includes time parameter happens practice latter needed requires history boa nodes storing history boas require maintenance primarily store changes made tree although storing history changes made tree done optimally question explicitly storing entire history boas nodes tree remains open enforcing bst property heuristic requires tree possess bst property let node bst node left subtree key key node right subtree key key satisfy bst property first see tree must general ttosom utilizes arbitrary number children per node one possibility bound value branching factor words tree trained ttosom restricted contain two children per node additionally tree must implicitly involve comparison operator two children discern branches thus perform search process comparison achieved defining unique key must maintained node tree turn allow course severe constraint forced require phenomenon achieving conditional rotations arbitrary trees unsolved research however currently undertaken lexicographical arrangement nodes leads different closely related concept concerns preservation topology som training process configuration tree change tree evolves positioning nodes accessed often closer root ordering hopefully preserved rotations particularly interesting case occurs imposed tree corresponds list neurons tree ttosom trained using tree node two children adaptive process alter original list rotations modify original configuration generating new state nodes might one two children case consequence incorporating enhancements ttosom imply results obtained significantly different shown shown optimal arrangement nodes tree obtained using probabilities accesses probabilities known priori heuristic offers solution involves decision whether perform single rotation towards root happens concept accessed node compatible corresponding bmu defined model neuron may accessed often others techniques take advantage phenomenon inclusion strategies add delete nodes implicitly stores information acquired currently accessed node incrementing counter node distant sense akin concept bmu counter adds delete nodes competitive networks training phase neuron frequent winner gains prominence sense represent points original data set phenomenon registered increasing bmu counter neuron propose training phase verify worth modifying configuration tree moving neuron one level towards root per algorithm consequently explicitly recording relevant role particular node respect nearby neurons achieves performing local movement node direct parent children aware neuron promotion neural promotion process neuron relocated privileged network respect neurons thus neurons born equal importance society neurons determined represent achieved explicit advancement rank position given premise nodes tree adapted way neurons bmus frequently tend move towards root reduction overall wpl obtained consequence promotion properties guarantee som bst tied together symbiotic manner one enhances vice versa adaptation achieved affecting configuration bst task performed every time training step som performed clearly task achieve integration far know aware research deals issue neural promotion thus believe concept pioneering bst som figure depicts main architecture used accomplish transforms structure som modifying configuration bst turn holds structure neurons figure architectural view adaptive som work constitutes first attempt constraint som using bst focus placed nodes sense unique identifiers nodes employed maintain bst structure promote nodes frequently accessed towards root currently examining ways enhance technique improve time required identify bmu well initialization initialization case ttosom accomplished two main steps involve defining initial value neuron connections among initialization codebook vectors performed manner basic ttosom neurons assume starting value arbitrarily instance placing randomly selected input samples hand major enhancement respect basic ttosom lays way neurons linked together basic definition ttosom utilizes connections remain static time beauty arrangement capable reflecting user perspective time describing topology able preserve configuration algorithm reaches convergence inclusion rotations renders dynamic required local information proposed approach codebooks som correspond nodes bst apart information regarding codebooks feature space neuron requires maintenance additional fields achieve adaptation besides node inherits properties bst node thus includes pointer left right children well make implementation easier pointer parent node also contains label able uniquely identify neuron company neurons identification index constitutes lexicographical key used sort nodes tree remains static time proceeds figure depicts fields included neuron som figure fields included som neuron neural state different states neuron may assume lifetime illustrated figure first node created assigned unique identifier rest data fields populated initial values codebook vector assumes starting value feature space pointers configured appropriately link neuron rest neurons tree bst configuration next significant portion algorithm enters main loop training effected training phase involves adjusting codebooks may also trigger optional modules affect neuron bmu identified neuron might assume restructured state means restructuring technique conrot algorithm applied alternatively neuron might ready accept queries part process mapping mode additionally option currently investigating involves case neuron longer necessary may thus eliminated main neural structure refer state deleted state depicted using dashed lines finally foresee alternative state referred frozen state neuron participate training mode although may continue part overall structure figure possible states neuron may assume training step ttoconrot training module ttoconrot responsible determining bmu performing restructuring calculating boa migrating neurons within boa basically done integrate conrot algorithm sequence steps responsible training phase ttosom algorithm describes details integration accomplished line performs first task algorithm involves determining bmu line invokes conrot procedure rationale following sequence steps parameters needed perform conditional rotation specified includes key element queried present context corresponds identity bmu stage algorithm bmu may rotated depending optimizing criterion given equations boa determined restructuring done performed lines algorithm respectively finally lines responsible neural migration oversee movement neurons within boa towards input sample algorithm train input sample signal pointer tree method ttosom find bmu ttosom calculate neighborhood radius update rule end end algorithm alternative restructuring techniques even though explained advantages conrot algorithm architecture proposing allows inclusion alternative restructuring modules conrot potential candidates used perform adaptation ones mentioned section include splay algorithms among others experimental results illustrate capabilities method experiments reported present work primarily focused lower dimensional feature spaces help reader geometrically visualizing results obtained however important remark algorithm also capable solving problems higher dimensions although graphical representation results illustrative know per results obtained ttosom capable inferring distribution structure data however present setting interested investigating effects applying neural rotation part training process render results comparable experiments section use schedule learning rate radius particular refinement parameters done specific data set additionally parameters follow rather slow decrement decay parameters allowing understand prototype vectors moved convergence takes place solving practical problems recommend refinement parameters increase speed convergence process ttoconrot structure learning capabilities shall describe performance ttoconrot data sets dimensions well experiments multidimensional domain specific advantages algorithm various scenarios also highlighted one dimensional objects since entire learning paradigm assumes data model first attempt see philosophy relevant unidimensional object curve really possesses linear topology thus prima facie case tested strength ttoconrot infer properties data sets generated linear functions plane figure shows different snapshots ttoconrot learns data generated curve random initialization used uniformly drawing points unit square observe original data points lie curve aim show algorithm could learn structure data using arbitrary initial values codebook vectors figures depict middle phase training process edges connecting neurons omitted simplicity interesting see hundred training steps original chaotic placement neurons rearranged fall within line described data points final configuration shown figure reader observe convergence achieved neurons placed almost equidistantly along curve even though codebooks sorted increasing numerical order hidden tree root denoted two concentric squares configured way nodes queried frequently tend closer root sense algorithm capturing essence topological properties data set time rearranging internal order neurons according importance terms probabilities access two dimensional data points demonstrate power including ads soms shall consider data sets studied first consider data generated distribution shown figures case initial tree topology unidirectional list although realistically quite inadvisable considering true unknown topology distribution words assume user priori information data distribution thus initialization phase tree employed tree structure respective keys assigned increasing order observe way providing minimal information algorithm root tree marked two concentric squares neuron labeled index figure also regards feature space prototype vectors initially randomly placed first iteration linear topology lost attributable randomness data points prototypes migrated iterations iterations iterations iterations figure tree list topology learns curve sake simplicity edges ommitted reallocated see figures tree modified consequence rotations transformation completely novel field soms finally figure depicts case convergence taken place tree nodes uniformly distributed entire triangular domain bst property still preserved rotations still possible training process continues experiment serves excellent example show differences current method original ttosom algorithm data set similar settings utilized case ttoconrot points effectively represent entire data set however reader must observe provide algorithm particular priori information structure data distribution learned training process shown figure thus specification initial tree topology representing perspective data space required ttosom longer mandatory alternative specification requires number nodes initial tree sufficient second experiment involves gaussian distribution gaussian ellipsoid learned using ttoconrot algorithm convergence entire training execution phase displayed figure experiment considers complete bst depth containing nodes simplicity labels nodes removed figure tree structure generated neurons suggest ellipsoidal structure data distribution experiment good example show nodes close root represent dense areas ellipsoid time node far root tree space occupy regions low density extremes ellipse ttoconrot infers structure without receiving priori information distribution structure experiment shown figures considers data generated irregular shape concave surface case experiments described earlier original tree includes neurons arranged unidirectionally list result training distribution learned iterations iterations iterations iterations figure tree list topology learns triangular distribution nodes accessed frequently moved closer root conditionally bst property also preserved figure tree learns gaussian distribution neurons accessed frequently promoted closer root tree adapted accordingly illustrated figure observe random initialization performed randomly selecting points unit square points thus necessarily fall within concave boundaries although initialization scheme responsible placing codebook vectors outside irregular shape reader observe training steps repositioned inside contour important indicate even though convergence algorithm line connecting two points passes outside overall unknown shape one must take account ttoconrot tree attempts mimic stochastic properties terms access probabilities user desires topological mimicry terms skeletal structure recommend use ttosom instead final distribution points quite amazing iterations iterations iterations iterations figure tree list topology learns different distributions concave object using ttoconrot algorithm set parameters examples three dimensional data points explain results obtained applying algorithm without conrot opt consider objects experiments utilize data generated contour unit sphere also initially involves chain neurons additionally order show power algorithm cases initialize codebooks randomly drawing points unit cube thus initially places points outside sphere figure presents case basic tto algorithm without conrot learns unit sphere without performing conditional rotations illustration presented figure show state neurons first iteration completed shown codebooks lie inside unit cube although neurons positioned outside boundary respective circumscribed sphere one want learn secondly figures depict intermediate steps learning phase algorithm processes information provided sample points neurons repositioned chain neurons constantly twisted adequately represent entire manifold finally figure illustrates case convergence reached case list neurons evenly distributed sphere preserving original properties object also presenting shape reminds viewer peano curve complimentary set experiments involved learning unit sphere tto scheme augmented conditional rotations conrot also conducted figure shows initialization codebooks starting positions neurons fall within unit cube case displayed figure figures show snapshots iterations respectively case tree configuration obtained intermediate phases differ significantly obtained corresponding configurations shown figure involved rotations case list rearranges per conrot modifying original chain structure yield iterations iter iter iter figure tree list topology learns sphere distribution algorithm utilize conditional rotation balanced tree finally results obtained convergence illustrated figure possible compare scenarios cases see tree accurately learned however first case structure nodes maintained list throughout learning phase case conrot applied configuration tree constantly revised promoting neurons queried frequently additionally experiments show dimensionality reduction property evidenced traditional som also present ttoconrot object domain successfully learned algorithm properties original manifold captured perspective tree iterations iter iter iter figure tree list topology learns sphere distribution multidimensional data points well known iris dataset chosen showing power scheme scenario dimensionality increased data set gives measurements centimeters variables sepal length sepal width petal length petal width respectively flowers species iris family species iris setosa versicolor virginica set experiments iris data set learned three different configurations using fixed schedule learning rate radius distinct tree configuration results experiments depicted figure involve complete binary tree depth respectively taking account dataset possesses high dimensionality present projection space facilitate visualization also removed labels nodes figure improve understandability using nodes using nodes using nodes figure three different experiments ttoconrot effectively captures fundamental structure iris dataset projection data shown experiment utilizes underlying tree topology complete binary tree different levels depth attempt show examples exactly parameters ttoconrot utilized learn structure data belonging also spaces executing main branches tree migrated towards center mass cloud points belonging three categories flowers respectively since ttoconrot unsupervised learning algorithm performs learning without knowing true labels samples however labels available one use evaluate quality tree sample assigned closest neuron tagging neuron class frequent table presents evaluation tree figure assigned neuron table cluster class evaluation tree figure using simple voting scheme explained possible see table instances incorrectly classified instances correctly classified additionally observe node contains instances corresponding class well known class linearly separable two classes algorithm able discover without providing labels find result quite fascinating experimental results shown table demonstrate potential capabilities ttoconrot performing clustering also suggest possibilities using pattern classification according several reasons performing pattern classification using unsupervised approach currently investigating classification strategy skeletonization general main objective skeletonization consists generating simpler representation shape object authors refer skeletonization plane process shape transformed one similar stick figure applications skeletonization diverse including fields computer vision pattern recognition explained traditional methods skeletonization assume connectivity data points case sophisticated methods required previous efforts involving som variants achieve skeletonization proposed remark ttosom one uses structure ttosom assumed shape object known priori rather learned accessing single point entire shape time instant results reported confirm actually possible focus conditional rotations affect skeletonization figure shows ttoconrot learned skeleton different objects domain cases schedule parameters used number neurons employed chosen proportionally number data points contained respective data sets important remark invoke edges minimum spanning tree skeleton observed exactly bstsom learned firstly figures illustrate shapes silhouette human rhinoceros representation head representation woman figures also show trees learned respective data sets additionally figures display data points opinion capable representing fundamental structure four objects way effectively final comment stress shapes employed experiments involve learning external structure objects case solid objects internal data points also provided ttoconrot able give approximation representation skeleton built inside solid object theoretical analysis according kiviluoto three different criteria evaluating quality map first criterion indicates continuous mapping implying input signals close input space mapped codebooks close output space well second criterion involves resolution mapping maps high resolution possess additional property input signals distant input space represented distant codebooks output space third criterion imposed accuracy mapping aimed reflect probability distribution input set exist variety measures quantifying quality topology preservation author surveys number relevant measures quality maps include quantization error topographic product topographic error trustworthiness neighborhood preservation although currently investigating quality som scheme quantified using metrics following arguments pertinent figure ttoconrot effectively captures fundamental structure four objects way figures show silhouette human rhinoceros representation head representation woman well respective trees learned figures show respective data points ordering weights respect position neurons som proved unidimensional topologies extending results higher dimensional configurations topologies leads numerous unresolved problems first question one means ordering higher dimensional spaces defined issue absorbing nature ordered state open budinich explains intuitively problems related ordering neurons higher dimensional configurations huang introduce definition ordering show even though position codebook vectors som ordered still possibility sequence stimuli cause disarrangement statistical indexes correlation measures weights distances related positions introduced regard topographic product authors shown power metric applying different artificial data sets also compared different measures quantify topology study concentrates traditional som implying topologies evaluated linear nature consequential extension means grids haykin mention topographic product may employed compare quality different maps even maps possess different dimensionality however also noted measurement possible dimensionality topological structure dimensionality feature space topologies considered study precise effort towards determining concept topology preservation dimensions greater unity specifically focused som define treelike topology measured define order topologies thus believe even tools analyze ttoconrot currently available experimental results obtained paper suggest ttoconrot able train preserve stimuli however order quantify quality topology matter defining concept ordering structure yet resolved although issue great interest rather ambitious task lies beyond scope present manuscript conclusions discussions concluding remarks paper proposed novel integration areas adaptive data structures adss maps soms particular shown som adaptively transformed employment underlying binary search tree bst structure subsequently restructured using rotations performed conditionally rotations nodes tree local done constant time performed decrease weighted path length wpl entire tree one main advantages algorithm user need priori knowledge topology input data set instead proposed method namely ttosom conditional rotations ttoconrot infers topological properties stochastic distribution time attempt build best bst represents data set incorporating data structure constraints ways achieved related approaches included premise regions accessed often promoted preferential spots tree representation yields improved stochastic representation experimental results suggest ttoconrot tree indeed able absorb stochastic properties input manifold also possible obtain tree configuration learn stochastic properties terms access probabilities time preserve topological properties terms skeletal structure discussions future work explained section work associated measuring topology preservation som including proof convergence unidimensional case performed traditional som questions unanswered topology measured defining order topologies thus believe even tools formally analyzing ttoconrot currently available experimental results obtained paper suggest ttoconrot able train neural network preserve stimuli concept ordering structures yet resolved even though principal goal obtain accurate representation stochastic distribution results also suggest special configuration tree obtained ttoconrot exploited improve time required identifying best matching unit bmu includes different strategies expand trees inserting nodes single neuron essentially based quantization error measure strategies error measure based hits number times neuron selected bmu principle type counter utilized conditional rotations conrot strategy ttoconrot asymptotically positions frequently accessed nodes close root might incorporate module taking advantage optimal tree bmu counters already present ttoconrot splits node root level thus splitting operation occur without necessity searching node largest assumption higher number hits indicates degree granularity particular neuron lacking refinement concept using root tree growing som pioneering far know design implementation details currently investigated references landis algorithm organization information sov math dokl akram khalid khan identification classification microaneurysms early detection diabetic retinopathy pattern recognition alahakoon halgamuge srinivasan dynamic maps controlled growth knowledge discovery ieee transactions neural networks allen munro binary search trees acm arsuaga uriarte topology preservation som international journal applied mathematics computer sciences astudillo self organizing maps constrained data structures phd thesis carleton university astudillo oommen using adaptive binary search trees enhance self organizing maps nicholson editors australasian joint conference artificial 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robust satisfaction temporal logic specifications via reinforcement learning oct austin derya zhaodan mac calin consider problem steering system unknown stochastic dynamics satisfy rich temporallylayered task given signal temporal logic formula represent system markov decision process states built partition statespace transition probabilities unknown present provably convergent reinforcement learning algorithms maximize probability satisfying given formula maximize average expected robustness measure strongly formula satisfied demonstrate via pair robot navigation simulation case studies reinforcement learning robustness maximization performs better probability maximization terms probability satisfaction expected robustness ntroduction consider problem controlling system unknown stochastic dynamics black box achieve complex task example controlling noisy aerial vehicle partially known dynamics visit set regions desired order avoiding hazardous areas consider tasks given temporal logic formulae extension first order boolean logic used reason state system evolves time stochastic dynamical model known exist algorithms find control policies maximizing probability achieving given specification planning stochastic abstractions however handful papers considered problem enforcing specifications system unknown dynamics passive active reinforcement learning used find policy maximizes probability satisfying given linear temporal logic formula paper contrast works reinforcement learning use propositional temporal logic use signal temporal logic stl rich predicate logic used describe tasks involving bounds physical parameters time intervals example work partially supported boston university onr grant number nsf grant numbers author mechanical engineering electrical engineering georgia institute technology atlanta usa austinjones authors mechanical engineering boston university boston usa cbelta daksaray author mechanical aerospace engineering university california davis davis usa zdkong author aeronautics astronautics stanford university stanford usa schwager author systems engineering boston university boston usa property within seconds region less reached regions larger avoided stl admits continuous measure called robustness degree quantifies strongly given sample path exhibits stl property real number rather providing yes answer measure enables use continuous optimization methods solve inference formal synthesis problems involving stl one difficulties solving problems formulae satisfaction instance specification requires visiting region region whether system steer towards region depends whether previously visited region linear temporal logic ltl formulae semantics broken translating formula deterministic rabin automaton dra model automatically takes care case stl construction difficult due timebounded semantics circumvent problem defining fragment stl progress towards satisfaction checked finite number state measurements thus define mdp called whose states correspond history system inputs finite collection control actions use reinforcement learning strategy called policy constructed taking actions observing outcomes reinforcing actions improve given reward algorithms either maximize probability satisfying given stl formula maximize expected robustness respect given stl formula procedures provably converge optimal policy case furthermore propose maximizing expected robustness typically effective maximizing probability satisfaction prove certain cases policy maximizes expected robustness also maximizes probability satisfaction however given specification satisfiable probability maximization return arbitrary policy robustness maximization return policy gets close satisfying policy possible finally demonstrate simulation case studies policy maximizes expected robustness cases gives better performance terms probability satisfaction expected robustness fewer training episodes available ignal emporal ogic stl stl defined respect continuously valued signals let denote set mappings define signal member signal denote value time sequence values moreover denote suffix time paper desired mission specification described stl fragment following syntax finite time bound stl formulae constants predicate signal function constant boolean operators negation conjunction respectively boolean operators defined usual temporal operators stand finally eventually globally always respectively note paper use version stl rather typical formulation semantics stl recursively defined iff iff iff iff iff iff plain english means within time units future true means times time units future true means exists time time units future true true stl equipped robustness degree also called degree satisfaction quantifies well given signal satisfies given formula robustness calculated recursively according quantitative semantics min max min supt min inft similar let hrz denote horizon length stl formula horizon length required number samples resolve future past requirements horizon length computed recursively hrz hrz hrz hrz hrz hrz max hrz hrz max hrz hrz max hrz hrz stl formulae example consider robot navigation problem illustrated figure specification visit regions visit regions every time mission horizon let components signal task formulated stl figure shows two trajectories system beginning initial location ending region satisfies inner specification given note barely satisfies slightly penetrates region appears satisfy strongly passes center region center region robustness degrees confirm horizon length inner specification hrz max max max iii odels einforcement earning system unknown stochastic dynamics critical problem synthesize control achieve desired behavior typical approach discretize state action spaces system use reinforcement learning strategy learning take actions trial error interactions unknown environment section present models systems amenable reinforcement learning enforce temporal logic specifications start discussion widely used ltl introducing particular model use reinforcement learning stl max use denote large positive would change large deviation order violate similarly large absolute value negative strongly violates reinforcement learning ltl one approach problem enforcing ltl satisfaction stochastic system partition statespace design control primitives nominally drive system one region another controllers stochastic dynamical model system quotient obtained partition used construct markob decision process mdp called bounded parameter mdp bmdp whose transition probabilities bmdps composed dra constructed given ltl formula form product bmdp dynamic programming applied product mdp generate policy maximizes probability satisfaction approaches problem include aggregating states given quotient mdp constructed transition probability considered constant bounded error optimal policy computed resulting mdp using approximate methods thus even stochastic dynamics system known logic encodes constraints timeabstract semantics problem constructing abstraction system amenable control policy synthesis difficult computationally intensive reinforcement learning methods enforcing ltl constraints make assumption underlying model control mdp implicitly procedures compute frequentist approximation transition probabilities asymptotically approaches true unknown value number observed sample paths increases since algorithm explicitly rely priori knowledge transition probability could applied abstraction system built propositionpreserving partition case uncertainty motion described intervals bmdp reduced via computation would instead described complete ignorance reduced via learning resulting policy would map regions statespace discrete actions optimally drive state system satisfy given ltl specification different partitions result different policies next section extend observation derive discrete model amenable reinforcement learning stl formulae reinforcement learning stl order reduce search space problem partition statespace system form quotient graph set discrete states corresponding regions statespace corresponds set edges edge two states exists neighbors share boundary partition case since stl semantics use automaton acceptance condition dra check satisfaction general whether given trajectory satisfies stl formula would determined directly using qualitative semantics stl fragment consists horizon length hrz modified either temporal operator means order update time whether given formula satisfied violated use previous state values reason choose learn policies mdp finite memory called whose states correspond sequences length regions defined partition example cont let robot evolve according dubins dynamics cos sin coordinates robot time forward speed time interval robot orientation given control primitives case given act right correspond directions grid noisy control primitive induces distribution support orientation robot facing desired cell motion primitive enacted robot rotates angle drawn distribution moves along direction time units partition statespace induced quotient shown figures respectively state quotient figure represents region partition statespace figure point lower left hand corner definition given quotient system finite set actions act decision process tuple act set finite states empty string state corresponds shorter path shorter paths length representing case system yet evolved time steps prepended times act probabilistic transition relation positive first states equal last states exists edge final state final state denote state time definition given trajectory original system define induced trace corresponds previous regions statespace state resided time time construction given quotient set actions straightforward details omitted due length constraints make following key assumptions quotient resulting defined control actions act drive system either point current region point neighboring region partition regions skipped transition relation markovian every state exists continuous set sample paths whose traces could state dynamics underlying system produces unknown distribution since robustness degree function sample paths length stl formula define distribution fig example robot navigation problem partitioned space subsection quotient example cont figure shows portion constructed figure states labeled corresponding sample paths length green blue states correspond green blue regions figure roblem ormulation paper address following two problems problem maximizing probability satisfaction let described previous section given stl formula syntax find policy act arg max problem maximizing average robustness let defined problem given stl formula syntax act find policy act arg max act furthermore probability satisfaction satisfiable arbitrary policy could solution problem policies result satisfaction probability unsatisfiable problem yields solution attempts get close possible satisfying formula optimal solution average robustness value least negative forms objective functions differ two different types formula case consider stl formula case objective function rewritten objective function rewritten max case consider stl formula objective function rewritten objective function rewritten min fig part constructed robot navigation mdp shown figure problems two alternate solutions enforce given stl specification policy found problem maximizes chance satisfied drives system policy found problem satisfy strongly possible average problems similar already considered literature however problem novel formulation provides advantages problem show achieves section special systems aximizing xpected robustness aximizing robability atisfaction demonstrate solution subsumes solution certain class systems due space limitations consider formulae type let act simplicity make following assumption assumption every state either every trajectory whose trace satisfies denoted every trajectory passes sequence regions associated satisfy denoted assumption enforced practice partitioning define set definition signed graph distance set min min length shortest path also make following two assumptions assumption signal let bounded rmin rmax assumption let two states rmin rmax define policies arg max act arg max act max proposition assumptions hold maximizes expected probability satisfacpolicy tion proof given policy associated reachability probability defined arg min let indicator function true false definition expected probability satisfaction given policy eps also expected robustness policy becomes max max max rmin max max rmin rmin rmax rmin rmin rmax rmin rmin since rmin constant maximizing equivalent max let satisfaction probability rewrite objective arg min rmax arg min arg min rmax arg min rmin thus policy increasing also leads increase since increasing equivalent increasing conclude policy maximizes robustness also achieves maximum satisfaction probability ontrol ynthesis aximize robustness policy generation since know dynamics system control priori predict given control action affect evolution system hence progress towards given specification thus use paradigm reinforcement learning learn policies solve problems reinforcement learning system takes actions records rewards associated pair rewards used update feedback policy maximizes expected gathered reward cases rewards collect related whether satisfied problem robustly problem solutions problems rely formulation let reward collected action act taken state define function act max max optimization problem cumulative objective function form optimal policy act found arg maxq applying update rule convergence batch given formula form objective maximizing expected robustness problem show applying algorithm converges optimal solution three cases discussed section proven similarly following analysis based optimal function derived max cause converges goes infinity batch reformulate problems form see section thus propose alternate formulation called batch solve problems instead updating action taken wait entire episode completed updating batch procedure summarized algorithm algorithm batch learning algorithm function batchqlearn sys probtype nep randominitialization initializepolicy nep sys updateqfunction probtype updatepolicy return algorithm function used update function used algorithm function updateqfunction probtype probtype maximumprobability qtmp max else qtmp max qnew qtmp return qnew function initialized random values computed initial values nep episodes system simulated using randomization used encourage exploration policy space observed trajectory used update function according algorithm new value function used update policy compactness algorithm written covers case case addressed similarly max max gives following convergence result proposition rule given max max converges optimal function sequence proof sketch proof proposition relies primarily proposition established rest proof varies slightly presentation note case ranges number episodes ranges time coordinate signal proposition optimal given fixed point contraction mapping max max proof contraction mapping fixed point consider max max max max define max wolog let define max max exist possibilities value trained performance trained performance trained performance trained performance robustness count count count robustness robustness thus means hence therefore contraction mapping vii ase tudy implemented learning algorithm algorithm applied two case studies adapt robot navigation model example case study solved problems compared performance resulting policies simulations implemented matlab performed ghz processor ram case study reachability first consider simple reachability problem given stl specification fig comparison policies case study histogram robustness values trained policies solution problem problem trajectory generated policies solution problem problem example trajectory example trajectory robustness example trajectory example trajectory count fig comparison policies case study subplots meaning figure generated system simulated using trained policies learning completed without randomization used learning phase note trained policies satisfied specification probability performance two algorithms similar mean robustness standard deviation probability maximization robustness maximization second row see trajectories simulated trained policies similarity solutions case study surprising state system deep within probability remain inside region next time steps satisfy higher edge region trajectories remain deeper interior region also high robustness value thus particular problem inherent coupling policies satisfy formula high probability satisfy formula robustly possible average case study repeated satisfaction stl subformula corresponding blue region plain english stated within time units reach blue region revisit blue region time results applying algorithm summarized figure used parameters nep probability iteration selecting action random constructing took algorithm took solve problem solve problem two approaches perform similarly first row show histogram robustness trials although conditions technically required prove convergence practice conditions relaxed without adverse effects learning performance second case study look problem involving repeatedly satisfying condition finitely many times specification interest plain english ensure every time units unit interval green region blue region results case study shown figure used parameters listed section except constructing took applying algorithm took problem problem first row see solution problem satisfies formula probability solution problem satisfies formula probability first seems counterintuitive proposition indicates policy maximizes probability would achieve probability satisfaction least high policy maximizes expected robustness however guaranteed infinite number learning trials performance terms robustness obviously better robustness maximization mean standard deviation probability maximization mean standard deviation second row see maximum robustness policy enforces convergence cycle two regions maximum probability policy deviates cycle discrepancy two solutions explained happens trajectories almost satisfy occur trajectory almost oscillates blue green region every four seconds encountered solving problem collects reward hand solving problem policy produces almost oscillatory trajectory reinforced much strongly resulting robustness less negative however since robustness degree gives partial credit trajectories close satisfying policy reinforcement learning algorithm performs directed search find policies satisfy formula since probability maximization gives partial credit reinforcement learning algorithm essentially performing random search encounters trajectory satisfies given formula therefore family policies satisfy formula positive probability small average take algorithm solving problem longer time converge solution enforces formula satisfaction viii onclusions uture ork paper presented new reinforcement learning paradigm enforce temporal logic specifications dynamics system priori unknown contrast existing works topic use logic signal temporal logic whose formulation directly related system statespace present novel convergent algorithm uses robustness degree continuous measure well trajectory satisfies formula enforce given specification certain cases robustness maximization subsumes established paradigm probability maximization certain cases robustness maximization performs better terms probability robustness partial training future research includes formally connecting approach abstractions linear stochastic systems eferences abate innocenzo benedetto approximate abstractions stochastic hybrid systems automatic control ieee transactions nov baier katoen principles model checking volume mit press cambridge brazdil chatterjee chmelik forejt kretinsky kwiatkowska parker ujma verification markov decision processes using learning algorithms cassez raskin editors automated technology verification analysis volume lecture notes computer science pages springer international publishing ding smith belta rus optimal control markov decision processes linear temporal logic constraints ieee transactions automatic control ding wang lahijanian paschalidis belta temporal logic motion control using methods robotics automation icra ieee international conference pages may dokhanchi hoxha fainekos monitoring temporal logic robustness runtime verification pages springer maler robust satisfaction temporal logic signals formal modeling analysis timed systems pages fainekos pappas robustness temporal logic specifications signals theoretical computer science topcu probably approximately correct mdp learning control temporal logic constraints corr jin donze deshmukh seshia mining requirements control models proceedings international conference hybrid systems computation contro pages jones kong belta anomaly detection systems formal methods approach ieee conference decision control cdc pages julius pappas approximations stochastic hybrid systems automatic control ieee transactions june kamgarpour ding summers abate lygeros tomlin discrete time stochastic hybrid dynamic games verification controller synthesis proceedings ieee conference decision control european control conference pages kong jones medina ayala aydin gol belta temporal logic inference classification prediction data proceedings international conference hybrid systems computation control pages acm lahijanian andersson belta temporal logic motion planning control probabilistic satisfaction guarantees robotics ieee transactions april lahijanian andersson belta approximate markovian abstractions linear stochastic systems proc ieee conference decision control pages maui usa lahijanian andersson belta formal verification synthesis stochastic systems ieee transactions automatic control luna lahijanian moll kavraki asymptotically optimal stochastic motion planning temporal goals workshop algorithmic foundations robotics istanbul turkey melo convergence simple proof http raman donze maasoumy murray sangiovannivincentelli seshia model predictive control signal temporal logic specifications proceedings ieee conference decision control cdc pages sadigh kim coogan sastry seshia learning based approach control synthesis markov decision processes linear temporal logic specifications corr sutton barto reinforcement learning introduction volume mit press cambridge svorenova chmelik chatterjee belta temporal logic control stochastic linear systems using abstraction refinement probabilistic games hybrid systems computation control hscc volume appear tsitsiklis asynchronous stochastic approximation qlearning machine learning

batched svd algorithms gpus applications hierarchical matrix compression jul wajih halim george hatem david abstract present high performance implementations singular value decomposition batch small matrices hosted gpu applications compression hierarchical matrices jacobi algorithm used simplicity inherent parallelism building block svd low rank blocks using randomized methods implement multiple kernels based level gpu memory hierarchy matrices reside show substantial speedups streamed cusolver svds resulting batched routine key component hierarchical matrix compression opening opportunities perform arithmetic efficiently gpus introduction singular value decomposition svd factorization general matrix form orthonormal matrix whose columns called left singular vectors diagonal matrix whose diagonal entries called singular values sorted decreasing order orthonormal matrix whose columns called right singular vectors compute reduced form matrix diagonal matrix one easily obtain full form reduced one extending orthogonal vectors zero block row without loss generality focus reduced svd real matrices discussions svd matrix crucial component many applications signal processing statistics well matrix compression truncating singular values smaller threshold gives approximation matrix matrix unique minimizer function context hierarchical matrix operations effective compression relies ability perform computation large batches independent svds small matrices low numerical rank randomized methods well suited computing truncated svd types matrices built three computational kernels factorization multiplications svds smaller matrices motivated task discuss implementation high performance batched svd kernels gpu focusing challenging svd tasks remainder paper organized follows section presents different algorithms used compute factorization svd well considerations optimizing extreme computing research center ecrc king abdullah university science technology kaust thuwal saudi arabia department computer science american university beirut aub beirut lebanon addresses batched svd algorithms algorithm householder procedure house gpus section discusses batched factorization compares performance existing libraries sections discuss various implementations svd based level memory hierarchy matrices reside specifically section describes implementation small matrix sizes fit registers section describes implementation matrices reside shared memory section describes block jacobi implementation larger matrix sizes must reside global memory section details implementation batched randomized svd routine discuss details application hierarchical matrix compression section conclude discuss future work section background section give review common algorithms used compute factorization svd matrix well discuss considerations optimizing gpu factorization factorization decomposes matrix product orthogonal matrix upper triangular matrix also compute reduced form decomposition matrix upper triangular common algorithm based transforming upper triangular matrix using series orthogonal transformations generated using householder reflectors algorithms modified produce factorization orthogonalizing column previous columns however methods less stable householder orthogonalization orthogonality resulting factor suffers condition number matrix another method based givens rotations entries subdiagonal part matrix zeroed form triangular factor rotations accumulated form orthogonal factor method stable parallelism householder method however expensive work challenging extract parallelism efficiently gpu implementation rely householder method due numerical stability simplicity method described algorithm svd algorithms implementations svd based approach popularized trefethen matrix first undergoes bidiagonalization form bqtv orthonormal matrices bidiagonal matrix matrix diagonalized using variant algorithm divide conquer method combination produce decomposition complete svd determined batched svd algorithms backward transformation methods require significant algorithmic programming effort become robust efficient still suffering loss relative accuracy alternative jacobi method pairs columns repeatedly orthogonalized sweeps using plane rotations columns mutually orthogonal process converges columns mutually orthogonal machine precision left singular vectors normalized columns modified matrix singular values norms columns right singular vectors computed either accumulating rotations solving system equations application need right vectors omit details computing algorithm describes jacobi method since pair columns orthogonalized independently method also easily parallelized simplicity inherent parallelism method make attractive first choice implementation gpu gpu optimization considerations gpu kernels launched specifying grid configuration lets organize threads blocks blocks grid launching gpu kernel causes short stall much microseconds kernel prepared execution kernel launch overhead prevents kernels complete work faster overhead executing parallel essentially serializing overcome limitation processing small workloads work batched single kernel call possible operations executed parallel without incurring kernel launch overhead grid configuration used determine thread work assignment warp group threads threads current generation gpus nvidia within block executes single instruction lockstep without requiring explicit synchronization occupancy kernel tells ratio active warps maximum number warps multiprocessor host metric dependent amount resources kernel uses register shared memory usage kernel launch configuration well compute capability card details requirement good performance generally good idea aim high occupancy memory gpu organized hierarchy memory spaces shown figure bottom global memory accessible threads plentiful slowest memory next space interest shared memory accessible threads within block configurable cache per thread block current generation gpus shared memory fast acts programmer controllable cache finally registers local threads registers fastest memory total number registers usable thread without performance implications limited kernel needs registers limit registers spilled local memory slow cached global memory making good use faster memories avoiding excessive algorithm jacobi svd converged pair columns aij atij aij rot aij aij batched svd algorithms registers shared memory cache cache cache global memory figure memory hierarchy modern gpu accesses slower ones key good performance gpu common use blocking techniques many algorithms block data brought global memory processed one faster memories related work batched gpu routines cholesky factorizations developed using block recursive approach increases data reuse leads good performance relatively large matrix sizes gpu routines optimized computing decomposition tall skinny matrices presented develop efficient transpose computation employed minor changes work hybrid algorithms batched svd using jacobi bidiagonalization methods introduced pair generation jacobi method solver phase bidiagonalization handled cpu work employs power method construct rank approximation filters convolutional neural networks routines handle svd many matrices gpus presented thread within warp computes svd single matrix batched decomposition section discuss implementation details batched kernel compare implementations magma cublas libraries implementation one benefit householder algorithm application reflectors trailing matrix line algorithm blocked together expressed multiplication level blas instead multiple multiplications level blas increased arithmetic intensity typically allows performance improve trailing matrix large however small matrix blocks overhead generating blocked reflectors vector form well lower performance multiplication small matrices hinder performance obtain better performance applying multiple reflectors vector form performing transpose multiplication efficiently within thread block first perform regular factorization column block called panel entire panel stored registers thread storing one row panel transpose product computed using series reductions using shared memory warp shuffles batched svd algorithms registers shu exor lane lane lane lane lane lane lane lane lane lane lane lane lane lane lane lane warp figure left matrix rows allocated thread registers warp right parallel warp reduction using shuffles within registers allow threads within warp read registers figure shows data layout theoretical warp size columns registers warp reduction using shuffles factor panel apply reflectors trailing separate kernel optimized performing core product update second kernel load factored panel panel trailing registers apply reflectors one time updating trailing panel registers let take example trailing panel reflector compute product mit flattening product reduction vector shared memory padded avoid bank conflicts reduction serialized reaches size partial reduction vector size take place steps final vector product mit quickly applied registers storing process repeated trailing panel within kernel maximize use reflectors stored registers figure shows one step panel factorization application reflectors trailing submatrix since threads limited per block current architectures use approach developed factorize larger matrices first factorize panels thread block limit single kernel call panels first factorized first loading triangular factor shared memory proceeding panel factorization taking triangular portion consideration computing reflectors updates keep occupancy small matrices devices resident block limit could reached thread limit assign multiple operations single thread block batch matrices dimensions kernels launched using thread blocks size thread block handles operations performance figures show performance batched square rectangular matrices panel width tuned gpu compare vendor implementation cublas well high performance library magma see proposed version performs well rectangular matrices column size starts losing ground magma larger square matrix sizes blocked algorithm starts batched svd algorithms figure one step factorization panel factored produce triangular factor reflectors used update trailing submatrix magma cublas magma cublas magma cublas magma cublas matrix size batched kernel performance square matrices matrix rows batched kernel performance rectangular matrices fixed column size figure comparing batched kernels matrices varying size gpu single double precision show performance benefits nested implementation kernel used factor relatively large panels blocked algorithm likely show additional performance improvements large square matrices leave future work register memory jacobi section discuss first batched svd kernel matrix data hosted registers analyze performance resulting kernel implementation implementation avoid repeated global memory accesses attempt fit matrix register memory using layout panel factorization one row per batched svd algorithms performance performance occupancy occupancy occupancy matrix size kernel performance achieved occupancy matrix size effect increasing matrix size occupancy register kernel figure performance batched register memory svd gpu matrices varying size single double precision arithmetics thread however number registers thread uses impact occupancy potentially lead lower performance addition register count exceeds limit set gpu compute capability registers spill local memory resides cached slow global memory since store entire matrix row registers one thread use serial jacobi algorithm compute svd column pairs processed threads one time bulk work lies computation gram matrix atij aij line algorithm update columns line since gram matrix symmetric boils three dot products executed parallel reductions within warp using warp shuffles computation rotation matrix well convergence test performed redundantly thread finally column update done parallel thread register data kernel keep occupancy smaller matrix sizes assigning multiple svd operations single block threads operation assigned warp avoid unnecessary synchronizations performance generate batches test matrices varying condition numbers using latms lapack routine calculate performance based total number rotations needed convergence figures show performance gpu batched svd kernel effect increased register usage occupancy profiling kernel see gram matrix computation takes cycles column rotations take cycles redundantly computed convergence test rotation matrices dominate cycles fact redundant portion computation dominates means preferable assign threads possible processing column pairs due low occupancy larger matrix sizes register spills local memory matrices larger obvious register approach suffice larger matrix sizes leads next implementation based slower shared memory warp warp step warp batched svd algorithms warp step step step step step step figure distribution column pairs warps step sweep shared memory jacobi register based svd performs well small matrix sizes need kernel handle larger sizes maintain reasonably high occupancy leads building kernel based shared memory next level gpu memory hierarchy section discusses implementation details kernel analyze performance compared register kernel implementation version matrix stored entirely shared memory limited per thread block current generation gpus using thread assignment register based kernel would lead poor occupancy due high shared memory consumption potentially warps active multiprocessor instead exploit inherent parallelism jacobi assign warp pair columns warps processing matrix stored shared memory total pairs columns must generate pairings steps step processing pairs parallel many ways generating pairs including round robin ring ordering implement round robin ordering using shared memory keep track column indexes pairs first warp block responsible updating index list step figure shows ordering matrix columns number matrix rows exceeds size warp assignment longer allows use fast warp reductions would force use even resources reductions would done shared memory instead assign multiple rows thread serializing portion reduction rows warp reductions used follows observation section assign threads possible process column pairs frees valuable resources increases overall performance reduction row padding used keep rows multiples warp size column padding used keep number columns even kernels launched using threads process matrix figures show examples thread allocation reductions matrix using theoretical warp size batched svd algorithms shared memory serial reduction shufflexor lane lane lane lane lane lane lane lane lane lane lane lane lane lane lane warp warp warp warp matrix columns assigned pairs multiple warps stored shared memory lane parallel reduction column data shared memory using register shuffles initial serial reduction step figure shared memory kernel implementation details performance figures show performance parallel shared svd kernel compared serial register svd kernel gpu see improved growth performance shared memory kernel due greater occupancy well absence local memory transactions looking double precision occupancy notice two dips occupancy matrix sizes number resident blocks become limited limits device dropping resident blocks performance increases steadily increase number threads assigned operation reach matrix size reach block limit threads handle larger sizes must use blocked version algorithm randomized svd see sections respectively global memory block jacobi longer store entire matrix shared memory operate matrix slower global memory instead repeatedly reading updating columns one time block algorithms facilitate cache reuse developed main benefit block jacobi algorithm high degree parallelism however since implement batched routine independent operations use serial block jacobi algorithm individual matrices rely parallelism batch processing parallel version multiple blocks processed simultaneously still used batch size small focus serial version section discuss implementation details two global memory block jacobi algorithms differ way block columns orthogonalized compare performance parallel streamed calls cusolver library routines gram matrix block jacobi svd block jacobi algorithm similar vector algorithm orthogonalizing pairs blocks columns instead vectors first method orthogonalizing pairs block columns based svd gram matrix sweep pair block columns batched svd algorithms reg occupancy reg occupancy register kernel smem kernel register kernel smem kernel occupancy smem occupancy smem occupancy matrix size shared memory kernel performance compared register kernel matrix size comparison occupancy achieved register shared memory kernels figure performance batched shared memory svd gpu matrices varying size single double precision arithmetics aij singular vectors gij updating apij uij orthogonalized forming gram matrix gij aij generating block rotation matrix uij computed left equivalently eigenvectors since symmetric positive definite orthogonalizes block columns since uij apij apij uij uij gij uij diagonal matrix singular values gij orthogonalizing pairs block columns entire matrix orthogonal give left singular vectors normalized columns singular values corresponding column norms right singular vectors needed accumulate action block rotation matrices identity matrix batched implementation use highly optimized batched syrk gemm routines magma compute apply block rotations svd computed shared memory batched kernel since different matrices converge different numbers sweeps keep track convergence operation computing norm entries scaled diagonal entries term inexact approximation terms full matrix sweep still good indication convergence cost extra cheap sweep since final sweep actually perform rotations within svd entire batched operation converge max convergence tolerance gives gram matrix path batched block jacobi algorithm compute svd batch matrices global memory worth noting computation gram matrix optimized taking advantage special structure since bulk computation svd result significant performance gains direct block jacobi svd gram matrix method indirect way orthogonalizing block columns may fail converge matrix matrices handled directly batched svd algorithms algorithm batched block jacobi svd pair block columns aij method gram batchsyrk aij else aij batchqr aij max scaledoffdiag batchsvd aij batchgemm aij max orthogonalizing columns using svd since block columns rectangular first compute decomposition followed svd triangular factor overwriting block column apij orthogonal factor multiplying left singular vectors scaled singular values give new block column apij qpij rij qpij uij vijp vij right singular vectors needed accumulate action vijp identity matrix batched implementation use batch routine developed section gemm routines magma multiply orthogonal factor left singular vectors svd computed shared memory batched kernel convergence test used gram matrix method used triangular factor since triangular factor close diagonal matrix pair block columns orthogonal gives direct path batched block jacobi algorithm compute svd batch matrices global memory performance figures show profiling different computational kernels involved batched block algorithms block width specifically percentages total execution time determining convergence memory operations matrix multiplications decompositions svd gram matrix gram matrix approach svd costly phase even larger operations svd decompositions take almost time larger matrices direct approach figure shows performance batched block jacobi svd matrices using methods figure compares performance batched svd routine batched routine uses cusolver svd routine using concurrent streams gpu increasing number streams cusolver showed little performance benefits highlighting performance limitations routines bound kernel launch overhead matrices generated randomly using latms lapack routine condition number gram matrix approach fails converge single precision types matrices whereas direct approach always converges however gram matrix approach performs better applicable larger matrices due strong performance multiplcations performance block algorithm improved preprocessing matrix using decompositions decrease number sweeps required convergence well adaptively selecting pairs block columns based batched svd algorithms misc gemm svd misc gemm svd total time total time computed offdiagonal norms gram matrices changes beyond scope paper focus future work matrix size matrix size gram matrix batched block jacobi svd profile direct batched block jacobi svd profile figure profile different phases block jacobi svd matrices varying size gpu double precision single precision exhibits similar behavior randomized svd mentioned section often interested approximation matrix compute approximation first determining singular value decomposition full matrix truncating smallest singular values corresponding singular vectors however matrix low numerical rank obtain approximation using fast randomization methods section discuss details gram direct direct time streamed cusolver streamed cusolver batched direct batched direct batched gram matrix size batched block jacobi svd performance matrix size comparison streamed cusolver batched block jacobi figure batched block jacobi performance matrices varying size gpu single double precision arithmetics batched svd algorithms algorithm batched randomized svd procedure rsvd size rand batchgemm batchqr batchgemm batchqr batchsvd batchgemm batchgemm algorithm compare performance full svd using block jacobi kernel implementation singular values matrix decay rapidly compute approximate svd using simple two phase randomization method first phase determines approximate orthogonal basis columns ensuring qqt numerical rank low sure small number columns well see drawing sample vectors random input vectors obtain reliable approximate basis orthogonalized boils computing matrix random gaussian sampling matrix computing decomposition qry desired approximate orthogonal basis second phase uses fact qqt compute matrix forming svd finalize approximation qub wide matrix first compute decomposition transpose followed svd upper triangular factor algorithm shows core computations randomized method multiplications decompositions singular value decompositions small matrices using batched routines previous sections straightforward form required randomized batched svd robust randomized svd algorithms would employ randomized subspace iteration methods obtain better basis columns rely core kernels discussed performance figure shows profiling different kernels used randomized batched routine determining top singular values vectors randomly generated low rank matrices using latms lapack routine miscellaneous portion includes random number generation using curand library default random number generator gaussian distribution batched transpose operations memory operations see performance kernels play almost equally important roles performance randomized routine matrix size grows keeping computed rank figure shows performance batched batched svd algorithms randomized svd operations figure compares runtimes direct block onesided jacobi routine randomized svd gpu set matrices showing significant time savings achieved even relatively small blocks total time misc gemm svd matrix size figure profile different phases batched randomized svd matrices varying size gpu double precision single precision exhibits similar behavior application hierarchical matrix compression application batched kernels presented consider problem hierarchical matrices problem significant importance building hierarchical matrix algorithms fact primary motivation development batched kernels hierarchical matrices received substantial attention recent years ability store perform algebraic operations near linear complexity rather regular dense matrices require effectiveness hierarchical matrices comes randomized svd randomized svd time randomized svd direct block svd randomized svd direct block svd matrix size batched randomized svd performance matrix size comparison batched block jacobi batched randomized svd figure batched randomized svd performance matrices varying size gpu single double precision first singular values vectors batched svd algorithms basis tree leaf nodes stored explicitly whereas inner nodes represented implicitly using transfer matrices leaves matrix tree simple hierarchical matrix red blocks represent dense leaves green blocks low rank leaves figure basis tree matrix tree leaves simple fact approximate matrix quad blocks many blocks regions rapidly decaying spectrum therefore numerically low rank representations low rank representations different levels hierarchical tree reduce memory footprint operations complexity associated matrix algorithms hackbush shows many large dense matrices appear scientific computing discretization integral operators schur complements discretized pde operators covariance matrices well approximated hierarchical representations reviewing analyzing hierarchical matrix algorithms beyond scope paper focus narrow task compressing hierarchical matrices compression task may viewed generalization compression low rank approximation large dense matrices case hierarchical matrices large dense matrices one way perform compression generate single exact approximate svd truncate spectrum desired tolerance produce truncated compressed representation hierarchical matrices equivalent operations involve batched svds small blocks one batched kernel call per level tree hierarchical representation size batch every call number nodes corresponding level tree compression algorithms controllable accuracy important practically often case hierarchical matrices generated analytical methods compressed significant loss accuracy even importantly performing matrix operations additiona multiplication apparent ranks blocks often grow recompressed regularly operations prevent superlinear growth memory requirements representation application use memory efficient variant hierarchical matrices exhibit linear complexity time space many core operations format hierarchical matrix actually represented three trees batched svd algorithms row column basis column trees organize row column indices matrix hierarchically node represents set basis vectors row column spaces blocks nodes leaves tree store vectors explicitly inner nodes store transfer matrices allow implicitly represent basis vectors terms children basis tree relationship nodes called nested basis example binary row basis tree transfer matrices explicitly compute basis vectors node children level figure shows example binary basis tree matrix tree hierarchical blocking formed dual traversal nodes two basis trees leaf determined block either small enough stored dense matrix low rank approximation block meets specified accuracy tolerance latter case node stored coupling matrix level tree rank level block ats matrix index set node row basis tree index set node column basis approximated ats sts vst figure shows leaves matrix quadtree simple hierarchical matrix case symmetric matrices trees identical numerical results symmetric covariance matrix compression compression symmetric represented two trees transfer transfer matrices matrices involves generating new optimal basis tree truncation phase new expresses contents matrix blocks new basis projection phase present version truncation algorithm generates memory efficient basis representation matrix given basis sophisticated algebraic compression algorithms involve use truncation phase order generate efficient basis subject future work truncation phase computes svd nodes basis tree level level explicit nodes level processed parallel produce new basis representation basis vectors leaves compute svd leaf nodes parallel batched kernels truncate singular vectors whose singular values lower relative compression threshold truncating node relative threshold using svd give approximation leaf new leaf nodes leaf level compute projection matrices tree node tid sweeping tree process inner nodes preserving nested basis property using relationship node children level forming matrices using batched multiplication compute svd qsw using batched svd kernel truncate leaves form batched svd algorithms truncated matrices sei block rows new transfer matrices level compressed nested basis projection matrices level key computations involved truncation phase consist one batched svd involving leaves tree followed sequence batched svds one per level tree involving transfer matrices data lower levels projection phase consists transforming coupling matrices matrix tree using generated projection matrices truncation phase coupling matrix sts compute new coupling matrix sets sts tst using batched multiplications phase operation consumes much less time truncation phase gpus substantial efficiencies executing regular arithmetically intensive operations results illustration effectiveness algebraic compression procedure generate covariance matrices various sizes spatial gaussian process observation points placed random perturbation regular discretization unit square isotropic exponential kernel correlation length hierarchical representations formally dense covariance matrices formed analytically first clustering points using mean split giving hierarchical index sets basis tree basis vectors transfer nodes generated using chebyshev interpolation matrix tree constructed using dual traversal basis tree coupling matrices generated evaluating kernel interpolation points approximation error constructed matrix controlled varying number interpolation points varying leaf admissibility condition dual tree traversal approximation error used following tests used maintain accuracy relative truncation error compressed matrices figure shows memory consumption compression hierachical covariance matrices leaf size initial rank corresponding chebyshev grid dense part remains untouched low rank part representation sees substantial decrease memory consumption compression minimal loss accuracy figure shows expected asymptotic linear growth time compression algorithm shows effect using randomized svd samples instead full svd computed shared memory kernel figure shows another example admissibility condition weakened generate coarser matrix tree increased rank corresponding chebyshev grid randomized svd samples also reduces compression time compared full svd using direct block jacobi kernels conclusions future work paper described implementation efficient batched kernels decomposition randomized singular value decomposition low rank matrices hosted gpu batched kernel provides significant performance improvements small matrices existing state art libraries batched svd routines first kind gpu performance exceeding batch matrices size batched svd algorithms dense portion original low rank compressed low rank dense portion original low rank compressed low rank full svd full svd compression time memory consumption randomized svd randomized svd problem size memory savings problem size compression time using randomized svd samples full svd using shared memory kernel figure compression results sample covariance matrices generated spatial statistics gpu single double precision using relative frobenius norm threshold initial rank full svd full svd randomized svd randomized svd compression time problem size figure compression time coarser matrix tree initial rank comparing randomized svd samples full svd precision illustrated power kernels problem involving algebraic compression hierarchical matrices stored entirely gpu memory demonstrated compression algorithm yielding significant memory savings practical problems future plan investigate alternatives jacobi algorithm svd small blocks randomized algorithm improve performance blocked algorithms using preconditioning adaptive block column pair selection also plan develop suite hierarchical matrix operations suited execution modern gpu manycore architectures batched svd algorithms acknowledgments thank nvidia corporation providing access gpu used work references halko martinsson tropp finding structure randomness probabilistic algorithms constructing approximate matrix decompositions siam review vol golub van loan matrix computations johns hopkins university press trefethen bau numerical linear algebra society industrial applied mathematics demmel veselic jacobi method accurate siam journal matrix analysis applications vol haidar dong tomov luszczek dongarra framework batched factorization algorithms applied block householder isc ser lecture notes computer science kunkel ludwig vol springer haidar dong luszczek tomov dongarra optimization performance energy batched matrix computations gpus proceedings workshop general purpose processing using gpus ser new york usa acm wilt cuda handbook comprehensive guide gpu programming pearson education volkov better performance lower occupancy proceedings gpu technology conference gtc vol charara keyes ltaief batched triangular dense linear algebra kernels small matrix sizes gpus submitted acm transactions mathematical software online available http anderson ballard demmel keutzer decomposition gpus parallel distributed processing symposium ipdps ieee international may kotas barhen singular value decomposition utilizing parallel algorithms graphical processors oceans kona sept kang lee improving performance convolutional neural networks separable filters gpu berlin heidelberg springer berlin heidelberg badolato paula farias many svds gpu image mosaic assemble international symposium computer architecture high performance computing workshop oct tomov nath ltaief dongarra dense linear algebra solvers 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analytical simplified models dynamic analysis short skew bridges moving loads feb nguyena goicoleaa group computational mechanic school civil engineering upm spain abstract skew bridges common highways railway lines non perpendicular crossings encountered structural effect skewness additional torsion bridge deck may considerable effect making analysis design complex paper analytical model following beam theory firstly derived order evaluate dynamic response skew bridges moving loads following simplified model also considered includes vertical beam bending natural frequencies eigenmodes orthogonality relationships determined boundary conditions dynamic response determined time domain using exact integration models validated numerical examples comparing results obtained models parametric study performed simplified model order identify parameters significantly influence vertical dynamic response skew bridge traffic loads results show grade skewness important influence vertical displacement hardly vertical acceleration bridge torsional stiffness really effect vertical displacement skew angle large span length reduces skewness effect dynamic behavior skew bridge keywords skew bridge bridge modelling modal analysis moving load corresponding author email addresses khanh nguyen goicolea preprint submitted engineering structures february introduction skew bridges common highways railway lines non perpendicular crossings encountered structural effect skewness additional torsion bridge deck may considerable effect making analysis design complex large research effort using analytical numerical well experimental approaches made last decades order better understand behavior type bridge static dynamic loadings special attention given researches related highway skew bridge subjected earthquake loadings fact first work subject reported ghobarah tso solution based beam model capable capturing flexural torsional modes proposed study dynamic response skewed highway bridges intermediate supports maragakis jennings obtained earthquake response skew bridge modelling bridge deck rigid body using finite element models socalled stick model firstly introduced wakefield stick model consists beam element representing bridge deck rigid flexible beam elements array translational rotational springs substructure bridge type model successfully used later works despite simplicity stick model provide reasonably good approximations preliminary assessment sophisticated models using shell beam elements also proposed study subject regarding behavior skew bridges traffic loads work subject performed models using combination shell beam elements assisted experimental testing models give good approximation require end user effort introduce information modelling structure element types sizes dimension material properties connection types etc therefore use limited determined case studies challenged parametric study monte carlo simulations large number case studies possible alternative develop analytical solution able capture behavior skew bridge give sufficient accuracy advantage analytical solution data input much simpler general information structure mass span length flexural torsional stiffness therefore use easy end user course able parametric study context main objective work derive analytical solution based beam theory skew bridge moving loads simplified model proposed order assimilate effect skewness support vertical vibration bridge exact integration time domain used solve differential equations models validated numerical examples comparing results obtained models parametric study performed simplified model order identify parameters significantly influence vertical dynamic response skew bridge traffic loads formulation problem skew bridge shown fig considered study work line abutment support forms orthogonal line centreline angle defined angle skewness length bridge taken length bridge idealized using following assumptions bridge deck modelled beam supported ends linear elastic behavior bridge deck stiff horizontal plane flexural deflection direction neglected bending stiffness torsional stiffness mass per unit length constant length warping distortion effects torsion bridge deck small enough neglected longitudinal axis deck width abutment bridge figure skew bridge plane view bridge model sketch assumptions bending bridge plane twisting axis principal types deformation bridge deck governing equations motion transverse torsional vibration transverse torsional loads radius gyration transverse deflection torsional rotation bridge deck transverse torsional loads applied bridge distance time respectively external damping mechanism introduced familiar term assumed proportional mass natural frequencies mode shapes using modal superposition technique solution free vibrations bridge deck decoupled infinite set modal generalized coordinates mode shapes nth flexural torsional mode shape generalized flexural torsional coordinates nth mode shape assumed governing equations free vibrations rewritten mode vibration solutions equations found many textbooks dynamic expressed following form sin cos sinh cosh sin cos six constants determined boundary conditions boundary conditions problem shown fig bridge ends abutments therefore support lines vertical displacement rotation axis bending moment axis using change coordinates shown fig following relationships obtained figure coordinate systems cos sin sin cos sin cos hence boundary conditions problem written sin sin cos sin cos sin cos cos six conditions homogeneous system equations obtained vector six constants determined matrix expressed sin cos sinh cosh cos sin cosh sinh sin cos sinh cosh sin cos cos sin cos sin eigenvalues calculated solving det noted determinant matrix expressed function unique variable extraction eigenvalues performed using symbolic matical program maple matlab fact study symbolic calculation implemented matlab used extract values desired modes used dynamic calculation eigenvector corresponding nht mode obtained applying singular value decomposition matrix orthogonality relationship order apply modal superposition technique solving forced vibration problems skew bridges necessary determine orthogonality relationship mode shapes basis equations equations reformulated multiplying sides arbitrary mode respectively integrating respect length one obtains cos cos cos sin means using integration parts side equations twice applying boundary conditions derived problem gives tan tan interchanging indices equation subtracting original form gives following relations tan tan next subtracting equation equation gives rise due fact condition established fulfilled corresponds orthogonality relationship skew bridge vibration induced moving load convoy moving loads natural frequencies associated mode shapes found orthogonality relationship modes known possible apply modal superposition technique obtaining response skew bridge due moving load vertical load twisting moment apply bridge deck determined cot magnitude moving load dirac delta function load eccentricity respect mass centre bridge deck section first part right side due skewness bridge second part due load eccentricity using modal superposition technique applying orthogonality relationship differential equations generalized coordinates uncoupled cot cot order solve differential equations several techniques applied work solution obtained using integration method based interpolation excitation advantage gives exact solution highly efficient numerical procedure solution time determined awi cqi velocity given cot cot coefficients depend structure parameters time step detail formulations found appendix moving load convoy moving loads figure moving loads case bridge forced convoy moving loads shown fig uncoupled differential equations generalized coordinates mode vibration given cot number moving loads distance first load load magnitude load solution obtained similar way case moving load attention needs paid determination modal loads right side loads enter bridge leave bridge modal loads associated loads zero simplified model part work simplified model developed order assimilate effect skewness support vertical vibration skew bridges well known skewness supports causes torsional moment bridge even vertical centric loads torsional moments turn certain influence bending moment particular negative bending moment introduced supports shown fig making purpose vertical flexure skew beam behaves like beam words beam rotational support stiffness shown fig noted negative bending moments supports change load position bridge therefore stiffness rotational support also changed different different supports order simplify calculation stiffness rotational support considered supports assumption stiffness rotational support determined additions previously adopted assumptions following additional assumptions used simplified model vertical vibration taken account model load eccentricity considered bridge deck modelled beam theory figure diagram bending moment skew bridge static load simplified model adopted skew bridge natural frequencies mode shapes governing equation free vibration simplified model similar solution equation given determination frequencies correspondent mode shapes solving homogeneous system equations vector containing four mode shape coefficients characteristic matrix determined applying boundary conditions simplified model proposed study boundary conditions vertical displacement supports equilibrium moments supports therefore characteristic matrix obtained cos cos sin sinh cosh sin cos cosh sinh sinh cosh procedure obtain eigenvalues eigenvector similar previously described section orthogonality relationship similar analysis section equation rewritten using boundary conditions simplified model interchanging indices subtracting resulting equation original form gives orthogonality relationship mode shapes simplified model moving load convoy moving loads dynamic response bridge moving loads obtained using way described analytical model section difference torsional response eliminated calculation numerical validations two numerical examples used order validate proposed models results obtained proposed models compared obtained finite element simulations example model developed program feap built beam element stick model moving load convoy moving loads applied nodal forces along centreline axis using amplitude functions dynamic responses models obtained solving time domain using modal superposition technique time step examples first five modes vibration considered calculation constant damping ratio assumed considered modes attention paid select total number modes vibration considered models since first five modes vibration obtained model always corresponding first five modes obtained analytical simplified models figure cross sections example example example skew slab bridge moving load skew slab bridge considered example skew angle bridge bridge cross section bridge shown fig following geometric mechanical characteristics used calculation elastic modulus poisson coefficient properties cross section damping ratio bridge subjected action moving load constant speed frequencies first five modes considered calculation extracted listed table models noted good agreement natural frequency analytical simplified models fact maximum difference frequency models exceed similar agreement also observed dynamic responses terms vertical displacement acceleration three models shown fig result remarked proposed simplified model enable simulate vertical dynamic response skew bridge table frequencies first five modes vibration different models modes anal model simpl model model description mode model mode model mode model mode model mode model example skew bridge convoy moving loads example attempts simulate dynamic response railway bridge hslm train desired application proposed displacement anal model simpl model model times acceleration anal model simpl model model times figure dynamic responses moving load displacement acceleration analytical simplified methods presented paper studied bridge typical bridge designed cross section shown fig skew angle considered bridge geometric mechanical properties bridge cross section used calculation elastic modulus poisson coefficient damping ratio train consists intermediate coaches power coach end coach either sides train total train axles load dynamic analysis carried different train speeds ranging increment vertical displacement acceleration obtained compared models envelope maximum vertical displacement acceleration also depicted models order validate proposed analytical simplified model presented paper table gives natural frequencies first five modes vibration considered calculation known bridge train velocities resonance estimated using following formula fundamental frequency regular distance load axles train according first three resonance peaks occur train velocities almost dynamic response train speed shown fig observed train speed near second critical speed responses amplified axle passing bridge envelope curves maximum vertical displacement acceleration shown fig noted fig considered range train velocities two peaks response displacement acceleration occur speeds closed predicted critical trains therefore remarked estimation train velocities resonance proposed still valid skew bridge furthermore figs concluded results obtained using analytical simplified model agree well ones obtained using model noted time consumed calculation using analytical simplified model approximately times faster ones using model cpu time required completing analysis using analytical model time model standard equipped intel xeon processor ghz ram table frequencies first five modes vibration different models modes anal model simpl model model description mode model mode model mode model mode model mode model parametric study part paper three parametric studies performed using simplified model order identify parameters influence significantly vertical dynamic response skew bridge moving loads study value studied parameter changed dynamic responses train corresponding value parameter obtained depicted function studied parameter basic properties skew bridge example adopted section effect skew angle figure shows maximum dynamic responses vary skew angle bridge forced train observed fig skewness important influence maximum vertical displacement bridge general displacement decreases skew angle increases sharp change slope observed skew angle value skew angle displacement decreases quickly furthermore changing train velocity resonance also observed skewness changed fact train velocity anal model simpl model model displacement times anal model simpl model model acceleration times figure dynamic responses train velocity displacement acceleration resonance increases skewness increases regarding maximum acceleration skew angle pronounced influence acceleration hardly increases skew angle grows effect torsional flexural stiffness ratio study torsional stiffness changed respect flexural stiffness ratio varies range figure shows variation maximum dynamic responses function torsional flexural stiffness ratio observed maximum vertical displacement increases slightly displacement anal model simpl model model velocity acceleration anal model simpl model model velocity figure envelope maximum response train displacement acceleration ratio increases maximum acceleration barely changed noted skew angle used study constant skew angle range skewness small influence dynamic response bridge mentioned preceding section shown fig result torsional stiffness pronounced influence vertical deflection small skew angles larger skew angle example torsional stiffness noticeable effect maximum vertical displacement shown fig maximum acceleration almost completely unaffected torsional stiffness acceleration displacement ocit skew angle ocit skew angle figure effect skewness dynamic responses displacement acceleration skew angles selected see fig displacement acceleration ocit ocit figure effect torsional flexural stiffness ratio dynamic responses skew angle displacement acceleration effect span length part paper influence span length dynamic response skew bridge carried span length changed increment order obtain consistent comparison results obtained parametric study cross section bridge redesigned span length using design criteria ratio depth cross section span length acceleration displacement ocit ocit figure effect torsional flexural stiffness ratio maximum dynamic responses skew angle displacement acceleration constant ratio usually applied railway bridge design depth cross section changed bridge length dimensions cross section considered unmodified basic properties cross section needed parametric study listed table table principal properties bridge parametric study first natural frequency corresponding span length obtained depicted fig different skew angles varying variation magnitude first natural frequency skew angle span length also obtained shown fig observed variation frequency span length generated skewness effect variation greater span length shorter decreases almost linearly span length therefore remarked span length decreases skewness effect bridge term natural frequency variation first natural frequency span length span length figure influence span length natural frequency skew bridge first natural frequency variation frequency well known dynamic response bridge traffic loads depends properties vehicle traveling bridge proper characteristics bridge parametric study traffic loads unmodified characteristics bridge changed span length therefore comparison dynamic responses term displacement acceleration determined train velocity consistent consistent comparison peak corresponding second train velocity resonance span length compared particular dynamic amplification factor daf vertical displacement maximum vertical acceleration used compare depicted fig observed daf decreases span length increases reduction variation magnitude daf displacement different skew angles span length increases however reduction variation magnitude maximum acceleration observed different skew angles remarked span length reduces skewness effect dynamic response bridge term vertical acceleration acceleration daf span length span length figure maximum dynamic responses skew bridge peak corresponding second velocity resonance different skew angles dynamic amplification factor displacement acceleration conclusions paper analytical model determining dynamic response skew bridge moving loads presented simplified model also proposed modal superposition technique used models decompose differential equation motions natural frequencies mode shapes orthogonality relationship determined boundary conditions modal equations solved exact integration therefore models highly accurate robust computationally efficient proposed models validated results obtained models using modal superposition method furthermore results obtained paper following conclusions made estimation train velocities resonance proposed still valid skew bridge grade skewness bridge plays important role dynamic behavior bridge term vertical displacement maximum vertical displacement decreases skew angle increases vibration bridge term vertical acceleration hardly affected skewness critical skew angle effect skewness noticeable cross section used parametric study critical skew angle torsional stiffness really important influence vibration bridge term vertical displacement skew angle larger critical skew angle vertical acceleration unaffected torsional stiffness span length reduces skewness effect dynamic behavior skew bridge term natural frequency acceleration appendix parameters exact integration sin cos sin cos sin sin cos sin sin cos sin cos sin cos acknowledgement authors grateful support mineco spanish government project edinpf ref support provided technical university madrid spain references kollbrunner basler torsion strucutres engineering approach berlin manterola bridges design calculation construction spanish colegio ingenieros caminos canales puertos madrid spain ghobarah tso seismic analysis skewed highway bridges intermediate supports earthquake engineering structural dynamics maragakis jennings analytical models rigid body motions earthquake engineering structural dynamics january wakefield nazmy billington analysis seismic failure skew bridge journal structural engineering meng lui seismic analysis assessment skew highway bridge engineering structures meng lui liu dynamic response skew highway bridges journal earthquake engineering nielson desroches analytical seismic fragility curves typical bridges central southeastern united states earthquake spectra pekcan seismic response skewed bridges earthquake engineering engineering vibration kaviani zareian taciroglu seismic behavior reinforced concrete bridges abutments engineering structures yang werner desroches seismic fragility analysis skewed bridges central southeastern united states engineering structures meng lui refined stick model dynamic analysis skew highway bridges journal bridge engineering nouri ahmadi influence skew angle continuous composite girder bridge journal bridge engineering deng phares greimann shryack hoffman behavior curved skewed bridges integral abutments journal constructional steel research mallick raychowdhury seismic analysis highway skew bridges nonlinear interaction transportation geotechnics bishara liu skew composite bridges journal structural engineering helba kennedy skew composite bridges ultimate load canadian journal civil engineering khaloo mirzabozorg load distribution factors simply supported skew bridges journal bridge engineering menassa mabsout tarhini frederick influence skew angle reinforced concrete slab bridges journal bridge engineering ashebo chan evaluation dynamic loads skew box girder continuous bridge part field test modal analysis engineering structures sheng scanlon linzell skewed concrete box girder bridge static dynamic testing analysis engineering structures chopra dynamics structures theory applications earthquake engineering edition prentice hall taylor element analysis program url http cen actions structures part traffic loads bridges rue stassart brussels

efficient pac learning crowd apr pranjal avrim nika yishay abstract recent years crowdsourcing become method choice gathering labeled training data learning algorithms standard approaches crowdsourcing view process acquiring labeled data separately process learning classifier gathered data give rise computational statistical challenges example cases known computationally efficient learning algorithms robust high level noise exists crowdsourced data efforts eliminate noise voting often require large number queries per example paper show interleaving process labeling learning attain computational efficiency much less overhead labeling cost particular consider realizable setting exists true target function consider pool labelers noticeable fraction labelers perfect rest behave arbitrarily show efficiently learned traditional realizable pac model learned computationally efficient manner querying crowd despite high amounts noise responses moreover show done labeler labels constant number examples number labels requested per example average constant perfect labelers exist related task find set labelers good perfect show identify good labelers least majority labelers good introduction last decade research machine learning seen tremendous growth partly due ease collect annotate massive amounts data across various domains rate data annotation made possible due crowdsourcing tools amazon mechanical turktm facilitate individuals participation labeling task context classification crowdsourced model uses large pool workers gather labels given training data set used purpose learning good classifier learning environments involve crowd give rise multitude design choices appear traditional learning environments include goal learning crowd differs goal annotating data crowd challenges high amount noise typically found curated data sets wais kittur ipeirotis pose learning algorithms learning labeling processes interplay many labels willing take per example much load labeler handle rutgers university carnegie mellon university avrim supported part nsf grants work done part author visiting simons institute theory computing carnegie mellon university nhaghtal supported part nsf grants microsoft research fellowship work done part author visiting simons institute theory computing blavatnik school computer science university mansour work done author microsoft research herzliya supported part grant science foundation isf grant united binational science foundation bsf israeli centers research excellence program center recent years many exciting works addressing various theoretical aspects questions slivkins vaughan reducing noise crowdsourced data dekel shamir task assignment badanidiyuru online offline settings karger role incentives paper focus one aspect namely efficiently learn generalize crowd minimal cost standard approach view process acquiring labeled data crowdsourcing process learning classifier isolation words typical learning process involves collecting data labeled many labelers via crowdsourcing platform followed running passive learning algorithm extract good hypothesis labeled data result approaches crowdsourcing focus getting high quality labels per example much task pipeline naive techniques taking majority votes obtain almost perfect labels cost per labeled example scales data size namely log queries per label training data size desired failure probability undesirable many scenarios data size large furthermore small fraction labelers crowd perfect approaches inevitably fail alternative feed noisy labeled data existing passive learning algorithms however currently lack computationally efficient pac learning algorithms provably robust high amounts noise exists crowdsourced data hence separating learning process data annotation process results high labeling costs suboptimal learning algorithms light initiate study designing efficient pac learning algorithms crowdsourced setting learning acquiring labels done tandem consider natural model crowdsourcing ask fundamental question whether efficient learning little overhead labeling cost possible scenario focus classical pac setting valiant exists true target classifier goal learn finite training set generated underlying distribution assume one access large pool labelers provide noisy labels training set seek algorithms run polynomial time produce hypothesis small error especially interested settings computationally efficient algorithms learning consistency model realizable pac setting additionally also want algorithms make label queries possible ideally requesting total number labels within constant factor amount labeled data needed realizable pac setting call overhead cost per labeled example furthermore realistic scenario labeler provide labels constant number examples hence ask many queries single labeler call number queries asked particular labeler load labeler perhaps surprisingly show noticeable fraction labelers pool perfect objectives achieved simultaneously efficiently pac learned realizable pac model efficiently pac learned noisy crowdsourcing model constant cost per labeled example words ratio number label requests noisy crowdsourcing model number labeled examples needed traditional pac model perfect labeler constant increase size data set additionally labeler asked label constant number examples load per labeler results also answer open question dekel shamir regarding possibility efficient noise robust pac learning performing labeling learning simultaneously perfect labelers exist related task find set labelers good perfect show identify set good labelers least majority labelers good overview results study various versions model described basic setting assume large percentage say labelers perfect always label according target function remaining labelers could behave arbitrarily make assumptions since perfect labelers strong majority straightforward approach label example majority vote randomly chosen labelers produce correct label every instance high probability however approach leads query bound log per labeled example size training set acceptable probability failure words cost per labeled example log scales size data set another easy approach pick labelers random ask label examples cost per labeled example constant approach infeasible crowdsourcing environment since requires single constant number labelers label entire data set yet another approach label example majority vote log labelers labeled sample set created way error still unsuitable used pac learning algorithms robust even small amounts noise noise heterogeneous computational challenges still persist nevertheless introduce algorithm performs efficient learning cost per labeled example load per labeler theorem informal let hypothesis class pac learned polynomial time error probability using samples learned polynomial time using samples crowdsourced setting cost per labeled example provided fraction labelers perfect furthermore every labeler asked label example notice theorem immediately implies example queried times average opposed data size dependent log cost incurred naive majority vote style procedures next extend result setting fraction perfect labelers significant might less say show efficiently pac learned using queries provided access expert correctly label constant number examples call queries made expert golden queries fraction perfect labelers close say show one golden query enough learn generally fraction perfect labelers show golden queries sufficient learn classifier efficiently describe results terms particularly interested regimes theorem informal let hypothesis class pac learned polynomial time error probability using samples learned polynomial time using samples crowdsourced setting cost per labeled example provided fraction labelers perfect constant furthermore every labeler asked label examples algorithm uses golden queries two theorems highlight importance incorporating structure crowd algorithm design oblivious labelers result noise models notoriously hard instance one assume example labeled single random labeler drawn crowd one would recover malicious misclassification noise rivest sloan getting computationally efficient learning algorithms even simple hypothesis classes long standing open problem space results highlight incorporating structure crowd one efficiently learn hypothesis class small overhead finally study scenario none labelers perfect assume majority labelers good provide labels according functions target function scenario generating hypothesis low error hard agnostic nonetheless show one detect good labelers using expected log queries per labeler target number labelers desired pool theorem informal assume target set labelers partitioned two sets good bad furthermore assume least good labelers always provide labels according happen instance labelers label according single function functions target function set bad labelers always provide labels according functions least away target polynomial time algorithm identifies probability least good labelers none bad labelers using expected log queries per labeler related work crowdsourcing received significant attention machine learning community mentioned introduction crowdsourcing platforms require one address several questions present traditional modes learning work dekel shamir shows use crowdsourcing reduce noise training set feeding learning algorithm results answer open question work showing performing data labeling learning tandem lead significant benefits large body work crowdsourcing focused problem task assignment workers arrive online fashion requester choose assign specific tasks specific workers additionally workers might different abilities might charge differently task goal requester point view finish multiple tasks within given budget maintaining certain minimum quality also significant work dynamic procurement focus assigning prices given tasks provide incentive crowd perform many possible within given budget badanidiyuru singla krause unlike setting goal works obtain generalization guarantee learn function rather complete many tasks possible within budget work karger also studies problem task assignment offline online settings offline setting authors provide algorithm based belief propagation infers correct answers task pooling together answers worker show approach performs better simply taking majority votes unlike setting goal get approximately correct set answers given data set generalize answers furthermore model assumes labeler makes error random independently certain probability hand make assumptions nature bad labelers another related model recent work steinhardt authors look problem extracting top rated items group labelers among constant fraction consistent true ratings items authors use ideas matrix completion design algorithm recover top rated items fraction noise provided every labeler rates items one access ratings trusted expert model incomparable since goal recover top rated items learn hypothesis generalizes test set results also shed insights notorious problem pac learning noise despite decades research pac learning noise tolerant polynomial time learning algorithms remain elusive substantial work pac learning realistic noise models massart noise tsybakov noise models bousquet however computationally efficient algorithms models known restricted cases awasthi contrast show using structure crowd one indeed design polynomial time pac learning algorithms even noise type mentioned generally interactive models learning studied machine learning community cohn dasgupta balcan koltchinskii hanneke zhang chaudhuri yan describe works appendix model notations let instance space set possible labels hypothesis function maps instance classification consider realizable setting distribution true target function hypothesis class formally consider distribution unknown hypothesis errd denote marginal error hypothesis respect distribution defined errd order achieve goal learning well respect distribution consider access large pool labelers label according formally labeler defined corresponding classification function say perfect errd consider distribution uniform labelers let errd fraction perfect labelers allow algorithm query labelers instances drawn goal design learning algorithms efficiently learn low error classifier maintaining small overhead number labels compare computational statistical aspects algorithms pac counterparts realizable setting traditional pac setting realizable distribution denotes number samples needed learning total number labeled samples drawn realizable distribution needed output classifier errd probability know theory anthony bartlett hypothesis class additional furthermore assume efficient algorithms assumptions realizable setting exist consider oracle set labeled instances returns function consistent labels one function exists outputs none otherwise given algorithm noisy setting define average cost per labeled example algorithm denoted ratio number label queries made algorithm number labeled examples needed traditional realizable pac model load algorithm denoted maximum number label queries answered individual labeler words maximum number labels queried one labeler infinitely large support number labelers fixed section define load simply number queries answered single labeler moreover allow algorithm directly query target hypothesis instances drawn call golden queries denote total number given set labelers instance define majl label assigned majority labelers moreover denote fraction labelers agree label majl given set classifiers denote maj classifier returns prediction majh given distribution labelers set labeled examples denote distribution conditioned labelers agree labeled samples consider small typically size note draw labeler first drawing labeler according querying labeled instances therefore infinitely large support load algorithm maximum size ever conditioned concepts total number queries load may seen analogous work depth parallel algorithms work total number operations performed algorithm depth maximum number operations one processor perform system infinitely many processors baseline algorithm improvement section briefly describe simple algorithm approach use improve consider simple baseline algorithm case baseline draw sample size label majl set randomly drawn labelers return classifier baseline algorithm queries enough labelers sample probability labels correct learns classifier using labeled set clear performance baseline far desirable first approach takes log labels requires samples leading average cost per labeled example increases size sample set moreover perfect labelers form small majority labelers number labels needed correctly label instance increases drastically perhaps even troubling perfect labelers minority may mislabeled may return classifier large error classifier work improve baseline aspects section improve log average cost per labeled example interleaving two processes responsible learning classifier querying labels particular baseline first finds high quality labels labels correct high probability learns classifier consistent labeled samples however interleaving process learning acquiring high quality labels make processes efficient high level given classifier larger desirable error one may able find regions performs particularly poorly classifications provided may differ correct label instances turn focusing effort getting high quality labels regions output correctly labeled sample set using less label queries overall additional correctly labeled instances regions performs poorly help improve error rate return section introduce algorithm draws ideas boosting probabilistic filtering approach develop work facilitate interactions learning querying section remove dependence label complexity using golden queries high level instances small majority labelers agree difficult label using queries asked labelers instances great test cases help identify large fraction imperfect labelers first ask golden query one instance get correct label consider labelers got label correctly words first test labelers one tests questions pass tests ask real label queries remainder algorithm never consider interleaving algorithm section improve average cost per labeled example baseline algorithm interleaving process learning acquiring high quality labels algorithm facilitates interactions learning process querying process using ideas classical pac learning adaptive techniques develop work ease presentation first consider case say introduce algorithm techniques work regime section show algorithm modified work value convenience assume analysis distribution discrete space fact without loss generality since using uniform convergence one instead work uniform distribution unlabeled sample multiset size drawn provide overview techniques ideas used algorithm boosting general boosting algorithms schapire freund freund schapire provide mechanism producing classifier error using learning algorithms capable producing classifiers considerably larger error rates typically error small particular early work schapire space shows one combine classifiers error get classifier error theorem schapire distribution consider three classifiers classifier errd classifier distributions denote distribution conditioned respectively classifier conditioned errd maj opposed main motivation boosting learner access learning algorithm error setting learn classifier desired error rate long sample set correctly labeled instances larger error rate smaller total number label queries needed producing correctly labeled set appropriate size use idea algorithm particular learn classifiers error using sample sets size labeled majority vote log labelers using fewer label queries overall baseline probabilistic filtering given classifier second step classical boosting algorithm requires distribution reweighed based correctness step done filtering process follows take large set labeled samples divide two sets depending whether instances mislabeled distribution instances mislabeled make half weight simulated picking set probability taking instance set uniformly random implement filtering setting however would need first get high quality labels set instances used simulating furthermore sample set typically large since least random samples needed simulate half weight points mislabels fraction total points getting high quality labels large sample set requires label queries large total number labels queried baseline algorithm ilter let log draw random labeler let odd maj break end let reaches step maj end return work introduce probabilistic filtering approach called ilter requires label queries cost per labeled example given classifier unlabeled sample set ilter returns set mislabeled probability least moreover correctly labeled likely included procedure described detail algorithm provide brief description working ilter queries one labeler time drawn random majority labels acquired far agree point ilter removes consideration hand majority labels never agree ilter adds output set consider correctly labeled since additional label agrees probability high probability majority labels agree point case ilter stops asking queries removes show lemma happens within queries time hand mislabeled labeler agrees probability clearly one set random labelers snapshot labels queried majority label agrees small probability show lemma even considering progression labels queried ilter throughout process probability majority label never agrees therefore added probability another key technique use work short means long correct label sampled points realizable setting samples never hurt algorithm although seems trivial first play important role approach particular probabilistic filtering procedure necessarily simulate distribution densities respectively high level sampling instances simulates process samples instances adds arbitrary instances formally stated proved appendix lemma given hypothesis class consider two discrete distributions absolute constant distributions labeled according exists constant probability labeled sample set size drawn error respect distribution techniques hand present algorithm high level algorithm proceeds three phases one classifier used theorem phase algorithm learns errd phase algorithm first filters set size set takes additional set samples queries log labelers instance get correct labels high probability next partitions instances two different sets based whether made mistake learns sample set drawn weighting two sets equally show lemma phase algorithm learns sample set drawn conditioned disagreeing finally algorithm returns maj case algorithm uses oracle runs time poly cost per labeled probability returns errd using log example golden queries load note log cost per labeled sample theorem start analysis algorithm stating abel labels correctly probability direct application hoeffding bound proof omitted lemma unlabeled sample set abel probability note direct consequence lemma phase algorithm achieves error lemma algorithm probability errd algorithm nterleaving oosting robabilistic iltering input given distribution class hypotheses parameters phase let abel set sample size let phase let ilter set samples size drawn let sample set size drawn let sall abel let sall let sall draw sample set size distribution equally weights let phase let abel sample set size drawn conditioned let return maj abel let set log labelers drawn maj end return next prove ilter removes instances correctly labeled good probability retains instances mislabeled least constant probability lemma given sample set classifier every ilter probability ilter probability proof first claim note maj consider time step since random query agrees probability independently majority log labels correct probability least therefore probability majority label disagrees every time step second claim interested probability exists maj probability return biased random walks also called probability ruin gambling feller given random walk takes step right probability takes step left remaining probability interested probability walk ever crosses origin left taking even infinitely probability many steps using probability return random see theorem walks maj ever therefore probability least remainder proof ease exposition assume errd per lemma fact errd assumption needed correctness results helps simplify notation analysis direct consequence lemma application chernoff bound deduce high probability size next lemma whose proof appears appendix formalizes claim lemma probability exp size next lemma combines probabilistic filtering techniques show desired error lemma let denote distribution conditioned respectively let probability proof consider distribution equal probability distributions induced let denote density point distribution relying technique see lemma sufficient show ease presentation assume lemma holds equality errd exactly probability let density instance distributions respectively note similarly let number occurrences sets respectively two cases exist absolute constants according lemma second sixth transitions sizes third transition fact exist absolute constants according lemma second sixth transitions sizes third transition fact fourth transition holds part lemma using guarantees lemma probability next claim shows probabilistic filtering step queries labels high level achieved showing instance contributes queries high probability hand instances mislabeled may get log queries points total number queries instances require lower order term lemma let sample set drawn distribution let errd probability exp ilter makes label queries proof using chernoff bound probability exp total number points disagrees number queries spent points log next show number queries taken majority agree constant let first show case expectation let expected number labels queried correct labels incorrect ones since probability least receive one correct label stop probability get wrong label case get two correct labels future moreover since get one correct label move one solving therefore expected total number queries next show random variable also let random variable indicates total number queries one correct label incorrect labels note unbounded random variable therefore concentration bounds hoeffding chernoff work instead show prove bernstein inequality see theorem holds show appendix bernstein inequality statisfied fact therefore instances probability exp finally ingredients needed proving main theorem proof theorem first discuss number label queries algorithm makes total number labels queried phases attributed labels queried abel abel log lemma almost surely therefore abel contributes log labels moreover showed lemma ilter queries labels surely total number labels queried algorithm log leads cost per labeled example log remains show maj error since abel abel return correctly labeled sets errd distribution conditioned showed lemma probability using boosting technique schapire described theorem conclude maj error general case section extend algorithm handle value necessarily satisfy show using golden queries possible efficiently learn function class small overhead two key challenges one needs overcome first longer assume taking majority vote random labelers get correct label instance therefore abel may return highly noisy labeled sample set problematic since efficiently learning using oracle crucially depends correctness input labeled set second ilter longer filters instances correctly based classification error particular ilter may retain constant fraction instances fact correct may throw instances incorrect high probability therefore guarantees lemma fall apart immediately overcome challenges using two key ideas outlined pruning alluded section instances small majority labelers agreement great identifying pruning away noticeable fraction bad labelers call instances good test cases particular ever encounter good test case ask golden query consider labelers got test correctly note make golden queries least fraction labelers would pruned repeated times number good labelers form strong majority case algorithm succeeds natural question would measure using label queries interestingly abel modified detect good test cases measuring empirical agreement rate set log labelers shown procedure rune abel part algorithm take majl label otherwise test prune labelers restart procedure ensures whenever use sample set labeled rune abel certain correctness labels stated following lemma proved appendix lemma unlabeled sample set probability either rune abel prunes set labelers rune abel immediate result first phase algorithm succeeds computing errd moreover every time rune abel prunes set labelers total fraction good labeler among remaining labelers increase show prunings set good labelers guaranteed form large majority case algorithm case used stated next lemma proved appendix lemma probability total number times algorithm restarted result pruning robust filtering step faces completely different challenge point good test case filtered wrong way however instances still strong majority labelers agree affected problem filtered correctly therefore first step ensure total number good test cases caught ilter starts small purpose start algorithm calling abel sample size log test points found set high probability total fraction good test cases underlying distribution since fraction good test cases small one show except fraction noisy distribution constructed filtering process purposes boosting satisfy conditions needed technique introduce robust version technique argue filtering step indeed produce error lemma robust lemma given hypothesis class consider two discrete distributions except fraction mass absolute constant distributions labeled according exists constant probability labeled sample set size drawn error respect combining techniques every execution algorithm ensure good test case ever detected prune small fraction bad labelers restart algorithm never detected algorithm returns classifier error theorem suppose fraction perfect labelers let small enough constant algorithm uses oracle runs time poly uses training set size algorithm oosting robabilistic iltering input given distribution class hypothesis parameters phase run algorithm quit let small enough draw log examples distribution rune abel phase let rune abel set sample size let phase let ilter set samples size drawn let sample set size drawn let sall rune abel let sall let sall draw sample set size distribution equally weights let phase let rune abel sample set size drawn conditioned let return maj rune abel let set log labelers drawn get golden query restart algorithm distribution else majl end end return size probability returns errd using golden queries load per labeler total number queries log log log note log cost per labeled query log proof sketch let set good test cases let total density points note high probability includes one point case rune abel identifies prunes set labelers therefore assume lemma easy see phase phase algorithm succeed producing errd remains show phase algorithm also produces consider filtering step phase first note guarantees ilter expressed lemma still hold let distribution equal probability distributions induced used simulating similarly lemma one show since therefore satisfy conditions robust lemma lemma fraction bad points hence argue remainder proof follows using boosting technique schapire described theorem perfect labelers section consider scenario pool labelers include perfect labelers unfortunately learning setting reduces notoriously difficult agnostic learning problem related task find set labelers good perfect section show identify set good labelers least majority labelers good consider setting fraction perfect labelers arbitrarily small assume least half labelers good others considerably worst performance formally given set labelers distribution unknown target classifier assume half labelers good error distribution hand remaining labelers call bad error rates distribution interested identifying good labelers high probability querying labelers unlabeled sample set drawn model presents interesting community structure two good labelers agree least fraction data bad good labeler agree data note rate agreement two bad labelers arbitrary due fact multiple bad labelers classification function case completely agree two bad labelers disagree classification every instance structure serves basis algorithm analysis provide overview working analysis algorithm ood abeler etection input given labelers parameters let graph vertices edges take set random pairs nodes disagree add edge end let set connected components nodes take one node disagree add edge end return largest connected component disagree take set samples return theorem informal suppose good labeler errd furthermore assume errd let number good labelers least algorithm returns set good labeler probability using expected load per labeler view labelers nodes graph edges start algorithm step algorithm takes random pairs labelers estimates level disagreement querying unlabeled sample set size measuring empirical disagreement application chernoff bound know probability isagree therefore pair good labelers tested algorithm isagree pair labelers one good bad isagree therefore connected components graph include labelers single community next show step algorithm probability exists least one connected component size good labelers see first prove two good labelers probability existing least let set nodes corresponding good labelers properties random graphs high probability component size random graph whose edges exists probability janson therefore probability component size vertices finally step algorithm considers smaller connected components tests whether join bigger components measuring disagreement two arbitrary labelers point good labelers form one single connected component size algorithm succeeds identifying good labelers next briefly discuss expected load per labeler algorithm labeler participates pairs disagreement tests expectation requiring queries expectation labeler labels instances references anthony bartlett neural network learning theoretical foundations cambridge university press pranjal awasthi maria florina balcan nika haghtalab ruth urner efficient learning linear separators bounded noise proceedings conference computational learning theory colt pages pranjal awasthi balcan nika haghtalab hongyang zhang learning compressed sensing asymmetric noise proceedings conference computational learning theory colt pages ashwinkumar badanidiyuru robert kleinberg yaron singer learning budget posted price mechanisms online procurement proceedings conference economics computation pages acm ashwinkumar badanidiyuru robert kleinberg aleksandrs slivkins bandits 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science wisdom crowds pages songbai yan kamalika chaudhuri tara javidi active learning imperfect labelers proceedings annual conference neural information processing systems nips pages chicheng zhang kamalika chaudhuri active learning weak strong labelers proceedings annual conference neural information processing systems nips pages additional related work generally interactive models learning studied machine learning community popular among area active learning cohn dasgupta balcan koltchinskii hanneke model learning algorithm adaptively query labels examples training set use produce accurate hypothesis goal use label queries possible number labeled queries used called label complexity algorithm known certain hypothesis classes learned model using much fewer labeled queries predicted theory particular many instances label complexity scales logarithmically opposed linearly however achieve computational efficiency algorithms model rely fact one get perfect labels every example queried would hard achieve model since worst case would lead labeler answering log many queries contrast want keep query load labeler constant hence techniques developed active learning insufficient purposes furthermore noisy settings work efficient active learning algorithms assumes existence empirical risk minimizer erm oracle minimize training error even instances labeled according target classifier however cases erm oracle hard implement improvements obtained label complexity less drastic noisy scenarios another line work initiated zhang chaudhuri models related notions weak strong labelers context active learning authors study scenarios label queries strong labeler reduced querying weak potentially noisy labelers often however discussed model yield relevant algorithms setting worst case one might end querying high quality labels leading prohibitively large load per labeler setting work yan studies model active learning labeler abstains providing label prediction often instances closer decision boundary authors show use abstentions order approximate decision boundary setting inherently different since make assumptions bad labelers proof lemma first notice labeled according errd therefore errd let errs errs errd claim follows fact proof lemma let first consider expected size sets using lemma similarly similarly claim follows chernoff bound remainder proof lemma prove bernstein inequality holds total number queries made majority agrees let random variable denoting number queries algorithm makes instance consider probability maj first time probability maj chernoff bound maj exp exp last inequality done integration satisfies bernstein condition stated theorem therefore exp therefore total number queries points high probability probability lemmas theorem probability ruin feller consider player starts dollars adversary dollars player bets one dollar gamble wins probability probability player ends money point game theorem bernstein inequality let independent random variables expectation supposed positive real number every exp omitted proofs section section prove theorem present proofs omitted section theorem restated suppose fraction perfect labelers let algorithm uses oracle runs time poly uses training set size size probability returns errd using golden queries load per labeler total number queries log log log note log cost per labeled query log proof lemma chernoff bound probability every set labelers rune abel queries hence identified set labelers pruned otherwise majl agrees good labelers gets labeled correctly according target function proof lemma recall small enough constant time rune abel called hoeffding bound guaranteed probability set labelers rune abel queries hence issue golden query prune away bad labelers guaranteed remove least fraction labelers furthermore good labeler ever removed hence fraction good labelers increases calls fraction good labelers surpasses switch using algorithm therefore probability overall total number golden queries proof lemma let set points satisfy condition notice labeled according errd therefore errd let errs errs errd claim follows fact proof theorem recall small enough constant let set good test cases let total density points note high probability includes one point case rune abel identifies prunes set labelers therefore assume lemma easy see errd analyze filtering step phase section goal argue consider distribution equal probability distributions induced let denote density point distribution show since therefore satisfy conditions robust lemma lemma fraction bad points hence show proof identical one lemma ease representation assume errd exactly let density instance distributions respectively note similarly let number occurrences sets respectively two cases exist absolute constants according lemma second sixth transitions sizes third transition fact exist absolute constants according lemma second sixth transitions sizes third transition fact fourth transition holds part lemma finally distribution conditioned using boosting technique schapire describe theorem conclude maj error label complexity claim follows fact restart algorithm times take additional log high quality labeled set run algorithm uses label complexity theorem getting restarted

automated identification trampoline skills using computer vision extracted pose estimation paul connolly guenole silvestre chris bleakley sep school computer science university college dublin belfield dublin ireland abstract novel method identify trampoline skills using single video camera proposed herein conventional computer vision techniques used identification estimation tracking gymnast body video recording routine frame open source convolutional neural network used estimate pose athlete body body orientation joint angle estimates extracted pose estimates trajectories angle estimates time compared labelled reference skills nearest neighbour classifier utilising mean squared error distance metric used identify skill performed dataset containing skill examples distinct skills performed adult male female gymnasts recorded used evaluation system system found achieve skill identification accuracy dataset introduction originating trampolining became competitive olympic sport sydney competition athletes perform routine consisting series skills performed number jumps skills scored human judges according trampoline code points fig although explicit objective judging criteria introduced recent years scores awarded still vary judges leading highly contentious final decisions eliminating human error means reliable automated judging trampoline routines desirable herein describe first step towards goal novel automated system identification trampoline skills using single video camera identification skills necessary prior judging since different skills scored different ways still challenging problem identification trampoline skills video enabled recent advances human pose estimation andriluka improved accuracy approaches achieved introduction convolutional neural network convnet based estimation estimators rely new convnet algorithms coupled recent gains gpu performance addition introduction larger varied general pose datasets sapp taskar johnson everingham leveraging annotation vastly increased quantity training data available best authors knowledge previous work reported identification trampolining skills video closest previous work identification trampoline skills required gymnast wear motion capture suit containing inertial sensors helten wearing special suits cumbersome allowed competition due strict rules regarding gymnast attire fig previous work automated judging rhythmic gymnastics video reported however method differs method used work algorithm proposed herein consists number stages bounding box gymnast extracted using conventional image processing techniques pose athlete subsequently determined allowing body orientation joint angles estimated angle trajectories time compared obtained reference skills skill performed identified nearest neighbour reference dataset based mean square error metric system evaluated using large number video recordings capturing movements male female gymnasts performing trampoline routines wide variety skills lighting conditions backgrounds recorded gymnasts wear special clothes markers camera placed performance position human judge structure paper follows section background information trampolining given section detail provided approaches analysis sporting movement pose estimation using video recordings proposed algorithm described section section discusses experimental procedure organisation dataset experimental results discussion provided section conclusions including future work follow section background trampoline routine consists sequence high continuous rhythmic rotational jumps performed without hesitation intermediate straight bounces routine show good form execution height maintenance height jumps demonstrate control body flying phase competition routine consists jumps referred work skills simplicity straight bounce taken skill competitor perform variable number straight bounces beginning routine called optional straight bounce taken completing routine control height gymnast required stop completely skills involve landing one four positions feet seat front back rotations body longitudinal lateral axes referred twist somersault rotations respectively skills combine rotations body shape tuck pike straddle straight landing positions shapes illustrated figure score performance calculated sum four metrics degree difficulty tariff execution horizontal displacement time flight degree difficulty scored based difficulty skill performed example full somersault awarded points somersault tariff assigned found simple based skill identification examples tariff scores seen table execution score allocated based well skill judged performed horizontal displacement time flight measured electronically using force plates legs trampoline related work one problems capture trampoline skills large performance space elite performers reach height tracking large volume prohibitively difficult many existing motion capture solutions including devices microsoft kinect helten inertial sensors used measure body point acceleration orientation gymnast required wear body suit containing ten inertial measurement units sensor data streams transformed feature sequence classification skill motion template learned feature sequence unknown trampoline motions compared set skill templates using variant dynamic time warping best accuracy achieved skill types survey methods general human motion representation segmentation recognition found weinland judging rhythmic gymnastics skills video investigated movement gymnast tracked using optical flow velocity field information extracted across frames skill projected velocity covariance eigenspace similar movements found trace unique similar trajectories new video recordings classified based distance reference trajectories system specificity approximately sensitivity approximately skills considered figure landing positions feet seat front back trampoline shapes tuck pike straddle straight human pose estimation process estimating configuration body typically single image robust pose estimation proven powerful starting point obtaining pose estimates human bodies overview pose estimation problem proposed methods found sigal poppe methods successful images limbs subject visible however unsuitable view trampoline routine inherent convnet based systems assume particular explicit body model since learn mapping image body pose machine learning based techniques provide greater robustness variations clothing accessories approaches mpii benchmark andriluka used access accuracy pose estimators approach described pishchulin achieved accuracy whereas convnet based method proposed newell achieved work described herein differs previous work system performs skill identification trampolining using single monocular video camera work takes advantage recently developed high accuracy open source convnet based pose estimators stacked hourglass network newell monocap zhou methods selected estimation filtering pose respectively stacked hourglass network pose estimates provided convnet architecture features processed across scales consolidated best capture spatial relationships body parts repeated steps pooling upsampling conjunction intermediate supervision previous methods monocap pose estimated via algorithm sequence images pose predictions conveniently joint location uncertainties marginalized inference proposed algorithm complete algorithm illustrated figure video recorded reduce resolution remove audio body extraction stage identifies tracks convex hull athlete video frames video segmented according detected bounces feature extraction stage estimates pose athlete body orientation joint angles frame based extracted feature angles classification performed identify skill experiments accuracy segment bounces record footage label ground truth skill downsample video subtract background track gymnast dim blur background save video frames estimate pose filter pose temporally calculate angles identify skill annotation classification body extraction feature extraction calculate accuracy evaluation figure flow chart illustrating proposed method system evaluated comparing detected skills manually marked ground truth algorithm stages explained detail following sections body extraction top trampoline identified based hue characteristics presented best guess user interface allows position fine tuned gymnast tracked assuming largest moving object trampoline background subtractor generates foreground mask frame static image components multiple frames taken part background camera assumed static without changing focus recording foreground mask eroded one iteration dilated ten iterations kernel largest segment morphed mask taken silhouette gymnast method moments used determine centroid silhouette video segmented individual skills based position centroid peak detection algorithm identifies local minima vertical position centroid local minima taken indicate start end frames skill threshold applied peaks local minima identify start end jumps routine convex hull silhouette used generate bounding box athlete image bottom bounding box compared position top trampoline detect contact phase bounce examples application method shown figure images body saved frames athlete contact trampoline maximum size bounding box across frames routine found image squarely cropped size centred centroid gymnast based extracted foreground mask background image blurred darkened helps reduce number incorrect pose estimates figure processed images original frame background model foreground mask body silhouette convex hull erosion dilation elbow shoulder hip knee leg torso twist table feature angles name index feature extraction stacked hourglass network monocap used pose estimation filtering respectively pose estimator generates pose predictions joint locations pose estimator used filter pose predictions across sequence images smoothed pose joint angles orientation angles represent athlete body position calculated feature angles denoted total number feature angles feature angles part time series frame number angles listed table example trajectories seen figure twist around body longitudinal axis estimated distance pose points labelled right left shoulder shoulder separation image maximum gymnast back front facing camera approximately zero sideways camera finding maximum separation whole routine separation normalised value way angle depend size performer right angle deg torso vertical left elbow shoulder hip knee leg vertical twist angle torso twist time figure motion sequence tuck jump estimated angles shown beneath classification feature angle trajectories compared labelled reference set calculation mean squared error mse observed skill identified equivalent reference giving minimum mse feature angle trajectories references aligned means interpolation number data points observed angle trajectory mse experimental procedure data acquisition procedure data collection submitted approved ucd office research ethics videos routines recorded training sessions competitions ucd trampoline club consent sought members ucd trampoline club prior recording video purposes project routines collected ucd sports centre normal sports hall lighting conditions background modified typically consisting brick wall nets routines recorded resolution frames per second fps using consumer grade camera shutter speed reduce motion blur camera positioned typical location viewing angle judging panel bounces within field view camera video subsequently downsampled maintaining aspect ratio audio removed steps significantly reduced data file size processing time maintaining usable resolution videos manually annotated labels means custom built web interface datasets resulting dataset consists routines adult athletes male female totalling minutes video contained distinct skills skill examples names distribution skills summarised table accuracy identification algorithm tested using cross validation skills fewer examples included test leaving distinct skills iteration evaluation subset examples skill randomly selected database subset split evenly give number reference examples number test examples total size reference set skill examples test set size average accuracy iterations evaluation reported herein results discussion average accuracy system distinct skills listed included classification table confusion matrix experiment shown figure noted subject identification sometimes incorrectly focus people background particularly seat front back landings gymnast becomes obscured trampoline bed causes errors trampoline contact detection resulting frames without obvious subject presented pose estimator resulting angles representative skill performed also cause errors jump segmentation due incorrect centroid extraction jump segmentation failed cases significant confusion skill identification occurs fpf pike jumps shown figure fsf straddle jumps shown figure view difficult distinguish movements another area confusion tuck pike shape barani skill bri features distinguish shapes angles hip knees tuck shape skill often performed loosely results angle hip similar pike shape identification angle knees becomes deciding feature may overwhelmed noise features use support vector machine might improve classification accuracy example difficulty estimating wrist ankle joints pose estimator lead noise angles elbows knees weighting features less important might improve overall accuracy tariff occurrences straight bounce tuck jump pike jump straddle jump half twist jump full twist jump seat drop half twist seat drop seat half twist seat feet seat half twist feet seat front drop feet front back drop feet back half twist feet back front somersault tuck front somersault pike barani tuck barani pike barani straight crash dive back somersault tuck back somersault pike back somersault straight back somersault seat tuck lazy back cody tuck back half barani ball tuck rudolph rudi full front full back ftf fpf fsf fsst fssp brit brip bris cdi bsst bssp bsss bstt lbk cdyt bha bbot rui ffr fub ftf fpf fsf brit brip cdi bsst bssp bsss bstt ftf fpf fsf brit brip cdi bsst bssp bsss bstt code ground truth skill skill name identified skill figure confusion matrix showing relative errors skill average iterations cross validation table skill dataset excluded classification likely accuracy could improved increasing amount data current pose estimation algorithms take single image input seems likely performance could improved tracking pose video sequence adding second video camera pointed towards front gymnast would likely improve accuracy allowing greater discrimination motion parallel axis subject first camera however issues regarding extra user effort setting second camera synchronisation two devices modern trampoline judging systems incorporate force plates detection centrality landing trampoline bed fusing information video data could possibly also result improved accuracy body extraction performed fps core intel ghz cpu estimation pose using stacked hourglass network ran fps ubuntu nvidia titan pascal gpu core intel ghz cpu default parameter settings execution monocap algorithm ran fps machine also default parameters conclusion system identifying trampolining skills using single monocular video camera developed system incorporated algorithms background subtraction erosion dilation pose estimation pose filtering classification system found provide accuracy identifying distinct skills present dataset contain skill examples future work plan extend classification algorithms perform automated execution judging references andriluka andriluka pishchulin gehler schiele human pose estimation new benchmark state art analysis ieee conference computer vision pattern recognition cvpr pages escalona olivieri automatic recognition scoring olympic rhythmic gymnastic movements human movement science fig fig trampoline code points accessed helten helten brock seidel classification trampoline jumps using inertial sensors sports engineering johnson everingham johnson everingham learning effective human pose estimation inaccurate annotation ieee conference computer vision pattern recognition cvpr pages newell newell yang deng stacked hourglass networks human pose estimation corr pishchulin pishchulin andriluka gehler schiele poselet conditioned pictorial structures ieee conference computer vision pattern recognition cvpr pages poppe poppe human motion analysis overview computer vision image understanding special issue vision interaction sapp taskar sapp taskar modec multimodal decomposable models human pose estimation ieee conference computer vision pattern recognition cvpr pages sigal sigal human pose estimation accessed weinland weinland ronfard boyer survey methods action representation segmentation recognition computer vision image understanding zhou zhou zhu leonardos derpanis daniilidis sparseness meets deepness human pose estimation monocular video ieee conference computer vision pattern recognition cvpr pages

parsing methods streamlined sep luca breveglieri stefano crespi reghizzi angelo morzenti dipartimento elettronica informazione bioingegneria deib politecnico milano piazza leonardo vinci milano italy email paper goals unifying parsing parsing yield single simple consistent framework producing provably correct parsing methods deterministic well tabular ones extended grammars ebnf represented networks departing traditional way presenting independent algorithms deterministic general tabular earley parsers unify coherent minimalist framework present simple general construction method ebnf elr parsers new category convergence conflicts added classical conflicts prove correctness show two implementations deterministic machines machines latter also used earley parsers beatty theoretical characterization grammars adapted derive extended ell parsing method first minimizing elr parser simplifying state information using notations elr case extended earley parser obtained since parsers operate compatible representations feasible combine mixed mode algorithms categories subject descriptors parsing formal languages theory general terms language parsing algorithm syntax analysis syntax analyzer additional key words phrases extended bnf grammar ebnf grammar deterministic parsing parser parser elr recursive descent parser parser ell tabular earley parser october see formal languages compilation crespi reghizzi breveglieri morzenti springer london edition planned parsing methods streamlined contents introduction preliminaries derivation machine nets grammar call sites machine activation parsing construction elr parsers base closure kernel elr condition elr versus classical definitions parser algorithm simplified parsing bnf grammars parser implementation using indexable stack related work parsing ebnf grammars deterministic parsing property pilot compaction merging properties compact pilots candidate identifiers pointers unnecessary ell condition discussion stack contraction predictive parser parser graph predictive parser computing left derivations parser implementation recursive procedures direct construction parser graph equations defining prospect sets equations defining guide sets tabular parsing string recognition syntax tree construction conclusion appendix parsing methods streamlined introduction many applications compilation program analysis document natural language processing language defined formal grammar processed using classical approach based syntax analysis parsing followed translation efficient deterministic parsing algorithms invented improved following decade described compiler related textbooks instance aho grune jacobs crespi reghizzi efficient implementations available widely used research last decade focused issues raised technological advances parsers datadescription languages efficient general tabular parsing grammars probabilistic parsing natural language processing mention leading research lines paper goals unifying parsing parsing minor extend also tabular earley parsing yield single simple consistent framework producing provably correct parsing methods deterministic also tabular ones extended grammars ebnf represented networks address first goal compiler language developers familiar technical aspects parsing invariably feel ought room improvement classical parsing methods annoyingly similar yet incompatible notions used tabular earley parsers moreover parsers presented independent algorithms indeed first invented without taking advantage known grammar inclusion properties tighten simplify constructions proofs may consequence excellent quality original presentations particularly exclusively knuth rosenkrantz stearns earley distinguished scientists made revision systematization less necessary first contribution conceptual economy clarity reanalyzed traditional deterministic parsers view provided beatty beatty rigorous characterization grammars special case allows show transform parsers parsers merging simplifying parser states anticipating parser decisions result one set technical definitions suffices present parser types moving second goal best known tabular parsers accept extended form bnf grammars known ebnf ecfg also regular rightpart rrpg make use regular expressions widely used language reference manuals popular among designers parsers using recursive descent methods systematically use graphic representation ebnf grammars means networks also known syntax charts brevity prevents discuss detail long history research parsing methods ebnf grammars suffices say existing deterministic method popular least since use elegant recursive descent pascal compiler wirth ebnf grammars represented network machines formalism already use since least lomet hand numerous interesting proposals methods ebnf grammars operate less general assumptions implemented various parsing methods streamlined types parsers section recent survey hemerik concludes rather negatively published parsing theory complex many feel tempted use striking phenomenon ideas behind recursive descent parsing ecfgs grasped applied immediately whereas literature parsing rrpgs difficult access tabular parsing seems feasible largely unexplored decided represent extended grammars transition diagram systems course equivalent grammars since regular expression easily mapped recognizer moreover transition networks dub machine nets often readable grammar rules stress constructions operate machine nets represent ebnf grammars unlike many past proposals make restrictive assumptions form regular expressions ebnf grammars past methods met difficulty formulate condition ensures deterministic parsing presence recursive invocations cycles transition graphs offer simple rigorous formulation adds two classical conditions neither conflicts third one convergence conflict parser presented two variants use different devices identifying right part handle typically substring unbounded length reduced deterministic pushdown automaton implementation named machine using unbounded integers pointers stack device also used last development tabular parser ebnf grammars last since parser types described operate uniform assumptions use compatible notations suggest possibility combine algorithms half century research parsing certain facts properties formally proved early studies become obvious yet endemic presence errors inaccuracies listed hemerik published constructions ebnf grammars warrants new constructions proved correct interest readability brevity first present enabling properties constructions also relying significant examples properties constructions new provide correctness proofs appendix paper organized follows section sets terminology notation grammars transition networks section presents construction elr parsers section derives ell parsers first transformation parsers also directly section deals tabular earley parsers preliminaries concepts terminology grammars automata classical see aho crespi reghizzi introduce specific notations bnf grammar specified terminal alphabet set nonterminal symbols set rules starting symbol axiom element called grammar symbol rule form left part nonterminal right part possibly empty denoted string parsing methods streamlined two rules called alternative shortened extended bnf ebnf grammar generalizes rule form allowing right part regular expression formula operators uses union written concatenation kleene star parentheses language defined denoted nonterminal assume without loss generality contains exactly one rule derivation strings relation possibly empty strings derivation leftmost respectively rightmost contain nonterminal resp contain nonterminal series derivations denoted derivation called derivation reverse relation named reduction denoted language generated grammar set language generated nonterminal set language nullable contains empty string nonterminal generates nullable language also called nullable following tradition dating least lomet going represent grammar rule graph graph finite automaton named recognizes regular expression collection graphs set nonterminals named network finite machines graphic representation grammar see fig advantages offers pictorial representation permits directly handle ebn grammars maps quite nicely parser implementation simple case contains terminal symbols machine recognizes language contains nonterminal machine edge labeled thought invocation machine associated rule nonterminals coincide invocation recursive convenient although necessary assume machines deterministic loss generality since nondeterministic finite state machine always made deterministic definition recursive net finite deterministic machines ebn grammar nonterminal set grammar rules denote regular languages alphabet respectively defined names finite deterministic machines accept corresponding regular languages usual assume machines reduced sense every state reachable initial state reaches final state set machines machine net denoted prevent confusion names states two machines made different appending machine name subscript set states machine avoid confusion call machines finite automata grammar rules reserve term automaton pushdown automaton accepts language parsing methods streamlined machine net ebnf grammar fig ebn grammar axiom machine network axiom running example initial respectively final states tagged incoming resp outgoing dangling dart initial state set final states state set net union states transition function every machine denoted symbol risk confusion state sets disjoint state machine symbol brevity denotes regular language alphabet accepted machine starting state initial state language includes every string labels path qualified accepting initial final state disallowing reentrance initial states simplify parsing algoc rithms stipulate every machine edge exists grammar symbol words edges may enter initial state normalization ensures initial state visited within computation stays inside machine clearly machine normalized adding one state transitions negligible overhead minor adjustment greatly simplifies reduction moves parsers arbitrary several algorithms described instance crespi reghizzi produce machine recognizing corresponding regular language practice used right parts grammars simple immediately translated hand equivalent machine neither assume forbid machines minimal respect number states facts translation always desirable use minimal machine different semantic actions may required two states would indistinguishable pure language theoretical definitions grammar purely right part form every alternative finite string therefore finite language machine acyclic graph made tree accept machine general case graph machine representing rule acyclic case mapping strings language set accepting paths machine therefore net essentially notational variant grammar witnessed common practice include ebn productions syntax diagrams language specifications indifferently denote language parsing methods streamlined need also terminal language defined net starting state machine possibly initial one formula string terminals nonterminals accepted machine starting state derivations originating produce terminal strings language particular previous definitions follows example running example ebn grammar machine net shown figure language generated viewed obtained language strings allowing character replace substring machines deterministic initial states reentered features needed exercise different aspects parsing present iteration branching multiple final states nullability nonterminal illustrate list language defined net component machines along aliases identify machine states alternative convention quite used grammars relies marked grammar rules instance states machine aliases bullet character need define set initial characters strings recognized starting given state definition set initials ini ini set computed applying following logical clauses fixed point reached let terminal nonterminals states machine clauses ini edge ini edge ini ini edge nullable ini illustrate ini ini parsing methods streamlined derivation machine nets machine nets ebn grammars preceding definition derivation models rule infinite set alternatives shortcomings derivation step replaces nonterminal string possibly unbounded length thus computation inside machine equated one derivation step application parsing analytical definition needed split large step series state transitions recall grammar rule form every machine represented equivalent grammar machine states nonterminal symbols grammar machine net replaced equivalent right linear grammar used provide rigorous semantic derivations constructed parsers straightforward write grammar named equivalent respect regular language nonterminals states axiom exists rule edge empty rule final state notice nonterminal original grammar therefore rule may form still since first symbol right part terminal symbol grammar provision identity viewed clearly holds next every grammar net replace symbol occurring rule thus obtain rules form resulting grammar denoted named grammar net terminal alphabet nonterminal set axiom right parts length zero two may contain two nonterminal symbols thus grammar obviously equivalent generate language example grammar running example said grammars choose name nonterminals alias states instance rule using instead obtain derivations steps elementary state transitions instead entire machine example suffice example derivation grammar classical leftmost derivation parsing methods streamlined expanded series truly atomic derivation steps grammar may also work consider reductions said grammar used proofs assign precise semantic parser steps otherwise use readable specification language parsed clearly grammar many rules original ebn grammar less readable syntax diagram machine net needs introduce plethora nonterminal names identify machine states call sites machine activation edge labeled nonterminal named call site machine corresponding return state parsing viewed process call sites activates machine graph performs scanning operations calls reaches final state performs reduction returns initially axiom machine activated program invokes parser step derivation nonterminal suffix derived string contains current state active machine followed return points suspended machines ordered right left according activation sequence example derivation machine return points looking derivation find machine active current state previously machine suspended resume state earlier activation also suspended resume state upon termination resumes return state collection first legal tokens scanned named set activation intuitive concept made precise following definition candidate inspecting next token parser avoid invalid machine call actions uniformity input string entirely scanned assume next token special character string terminator simply candidate since exclusively deal lookahead length pair hqb intended meaning token legal token current activation machine reformulate classical knuth notion closure function machine net use compute set legal candidates definition closure functions initial activation machine encoded candidate let set candidates initialized closure function parsing methods streamlined defined applying following clauses fixed point reached closure closure closure edge ini thus closure functions compute set machines reachable given call site one invocations without intervening state transition conciseness group together candidates state write instead collection termed set definition empty list values closure function grammar function closure parsing show construct deterministic parsers directly ebn grammars represented machine nets deviate classical knuth method operates pure grammars call method elr instead brevity whenever passages identical immediately obtainable classical ones spend much time justify hand include correctness proofs main constructions past works extended parsers found rarely flawed hemerik end section briefly compare method older ones elr parser deterministic pushdown automaton equipped set states named macrostates short avoid confusion net states consists set brevity candidate automaton performs moves two types shift action reads current input character token applies function compute next token next pushed stack reduce action applied grammar symbols stack top match recognizing path machine current token admitted set parser deterministic configuration shift permitted reduction impossible configuration one possible reduce action grows syntax forest pops matched part stack pushes nonterminal symbol recognized next accepts input string last move reduces axiom input exhausted latter condition expressed saying special character current token presence convergent paths machine graph complicates reduction moves two paths may require pop different stack segments reduction handles difficulty acknowledged past research methods ebnf grammars proposed solutions differ technique used generality implement reduction moves enrich stack organization pointers enable parser trace back recognizing path popping stack pointer parsing methods streamlined implemented two ways bounded integer offset identifies candidate previous stack element unbounded integer pointer distant stack element former case parser still qualifies stack symbols taken finite set latter case pointers unbounded integers organization indexable stack called also used earley parsers construction elr parsers given ebn grammar represented machine net show construct elr parser certain conditions met method operates three phases net construct called pilot pilot state named includes non empty set candidates pairs states terminal tokens set pilot examined check conditions deterministic parsing check involves inspection components transitions outgoing three types failures may occur conflicts respectively signify parser configuration shift reduction possible multiple reductions convergence conflict occurs two different parser computations share character lead machine state test passed construct deterministic parser using pilot control adding operations needed managing reductions unbounded length last would simple exercise encode programming language candidate hpa terminal nonterminal symbol qualified depending terminal hpa hqa edge exists empty set otherwise set candidates shift symbol union shifts candidates algorithm construction elr pilot graph pilot named defined set pilot alphabet union terminal nonterminal alphabets initial set closure set computed starting following steps traditional lengthy name recognizer viable prefixes known function named theta avoid confusion traditional name delta transition function net also parsing methods streamlined symbols closure add graph add set end end end base closure kernel every set candidates partitioned two subsets base closure base includes candidates initial state clearly computed line algorithm base coincides pairs computed closure contains remaining candidates initial state initial empty base definition base closure may empty kernel projection first component particular condition may affect determinism occurs two states belong outgoing transitions defined grammar symbol definition multiple transition property convergence pilot multiple transition property includes two candidates grammar symbol transitions defined pilot transition called convergent convergent transition convergence conflict sets overlap illustrate consider two examples example pilot running example pilot graph ebn grammar net example see figure shown figure top bottom parts contain base closure respectively either part missing side shows part present base closure tokens grouped state final states evidenced encircling none edges graph convergent parsing methods streamlined fig elr pilot graph machine net figure two kernel differing sets called simplified parser constructions later introduced rely kernel equivalence reduce number observe two grammar symbol either defined neither one kernelequivalent illustrate notion convergent transition without conflict refer figure convergent transitions latter conflict parsing methods streamlined elr condition presence final candidate tells parser reduction move ought considered set specifies tokens occur next confirm decision reduce machine net one reduction may applied final state choose correct one parser stores additional information stack later explained formalize conditions ensuring parser decisions deterministic definition elr condition grammar machine net meets condition elr corresponding pilot satisfies following conditions condition every satisfies next two clauses conflict candidates final edges conflict candidates final condition transition pilot graph convergence conflict pilot figure meets conditions edge convergent elr versus classical definitions first discuss relation definition classical one knuth case grammar nonterminal finitely many alternatives since alternatives contain star union operations straightforward nondeterministic machine acyclic graph shaped tree many legs originating initial state alternative rules clearly graph satisfies reentrance hypothesis initial state general minimal classical pilot machine exhibit multiple transition property requirement parser determinism comes clauses def representation machine may differ two ways first assume machine deterministic convenience necessity thus consider nonterminal two alternatives else determinization effect normalizing alternatives left factoring longest common prefix using equivalent ebn grammar else second allow actually recommend graph minimal respect number states particular final states merged together state reduction undistinguishable also state may correspond multiple states state reduction may cause pilot edges become convergent therefore addition checking conditions def imposes convergent edge free conflicts since point quite subtle illustrate next example example convergent transitions consider equivalent ebn grammars corresponding machines fig determinizing states machine equivalent parsing methods streamlined grammars bnf ebnf abc abd machine nets net common part fig ebn grammars networks merged state machine turning attention elr conditions find grammar conflict caused derivations hand ebn grammar pilot shown fig two convergent edges highlighted arrows one conflict without arc violates elr condition disjoint notice explanatory purposes two candidates deriving convergent transition conflict kept separate depicted one candidate single usually done candidates state observe general machines net effect pilot automaton transform violations convergence conflicts next prove essential property justifies practical value theoretical development ebn grammar elr equivalent rightlinearized grammar defined sect heorem let ebn grammar represented machine net let equivalent grammar net meets elr condition grammar meets condition proof appendix parsing methods streamlined convergent conflictual fig pilot graph machine net fig edges convergent although proposition may sound intuitively obvious knowledgeable readers believe formal proof due past proposals extend definitions ebn albeit often restricted types regular expressions omitted formal proofs later found inaccurate see sect fact conflicts preserved two pilots quite easy prove less obvious part proof concerns correspondence convergence conflicts elr pilot conflicts grammar grasp without reading proof convergence conflict fig corresponds conflict fig address possible criticism significance theorem starting ebn grammar several equivalent grammars obtained removing regular expression operations different ways grammars may may fact would seem make somewhat arbitrary definition elr based highly constrained form defend significance generality choice two grounds first original grammar specification set set machines choice transform grammar standard almost obliged already shown heilbrunner standard form would exhibit conflicts cases second author proves grammar equivalent every grammar equivalent provided ambiguous besides shows definition elr grammar dominates parsing methods streamlined preexisting alternative definitions believe also new definitions later years dominated present one illustrate discussion helps consider simple example machine net elr theorem yet another equivalent grammar obtained natural transformation conflicts example phrase structure construct either form language defined elr net contrary conflict equivalent grammar bef caused indecision whether reduce shift grammar postpones reduction decision long possible avoids conflicts parser algorithm given pilot elr grammar machine net explain obtain deterministic pushdown automaton recognizes parses sentences cost repetition recall three sorts abstract machines involved net state set states drawn circular nodes pilot set states drawn rectangular nodes next defined said stores stack series entered computation enriched additional information used parsing steps moreover interleaved terminal nonterminal grammar symbols current one top stack determines next move either shift scans next token reduction topmost stack segment also called reduction handle nonterminal identified final candidate included current absence conflicts makes choice shift reduction operations deterministic similarly absence conflicts allows parser uniquely identify final state machine however leaves open problem determine stack segment reduced two designs presented first uses finite pushdown alphabet second uses unbounded integer pointers strictly speaking longer qualifies pushdown automaton first specify pushdown stack alphabet since given net finitely many different candidates number bounded number candidates also bounded cmax stack elements parsing methods streamlined two types grammar symbols stack sms sms denoted contains ordered set triples form state candidate identifier cid named stack candidates specified hqa cidi cid cmax cid readability cid value prefixed marker parser makes use surjective mapping set sms set denoted property set stack candidates deprived candidate identifiers equals set candidates notational convenience stipulate identically subscripted symbols related said stack sms interleaved grammar symbols algorithm elr parser let current stack grammar symbol top element initialization analysis starts pushing stack sms every candidate thus initial pilot shift move let top sms current token assume inspecting pilot decided shift let shift move push token stack get next token push stack sms computed follows hqa hqa position thus notice last condition implies state base reduction move state stack let corresponding assume inspecting pilot chooses reduction candidate hqa final state let hqa stack candidate current token cid chain starts links stack candidate hpa stack candidate reached cid therefore state initial reduction move grow syntax forest applying reduction pop stack symbols following order execute nonterminal shift move see reduction move initial state differs preceding case chosen candidate parser move grows syntax forest reduction performs nonterminal shift move corresponding parsing methods streamlined machine pilot pushdown stack hqa hqa hqa hqa plus completion parser stack pointers fig schematization shift move nonterminal shift move shift move except shifted symbol nonterminal difference parser read next input token line shift move acceptance parser accepts halts stack move nonterminal shift defined current token shift moves note computed alg may contain multiple stack candidates state happens whenever edge convergent may help look situation shift move schematized figure example parsing trace step step execution parser input string produces trace shown figure clarity two parallel tracks input string progressively replaced nonterminal symbols shifted stack stack stack stack one entry scanned prefix suffix yet scanned right stack inside stack element identified ordinal position starting first value cid field element encodes pointer ease reading parser trace simulation number appears framed stack element denoted etc final candidates encircled etc avoid clogging sets shown needed convergent transitions occur always found inspecting pilot graph fig highlights shift moves dashed forward arrows link two topmost stack candidates instance first figure top terminal shift first nonterminal shift third one null reduction similarly shifts fig highlights reduction handles means solid backward arrows link candidates involved instance see reduction stack configuration shows chain three pointers three stack elements form handle popped finally initial pointer stack element reduction origin popped initial pointer marks initial candidate obtained means closure operation applied candidate stack element see dotted arrow links subsequent shift starts see dashed arrow stack configuration solid candidate parsing methods streamlined string parsed stack contents stack base effect initialisation stack shift reduction shift reduction shift reduction shift shift reduction shift shift shift shift reduction shift reduction accept without shifting fig parsing steps string grammar elr pilot figures name sms maps onto corresponding shift token see effect line highlights null reduction pop anything stack immediately followed shift observe parser must store stack scanned grammar symbols parsing methods streamlined reduction move step may necessary selecting correct reduction handle build subtree added syntax forest returning example execution order reductions pasting together reductions obtain syntax tree order reductions displayed clearly reduction order matches rightmost derivation reversed order worth examining closely case convergent edges returning figure notice stack candidates linked cid chain mapped onto state candidates machine state stack candidate hqa linked via hqa set candidates general superset set included stack candidate due possible presence convergent transitions two sets coincide convergent transition taken pilot automaton parsing time elr grammar convergent edges studied next example net pilot graph trace parse shown figure readability cid values stack candidates visualized backward pointing arrows stack contains two candidates differ lookahead sets corresponding pilot two candidates targets convergent transition highlighted pilot graph simplified parsing bnf grammars grammars use regular expressions rules features elr parsing algorithm become superfluous briefly discuss highlight differences extended basic parsers since graph every machine tree edges entering machine state rules presence convergent edges pilot moreover alternatives nonterminal recognized distinct final states machine therefore candidate chosen parser simply pops stack elements performs reduction since pointers preceding stack candidate longer needed stack coincide pilot ones second related reason interleaved grammar symbols longer needed stack pilot grammar property edges entering carry label therefore reduction handle uniquely determined final candidate current parsing methods streamlined machine net pilot graph convergent edge parse traces reductions fig elr net pilot convergent edges double line parsing trace string parsing methods streamlined simplifications effect formal notation grammars use machine nets states becomes subjectively less attractive classical notation based marked grammar rules parser implementation using indexable stack finishing parsing present alternative implementation parser ebn grammars algorithm memory analyzer array elements element directly accessed means integer index named reason presenting new implementation twofold technique compatible implementation tabular parsers sect potentially faster hand general pushdown stack therefore parser viewed pure elements two alternating types vsms grammar symbols vsms denoted set triples named candidates form hqa elemidi simply differs earlier stack candidates third component positive integer named element identifier instead cid notice also set ordered surjective mapping pilot denoted elemid points back element containing initial state current machine reduction move performed length string reduced reduction handle obtained directly without inspecting stack elements top one clearly value elemid ranges maximum height algorithm elr parser automaton using let current stack grammar symbol top element initialization analysis starts pushing stack sms every candidate thus initial pilot shift move let top vsms current token assume inspecting pilot decided shift let shift move push token stack get next token push stack sms precisely computed follows hqa hqa thus reduction move state stack let corresponding assume inspecting pilot chooses reduction candidate hqa final state let hqa stack candidate current token parsing methods streamlined reduction move grow syntax forest applying reduction pop stack symbols following order execute nonterminal shift move see reduction move initial state differs preceding case chosen candidate parser move grows syntax forest reduction performs nonterminal shift move corresponding nonterminal shift move shift move except shifted symbol nonterminal difference parser read next input token line shift move acceptance parser accepts halts stack move nonterminal shift defined current token although algorithm uses stack viewed stack alphabet unbounded since vsms contains integer values example parsing trace figure compared figure shows step step execution parser example input string graphical conventions every stack element type vsms second field candidate elemid index points back inner position stack elemid equal current stack position candidates closure part stack element points previous position candidates base figure reduction handles highlighted means solid backward pointers dotted arrow locate candidate shifted soon reducing notice arrows span longer distance figure elemid goes directly origin reduction handle forward shift arrows fig shown results analysis execution order reductions obtained syntax tree identical example illustrate use previous example net featuring convergent edge net figure parsing traces shown figure integer pointers represented backward pointing solid arrows related work parsing ebnf grammars years many contributions published extend knuth method ebn grammars number papers one purporting improve previous attempts testifies optimal solution found papers usually start critical reviews related proposals grasp difficulties motivations perceived following discussion particularly draws later papers morimoto sassa kannapinn hemerik first dichotomy concerns format ebn specification taken input either grammar regular expressions right parts grammar finiteautomata right parts since nowadays perfectly clear interchangeable notations regular languages distinction longer relevant yet authors insisted language designer allowed specify syntax constructs arbitrary even ambiguous ones allegedly permit flexible parsing methods streamlined stack base string parsed stack contents indices effect initialisation stack shift reduction shift reduction shift reduction shift shift reduction shift shift shift shift reduction shift reduction accept without shifting fig tabulation parsing steps parser using string generated grammar figure elr pilot figure parsing methods streamlined pilot graph convergent edge parse traces reductions fig parsing steps parser using string recognized net figure pilot reproduced convenience mapping syntax semantics view share transforming original entirely satisfactory others imposed restrictions instance limiting depth kleene star nesting forbidding common subexpressions although original motivation simplify parser construction since vanished fair say used language reference manuals typically simple reason avoiding parsing methods streamlined obscurity others prefer specify right parts using readable graphical notations syntax diagrams pictorial variant diagrams whether deterministic really make difference either terms grammar readability ease parser generation even source specification includes simple transform standard construction leave responsibility removing algorithm constructs automaton pilot hand found inexpensive normalization disallowing reentrance initial states def pays terms parser construction simplification assuming grammar specified net two approaches building parser followed transform grammar grammar apply knuth construction directly construct elr parser given machine net generally agreed approach better approach transformation adds inefficiency makes harder determine semantic structure due additional structure added transformation morimoto sassa since approach leverages existing parser generators bison quite common language reference manuals featuring syntax chart notations include also equivalent even lalr grammar celentano systematic transformation ebn used obtain ebn grammar elr parser simulates classical knuth parser grammar technical difficulty approach authors deal identify left end reduction handle since length variable possibly unbounded list different solutions found already cited surveys particular many algorithms including use special shift move sometimes called stackshift record stack left end handle new computation net machine started whenever algorithms permit initial state reentered conflict normal shift unavoidable various devices invented arbitrate conflict add states control parser dig stack chapman lalonde others sassa nakata use counters purpose mention proposed devices unfortunately shown kannapinn several proposals precisely characterize grammars apply cases may fall unexpected errors motivated mentioned flaws previous attempts paper lee kim aims characterizing property ecf grammars defined network although definition intended ensure grammars parsed left right symbols authors admit subject efficient techniques locating left end handle beyond scope paper long history interesting proposals finding rather simple formulation elr condition leading naturally corresponding parser rather unexpected definition simply adds treatment convergent edges knuth definition technical difficulties well understood since long combined existing ideas simple provably correct solution course experimental work would needed evaluate performance algorithms parsing methods streamlined grammars deterministic parsing simpler flexible parsing method traditionally called ell applies elr grammar satisfies conditions although less general elr method several assets primarily ability anticipate parsing decisions thus offering better support translation implemented neat modular structure made recursive procedures mirror graphs network machines next sections presentation rigorously derives step step properties topdown deterministic parsers add one one simple restrictions elr condition first consider single transition property simplifies shiftreduce parser number reduced number net states convergent edges longer possible chain stack pointers disposed second add requirement grammar obtain traditional predictive parser constructs syntax tree last direct construction ell parsers sums historical note contrast twisted story elr methods early efforts develop parsing algorithms ebn grammars met remarkable success need critically discuss cite main references explain work adds value deterministic parsers operating topdown among first constructed compilation pioneers theory grammars shortly developed rosenkrantz stearns knuth sound method extend parsers ebn grammars popularized wirth compiler systematized book lewi included widely known compiler textbooks aho however books deterministic parsing presented methods independently presumably easier understand contrary section shows parsing ebn grammars corollary elr parser construction presented course pure grammars relationship grammar language families carefully investigated past see particular beatty building concept multiple transitions introduced elr analysis extend beatty characterization ebn case derive minimalist provably correct way ell parsing algorithms unified approach mention conclusion use heterogeneous parsers different language parts property pilot compaction given elr net elr pilot recall multipletransition property two identically labeled state transitions originate two candidates present brevity also say violates property next example illustrates several cases violation example violations extended left right leftmost length equal one parsing methods streamlined machine net pilot graph fig elr net multiple candidates base three cases examined first grammar generating deterministic language represented net fig top presence two candidates also reveals parser carry two simultaneous attempts parsing reduction takes place unique since pilot satisfies elr condition neither conflicts convergence conflicts second net figure illustrates case multiple candidates base entered convergent edge third grammar pilot violates yet contains one candidate base hand every base contains one candidate parser configuration space reduced well range parsing choices furthermore entails convergent edges pilot next show kernel qualified brevity kernelidentical safely coalesced obtain smaller pilot equivalent original one bears closer resemblance machine net hasten say transformation work general elr pilot proved correct hypothesis parsing methods streamlined merging merging operation coalesces two kernelidentical suitably adjusts pilot graph possibly merges defined follows algorithm erge replace new denoted lookahead set union corresponding ones merged becomes target edges entered pair edges labeled target clearly merged call erge clearly merge operation terminates graph fewer nodes set equivalence class applying merge algorithm members every equivalence class construct new graph called compact pilot denoted example compact pilot reproduce figure machine net original elr pilot bottom part compact pilot convenience renumbered notice sets expanded first row union corresponding merged going prove loss precision harmful parser determinism thanks stronger constraints imposed anticipate section compact pilot directly constructed machine net saving work compute larger elr pilot merge nodes properties compact pilots going show safely use compact pilot parser controller roperty let respectively elr pilot compact pilot net satisfying elr parsers controlled equivalent recognize language construct syntax tree every roof show every compact pilot elr conflicts since merge operation change kernel obvious every compact pilot number lalr pilots historical simpler variants parsers considered see instance crespi reghizzi however neither lalr pilots comply condition weaker hypothesis would suffice every one candidate base simplicity preferred assume also convergent edges present parsing methods streamlined machine net pilot graph compact pilot graph fig top bottom machine net elr pilot graph compact pilot equivalence classes named evidence correspondence states net parsing methods streamlined satisfies therefore conflicts excluded since involve two candidates base condition ruled next suppose contradiction conflict final occurs neither since hqa state shift base must also bases labeled edge originates therefore conflict already one next suppose contradiction new initial conflict outgoing labeled edge candidate definition merge already set say moreover two symbol either defined neither one therefore labeled edge originates thus conflict contradiction last suppose contradiction contains new initial conflict terminal character sets clearly one merged say closure candidates closure candidates show contradiction recall computed let bases respectively hqc hqc character say comes two possible ways present character follows state closure focusing character second possibility excluded whereas elements brought case necessarily remains presence comes presence hence must also contradiction parser algorithms differ controllers first take string accepted parser controlled since created merge encodes exactly cases reduction shift original merged parsers perform exactly moves pilots moreover chains candidate identifiers clearly identical since candidate offset affected merge therefore time parser stacks store elements merge relation compact parser recognizes strings constructs tree second suppose contradiction illegal string recognized using compact parser string consider first parsing time stops error whereas able move move terminal shift shift necessarily present reduction since candidate identifier chains identical string reduced nonterminal therefore also following nonterminal shift operated legal also contradiction thus established parser controlled compact pilot equivalent original one candidate identifiers pointers unnecessary thanks property parser simplified remove need cid stack pointers recall cid needed find reach reduction move stack elements parsing methods streamlined popped cid chain reached initial state sentinel hypothesis test replaced simpler device later incorporated final ell parser alg reference alg shift reduction moves modified first let focus situation old parser top stack hqa performs shift terminal state necessarily shift would require compute record cid sms pushed stack situation pointerless parser cancels element candidates since correspond discarded parsing alternatives notice canceled candidates necessarily hence contain initial states elimination stack allows parser uniquely identify reduction made final state machine entered simple rule keep popping stack first occurrence initial state found second consider shift initial state necessarily case pointerless parser leaves unchanged pushes stack candidates present canceled may origin future nonterminal shifts since cid used parser stack element identical compact pilot thus shift move first updates top stack element pushes input token next specify moves differ alg algorithm pointerless parser let pilot compacted denoted stack symbols set candidates weakly included shift move let current character top stack element containing candidate hqa let respectively state transition transition applied shift move initial eliminate candidates set equal push stack get next token push stack reduction move state let stack assume pilot chooses reduction candidate hqa final state let topmost stack element move grow syntax forest applying reduction pop stack symbols execute nonterminal shift move reduction move initial state differs preceding case chosen reduction candidate state initial final reduction applied grow syntax forest parser performs nonterminal shift move nonterminal shift move shift move except shifted symbol nonterminal difference parser read next input token line shift move clearly reorganization removes need cid pointers preserving correctness parsing methods streamlined roperty elr pilot ebn grammar machine net satisfies condition pointerless parser alg equivalent elr parser alg roof two parts proof straightforward check first parsing string stacks parsers contain number stack elements respectively every pair corresponding elements set states included subset set states included alg may discarded candidates furthermore claim next relation pair stack elements position candidate hqa points hpa hqa candidate hpa candidate hqa points hqa equals projection hstate specification reduction moves performs reduction performs reduction example pointerless parser trace parser characterized stack element one machine state plus possibly initial ones therefore parser explores one possible reduction time candidate pointers needed given input string figure shows execution trace pointerless parser input figure parser using cid graphical conventions unchanged compact pilot cell framed denoted etc final candidates encircled etc omitted avoid clogging candidate appears pure machine state course pointers instead initial candidates canceled alg mstate striked observe upon starting reduction initial state active machine might principle show stack elements popped instance first figure top reduction machine pops three stack elements namely one popped last contains striked candidate would initial machine real initial state reduction remains instead unstriked stack element fact popped thus origin shift soon executed also point alg cancel initial candidates stack element shift move executed one candidate see shift move case alg motivation canceling twofold first candidates cause early stop series pop moves reductions may come later said differently break reduction handle instance first figure top shift machine keeps initial candidates parsing methods streamlined string parsed stack contents stack base effect initialisation stack shift reduction shifts reduction shift shift shift reduction shift reduction shifts reduction shift reduction accept without shitting fig steps pointerless parsing algorithm apl candidates canceled shift moves explained algorithm parsing methods streamlined stack shift originates initial candidate candidate instead canceled show striked shift executed soon reduction shift originates candidate second initial candidates may needed subsequent nonterminal shift move sum shown condition permits construct simpler parser reduced stack alphabet need pointers manage reductions parser simplified make another hypothesis grammar ell condition definition deterministically parsable grammar network comes next definition ell machine net meets ell condition following three clauses satisfied derivations net meets elr condition net single transition property condition easily checked drawing graph denoted nodes initial states net edge machine exists edge generally path nonterminals nullable net graph contains circuit actually left recursive derivations cause one situations violation clause def case would remain undetected derivation involving axiom caused form three different types derivations illustrated figure first two cause violations clause third needs checked graph graph contains node would difficult formalize properties illustrated examples restate clause definition follows net derivation form involving axiom using apparently weaker indeed equivalent condition stated beatty definition grammars discussion sum ell grammars defined elr grammars allow derivations satisfy property multiple candidates bases hence convergent transitions historical acronym ell introduced past several authors slight differences hope reusing acronym considered abuse parsing methods streamlined conflict caused derivation makes use derivation nonterminal axiom effect creating two candidates base thus violates clause clauses met derivation entirely contained fig nets violating ell conditions top violates clause middle axiom violates clause bottom axiom undetected clauses precise reformulation definitions ell grammars accumulated half century fair perhaps popular definition grammar contrast marginal contrived examples parsing methods streamlined violation caused presence multiple candidates base hinder parser working deterministically beatty typical case grammar one two candidates base choice alternatives determined following character yet grammar violates easy see necessary condition situation occur language derived nonterminal grammar question consists empty string however case usually removed without penalty loss generality grammar applications fact grammar simplified equivalent grammar obtained complies stack contraction predictive parser last development next presented transforms already compacted pilot graph control flow graph predictive parser way latter parser uses stack differs previous models pointerless parsers precisely terminal shift move always executes push operation sometimes implemented without push sometimes multiple pushes former case happens shift remains inside machine predictive parser push element upon performing terminal shift updates top stack element record new state multiple pushes happen shift determines one transfers current machine others predictive parser performs push transfer essential information kept stack sequence machines activated reached final state reduction occurs parsing time current active machine one analysis current state kept top stack element previous activations machines suspended state suspended machine stack entry needed store state machine resume computation control returned performing relevant reductions main advantage predictive parsing construction syntax tree anticipated parser generate left derivation input parser graph moving considerations first slightly transform compact pilot graph make isomorphic original machine net new graph named parser graph represents blueprint parser code first step transformation splits every contains multiple candidates nodes contain one candidate second nodes coalesced original sets combined one third step creates new edges named call edges whenever machine transfers control another machine last call edge labeled set characters named guide set summary information needed parsing decision transfer control another machine definition parser graph every node denoted identified state machine net parsing methods streamlined denoted without ambiguity moreover every node final consists pair hqa set named set union sets every candidate hqa existing compact pilot graph hqa edges two types named shift call exists shift edge terminal edge machine exists call edge nonterminal possibly differa ent hence necessarily exist candidates hqa contains candidate hra call edge label named guide recursively defined follows holds conditions hold ini nullable ini nullable call edge relations recursive respectively consider generated called starting state follows rel recursive traverses net far chain call sites activated observe rel determines inclusion relation two concatenated call edges also write gui instead next extend definition guide set terminal shift edges dangling darts final nodes terminal shift edge labeled set gui dart tags final node containing candidate hfa set gui edges except nonterminal shifts interpreted conditional instructions enabled current character belongs associated guide set terminal shift edge labeled enabled predicate uniformity call edge labeled represents conditional procedure invocation enabling predicate final node dart labeled interpreted conditional instruction executed remaining edges nonterminal shifts interpreted unconditional instructions show state predicates never conflict one another roperty every node grammar satisfying ell condition guide sets two edges originating disjoint although traditionally word used parsers set definitions differ prefer differentiate names predictive parsing table element aho parsing methods streamlined roof since every machine deterministic identically labeled shift edges originate node remains consider cases edge pairs consider shift edge two call edges first assume contradiction comes rel pilot two base candidates condition ruled comes rel base conflict shift reduction owes path machine comes rel conflict though reduction owes ahead machine previously invoked machine finally may come rel recursive case defers three cases machine immediately invoked machine either violation conflict second assume contradiction since comes one four relations twelve combinations examined since similar previous argumentation deal one namely case clearly violates converse property also holds makes condition disjoint guide sets characteristic property ell grammars see condition checked easily need build elr pilot automaton roperty guide sets disjoint net satisfies ell condition definition proof appendix example running example represented figure layout machine net comparability new nodes node example derive initial candidates excluding containing axiom extracted closure part added nodes closure part nodes except node becomes redundant eliminated node contents said prospect sets needed final states following properties final states initial prospect set coincides corresponding set compact pilot case nodes state prospect set union sets every candidate occurs instance takes solid edges represent shift ones already present machine net pilot graph dashed edges represent call ones labeled guide sets compute next illustrated guide set call edge since state characters shifted guide set edge includes terminals since language nullable ini since call edge goes prospect set parsing methods streamlined compact pilot graph machine network parser graph fig parser graph running example net compact pilot accordance prop terminal labels edges originate node overlap predictive parser straightforward derive parser nodes pushdown stack elements top stack element identifies state active machine inner stack elements refer states suspended machines correct order suspension four sorts moves scan move associated terminal shift edge reads current character corresponding machine would call move associated call edge checks enabling predicate saves stack return state switches invoked machine without consuming return move triggered active machine enters final state whose prospect set includes active state set return state recently suspended machine recognizing move terminates parsing algorithm predictive recognizer stack elements states beginning stack contains initial candidate hqa top element meaning active machine state moves next specified way parsing methods streamlined move shift edge exists scan next character replace stack top hra active machine change move exists call edge let corresponding nonterminal shift edge pop push element hra push element move final state prospect set associated pop move axiom machine final state accept halt case reject string halt prop follows every parsing configuration one move possible algorithm deterministic computing left derivations construct syntax tree algorithm next extended output function thus turning pushdown transducer computes left derivation input string using grammar output actions specified table straightforward need prove correctness moreover recall syntax tree grammar essentially encoding syntax tree original ebn grammar node two child nodes table derivation steps computed predictive parser current character parser move output derivation step scan move transition call move call edge transition return move state example running example trace predictive parser input parsing methods streamlined stack predicate left derivation scan scan scan accept original grammar corresponding derivation parser implementation recursive procedures predictive parsers often implemented using recursive procedures machine transformed syntactic procedure matching corresponding subgraph current state machine encoded runtime program counter parsing starts axiom procedure successfully terminates input exhausted unless error occurred standard runtime mechanism procedure invocation return automatically implements call return moves example suffice show procedure mechanically obtained example recursive descent parser figure syntactic procedures shown figure optimized several ways direct construction parser graph presented justified series rigorous steps lead elr pilot compact parser finally parser graph however human wishing design predictive parser would tedious perform steps therefore provide simpler procedure checking ebn grammar satisfies ell condition procedure operates directly grammar parser control flow graph require construction elr pilot uses set recursive equations defining prospect guide sets states edges equations interpreted instructions compute iteratively guide sets ell check simply verifies according property guide sets disjoint equations defining prospect sets parsing methods streamlined recursive descent parser machine network procedure state next else next state call else error end state next else error end else error end state return else error end end procedure recursive descent parser program ell arser next call accept else reject end end program procedure optimized call end return else error end end procedure fig main program syntactic procedures recursive descent parser fig function next programming interface lexical analyzer scanner function error messaging interface net includes shift edges kind prospect set state net includes nonterminal shift edge corresponding call edge pcfg prospect set initial state machine ini ullable else notice two sets rules apply exclusive way disjoints sets nodes normalization machines disallows initial states equations defining guide sets call edge associated nonterminal shift edge parsing methods streamlined possibly call edges depart state guide set gui call edge defined follows see also conditions ini ullable ini else gui ullable ullable else gui final state guide set tagging dart equals prospect set gui terminal shift edge guide set simply shifted terminal gui computation starts assigning prospect set initial state axiom machine sets initialized empty rules repeatedly applied fixpoint reached notice rules computing prospect sets consistent definition set given section furthermore rules computing guide sets consistent definition parser control flow graph provided section example running example computing prospect guide sets following table shows computation prospect guide sets figure computation completed third step prospect sets guide sets example guide sets grammar grammar example violates single transition property shown figure hence ell verified computing guide sets gui gui gui guide sets edges departing states disjoint tabular parsing parsing methods inadequate dealing nondeterministic ambiguous grammars seminal work earley introduced algorithm parsing methods streamlined recognizing strings grammar even ambiguous one though explicitly present parsing method means constructing potentially numerous parsing trees accepted string later efficient representations parse forests invented duplicate common subtrees thus achieve complexity tree construction algorithm recently aycock borsotti earley parsers performed complete recognition input constructing syntax tree grune jacobs concerning possibility directly using extended grammars earley already gave hints later parser generators implemented authoritative work exists best knowledge another discussion concerns pros cons using since issue scope experimental studies see aycock horspool indicated algorithms faster least programming languages present simpler version use line classical theoretical presentations earley parsers grammars one actually focus present work programming languages nonambiguous formal notations unlike natural languages therefore variant earley algorithm well suited ebn grammars though possibly nondeterministic related procedure building parse trees deal multiple trees forests string recognition algorithm straightforward understand one comes vector stack implementation elr parser section analyzing string algorithm uses vector respectively elements called earley vector every vector element contains set pairs consist state machine nonterminal integer index points back element contains corresponding pair initial state machine index marks position input string currently assumed derivation nonterminal may started introducing variant earley algorithm ebn grammars preliminarily define operations completion terminalshift completion index loop computes closure operation pair launches machine pair add pair element end nested loops compute nonterminal shift operation final pair enables shift nonterminal parsing methods streamlined pair pair shifts nonterminal pair add pair element end end pair added notice completion operation nullable nonterminals dealt combination closure nonterminal shift operations terminalshift index loop computes terminal shift operation preceding pair shifts terminal pair add pair element end algorithm earley syntactic analysis uses completion erminalshif algorithm earley syntactic analysis analyze terminal string possible acceptance define earley vector initialize first elem initialize end completion complete first elem vector finished previous elem empty terminalshift put current elem completion complete current elem end example ebn grammar nullable nonterminal parsing methods streamlined earley acceptance condition following string belongs language earley acceptance condition true figure lists ebn grammar shows corresponding machine net string analyzed syntax tree analysis trace figures respectively edges latter figure ought ignored related syntax tree construction discussed later machine network extended grammar ebnf bba fig ebn grammar network example fig syntax tree string example following lemma correlates presence certain pairs earley vector elements existence leftmost derivation string prefix analyzed parsing methods streamlined fig tabular parsing trace string machine net figure point together associated corollary provides proof correctness algorithm emma holds implies inequality state belongs machine nonterminal holds grammar admits leftmost derivation proof appendix orollary earley acceptance condition satisfied ebnf grammar admits derivation string belongs language following lemma converse lemma states completeness earley algorithm emma take ebnf grammar string length belongs language grammar consider leftmost derivation prefix two points apply holds holds machine net arc grammar admits parsing methods streamlined two leftmost derivations derivation decomposes follows arc net maps rule grammar point split two steps second crucial one holds holds nonterminal axiom also holds prefix also belongs language holds state final axiomatic machine prefix accepted earley algorithm limit cases holds holds holds holds holds cases hold prefix coincides whole string step implies string hypothesis belongs language accepted earley algorithm therefore complete proof appendix syntax tree construction next procedure buildtree builds parse tree recognized string processing vector constructed earley algorithm function recursive four formal parameters nonterminal state two nonnegative indices nonterminal root sub tree built state final machine end computation path corresponds analyzing substring generated indices satisfy inequality respectively specify left right ends substring generated grammar admits derivation earley algorithm accepts string thus element contains final axiomatic pair build tree string root node function called parameters function recursively build subtrees assemble final tree function returns syntax tree form parenthesized string brackets labeled root nonterminal sub tree commented code follows buildtree nonterminal final state return parenthesized string syntax tree rooted node node list terminal nonterminal child nodes either list remain empty filled right left set list child nodes parsing methods streamlined set state machine set index vector walk back sequence term nonterm shift current state initial try backwards recover terminal shift move check node terminal current child leaf net concatenate leaf list end try backwards recover nonterm shift oper check node nonterm current child node net recursively build subtree derivation concatenate list subtree node buildtree end shift current state back drag current index back end return return tree rooted node figure reports analysis trace example also shows solid edges correspond iterations loop procedure dashed edges match recursive calls notice calling function equal indices means building subtree empty string may made one nullable nonterminals happens figure witch shows tree calls returned subtrees example call notice since leftmost derivation uniquely identifies syntax tree conditions two mutually exclusive conditionals inside loop always satisfied one way otherwise analyzed string would admit several distinct left derivations therefore would ambiguous previously remarked nullable terms dealt earley algorithm chain closure nonterminal shift operations optimized version algorithm defined perform analysis nullable terminals single step along parsing methods streamlined fig calls return values buildtree example lines defined aycock horspool work authors also defines optimized procedures building parse tree presence nullable nonterminals also adjusted applied version earley algorithm conclusion hope extension conceptual compaction classical parser construction methods appreciated compiler language designers well instructors starting syntax diagrams readable representation grammars language reference manuals method directly constructs deterministic parsers general accurate preceding proposals extended ebn case beatty old theoretical comparisons versus grammars exploited derive general deterministic parsers stepwise simplifications parsers evidenced simplifications correct derivations excluded completeness address needs ebn grammars included accurate presentation tabular earley parsers including syntax tree generatio goal coming minimalist comprehensive presentation parsing methods extended grammars thus attained finish mention practical development circumstances suggest impose use separate parsers different language parts identified grammar partition sublanguages generated certain subgrammars idea grammar partition dates back korenjak wanted reduce size pilot decomposing original parser family subgrammar parsers thus reduce number candidates parser size reduction remains goal parser partitioning domain natural language processing see meng projects component parsers homogeneous whereas interested heterogeneous partitions use different parsing algorithms one want diversify algorithms used different language parts first language may contain parsing methods streamlined parts harder parse others thus simpler ell parser used whenever possible limiting use elr parser even earley one sublanguages warrant powerful method second examples language embedding sql inside two languages may biased towards different parsing methods although heterogeneous parsers built top legacy parsing programs past experimentation parsers psaila met practical rather conceptual difficulties caused need interface different parser systems unifying approach looks promising building seamless heterogeneous parsers switch algorithm another proceed due homogeneous representation parsing stacks tables exact formulation conditions enable switch less general algorithm within current approach mixed mode parsing little implementation overhead viewed pragmatic technique achieving greater parsing power without committing general less efficient algorithm generalized parsers initiated tomita aknowledgement grateful students formal languages compiler courses politecnico milano bravely accepted study theory progress also thank giorgio satta helpful discussions earley algorithms references ethi llman compilers principles techniques tools prenticehall englewoof cliffs aycock orsotti early action earley parser acta informatica aycock orspool practical earley parsing comput eatty relationship grammars journal acm elentano parsing technique extended grammars computer languages hapman lalr parser generation regular right part grammars acta inf respi eghizzi formal languages compilation springer london respi eghizzi saila grammar partitioning modular deterministic parsing computer languages arley efficient parsing algorithm commun acm note proposed lalr parser extended grammars inf process lett rune jacobs parsing techniques practical guide vrije universiteit amsterdam rune jacobs parsing techniques practical guide springer london eilbrunner definition elr ell grammars acta informatica emerik towards taxonomy ecfg rrpg parsing language automata theory applications dediu ionescu eds lncs vol springer annapinn reconstructing theory eliminate redundance application construction elr parsers german thesis technical university berlin nuth translation languages left right information control nuth syntax analysis acta informatica orenjak practical method constructing processors commun acm onde constructing parsers regular right part grammars acta informatica characterization extended grammars inf process lett parsing methods streamlined ewi laminck uens uybrechts programming methodology compiler construction amsterdam omet formalization transition diagram systems acm eng eng glr parsing multiple grammars natural language queries acm trans asian language information processing june orimoto assa yet another generation lalr parsers regular right part grammars acta informatica introduction formal languages dover new york rosenkrantz tearns properties deterministic parsing information control assa akata simple realization regular right part grammars inf process lett omita efficient parsing natural language fast algorithm practical systems kluwer boston irth algorithms data structures programs publishers englewood cliffs parsing methods streamlined appendix proof theorem let ebn grammar represented machine net let equivalent grammar net meets elr condition grammar meets condition roof let set states clearly rule holds let elr pilots respectively preliminarily study correspondence transition functions helps compare pilot graphs running example figures denoted example figure notice pilot candidates denoted marked rules observe since grammar graph property edges entering identical labels contrast identicallabel property form locality hold therefore possibly split several due special grammar form following mutually exclusive classification exhaustive initial intermediate every candidate form sink reduction every candidate form instance figure intermediate numbered sink reduction say candidate hqx corresponds candidate form hpx sets identical two mstates respectively called correspondent candidates correspond moreover arbitrarily define correspondent initial illustrate running example pairs correspondent following straightforward properties correspondent needed emma mapping defined correspondence relation set containing intermediate set total onto surjective terminal nonterminal symbol correspondent transition defined transition defined intermediate moreover correspondent let state final candidate hfa actually base correspondent contains candidates hpa hfa let state final initial candidate correspondent contains candidate let state final candidate candidate parsing methods streamlined fig pilot graph grammar running example see parsing methods streamlined initial state holds pair correspondent every alternative roof lemma first prove mapping onto assume contradiction hqa correspondent intermediate clearly exists kernels identical grammar derivations sect remains correspondent candidate differ definition set grammar included traditional one easy check def closure function computes exactly sets contradiction item observe bases identical therefore candidate computed computed def matches candidate traditional closure function therefore defined also defined yields intermediate moreover next clearly sink reduction spondent since bases identical hand undefined item consider edge entering state item consider correspondent include predecessor state resp within candidate includes hpa hqa includes hfa hpa hence also includes hfa item candidate thus contains candidate hpb closure hpb therefore intermediate contains candidate hqb base candidate closure converse reasoning analogous item case obvious item consider case initial state results closure operation applied candidate cases initial state results closure operation applied candidate similarly dealt machine includes edge every correspondent state suitable every alternative nonterminal concludes proof lemma part theorem argue violation elr condition implies conflict three cases need examined shift reduce conflict consider conflict hfb final defined lemma items exists correspondent thus proving conflict defined hfb similarly consider conflict final initial defined parsing methods streamlined lemma items exists correspondent thus proving conflict defined reduce reduce conflict consider conflict hfa hfb final item lemma conflict exists hfa hfb similarly conflict hfb final corresponds items conflict hfb item holds true special case convergence conflict consider convergence conflict hpa hqa hra first neither initial state candidates base item correspondent intermediate transition hpa hpa therefore contains two reduction candidates identical conflict quite similarly arbitrarily candidate hpa base necessarily contains candidate closure therefore exists correspondent hence holds part theorem argue every conflict entails violation elr condition shift reduce conflict conflict occurs hfb defined items lemma correspondent contains hfb move defined thus resulting conflict reduce reduce conflict first consider form hfa hfb final item lemma correspondent contains candidates hfa hfb conflict second consider form hfa final items conflict correspondent parsing methods streamlined last consider conflict sink reduction hpa hqa contains candidates hpa exist transition hqa therefore correspondent contains candidate hra transition holds hpa hqa since sink reduction state let call correspondent state initial hpa virtue item initial exists candidate hta notice still machine hpa similar reasoning applies state therefore hpa convergence conflict concludes parts theorem second example illustrate convergence conflicts pilot graph equivalent figure shown figure proof property guide sets disjoint machine net satisfies ell condition definition roof since ell condition consists three properties elr pilot absence left recursion absence multiple transitions absence conflicts prove presence disjoint guide sets implies three conditions hold prove grammar represented net left recursive guide sets disjoint grammar left recursive call edges since holds shift edge gui hence guide sets two edges disjoint notice presence left recursion due rules kind nullable nonterminal ruled kind left recursion leads conflict discussed section shown figure prove presence multiple transitions implies guide sets disjoint done induction starting initial pilot automaton empty base showing reachable pilot automaton satisfy single transition property unless guide sets disjoint also note transitions satisfying lead base singleton set call singleton base say satisfy singleton base property sbp induction base assume exists multiple transition initial closure hence includes candidates parsing methods streamlined sink grammar sink conflict convergence conflict fig part traditional marked rules pilot grammar reducec reduce conflict sink matches convergence conflict edge fig machine net inx cludes transitions pxh pxk terminal symbol nonterminal one let first consider case assume candidate derives candidate possibly iterated closure operation includes call edges inclusions gui gui hold two nona disjoint guide sets gui pxh gui assuming instead candidate derived closure operation distinct paths includes call edges parsing methods streamlined thus following inclusions hold gui gui gui gui therefore two guide sets gui gui disjoint let consider case candidate derives includes call edges well hence ini gui ini gui gui two guide sets gui gui disjoint case candidates deriving common candidate distinct sequences closure operations similarly dealt leads conclusion two call edges originating state guide sets disjoint concludes induction base inductive step consider singleton base multiple transition defined since singleton base two candidates multiple transition originates least one case candidates closure similar one treated base case induction therefore consider case one candidate base state one closure initial state states shift edges call edges ini gui gui gui thus two guide sets gui gui disjoint otherwise includes nonterminal shift edges ryn ryn qyn call edges also two call edges holds gui gui ini two guide sets gui gui disjoint concludes induction prove presence conflicts implies guide sets disjoint assume singleton base hence two conflicting candidates one base point first consider conflicts two cases candidates closure one candidates closure candidates candidates conflicting hence holds gui gui candidate derives candidate sequence closure operations includes chain call edges gui gui therefore gui gui two guide sets disjoint instead candidate obtained candidate distinct paths shown two call edges departing state guide sets parsing methods streamlined disjoint point case one candidate base one closure quite similar one previous point candidate hfa candidates candidates hfa conflicting hence holds gui gui includes call edges whence gui gui two guide sets disjoint consider conflicts three cases arise depending whether conflicting candidates closure one whether one candidate shift reduction two conflicting candidates closure either closure includes candidates includes call edges closure includes three candidates derive distinct chains closure operations first consider linear chain closure operations let first consider case reduction candidate hence final state qxn holds gui gui therefore gui gui two guide sets disjoint let consider symmetric case final state hence reduction candidate holds gui gui therefore gui gui two guide sets disjoint case two candidates derive third one distinct chains closure operations treated similar way leads conclusion two call edges departing state guide sets disjoint two conflicting candidates one base one closure consider two cases arise depending whether reduction candidate closure base similarly point let first consider case base contains candidate hpa closure includes candidates state final hence reduction candidate since final holds ullable gui whence gui gui two guide sets disjoint let consider symmetric case base contains candidate hfa hence hfa reduction candidate closure includes candidates parsing methods streamlined qxn holds gui gui therefore gui gui two guide sets disjoint iii one candidate hpa shift reduce holds gui gui two guide sets edges departing node disjoint concludes proof property proof lemma holds implies inequality state belongs machine nonterminal holds grammar admits leftmost derivation roof induction sequence insertion operations performed vector analysis algorithm base property stated lemma trivially satisfied induction examine three operations terminalshift hqa results terminalshift operation holds hqa well inductive hypothesis hence closure property trivially satisfied nonterminalshift suppose first hqfa qfa inductive hypothesis furthermore hqb inductive hypothesis also since hqb added eventually proved hqb implies qfa reasoning applies terminalshift since derivation generate terminal particular nonterminalshift case hqfa qfa inductive hypothesis rest follows reader may easily complete two remaining proof cases combinations concludes proof lemma parsing methods streamlined proof lemma take ebnf grammar string length belongs language grammar consider leftmost derivation prefix two points apply holds holds machine net arc grammar admits two leftmost derivations derivation decomposes follows arc net maps rule grammar point split two steps second crucial one holds holds nonterminal axiom also holds prefix also belongs language holds state final axiomatic machine prefix accepted earley algorithm limit cases holds holds holds holds holds cases hold prefix coincides whole string step implies string hypothesis belongs language accepted earley algorithm therefore complete roof induction length derivation base since thesis case satisfied taking induction examine cases closure operation adds item hence thesis case holds taking value value value value operation terminalshift adds item hra hence thesis case holds taking value value current derivation step last one string accepted pair hfs inductive hypothesis applying inductive hypothesis following facts hold parsing methods streamlined vector hpc hpb hqa nonterminal shift operation completion procedure adds pair hqb first notice follows next since holds therefore thesis holds taking value value value value situation schematized figure hpc hpb hpa hqb fig schematic trace tabular parsing concludes proof lemma

similarity rasmus pagh ninh pham francesco morten mar university copenhagen denmark abstract present algorithm computing similarity joins based hashing lsh contrast filtering methods commonly suggested method provable subquadratic dependency data size contrast straightforward implementations known algorithms external memory approach able take significant advantage available internal memory whereas time complexity classical algorithms includes factor parameter lsh used complexity algorithm merely includes factor data size size internal memory algorithm randomized outputs correct result high probability simple recursive procedure believe useful also computational settings parallel computation keywords similarity join locality sensitive hashing cache aware cache oblivious introduction ability handle noisy imprecise data becoming increasingly important computing database settings kind capability often achieved using similarity join primitives replace equality predicates condition similarity make precise consider space distance function similarity join sets following given radius compute set problem occurs numerous applications web deduplication document clustering data cleaning applications arise datasets problem scaling similarity join different metric distances getting important challenging many known similarity join techniques prefix filtering positional filtering inverted filtering based filtering techniques research leading results received funding european research council framework programme erc grant agreement part supported university padova project miur italy project amanda working university padova supported danish national research foundation sapere aude program often always succeed reducing computational costs let techniques generally require comparisons worstcase data another approach hashing lsh candidate output pairs generated using collisions carefully chosen hash functions lsh defined follows definition fix distance function positive reals family functions uniformly chosen say monotonic function distance function also say uses space function stored evaluated using space lsh able break barrier cases constant number pairs large words many pairs distance within factor threshold reason pairs likely become candidates yet considering contribute output notational simplicity talk far pairs distance greater reported near pairs distance reported pairs distance reported lsh provides collision guarantees contribution paper study similarity join methods based lsh interested minimizing number operations block points transferred external memory internal memory capacity points main result first algorithm similarity join provably dependency data size time inverse polynomial dependency essence previous methods overhead factor either obtain overhead parameter lsh employed strictly improving show theorem consider let assume log exists monotonic family functions respect distance measure using space let log log exists randomized algorithm computing probability using hides polylog factors conjecture bound theorem close best possible class signature based algorithms work generating set lsh values monotonic family checking pairs collide conjecture based informal argument given full section describe input seems significant advances required beat theorem asymptotically observe bound coincides optimal bound reading input bound coincides bounds best known internal memory algorithms worth noting whereas methods literature focus single distance measure method works arbitrary space distance measure allows lsh hamming manhattan euclidean jaccard angular metric distances since approach makes use lsh black box problem reporting complete join result certainty would require major advances lsh methods see recent progress direction primary technical hurdle paper use kind strong concentration bounds number points particular value since hash values lsh family may correlated definition another hurdle duplicate elimination output stemming pairs multiple lsh collisions however context algorithms natural require listing near pairs rather simply require algorithm enumerates near pairs precisely algorithm calls near pair function emit natural assumption external memory since reduces complexity addition desired many applications join results intermediate results pipelined subsequent computation required stored external memory upper bound easily adapted list instances increasing complexity unavoidable additive term organization organization paper follows section briefly review related work section describes algorithms including approach main results solution analysis randomized approach remove duplicates section provides discussions algorithms real datasets section concludes paper related work section briefly review lsh computational model similarity join techniques hashing lsh lsh originally introduced indyk motwani similarity search problems high dimensional data technique obtains sublinear time complexity increasing gap collision probability near points far points using lsh family defined definition gap collision probability polynomial exponent log log dependent worth noting standard lshs metric distances including hamming jaccard angular distances monotonic common lshs use space comparable required store point except lsh requires space explicitly require hash values particularly small however using universal hashing always map small bit strings introducing new collisions high probability thus assume hash values fit one memory block computational model study algorithms similarity join external memory model widely adopted literature see survey vitter external memory model consists internal memory words external memory unbounded size processor access data stored internal memory move data two memories blocks size simplicity measure block internal memory size units points contain points points respectively complexity algorithm defined number blocks moved two memories algorithm approach makes explicit use parameters achieve complexity whereas one explicitly use model parameters latter approach desirable implies optimality levels memory hierarchy require parameter tuning executed different physical machines note model assumes internal memory ideal sense optimal cachereplacement policy policy evict block used furthest future place block anywhere cache full associativity similarity join techniques review similarity join techniques closely related work similarity join popular approach make use indexing techniques build data structure one relation perform queries using points relation indexes typically perform kind filtering reduce number points given query point compared see indexing space consuming particular lsh context similarity join big concern since many queries thus afford construct hash table fly hand clear similarity join techniques able take significant advantage internal memory indeed query complexity stated thus complexity using indexing similarity join high indexing technique adapted compute similarity joins efficiently using fact many points looked hash tables means lookups done batched fashion using sorting results dependency parameter lsh family generic joins close improved using general join operators optimized case easy see integer nested loop join requires algorithm make use following result cacheoblivious nested loop joins theorem luo given similarity join condition join relations computed algorithm number suffices generate result memory may suffice write disk note similarity join part join involving two relations class acyclic joins variables compared join conditions organized tree structure one initially apply full reducer removes tuples part output efficiently reduces acyclic join sequence binary joins handling cyclic joins much harder see outside scope paper algorithms section describe efficient algorithms start section algorithm uses lsh family value collision probability set function internal memory size section presents main result recursive algorithm uses lsh approach make assumption value collision probability section describes analysis section shows reduce expected number times emitting near pairs algorithm asimjoin describe simple algorithm called asimjoin achieves worst case bounds stated theorem asimjoin relies family hash functions following properties suitable value given arbitrary monotonic family family built concatenating hash functions simplicity assume integer thus probabilities exactly obtained nevertheless algorithm analysis extended algorithm asimjoin input sets repeat log times associate point counter initially set repeat times choose uniformly random use partition buckets points hash value hash value generated previous step simplicity assume split chunks size every chunk load memory every chunk load memory compute emit near pairs far pair increment associated counters remove points associated counter larger write back external memory write back external memory general case increasing complexity factor worst case practical scenarios factor small constant asimjoin assumes point associated counter initially set counter thought additional dimension point hash functions comparisons take account algorithm repeats times following procedure hash function randomly drawn family used partitioning sets buckets points hash value let denote buckets respectively containing points hash value algorithm iterates every hash value hash value uses double nested loop generating pairs points double nested loop loads consecutive chunks size outer loop runs smaller set say inner one runs larger one say pair algorithm emits pair increases counters associated ignores pair every time counter point exceeds point considered far away points removed bucket chunks moved back memory needed entire asimjoin algorithm repeated log times find near pairs high probability following theorem shows bounds approach theorem consider let sufficiently large assume exists monotonic family functions respect distance measure suitable value probability asimjoin algorithm enumerates near pairs using proof observe cost steps according hash function sort repetitions consider cost iteration loop step given hash value size one bucket say smaller able load whole internal memory load consecutive blocks execute join operations hence cost step total cost log iterations step among possible hash values least one bucket size smaller cost step buckets larger means amortized cost pair therefore amortized cost iterations step bucket size less upper bounded multiplying total number generated pairs based observation classify enumerate generated pairs three groups near pairs pairs far pairs denote ccn respective size group upper bound quantities derive proof number near pairs definition lsh gives lower bound probability collision near pairs may happen collision probability near pairs thus two near points might collide repetitions step log repetitions step means log note bound deterministic worst case bound number pairs pair appears bucket probability due monotonicity lsh family since repetitions pair collides expectation words expected number pair collisions among repetitions using chernoff bound exercise log independent repetitions step ccn log ccn log let sort log shorthand complexity sorting points number far pairs far away points expected number collisions hash table including duplicates since point removed step hence total number examined far pairs log therefore summing number near pairs pairs ccn far pairs multiplying quantities amortized complexity upper bound cost iterations step buckets size less probability least summing previous bounds get claimed bound high probability analyze probability enumerate near pairs consider one iteration step near pair emitted least one following events happen two points collide bucket iterations step happens probability one two points removed step collides far points markov inequality since far points probability collides least points iterations event happens probability therefore near pair collide one iteration step probability never collides log iterations probability log union bound follows near pairs collide probability least theorem follows already mentioned introduction near pair emitted many times algorithm since points hashed value rounds step denotes actual collision probability simple approach avoiding duplicates following near pair found iteration step pair emitted two points collide hash functions used previous rounds check starts hash function used previous round backtracks collision found hash functions approach increases worst case complexity factor section shows efficient randomized algorithm reduces number replica per near pair constant technique also applies algorithm described next section algorithm osimjoin input sets recursion depth swap references sets compute using algorithm theorem return pick random sample points points compute containing points distance smaller least half points compute using algorithm theorem repeat times choose uniformly random use partition buckets points hash value nonempty recursively call osimjoin algorithm osimjoin algorithm uses family functions partitioning initial sets smaller buckets efficiently processed internal memory using nested loop algorithm know internal memory size lsh family constructed concatenating hash functions given primitive family without knowing setting family built therefore propose osimjoin algorithm efficiently computes similarity join without knowing internal memory size block length osimjoin uses given monotonic family value considered constant practical scenario common settings use recursive approach splitting problem smaller smaller subproblems point fit internal memory although point known algorithm first give high level description algorithm intuitive explanation provide detailed description analysis osimjoin receives input two sets similarity join parameter denoting depth recursion tree initially used recognizing base case let denote log two global values kept invariant recursive levels computed using initial input size simplicity assume integers assume without loss generality initial size power two note monotonicity requirement relaxed following every two pairs monotonic lsh family clearly satisfies assumption integer last iteration step performed random variable osimjoin works follows problem currently recursive level recursion ends problem solved using nested loop described theorem otherwise following operations executed exploiting sampling algorithm identifies subset containing almost points near constant fraction points steps compute using cacheoblivious theorem remove points step subsequently algorithm repeats times following operations hash function extracted family used partitioning buckets denoted hash value steps join computed recursively osimjoin step explanation approach following recursively partitioning input points hash functions algorithm decreases probability collision two far points particular collision probability far pair recursive level hand repeating partitioning times level algorithm guarantees near pair enumerated constant probability since probability near pair collide recursive level deserves noticed collision probability far near pairs recursive level respectively asymptotically equivalent values algorithm words partitioning points level equivalent one algorithm collision probability far pair finally observe point becomes close many points efficient detect remove instead propagating base cases due fact collision probability near pairs always large close algorithm able split subproblems fit memory complexity correctness osimjoin analysis complexity bound expected number algorithm rather worst case converted high probability bound running log parallel instances algorithm without loss generality assume optimal cache replacement splits cache log parts assigned instance total execution stops first parallel instance terminates probability least within logarithmic factor expected bound logarithmic factors absorbed notational simplicity section let denote initial input sets let denote subsets given input particular recursive subproblem note due step denote subset also similarly also let denote sampling step subset computed step lemma says two properties choice random sample step almost certain proof relies chernoff bounds choice remainder paper assume lemma holds refer event holding probability lemma probability least random choices step following bounds hold every subproblem osimjoin proof let point one sixth points point enters least cnear points happens chernoff bound theorem probability point appears subproblems points therefore probability every subproblem osimjoin point points hence point least points bound equation follows similarly show probability every subproblem osimjoin points least points point far points equation follows analyze number subproblems size bound cost terms different types collisions pairs end subproblem recursion say particular subproblem osimjoin observe pair subproblem colliding hash values every step call path initial invocation osimjoin definition given let number times pair call osimjoin level recursion also let log denote number times pair call osimjoin level recursion smallest input set size log log count pairs multiplicity several subproblems level counted next bound complexity osimjoin terms later upper bound expected size quantities lemma get claim theorem lemma let log given holds complexity osimjoin log proof ease analysis assume internal memory used store blocks containing elements respectively since model assumes optimal cache replacement policy decrease complexity also internal memory space used things data input output buffers recursion stack size less assumption log consequence number solving subproblem osimjoin including recursive calls space dedicated input sets reading input required charging cost subproblems writing inputs parent problem focus subproblems largest set size notice cost steps dominated costs assumption set fits internal memory implies suffices scan data implement steps cost clearly negligible respect remaining steps thus ignore first provide upper bound complexity required subproblems recursive level let osimjoin recursive call recursive level cost loop join step osimjoin theorem ignore term since asymptotically negligible respect cost iteration step upper bounded later pairs thus equation contains cost step osimjoin means bound total cost executions step level recursion since near pair appears subproblems level second major part complexity cost preparing recursive calls osimjoin steps fact iteration step cost includes cost hashing sorting form buckets since point replicated subproblems step point initial sets replicated times level since average cost per entry total cost preparing recursive calls level summing terms total complexity subproblems recursive level upper bounded focus analysis bound complexity required subproblems recursive level let osimjoin recursive call recursive level observe part cost subproblem level upper bounded suitable function collisions among far points osimjoin specifically consider iteration step subproblem level cost preparing recursive calls performing step subproblem level generated iteration upper bounded since near pair found step one subproblem level generated iteration since easily get bound ten min observe bound holds even case cost includes required solving subproblems level called iteration solved using nested loop theorem see step lemma quantity upper bounded number far collisions getting min recall denotes number times pair call osimjoin level recursion smallest input set size log log total cost preparing recursive calls steps subproblems level performing step subproblems level log factor bound follows since far collisions level used amortizing cost step one iterations step get total complexity algorithm sum complexity required recursive level bound cost level follows level use bound equation level use bound note true input size subproblem however expected value computed assuming worst case close pairs thus equation level use bound given equation add first term equation since cost step level included equation note addition equations gives weak upper bound level lemma follows analyze expected sizes terms lemma clearly pair top level call number collisions lower levels show expected number times pair collides either decreases increases geometrically depending whether collision probability smaller larger equivalently depending whether distance greater smaller radius lemma follows expressing number collisions pairs recursive level branching process lemma given holds log proof let interested upper bounding number collisions pair recursive level envision problem branching process specifically galtonwatson process see expected number children recursive calls preserve particular collision random standard fact theory expected population size generation number times problem recursive level theorem far pair appears times expectation level follows equation moreover since probability collisions monotonic distance follow equations order get last bound observe entry replicated total times level thus maximum number far collisions subproblems level smallest input set size one collisions survives level probability thus expected number collisions ready prove complexity osimjoin claimed theorem linearity expectation lemma get expected complexity osimjoin log note log plugging bounds expected number collisions given lemma get claimed result analysis correctness following lemma shows osimjoin outputs probability near pairs claimed theorem lemma let executing dent repetitions osimjoin outputs probability least argue pair output probability log let number subproblems level containing applying branching process get fact positive constant probability survives indefinitely extinct since every branch recursion eventually compare points collide hash functions path root call implies reported positive constant probability critical case need consider variance theorem equal variance number children hash collisions recursive calls integer number children branching process follows binomial distribution mean implies also case integer easy see variance bounded var chebychev inequality means integer var since since implies furthermore recursion depth log implies probability near pair found log thus repeating times make error probability particular pair entire output applying union bound removing duplicates given two near points definition lsh requires collision probability osimjoin algorithm emit many times example suppose algorithm ends one recursive call pair expected bucket iterations step thus emitted times expectation moreover pair emitted first recursive level expected number emitted pairs increases since pair contained subproblems recursive level simple solution requires store emitted near pairs external memory using sorting algorithm removing repetitions expected however approach requires average replication emitted pair dominate complexity osimjoin similar issue appears algorithm asimjoin well near pair emitted times since recursion partitioning two input sets repeated times collision probability explicitly computed time pair possible emit near pair expectation without storing near pairs external memory note collision probability computed many metrics including hamming jaccard angular distances algorithm approach following near pair found recursive level pair emitted probability otherwise ignore algorithm idea near pair emitted probability theorem approaches guarantee near pair emitted constant probability asimjoin osimjoin proof claim easily follows algorithm indeed two points near pair hash value expectation repetitions step therefore emitting pair probability get claim focus algorithm claim requires articulated proof given near pair let random variables denoting respectively number subproblems level containing pair number subproblems level found nested loop join algorithm theorem let also random variable denoting actual number times pair emitted level followings properties since near pair emitted probability subproblems pair found join algorithm since near pair bucket probability follows previous analysis based standard branching since pair exists beginning algorithm since pair surviving last recursive level found nested loop join algorithm interested upper bounding induction first call osimjoin note equality verified since consider generic level since pair propagates lower recursive level probability thus exploiting inductive hypothesis get since claim follows observe proposed approach equivalent use lsh near pair finally remark approach avoid replica near pair algorithm repeated increasing collision near pairs thus probability emitting probability pair least shown second part section repetitions osimjoin suffices find pairs high probability however expected number replica given near pair becomes even proposed approach discussion argue informally complexity theorem close optimal simple arguments split complexity algorithms two parts argue necessary first notice need per hash function transferring data memories computing writing hash values disk find collisions second since brings points order compute distance points residing internal memory need examine pairs means collision probability far pairs number collisions far pairs expectation need detect far pairs consider case pairs distance due monotonicity lsh family collision probability pair must ensure suffices examine pairs turn means collision probability near pairs within distance must need repetitions different hash functions expect least one collision near pair data set given might need examine hash functions constant fraction pairs whose collision probability constant example happen include two clusters near points one could speculate pairs could marked finished computation compute distances however seems hard make idea work arbitrary distance measure may little structure output set hence additional per repetition needed order argue term needed consider case pairs distance value small enough make collision probability pair distance indistinguishable collision probability pair distance every pair must brought internal memory ensure correct result requires holds algorithm enumerating listing near pairs therefore exist algorithm beats quadratic dependency input sets unless distribution input known beforehand however subquadratic regarding potential approach achieve subquadratic dependency expectation similarity join problem filtering invalid pairs based distances currently method way cdf pairwise distances cdf pairwise similarities jaccard similarity cosine similarity distance distance jaccard cosine similarity distance enron email dataset mnist dataset fig cumulative distributions pairwise similarities pairwise distances samples points enron email mnist datasets note values decrease figure increase figure note cost would expect since reading input optimal extreme bound matches time complexity internal memory techniques bounded algorithm achieves subquadratic dependency assumption realistic datasets shown experimental evaluation section complement discussion evaluate complexity computing explicit constants evaluating total number spent analyzing real datasets performing simulated experiments advantage real experiments impacted properties physical machine split complexity algorithms two parts carry experiments demonstrate first term often dominates second term real datasets particular depict cumulative distribution function cdf scale pairwise distances pairwise similarities jaccard cosine two commonly used datasets enron shown figure since enron data set fixed data size per point consider version data set dimension reduced vector fixed size https http data set metric enron jaccard enron cosine mnist standard lsh nested loop asimjoin fig comparison cost similarity joins standard lsh nested loop asimjoin algorithms figure shows inverse polynomial relationship small exponent similarity threshold number pairwise similarities greater degree polynomial particularly low setting commonly used many applications jaccard cosine similarities similarly figure also shows monomial relationship distance threshold number pairwise distances smaller turn means number pairs much greater words second term often much smaller first term finally data sets metrics simulated algorithm explicit constants examined cost compare standard nested loop method section lower bound standard lsh method section set cache size reasonable judging number cache misses since size ratio cpu caches ram order magnitude general setting allows investigate happens data size much larger fast memory simplicity use since methods contain multiplicative factor complexity values computed using good lsh families specific metric parameters parameters picked according figure number pairs order magnitude larger number near pairs complexity used nested loop join assume sets size complexity standard lsh approach lower bounded sort complexity lower bound standard sorting based approaches lacks additional cost depends lsh distributes points since bound sorting complexity use sort since points read written twice complexity approach stated theorem computed figure show complexity algorithm lower instances examined nested loop suffers quadratic dependency standard lsh bounds lack dependency overall cost indicates algorithm practical examined data sets conclusion paper examine problem computing similarity join two relations external memory setting new algorithm section algorithm section improve upon current state art around factor unless number pairs huge believe first algorithm similarity join importantly first subquadratic algorithm whose performance improves significantly size internal memory grows would interesting investigate approach also practical might require adjusting parameters bound probably easy improve significantly interesting open problems remove error probability algorithm improve implicit dependence dimension note work assumes simplicity unit number points general may get tighter bounds taking account gap space required store point space hash values also result paper made general spaces mind interesting direction examine dependence dimension could made explicit improved specific spaces references alexandr andoni piotr indyk hashing algorithms approximate nearest neighbor high dimensions proceedings focs pages arvind arasu venkatesh ganti raghav kaushik efficient exact joins proceedings vldb pages roberto bayardo yiming ramakrishnan srikant scaling pairs similarity search proceedings www pages andrei broder steven glassman mark manasse geoffrey zweig syntactic clustering web computer networks moses charikar similarity estimation techniques rounding algorithms proceedings stoc pages surajit chaudhuri venkatesh ganti raghav kaushik primitive operator similarity joins data 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jan completion derived double centralizer marco porta liran shaul amnon yekutieli abstract let commutative ring let weakly proregular ideal noetherian ideal weakly proregular suppose compact generator category cohomologically complexes prove derived double centralizer isomorphic completion proof relies mgm equivalence psy derived morita equivalence result extends earlier work dgi efimov introduction let commutative ring denote mod derived category given mod define exta homd mod graded yoneda multiplication call ext algebra suppose choose resolution resulting aalgebra enda called derived endomorphism algebra turns see proposition algebra unique quasil isomorphism course cohomology canonically isomorphic exta graded consider derived category dgmod left view object dgmod thus like get graded aalgebra extb derived double centralizer algebra corollary graded algebra extb independent resolution isomorphism let ideal functor right derived giving triangulated functor mod complex mod called cohomologically complex canonical morphism isomorphism full triangulated category cohomologically torsion complexes denoted mod known finitely generated category mod compactly generated instance koszul complex associated finite generating sequence date december key words phrases adic completion derived functors derived morita theory mathematics subject classification primary secondary research supported israel science foundation center advanced studies bgu marco porta liran shaul amnon yekutieli completion commutative let denote finitely generated ring ideal complete weakly proregular sequence finite sequence elements whose koszul cohomology satisfies certain vanishing conditions see definition concept introduced ajl schenzel ideal called weakly proregular generated weakly proregular sequence important note noetherian finite sequence weakly proregular ideal weakly proregular fairly natural examples see ajl example psy example main result repeated theorem body paper theorem let commutative ring let weakly proregular ideal let compact generator mod choose resolution define enda unique isomorphism graded extb result extends earlier work dgi efimov see remark discussion let say words proof theorem use derived morita theory find isomorphism graded algebras extb exta necessary facts derived morita theory recalled section use mgm equivalence recalled section prove exta exta acknowledgments wish thank bernhard keller john greenlees alexander efimov maxim kontsevich vladimir hinich peter helpful discussions also grateful anonymous referee careful reading paper constructive remarks weak proregularity mgm equivalence let commutative ring let ideal assume noetherian complete two operations associated data completion completion element called element elements form submodule let denote mod category additive functors mod functor left exact whereas neither left exact right exact called complete canonical homomorphism bijective texts would say complete separated canonical homomorphism bijective ideal finitely generated functor idempotent namely module completion complete counterexamples infinitely generated ideals see example derived category mod denoted mod derived functors mod mod completion derived double centralizer exist left derived functor constructed using resolutions right derived functor constructed using resolutions means complex canonical morphism isomorphism complex canonical morphism isomorphism relationship derived functors first studied ajl duality established following paper complex mod called cohomologically complex canonical morphism isomorphism complex called cohomologically complete complex canonical morl phism isomorphism denote mod mod full subcategories mod consisting cohomologically atorsion complexes cohomologically complete complexes respectively triangulated subcategories little said functors corresponding triangulated categories mod mod general however know lot ideal weakly proregular defining weak proregularity talk koszul complexes recall element koszul complex concentrated degrees given finite sequence elements koszul complex associated sequence complex finitely generated free concentrated degrees canonical isomorphism ideal generated sequence let canonical homomorphism complexes corresponds surjection thus every get inverse system transition homomorphisms course inverse limit equals completion turns crucial behavior inverse system details please see psy section inverse system abelian groups transition maps called every exists zero definition let finite sequence sequence called weakly proregular sequence every inverse system ideal called weakly proregular ideal generated weakly proregular sequence etymology name weakly proregular sequence history related concepts explained ajl marco porta liran shaul amnon yekutieli regular sequence weakly proregular important following result theorem ajl noetherian every finite sequence weakly proregular every ideal weakly proregular another useful fact theorem let weakly proregular ideal ring finite sequence generates weakly proregular theorems repeated different proofs psy theorem psy corollary respectively next theorem shows weak proregularity correct condition derived torsion functor suppose finite sequence generates ideal consider infinite dual koszul complex lim homa given complex canonical morphism mod theorem sequence weakly proregular iff morphism isomorphism every mod following theorem psy theorem plays central role work theorem mgm equivalence let commutative ring weakly proregular ideal mod one mod mod functor mod mod equivalence remark slightly weaker versions theorem appeared previously ajl theorem theorem difference earlier results assumed ideal generated sequence weakly proregular moreover bounded torsion extra condition certainly holds noetherian sake convenience present paper quote psy regarding derived completion torsion tacitly understood noetherian case results ajl suffice derived double centralizer section define derived double centralizer module see remarks discussion concept related literature let commutative ring let associative unital necessarily commutative given left completion derived double centralizer denote homia homomorphisms degree get homa homia usual differential object enda homa since left actions enda commute see left module algebra enda category left denoted dgmod set morphisms homdgmod precisely set homa note dgmod abelian category let dgmod homotopy category dgmod dgmod homa derived category dgmod gotten inverting dgmod categories dgmod dgmod triangulated happens ring dgmod mod category complexes mod dgmod mod usual derived category dgmod define extia dgmod exta extia definition let dgmod define exta exta graded yoneda multiplication composition morphisms dgmod call exta ext algebra canonical homomorphism graded enda exta either homomorphism bijective definition let choose resolution dgmod enda called derived endomorphism algebra note isomorphisms graded exta exta dependence derived endomorphism algebra enda resolution explained next proposition proposition let let resolutions dgmod define enda enda marco porta liran shaul amnon yekutieli proof choose dgmod lifting given quasiisomorphisms done course let cone dgmod mapping cone graded differential viewed degree homomorphism course acyclic module takeh let triangular matrix graded algebra obvious matrix multiplication makes sense canonical isomorphism algebras enda note subalgebra enda make algebra differential denda projections diagonal entries algebra kernels acyclic complexes homa homa respectively restriction functor dgmod dgmod likewise consider exact sequence dgmod induced distinguished triangle dgmod acyclic isomorphism finally let choose resolution dgmod induces dgmod corollary situation proposition isomorphism graded extb extb proof since algebras follows restriction functor dgmod dgmod equivalence triangulated categories therefore get induced isomorphism graded algebras extb extb similarly get graded isomorphism extb extb definition let let resolution dgmod graded extb called derived double centralizer remark uniqueness graded extb provided corollary sufficient purposes paper see theorem possible show detailed calculation isomorphism provided corollary fact canonical depend choices made proof proposition let choose resolution dgmod define algebra endb called double endomorphism algebra course extb graded algebras canonical algebra homomorphism tried work comprehensive treatment derived endomorphism algebras iterates using methods get far hence included paper expect full treatment possible terms completion derived double centralizer remark derived endomorphism algebras double derived ones treated several earlier papers including dgi papers mention uniqueness properties algebras indeed far tell pick convenient resolution work algebra enda subsection dgi issue briefly discussed detailed treatment derived endomorphism algebras know keller paper section concept lift module introduced pair definition called standard lift proved lifts unique basically done proposition statement regarding uniqueness also discussion derived double centralizers supplement derived morita equivalence derived morita theory goes back rickard paper dealt rings tilting complexes generalizations found purposes section need know certain precise details derived morita equivalence case algebras compact generators specifically formula functor appearing theorem hence give full proof let triangulated category infinite direct sums recall object called compact small collection objects canonical homomorphism home home bijective object called generator nonzero object home section consider commutative ring next lemma seems known could find reference lemma let triangulated category infinite direct sums let dgmod triangulated functors commute infinite direct sums let morphism triangulated functors assume isomorphism isomorphism proof suppose given distinguished triangle dgmod two three morphisms isomorphisms third also isomorphism since functors commute shifts direct sums since isomorphism follows isomorphism free next consider module choose gives rise exhaustive ascending filtration submodules every distinguished triangle dgmod inclusion since free module induction conclude isomorphism every marco porta liran shaul amnon yekutieli telescope construction see remark gives distinguished triangle shows isomorphism finally module admits semifree therefore isomorphism let full triangulated subcategory dgmod closed infinite direct sums let fix resolution dgmod let enda derived endomorphism algebra definition since dgmod triangulated functor dgmod dgmod calculated resolutions dgmod warning usually functor commutes infinite direct sums dgmod therefore every dgmod triangulated functor dgmod dgmod homa dgmod lemma functor dgmod commutes infinite direct sums compact object proof know dgmod rhoma functorially dgmod compact relative functors commute direct sums asking commute direct sums theorem let let full triangulated subcategory dgmod closed infinite direct sums let compact generator choose resolution dgmod define enda functor dgmod equivalence triangulated categories functor proof let write dgmod etc begin proving functors adjoints take construct bijection homd homd completion derived double centralizer bifunctorial choose resolution dgmod since sequence isomorphisms homd rhoma homb homa homa rhomb homd choice made resolution bifunctorial corresponding morphisms denoted respectively next prove fully faithful showing every morphism isomorphism know factors via full subcategory therefore using lemma know functor commutes infinite direct sums lemma suffices check case canonical homomorphism homa clearly bijective remains prove essential image functor take consider distinguished triangle mapping cone applying using get distinguished triangle therefore rhoma therefore homd every since generator get hence isomorphism essential image main theorem interpretation completion appearing efimov recent paper attributed kontsevich remark comparison similar results recent literature setup section commutative ring weakly proregular ideal assume completion noetherian complete let let ideal since ideal finitely generated complete hence ring follows complete full subcategory mod mod triangulated closed infinite direct sums results sections invoked recall koszul complex associated finite sequence see section bounded complex free hence next result proved several authors see proposition corollary proposition proposition let finite sequence generates koszul complex compact generator mod course compact generators mod theorem let commutative ring let weakly proregular ideal let compact generator mod choose resolution mod let enda extib unique isomorphism marco porta liran shaul amnon yekutieli recall derived endomorphism algebra definition graded extb derived double centralizer definition need lemmas proving theorem lemma let compact object mod also compact mod perfect complex proof choose finite sequence generates psy corollary isomorphism functors infinite dual koszul complex therefore functor commutes infinite direct sums let mod consider function hom homd mod homd mod given morphism mod define since functor idempotent theorem function inverse hom latter bijective let collection objects mod due fact compact object mod observations get isomorphisms homd mod homd mod homd mod homd mod homd mod see also compact mod consider contravariant functor mod mod defined choosing injective resolution letting homa lemma functor induces duality contravariant equivalence full subcategory mod consisting objects perfect full subcategory mod consisting objects perfect proof take mod perfect enough show canonical homomorphism homa homa forget structure view homomorphism choose resolution bounded complex finitely generated projective replace equation replace clear lemma let complexes write completion derived double centralizer isomor morphisms phisms homomorphism homd mod hom homd mod bijective proof morphism isomorphism psy proposition theorem complex cohomologically complete therefore also cohomologically complete means isomorphism take morphism mod part know isomorphisms define function inverse hom proof theorem shall calculate extb indirectly lemma know hence also perfect according lemma isomorphism graded extb extb next note homa homa dgmod functor therefore get isomorphism graded extb extb let mod claim dgmod see first note canonical morphism mod represented actual module homomorphism say replacing resolution consider induced homomorphism homa homa like proof lemma suffices show true since duality psy theorem canonical morphism rhom rhoma rhoma mod isomorphism conclude graded isomorphism extb extb take mod theorem since dgmod equivalence full mod see induces isomorphism graded exta extb next step use mgm equivalence know mod functor induces isomorphism graded exta exta marco porta liran shaul amnon yekutieli lemma homoit remains analyze graded exta morphism hom homd mod rhomd mod bijective every therefore extia homomorphism exta bijective combining steps see extib bop commutative bop isomorphism regarding uniqueness since image ring homomorphism dense complete follows automorb identity therefore isomorphism phism produced unique remark explain surprising theorem take case koszul complex associated sequence generates ideal free forgetting grading differential grading depends exterior algebra differential place sequence enters similarly algebra enda graded matrix algebra size differential expressed forgetting differentials working graded classical morita theory tells endb graded furthermore projective even extb however theorem tells structure thus get transcendental outcome completion extb homological operation finite input basically finite linear algebra together differential remark motivation work completion derived double centralizer came looking recent paper efimov main result theorem completion category qcoh noetherian scheme along closed subscheme idea attributed kontsevich corollary special case theorem extra assumptions ring noetherian regular finite global cohomological dimension writing first version paper learned similar result proved dgi paper authors continue work derived completion torsion main result theorem combination mgm equivalence derived morita equivalence abstract setup includes algebra topology manifestation main result commutative algebra dgi proposition also special case theorem ring noetherian quotient ring regular recall theorem requires ideal weakly proregular regularity condition rings word regular double meaning quite possible methods dgi pushed remove regularity conditions rings completion derived double centralizer however less likely methods handle case assuming ideal weakly proregular references ajl dgi psy alonso jeremias lipman local homology cohomology schemes ann sci ens correction availabe online http bokstedt neeman homotopy limits triangulated categories compositio math bondal van den bergh generators representability functors commutative noncommutative geometry moscow math dwyer greenless complete modules torsion modules american math dwyer greenlees iyengar duality algebra topology advances math efimov formal completion category along subcategory eprint http greenlees may derived functors completion local homology algebra recollement differential graded algebras algebra keller deriving categories ann sci ecole norm sup lipman neeman boundedness twisted inverse image functor illinois math number porta shaul yekutieli homology completion torsion appear algebras representation theory rickard derived equivalences derived functors london math soc rouquier dimensions triangulated categories journal schenzel proregular sequences local cohomology completion math scand yekutieli flatness completion infinitely generated modules noetherian rings comm algebra issue department mathematics ben gurion university sheva israel address porta shaul shlir yekutieli amyekut

ndt neual decision tree towards fully functioned neural graph han xiao dec abstract though traditional algorithms could embedded neural architectures proposed principle xiao variables occur condition branch could updated special case tackle issue multiply conditioned branches dirac symbol approximate dirac symbol continuous functions way gradients variables could worked process approximately making fully functioned neural graph within novel principle propose neural decision tree ndt takes simplified neural networks decision function branch employs complex neural networks generate output leaf extensive experiments verify theoretical analysis demonstrate effectiveness model introduction inspired brain science neural architectures proposed mcculloch pitts branch artificial intelligence develops single perception casper deep complex network lecun achieving several critical successes alphago silver notably operators matrix multiply function convolution etc traditional neural networks numerical continuous could benefit algorithm rumelhart recently methods hungarian algorithm algorithm searching embedded neural architectures dynamically manner opening new chapter intelligence system xiao state key laboratory intelligent technology systems national laboratory information science technology department computer science technology tsinghua university beijing china correspondence han xiao proceedings international conference machine learning stockholm sweden pmlr copyright author generally neural graph defined intelligence architecture characterized logics neurons proposed principle seminal work attempt tackle image classification specifically regarding task overfull categories make much burden classifiers normal issue datasets imagenet deng conjecture would make effects roughly classify samples decision tree category corresponding samples strong neural network leaf leaf much fewer categories predict attribute split traditional decision trees random forest etc oversimplified precise zhou feng thus propose method neural decision tree ndt applies neural network decision function strengthen performance regarding calculus procedure ndt basic principle treat logic flow sense programming language dynamic process illustrated figure figure demonstrates classification four categories sun moon car pen structure employed split samples two branches fully connected networks generate results respectively forward propagation methodology activates branch according condition dynamically constructs graph according instructions activated branch way calculus graph constructed continuous structure backward propagation could performed conventionally demonstrated figure generally note repeat could treated performing multiple times could also tackled proposed principle thus traditional algorithms could embedded neural architectures details please refer xiao however special challenge paper variables introduced condition branch could updated backward propagation outside dynamically constructed graph example figure thus make completely functioned neural graph paper attempts tackle issue ndt neual decision tree towards fully functioned neural graph cantly illustrates effectiveness methodology important conclusion model differentiable verifies theory provides novel methodology fully functioned neural graph contributions complete principle neural graph characterizes intelligence systems logics neurons also provide proof neural graph turing complete makes learnable turing machine theory computation tackle issue overfull categories propose method neural decision tree ndt takes simplified neural networks decision function branch employs complex neural networks generate output leaf model outperforms baselines extensively verifying effectiveness theory method organization section methodology neural architecture discussed section specific implementation fully functioned neural graph detail section provide proof neural graph turing complete section conduct experiments performance verification section briefly introduce related work section list potential future work developing perspective finally section conclude paper publish codes figure illustration logic flow processed methodology referring process structure dynamically manner theoretically construct graph according active branch forward propagation forward propagation constructed graph according logic instructions backward propagation would performed usual continuous graph practically dynamically constructed process corresponds batch operations samples activated branch tackled else branch end instruction hidden representations joined classified results proximated manner simply multiply symbols inside branch dirac function specifically regarding figure reform etwork img etwork img branch perform corresponding transformation else branch multiplication forward propagation would modified reformulation backward process approximate dirac symbol continuous function work gradients condition solves issue noted paper continuous function conduct experiments public benchmark datasets mnist cifar experimental results illustrate model outperforms baselines extensively methodology first introduce overview model discuss component specifically last discuss model ensemble perspective architecture architecture illustrated figure composed three customized components namely feature condition target network firstly input transformed feature network hidden features classified decision tree component composed hierarchal condition networks secondly target networks predict categories sample leaf finally targets joined work cross entropy objective process exemplified algorithm feature network extract abstract features deep neural structures introduce feature network often stacked cnn lstm condition network exactly sample employ simplified neural network condition network usually function tanh layer applied inner nodes decision tree actually effectiveness traditional decision tree stems information gain ndt neual decision tree towards fully functioned neural graph plef pright pntotal pntotal pntotal pntotal short condition network adhoc label vector sample true label position otherwise simple computations pntotal plef ntotal pntotal pright ntotal short inf ogain discussed introduction approximate dirac symbol continuous function specifically thus gradient condition network could deducted figure neural architecture ndt depth input classified decision tree component condition networks target networks predict categories sample leaf notably tree component takes advantages subbatch technique targets joined batch compute cross entropy objective splitting rules could learned condition networks directly thus involve objective item decision node maximize information gain max inf ogain nright right pright ntotal corresponding count feature number corresponding probabilistic distribution features regarding derivatives relative dirac symbol firstly reformulate information gain form dirac symbol nlef nright total actually reduction could performed automatically within proposed principle multiply symbols inside branch dirac function pntotal specifically example count nlef target network finally predict category sample apply complex network target network often stacked convolution one image lstm sentence analysis ensemble perspective nlef lef plef ntotal total sign function ndt could treated ensemble model ensembles many target networks hard branching condition networks currently exist two branches ensemble methods namely split features split samples increases difficulty single classifier however ndt splits data categories means single classifier deals simpler task key point split purity condition networks branching reduces sample numbers leaf relatively single classifier model keeps sample number per category ndt could make effects example one leaf sample number reduces category number reduces similarly sufficient samples model deals much easier thus model benefits strengthen single classifier ndt neual decision tree towards fully functioned neural graph algorithm neural decision tree ndt implement logic component propose two batch operations namely operation take example figure begin five samples batch forward pass processing branch according condition batch split two respectively tackled instructions corresponding branch simultaneously processed two branches joined one batch according original order backward propagation gradients joined batch split two parts correspond two process propagated two branches gradients two joined form gradients stacked cnn theoretically sample means corresponding branch activated sample branch deactivated word hidden representations sample connect activated branch rather deactivated one thus logic components perform proposed principle manner operation notably variable introduced condition unnecessary update condition makes corresponding neural graph exact method dynamical graph construction previously introduced neural graph intelligence architecture characterized logics neurons mathematically component neurons continuous functions matrix multiply hyperbolic tangent tanh convolution layer etc could implemented mathematical operations obviously simple principal implementation mode easy direct practically latest training methods take advantages batched mode hence focus batched implementation neural graph section conventionally neural graph composed two styles variable namely symbols figure atomic types integer algorithm line essence symbolic variables originate weights neurons atomic types introduced embedded traditional algorithms therefore regarding component logics exist two styles logic components differentiated implementation symbolspecific logics indicates condition involves symbols line algorithm logics means atomic types condition line algorithm however proposed principle dynamically constructing neural graph could process situations implement logic component propose flexible batch operation namely allocatebatch take example hungarian layer xiao hungarian algorithm deals similarity matrix provide alignment information according dynamic links symbols dynamically allocated shown figure xiao thus forward backward propagation could performed continuous calculus graph simply forward pass record allocated dynamic links sample batch backward pass propagate gradients along dynamic links obviously logic components perform proposed principle manner operation traditional algorithms combination branch repeat repeat could treated performing branch multiple times thus three batch operations namely operation could process traditional algorithms resolution method searching label propagation pca bandit mab adaboost neural graph turing complete actually neural graph could simulate turing machine turing complete turing machine composed four parts tape head state register finite table instructions correspondingly ndt neual decision tree towards fully functioned neural graph symbols based tensor arrays simulate celldivided tape process indicate variables record state last logic flow constructs finite instruction table summary neural graph turing complete specifically neural graph learnable turing machine rather static one learnable turing machine could adjust according data environment traditional computation models focus static algorithms neural graph takes advantages data perception strengthen rationality behaviors experiment section verify model two datasets mnist lecun cifar krizhevsky first introduce experimental settings section section conduct performance experiments testify model last section verify theoretical analysis ndt could reduce category number leaf nodes perform case study justify assumption experimental setting exist three customized networks model feature condition target network simply apply identify mapping feature network regarding condition network apply fully connected perceptions mnist cifar regarding target network also employ fully connected perceptions mnist train model leverage adadelta zeiler optimizer moment factor train model convergence rounds regarding batch size always choose largest one fully utilize computing devices notably approximated continuous function performance verification mnist mnist dataset lecun classic benchmark dataset consists handwritten digit images pixels size organized classes training test samples select representative competitive baselines modern know feature target network oversimplified task version targets exemplified model still could verify conclusions perform complex feature target network version table performance evaluation mnist dataset methods single target network cnn deep belief net svm rbf kernel random forest accuracy ndt depth architecture dropout relus classic linear classifier svm rbf kernel deep belief nets standard random forest trees could observe ndt beat baselines verifying theory justifying effectiveness model compared single target network ndt promotes point illustrates ensemble target network effective compared random forest also method ndt promotes point demonstrates neurons indeed strengthen decision trees cifar dataset krizhevsky also classic benchmark overfull category classification consists color natural images pixels size classes training test images several representative baselines selected network network nin lin fitnets rao deep supervised network dsn lee srivastava springenberg exponential linear units elu clevert fitresnets mishkin matas gcforest zhou feng deep resnet could conclude ndt beat strong baselines verifies effectiveness neural decision trees justifies theoretical analysis compared single target network ndt promotes point illustrates ensemble target network effective compared gcforest performance improves points illustrates neurons empower decision trees effectively direct ensembles compared resnet strongest baseline promote results points justifies ndt neual decision tree towards fully functioned neural graph figure case study ndt mnist depth depth left tables test sample numbers correspond leaf node category example sliced means test samples category leaf node slice main component leaf draw corresponding decision trees right panel notably indicates empty class table performance evaluation error cifar methods nin dsn fitnets elu fitresnets resnet gcforest random forest single target network ndt depth assumption ndt could reduce category number leaf nodes enhance intelligence systems case study testify assumption ndt could reduce category number leaf nodes perform case study mnist make statistics test samples leaf node illustrated figure item table means leaf node many samples category example first row second column means test samples category leaf node correspondingly draw decision trees right panel labeled categories specifically illustrates decision process ndt complete verification vary depth ndt firstly could clearly draw conclusion figure leaf node needs predict less categories justifies assumption example bottom figure node needs predict category single classification node needs predict categories four classification small classification less difficult large one target network leaf could perform better leads performance promotion manner secondly figure split purity could worked generally tanh achieves decent split purity indeed difficult leaf nodes top bottom perfect others gain competitive split purity statistically main component sliced grid takes share total samples large probability ndt would perform better accuracy case ndt neual decision tree towards fully functioned neural graph finally discuss depth top bottom figure categories split example node top split bottom means category way deep neural decision tree advantageous much deeper ndt makes less sense categories already split well would mostly difference however considering efficiency consuming resources suggest apply suitable depth theoretically total category number related work section briefly introduce three lines related work image recognition decision tree neural graph convolution layer necessary current neural architectures image recognition almost every model convolutional model different configurations layers springenberg dsn lee empirically deeper network produces better accuracy difficult train much deeper network issue gradients glorot bengio recently emerge two ways tackle problem srivastava residual network inspired lstm network applies layer allow information flow across layers along computation path without attenuation direct manner residual network simply employs identity mappings connect relatively top bottom layers propagates gradients effectively whole network notably achieving performance residual network resnet strongest model image recognition temporarily decision tree classic paradigm artificial intelligence random forest representative methodology branch recent years completely random tree forest proposed iforest liu anomaly detection however popularity deep neural network lots researches focus fusion neurons random forest example richmond converts cascaded random forests convolutional neural network welbl leverages random forests initialize neural network specially model gcforest zhou feng allocates deep architecture forests experimentally verified several tasks notably branch could jointly train neurons decision trees main disadvantage jointly fuse neurons logics xiao proposes basic principle neural graph could embed traditional algorithms neural architectures seminal paper merges hungarian algorithm neurons hungarian layer could effectively recognize sentence pairs however special case variables introduced condition could updated disadvantage characterizing complex systems thus paper focuses issue make fully functioned neural graph future work list three lines future work design new components neural graph implement script language neural graph analyze theoretical properties learnable turing machine paper exemplifies approach embed decision tree neural architectures actually many traditional algorithms could promote intelligence system neurons example neural searching could learn heuristic rules data could effective less resource consuming example could represent data deep neural networks conduct label propagation upon hidden representations propagation graph constructed method label propagation deep neural networks trained jointly performance promotion could expected fact fully functioned neural graph may extremely hard complex implement thus expect publish script language modeling neural graph also library includes mainstream intelligence methods based instruments neural graph could convenient practical usage finally discussed neural graph turing complete making learnable turing machine believe theoretical analysis necessary compilation ability neural graph take example learnable static turing machine ability take example could brain excel turing machine excellent neural graphs may gain advantages biological brain learnable turing machines could theoretical foundations intelligence reformed take final example best computation model intelligence conclusion paper proposes principle fully functioned neural graph based principle design neural decision tree ndt image recognition experimental results benchmark datasets demonstrate effectiveness proposed method ndt neual decision tree towards fully functioned neural graph references casper mengel fuhrmann herrmann appenrodt schiedermaier reichert bruns engelmann grnhage perceptrons introduction computational geometry clevert djorkarn unterthiner thomas hochreiter sepp fast accurate deep network learning exponential linear units elus computer science deng jia dong wei socher richard kai imagenet hierarchical image database computer vision pattern recognition cvpr ieee conference ieee glorot xavier bengio yoshua understanding difficulty training deep feedforward neural networks proceedings thirteenth international conference artificial intelligence statistics kaiming zhang xiangyu ren shaoqing sun jian deep residual learning image recognition kaiming zhang xiangyu ren shaoqing sun jian deep residual learning image recognition computer vision pattern recognition krizhevsky alex learning multiple layers features tiny images lecun bottou bengio haffner gradientbased learning applied document recognition proceedings ieee lecun yann bengio yoshua hinton geoffrey deep learning nature lee chen xie saining gallagher patrick zhang zhengyou zhuowen nets eprint arxiv lin min chen qiang yan shuicheng network network computer science liu fei tony kai ming ting zhou zhi hua isolation forest eighth ieee international conference data mining mcculloch warren pitts walter logical calculus ideas imminent nervous activity mishkin dmytro matas jiri need good init rao jinfeng hua lin jimmy estimation answer selection deep neural networks acm international conference information knowledge management richmond david kainmueller dagmar yang michael myers eugene rother carsten relating cascaded random forests deep convolutional neural networks semantic segmentation computer science rumelhart hinton williams learning internal representations error propagation mit press silver david huang aja maddison chris guez arthur sifre laurent driessche george van den schrittwieser julian antonoglou ioannis panneershelvam veda lanctot marc mastering game deep neural networks tree search nature springenberg jost tobias dosovitskiy alexey brox thomas riedmiller martin striving simplicity convolutional net eprint arxiv srivastava rupesh kumar greff klaus schmidhuber jrgen training deep networks computer science welbl johannes casting random forests artificial neural networks profiting german conference pattern recognition springer xiao han hungarian layer logics empowered neural architecture arxiv preprint zeiler matthew adadelta adaptive learning rate method computer science zhou zhi hua feng deep forest towards alternative deep neural networks

alignment elimination adams grammars institute computer science university tartu liivi tartu estonia jun abstract adams extension parsing expression grammars enables specifying indentation sensitivity using two grammar constructs indentation binary relation alignment paper proposes transformation adams grammars elimination alignment construct grammar idea alignment could avoided suggested adams process achieving aim described acm subject classification formal definitions theory processors grammars rewriting systems keywords phrases parsing expression grammars indentation grammar transformation introduction parsing expression grammars peg introduced ford serve modern framework specifying syntax programming languages alternative classic grammars cfg core difference cfg peg descriptions cfg ambiguous pegs inherently deterministic syntax specification written peg principle interpreted parser syntax case left recursion treatment straightforward doable see formally peg quadruple finite set finite set terminals function mapping replacement corresponding set productions cfg start expression corresponding start symbol cfg set parsing expressions writable defined inductively follows empty string every terminals every whenever concatenation whenever choice whenever negation lookahead whenever repetition constructs peg except negation direct analogues constructs ebnf form cfg semantics always deterministic repeats parsing failure always tries parse first parsed fails example expression consumes input string entirely consumes first character corresponding ebnf expressions equivalent match either input string negation tries parse fails succeeds nestra licensed creative commons license alignment elimination adams grammars fails succeeds consuming input constructs ebnf like repetition optional occurrence introduced peg syntactic sugar languages like python haskell allow syntactic structure programs shown indentation alignment instead conventional braces semicolons handling indentation alignment python specified terms extra tokens indent dedent mark increasing decreasing indentation must generated lexer haskell rules handling indentation alignment sophisticated languages enable locally use different layout mode indentation matter additionally complicates task formal syntax specification adams proposed extension peg notation specifying indentation sensitivity argued considerably simplifies task python haskell many languages extension expression example denotes parsing assuming greater indentation surrounding block general parsing expressions may equipped binary relations example must hold baselines local current indentation block addition denotes parsing assuming first token input aligned positioned current indentation baseline example expressions haskell specified doexp istmts stmts istmts stmts stmt stmt stmt istmts stmts stand statement lists indentation relaxed mode respectively indentation mode statement list indented marked second production statements aligned marked relaxed mode however relation used indicate indentation baseline contents anything technically binary relation containing pairs natural numbers terminals also equipped meet haskell requirement subsequent tokens aligned blocks must indented first token alignment construct provides fulcra disambiguating often large variety indentation baseline candidates besides simplicity grammar extension use strength lies fact grammars still serve parsers rest paper organized follows section formally introduces additional constructs peg specifying code layout defines semantics studies semantic properties sect process eliminating alignment construct grammars described section refers related work sect concludes indentation extension peg adams extend pegs indentation alignment constructs propose slightly different extension three rather two extra constructs approach agrees implemented adams indentation package haskell whence calling grammars approach adams grammars justified differences definitions paper listed discussed subsect let denote set natural numbers let boolean domain denote set subsets set let denote set binary relations set standard examples consisting pairs natural numbers identity nestra relation consisting pairs equal natural numbers indentation extension also makes use relation containing pairs natural numbers whenever denote image relation inverse relation defined composition relations finally denote adams grammars extend definition given sect following three additional clauses every indentation every token position every alignment parsing expression means parsing assuming part input string corresponding forms new indentation block whose baseline relation baseline surrounding block baselines identified column numbers position construct missing determines tokens input situated current indentation baseline finally parsing expression means parsing assuming first token input positioned current indentation baseline unlike position operator construct affect processing subsequent tokens inspired indentation package call relations determine token positioning indentation baseline token modes token mode example tokens may appear right indentation baseline applying position operator relation parts haskell grammar parsed indentation mode avoids indenting every single terminal example sect also indenting terminals inadequate expressions occurring inside block relaxed mode position construct easily used change token mode blocks call peg extended three constructs peg recall sect denote set terminal symbols grammar respectively production function concerning semantics peg expression parses input string terminals context current set indentation baseline candidates current alignment flag indicating whether next terminal aligned assuming certain token mode parsing may succeed fail diverge parsing succeeds returns result new triple containing rest input new set baseline candidates updated according information gathered parsing new alignment flag result denoted parsing fails result triple form failure denoted triples form behaving operation states parsing parsing step may use data update write state never deal different terminal sets dependence explicitly marked denote state set possible results parsing state assertion parsing expression grammar input string context assuming token mode results state denoted formal definition must interpreted inductively assertion form valid iff finite derivation following ten rules every holds two cases alignment elimination adams grammars denotes occurring column either either every holds holds every holds two cases exists triple every holds two cases exists triple every holds two cases exists triple every holds two cases exists triple every holds two cases exists triple every holds holds every holds two cases exists triple idea behind conditions occurring clause column token may appear relation current indentation baseline known alignment flag set coincide indentation baseline otherwise reason consuming token column restricts set allowed indentations depending alignment flag cases alignment flag set clause set allowed indentation replaced local indentation baseline must relation current indentation baseline known successful parsing resulting set allowed local indentations set allowed indentations surrounding block restricted clause similarly operates alignment flag toy example consider parsing operation state assuming token mode must parse clause since turn must parse clause clause therefore clause finally clauses set final state shows still candidates indentation baseline outside parsed part input parsing candidate set whole note definition involves circular dependencies instance clause result parsing cases even call behaviour divergence nestra properties semantics ford proves parsing peg unambiguous whereby consumed part input string always prefix theorem analogous result peg besides uniqueness result parsing states consider relations whole operation state setting certain sense decreasing parsing denote suffix order strings iff implication order truth values denote pointwise order operation states iff theorem let peg state one whereby implies also implies implies proof induction shape derivation tree assertion theorem enables observe common pattern semantics indentation alignment denoting either clauses following form parametrized two mappings state state holds two cases exists state meanings indentation alignment constructs distinguished solely many properties proofs rely abstract common definition carried assuming monotone preserves largest element follows together axiom class meet semilattices top element equipped mappings satisfying three conditions contains identities semilattices idl closed compositions different defined semilattice direct products conditions hold similarly case direct product identities gives indentation case direct product identities boolean lattice gives alignment case satisfy conditions since adding dual conditions monotone would make galois connection cases dual axioms hold semantic equivalence definition let peg say semantically equivalent denote iff every state state example one easily prove particularly interested equivalences involving additional operators peg sect useful eliminating alignment position operators following theorem states distributivity laws three new operators peg constructs theorem let peg alignment elimination adams grammars proof equivalences claim hold token mode steadily distributes case semantics definition claims straightforward proofs using joint form semantics indentation alignment axioms note indentation distribute concatenation assumes one indentation block baseline common tolerates different baselines example take let token mode input state recall means terminal occurring column since therefore hand implies since analogously since consequently however prove following facts theorem let peg identity indentation law composition law indentations distributivity indentation alignment idempotence alignment cancellation outer token modes terminal alignment property proof claim note indentation identity relation corresponds identity mappings hence replaced theorem concerning claims let two constructs whose semantics follow common pattern indentation alignment mapping pairs respectively nestra monotonicity fact third axiom also whence consequently replaced hence semantics composition follows pattern semantics indentation alignment mappings prove claim suffices observe mappings semantics equal compositions corresponding mappings semantics claim suffices observe mappings given indentation alignment modify different parts operation state whence order application irrelevant claim holds mappings alignment semantics idempotent finally claim trivial claim follows straightforward case study theorems enact bringing alignments syntactic constructs except concatenation alignment distribute concatenation parsing expression form terminal aligned part input consumed parsing succeeds consuming input alignment nevertheless moved concatenation successful parsing first expression concatenation either never consumes input always consumes input theorem let peg implies state implies state proof straightforward case study theorem holds also indentation instead alignment proof terms valid finally following theorem states position indentation terminals equivalent alignment flag false token mode identity theorem let peg let state proof straightforward case study differences approach previous work specification peg differs definition used adams three essential aspects listed last two discrepancies understood bugs original description corrected haskell indentation package adams package also provides means locally changing token mode modifications fully agree indentation package position operator missing treatment assumes one default token mode applying whole grammar whence token positions deviating default must specified using indentation operator benefits position operator shortly discussed subsect according grammar semantics provided alignment flag never changed end parsing expression form appropriate succeeds without consuming token alignment flag would unexpectedly remain true parsing next token scope alignment operator value alignment flag starting parsing restored case purpose conjunction alignment semantics described paper alignment elimination adams grammars alignment interpreted indentation baseline block corresponds parsing expression alignment operator applied indentation operators occurring inside expression processed alignment flag true neglected semantics described paper raising alignment flag suppress new indentations alignments interpreted indentation baseline force aligned token site seems appropriate former approach indentations cancelled alignment apply even subsequent tokens distributivity indentation alignment fails semantics note alignment block nevertheless suppresses influence position operators whose scope extend first token block grammar semantics also two purely formal deviations semantics used adams ford keep track rest input operation state expose consumed part input instead difference introduced simplicity achieve uniform decreasing operation states theorem explicit step counts used compose proofs induction provide analogous proofs induction shape derivation trees elimination alignment position operators adams describes alignment elimination context cfgs adams claim alignment elimination process pegs difficult due lookahead construct knowledge concrete process alignment elimination described pegs provide one grammars rely existence position operators grammar issue since also show position operators eliminated grammars approximation semantics expressions defining first need introduce approximation semantics consists assertions form semantics decidable extension predicate tells whether parsing may succeed consuming input result succeed consuming input result fail result particular input strings indentation sets etc involved whence semantics deterministic following set clauses define approximation semantics inductively every every every holds four cases exist exists every holds two cases every holds two cases nestra exists every holds two cases every every every peg constructs definition basically copies given ford except case definition requires besides sound since parsing never fails parsing terminate difference matter grammar transformations assume grammars theorem let peg assume state state proof induction shape derivation tree assertion decidable conservative approximation predicate true iff parsing never diverges definitely excludes grammars left recursion exclude also safe grammars pegs introduced ford following set clauses inductive definition predicate expressions peg every every every addition implies every addition implies every every every every every definition rejects directly indirectly left recursive rules since concatenation must leading infinite derivation case kind left recursion hand requiring clause would restrictive since would reject meaningful recursive productions like clauses peg constructs mostly copy definition given ford time choice case exception considered needlessly rejects safe recursive rules like require could possibly executed grammar called every expression occurring subexpression ford proves induction length alignment elimination adams grammars input string grammars parsing every expression whose subexpressions terminates every input string prove analogous result similar way prefer generalize statement stricter semantics enables occasionally construct easier proofs later new semantics call strict defined replacing choice clause definition subsect following every holds two cases exists triple addition implies state new semantics restrictive since finish parsing expression form parsing successfully also must parsed happens case standard semantics parsing try parsing successful parsing expression terminates strict semantics terminates result standard semantics necessarily vice versa therefore proving parsing always gives result strict semantics establish also standard semantics rest sign strict semantics exclamation mark parsing assertions form theorem let peg let assume subexpressions every state exists state proof induction length input string first component induction step uses induction shape derivation tree assertion splitting repetition operator always eliminated adding new pap subexpression occurs star operator may assume input grammar first stage process also assumes negations applied atomic expressions choices disjoint choice expression called disjoint parsing succeed input state token mode achieving last two preconditions considered preparatory previously studied stage negation elimination step process issues concerning discussed briefly subsect use principle splitting algorithm stage negation elimination process described ford adding clauses extra operators peg approach defines two functions follows metavariable denoting expression always fails otherwise otherwise otherwise nestra correctness definition follows induction shape derivation tree assertion negation case use negations applied atomic expressions whence reference eliminated replacement definition definition sound induction shape expression new grammar defined using equations equivalence input output grammars relies splitting invariant established theorem allows instead parsing expression negations front atoms disjoint choices equivalently use parsing expression claim analogous splitting invariant used provide simpler proof using strict semantics analogous proof using standard semantics would fail choice case theorem let state state assuming choices rules expression disjoint negations applied atoms following holds proof use repetition operator whence expressions grammars fact follows easy induction expression structure theorems desired result follows induction shape derivation tree using disjointness assumption choice case result transformation sizes sides productions grow exponentially though number productions stays unchanged preprocessing grammar via introducing new way concatenations applied atoms similarly ford would hinder growth size worst case remains exponential subsequent transformations cause linear growth sides alignment elimination grammar obtained via splitting eliminate alignments using following three steps introduce copy define sides productions start expression apply distributivity laws theorem theorem theorem idempotence theorem bring alignment operators terminals replace alignment terminals position theorem sides productions start expression replace subexpressions form corresponding new establishing equivalence original obtained grammar following general theorem used theorem let peg every always implies proof easy induction shape derivation tree alignment elimination adams grammars denote function defined performs transformations steps distributes alignment operators replaces aligned corresponding new denote grammar obtained step note step change semantics expressions written original grammar steps replace sides productions expressions semantically equivalent grammar obtained step theorem implies whenever parsing final grammar produces result result obtained parsing input state token mode original grammar order able apply theorem grammars interchanged need equivalence sides productions also grammar sufficient show every turn would follow statement consequently equivalence initial final grammars implied following theorem theorem every proof claim direct consequence following two lemmas holding arbitrary state state lemmas proven induction shape derivation trees assertion two alignments outside inside needed case form elimination position operators peg get rid position operations using process largely analogous alignment elimination consisting following four steps introduce new existing relation used position operator apply distributivity laws theorem cancellation theorem bring position operators terminals replace subexpressions form corresponding new replace subexpressions form denote function defined performs transformations steps distributes position operators terminals replaces position operators corresponding new denote grammar obtained step theorem applies well whence equivalence grammar obtained step initial grammar implied following theorem theorem every proof claim direct consequence following two lemmas holding arbitrary state state lemmas proven induction shape derivation trees claim position operator outside inside needed case application position operator nestra correctness step proven induction shape derivation trees using theorem note must assume parsing according final grammar performed alignment flag false natural assumption grammar token mode discussion preconditions alignment elimination correctly defined assumption input grammar negations front atoms disjoint choices conditions needed stage second assumption easily established introducing new expression occurs productions start expression done lines first stage negation elimination process described ford transformation preserves grammar achieving disjoint choices subtle topic straightforward way would replacing choices form disjoint choices seems work well equivalent standard semantics alas equivalent approximation semantics due replacing break take due alone recursive call arises however whence recursively requires thus argument similar shows first stage negation elimination process ford also break second stage correctly defined grammars whole process fails one solution would changing approximation semantics adding inductive definition subsect general clause forces hold whenever assertion form holds particular becomes equivalent replacing preserves although predicate becomes restrictive rejects safe grammars loss seems little acceptable practice expressions seem occur commonly influenced productions investigation needed clarify related work pegs first introduced studied ford also showed closely related system tdpl well generalized forms adams adams provide excellent overview previous approaches describing languages attempts building indentation features parser libraries work theoretical study approach proposed details semantics used paper corrected lines adams indentation package haskell package enables specifying indentation sensitivity within parsec trifecta parser combinator libraries process alignment operator elimination previously described cfgs adams matsumura kuramitsu develop general extension peg also enables specify indentation framework powerful complicated approach proposed alignment elimination adams grammars followed contrast focusing indentation aiming maximal simplicity convenience usage conclusion studied extension peg proposed adams indentationsensitive parsing extension uses operators marking indentation alignment besides classic ones added one operator position convenience found lot useful semantic equivalences valid expressions written extended grammars applied equivalences subsequently defining process algorithmically eliminates alignment position operators grammars references michael adams indentation package url http michael adams principled parsing languages revisiting landin offside rule roberto giacobazzi radhia cousot editors annual acm symposium principles programming languages popl rome italy january pages acm url http michael adams parsing parsec wouter swierstra editor proceedings acm sigplan symposium haskell gothenburg sweden september pages acm url http alfred aho jeffrey ullman theory parsing translation compiling upper saddle river usa alexander birman jeffrey ullman parsing algorithms backtrack information control url http bryan ford parsing expression grammars syntactic foundation neil jones xavier leroy editors proceedings acm symposium principles programming languages popl venice italy january pages acm url http tetsuro matsumura kimio kuramitsu declarative extension parsing expression grammars recognizing programming languages jip url http medeiros fabio mascarenhas roberto ierusalimschy left recursion parsing expression grammars francisco heron carvalho junior soares barbosa editors programming languages brazilian symposium sblp natal brazil september proceedings volume lecture notes computer science pages springer url http

bridging static dynamic program analysis using fuzzy logic jacob lidman josef svenningsson chalmers university technology lidman josefs static program analysis used summarize properties dynamic executions unifying approach based logic properties either assigned definite value unknown summarizing set executions property accurately represented biased towards true towards false compilers use program analysis determine benefit optimization since benefit performance justified based common case understanding bias essential guiding compiler furthermore successful optimization also relies understanding quality information plausibility bias quality static information low form decision would like mechanism improves dynamically consider problem building reasoning framework present fuzzy analysis approach generalize previous work use logic derive fuzzy extensions analyses used lazy code motion optimization unveil opportunities previous work would detect due limited expressiveness furthermore show results analysis used adaptive classifier improve application executes introduction one reconcile static dynamic program analysis two forms analysis complement static analysis summarizes possible runs program thus provide soundness guarantees dynamic analysis provides information particular runs program actually happen practice therefore provide relevant information able combine two paradigms applications many forms analyses alias analysis dependence analysis compilers use program analysis frameworks prove legality well determining benefit transformations specifications legality composed safety liveness assertions universal existentially quantified properties specifications benefit use assertions hold common case reason adopting common case transformations improve performance general every input environment similarly transformations could potentially improve performance least one case compiler optimizations instead motivated based approximation majority case weighted mean determining legality improved due advances verification community progress establishing benefit slow paper introduce fuzzy analysis framework static program analysis based fuzzy logic salient feature framework naturally incorporate dynamic information still static analysis ability comes thanks shift crisp sets membership binary employed conventional static analysis fuzzy sets membership gradual vink wiklicky eds qapl eptcs lidman svenningsson work licensed creative commons attribution license bridging static dynamic program analysis using fuzzy logic make following contributions section introduces main contribution fuzzy framework section demonstrates benefit framework presenting generalization wellknown code motion algorithm show generalization provides new opportunities optimizations previous approaches would discover section shows fuzzy logic benefit program analysis using fuzzy sets separate uncertainty hence improve analysis using fuzzy regulators refine results static analysis hence improving precision dynamically preliminaries introduce define fuzzy sets operators form fuzzy logic concepts used section define transfer functions analysis fuzzy set elements crisp either members universe discourse fuzzy set instead allow partial membership denoted number unit interval membership degree typically denotes vagueness process convert crisp membership fuzzy grades called fuzzification inverse called defuzzification following dubois let crisp set membership function fuzzy set convention understood context sometimes refer fuzzy set membership function formalizes fuzzification fuzzy sets ordered accommodate notion uncertainty vagueness considering fuzzy set membership degree fuzzy set membership functions composed primary secondary membership uncertainty represented secondary membership define possibility primary membership holds called interval gehrke showed equivalently described interval valued fuzzy sets ivfs ivfs special case lattice valued fuzzy sets sets membership domain forms lattice defuzzification often proceeds two phases first phase applies type reduction transform second phase applies defuzzification fuzzy logic fuzzy logic defines formal systems reason truth presence vagueness contrary classical logic law excluded middle law general hold systems fuzzy logic uses norms generalize logical operators compactly represent fuzzy logic sometimes called morgan system satisfies generalization morgans laws context fuzzy logic crisp boolean set refer classical set avoid confusion fuzzy sets one would expect definition fuzzy logic include fuzzy implication operator work consider although lidman svenningsson fuzzy logic algebraic lukasiewicz nilpotent min max min otherwise max min max otherwise table common instantiations fuzzy logics definition let binary function commutative associative increasing identity element triangular norm triangular conorm definition unary function decreasing involutory boundary conditions standard examples fuzzy logics shown table examples special cases limits frank family fuzzy logics central work formally defined definition definition let frank family defined min max otherwise set intervals forms bounded partial order per gehrke lift fuzzy logic ivfs fuzzy logic fuzzy analysis static analyses deduce values semantic properties satisfied dynamics application dynamics formalized system monotone transfer functions collector functions transfer functions describe blocks alter semantic properties collectors functions merge results different possibly mutual exclusive branches application solution analysis obtained kleene iteration unique system equations classical framework domain values binary either true false interpretation values depends type analysis value true means property possibly hold impossible value always false means property always holds value false could mean either opposite true result inconclusive fuzzy analysis instead computes partial truth property values elements value closer means property biased towards false vice versa furthermore transfer functions logical formulas frank family fuzzy logic collector general concept allowing called uninorm either orlike andlike work require full generality confused partial order used interval abstraction bridging static dynamic program analysis using fuzzy logic functions weighted average functions constant determined prior performing analysis contrast classical framework kleene iteration proceeds results differ constant maximal error allowed solution error made arbitrarily small section introduces fuzzy framework prove termination using continuity properties banach theorem section presents example analysis demonstrate benefit framework analysis performed weighted set logical formulas denoting transfer function block set edges denoting control transfers denotes normalized contribution edge running example use figure left shows four nodes corresponding logical formula flow graph four control edges denoting contributions nodes instance block receives contribution min max figure example left corresponding equation system middle analysis result error function iteration right definition let finite set properties valuation property use denote interpretation fuzzy formula given given unique start node vstart map describes value property node fuzzy analysis kleene iteration vstart vstart otherwise figure middle shows equation system implied definition interpreted fuzzy logic example red colored text corresponds collector function weighted average normal text interpretation logical formula order prove termination fuzzy analysis need introduce continuity property definition function iff absolute value called contraction mapping called mapping sequence applications contraction mapping difference two consecutive applications decrease limit reach zero imposing bounded error guarantee definition restricts domain metric metric spaces domain compared general common definition lipschitz continuous function used restrict logic fuzzy logic lidman svenningsson sequence terminates bounded amount time analysis error result function iteration example shown figure right note error red line decreasing value blue line tends towards final value next proceed prove fuzzy analysis iteratively computes precise results terminates bounded amount time finite maximum error let denote maximal congruence set elements least apart set intervals defined analogously prove property fuzzy formulas theorem let functions min abs constants composition finally formulas defined frank family fuzzy logic inq satisfies inq summary per theorem transfer functions frank family fuzzy logic mappings vstart constant hence contraction mapping composition two functions function nonexpansive contraction function contraction function analysis performed unit interval together forms complete metric space guarantee termination banach theorem theorem banach theorem let complete metric space contraction unique concludes development fuzzy analysis lazy code motion improving performance often means removing redundant computations computations said fully redundant dead operands points remain two computations enough keep one store away result later eliminate redundancy using global common elimination gcse furthermore computation change paths said partially redundant loop invariant code motion licm finds partially redundant computations inside loops move entry block loop lazy code motion compiler optimization eliminate fully partially redundant computations hence subsumes cse licm krs algorithm performs lcm production compilers gcc optimizing speed bridging static dynamic program analysis using fuzzy logic diffpcm transform incrate abs transform incrate diffpcm anfis decision updating anfis decision leaving update leave decision error else update leave decision error transform incrate figure diffpcm function left corresponding middle version used section annotated anfis classifier invocations right consists series analyses summarized four steps solve busy available expression problem introduce set describes earliest block expression must evaluated determine latest control flow edge expression must computed introduce insert delete sets describe expressions evaluated target domain analysis set static expressions program input analysis three predicates determining properties expressions different blocks expression downward exposed produces result evaluated end block defined use dee denote downward exposed block expression upward exposed produces result evaluated start block defined use uee denote expression killed block variable appearing updated use kill denote busy expression analysis analysis depends uee kill computes set expressions guaranteed computed time future similarly available expression analysis analysis depends dee kill deduces set previously computed expressions may reused system two analyses shown figure beyond scope paper elaborate details analyses interested reader consider nielson lcm algorithm analyses depends applications use demonstrate benefit framework rudimentary understanding sufficient consider simplified differential modulation routine diffpcm figure left assume relative number times block denoted statically iteration diffpcm invokes pure functions transform encode differential output incrate get quantification rate use algorithm determine invocations made prior entering loop contrast situation analyses performed knoop refer anticipatable expression problem demonstration let conclusions hold increases approaches lidman svenningsson block lcm available expression avin avout avout dee avin busy expression anin anout anout uee anin dee uee block anin kill anout earliest anin otherwise laterin laterout laterout earliest laterin dee uee kill block insert laterout delete uee block edge insert block expression index delete abs transform incrate edge kill insert block delete figure lcm formulation middle using classical left fuzzy analysis bridging static dynamic program analysis using fuzzy logic fuzzy framework show fuzzy allows uncover opportunites classical would miss static analysis problems krs algorithm use expressions domain mapping expressions diffpcm indexes listed figure bottom together values dee uee kill block top right classical krs algorithm conclude calls must evaluated bottom light gray box delete matrix column fuzzy analyses use fuzzy logic corresponding fuzzy sets dee uee kill given figure top dark gray box step fuzzy hence system equations available expression analysis system avout avout avout dee avout dee avout avout dee avout dee pavout avout avout dee avout busy expression analysis system uee anout anout anout uee uee panout anout anout uee anout anout uee anout anout steps introduce constant predicates performed outside analysis framework step done similarly step figure bottom dark gray box shows result step contrast classical lcm result implies plausible delete invocation transform delete matrix column block instead add end start however result invocation incrate remains invocation depends value updated end static analysis increase analysis precision function call sometimes inlined call site improvement however reduced analysis inaccurate multiple targets considered particular call site show uncertainty quantified two different dimensions using interval fuzzy sets per section lift arbitrary fuzzy predicate intervals assume knowledge relative number calls target treat different calls assume two different incrate functions figure left determined targets respective uee kill entries since updated end block dee entry differ result depends variable therefore dee contrast entry dee new lidman svenningsson transform incrate incrate incrate return block edge kill dee uee insert block delete figure implementations incrate inlined block left dee uee kill vectors block delete insert analysis result expression incrate right dee new entry block given dee dee kill dee uee sets given figure right applying fuzzy fuzzy logic lifted interval minmax fuzzy logic gives values delete insert expression incrate figure right result invoking incrate prior loop opposed analysis section added dimension result fuzzy analysis allows differentiate uncertain results pessimistic results given example showed result section pessimistic two paths need considered seperately increase precision hybrid analysis result fuzzy analysis set fuzzy membership degrees section shows result automatically improved following static analysis using fuzzy specific information provided later point classifier fuzzy inference system shown figure composed five layers lookup fuzzy membership degree input value compute firing strength rule conjunction membership degrees rule normalize firing strengths weight normalized firing strength consequent output rule combine rule classifiers classifier uses polynomial consequent part adaptive rules decide output membership order order polynomial classification accuracy improved fitting polynomial input data implemented follows offline affine least square optimization convex optimization problem finds affine function minimizes input output vectors training set online least mean square lms adaptive filter gradually steps given constant minimizes sample bridging static dynamic program analysis using fuzzy logic figure anfis two rules two variables left four example fuzzy sets right exemplify functionality consider classification using two rule figure left let membership functions given figure right membership degrees marked figure first rule second rule hence weight first rule second rule normalized weights consequence functions output produce prediction return diffpcm function consider invoke transform prior entering loop saw section fuzzy membership degree improve classification accuracy let also use variable first input value variables part analysis conservatively assume fuzzy membership degree value variables experiments shown figure right inserted calls compute anfis decision updating keeping variable constant diffpcm function incorrect decision made error noted error rate computed handling input samples consider invoking diffpcm function four different input sets input set defined periods input values period input sets given figure left use lms incorrect classification algorithm error rate period larger equal note values period always perfectly representable linear classifier sometimes varies different periods although periods similar hence expect classifier monotonically improving increasing period shown result figure right classification error decreases fast period input sample two cases small residual error remains final period show improve analysis result dynamically hence increase accuracy transform invoked prior entering loop constant four different runs set respectively error rate input value lidman svenningsson sample period error rate input value sample period error rate input value sample period error rate input value sample period figure input values left corresponding classification error rate right bridging static dynamic program analysis using fuzzy logic related work systems include elements input values environment state information limited probabilistic uncertainty formulated systems likely even quantitative analysis properties possible often analysis relies probability theory logical soundness cousot monerau introduced unifying framework probabilistic abstract interpretation much work since although perhaps implicitly relied formulation often probabilistic descriptions known imprecision manifests uncertainty adje introduced abstraction based zonotope abstraction structures pierro developed probabilistic abstract interpretation framework demonstrated alias analysis algorithm could guide compiler decision later formulated problems liveness analysis framework important distinction similar probabilistic frameworks classical frameworks definition confluence operator contrast classical must framework use weighted average similar work ramalingam showed mop solution exists confluence operator transfer function defined terms min max negation fuzzy logic work extends allow transfer functions integrates static analysis dynamic refinement mechanism fuzzy control theory conclusion major problem static program analysis limited input information hence conservative results alleviate situation dynamic program analysis sometimes used accurate information available contrast static results cover single runs bridge gap find promising program analysis frameworks proposed frameworks considered intersect static program analysis uses dynamic information introduced framework based fuzzy sets supports analyses solved problems use speculative compilation showed analysis unveils opportunities previous approaches could express reason furthermore showed framework based fuzzy sets admit mechanisms fuzzy control theory enhance analysis result dynamically allowing hybrid analysis framework references assale adje olivier bouissou jean eric goubault sylvie putot static analysis programs imprecise probabilistic inputs verified software theories tools experiments lecture notes computer science springer berlin heidelberg patrick cousot radhia cousot abstract interpretation past present future proceedings joint meeting eacsl annual conference computer science logic csl annual symposium logic computer science lics acm lower upper bounds cumulative probability distribution functions lidman svenningsson patrick cousot monerau probabilistic abstract interpretation european symposium programming esop lecture notes computer science pierro wiklicky probabilistic data flow analysis linear equational approach proceedings fourth international symposium games automata logics formal verification alessandra pierro chris hankin herbert wiklicky systematic approach probabilistic pointer analysis programming languages systems lecture notes computer science springer berlin heidelberg drechsler manfred stadel variation knoop steffen lazy code motion sigplan dubois prade fuzzy sets systems theory applications academic press new york dubois prade prade fundamentals fuzzy sets handbooks fuzzy sets springer mai gehrke carol walker elbert walker comments interval valued fuzzy sets international journal intelligent systems sici jang anfis fuzzy inference system systems man cybernetics ieee transactions roger jang sun soft computing computational approach learning machine intelligence upper saddle river usa jens knoop oliver bernhard steffen lazy code motion proceedings acm sigplan conference programming language design implementation pldi acm maleki yaoqing gao garzaran wong padua evaluation vectorizing compilers parallel architectures compilation techniques pact international conference mesiarov international journal approximate reasoning special section aggregation operators markus mock manuvir das craig chambers susan eggers dynamic sets comparison static analyses potential applications program understanding optimization proceedings acm workshop program analysis software tools engineering paste acm flemming nielson hanne nielson chris hankin principles program analysis springerverlag new york petersen padua static dynamic evaluation data dependence analysis techniques parallel distributed systems ieee transactions ramalingam data flow frequency analysis proceedings acm sigplan conference programming language design implementation pldi acm constantinog ribeiro marcelo cintra quantifying uncertainty relations languages compilers parallel computing lecture notes computer science springer berlin heidelberg bridging static dynamic program analysis using fuzzy logic appendix omitted proofs theorem let functions min abs constants let abs abs definition definition triangle inequality distributivity definition triangle inequality distributivity substitution definition triangle inequality distributivity min min definition min min triangle inequality lidman svenningsson definition associativitiy commutativity triangle inequality distributivity composition definition definition definition formulas defined frank family fuzzy logic follows structural induction height parse tree predicate morgan laws enough show induction step base case induction step base case constants cases assumption min theorem case theorem case max equal min using morgans law expression theorem case inq satisfy inq inq decomposed two functions inq inq gives infimum gives supremum show continuous inf assume since finite rewrite operation wise applications min min min min per case min similarly composition two functions also extension finite number compositions bridging static dynamic program analysis using fuzzy logic sup max equivalent min case proof follows way case

complexity manipulation partial information voting jul palash dey tata institute fundamental research mumbai neeldhara misra indian institute technology gandhinagar mail narahari indian institute science bangalore hari january abstract coalitional manipulation problem studied extensively literature many voting rules however studies focused complete information setting wherein manipulators know votes assumption reasonable purposes showing intractability unrealistic algorithmic considerations scenarios impractical assume manipulators accurate knowledge votes work investigate manipulation incomplete information framework manipulators know partial order voter consistent true preference voter setting formulate three natural computational notions manipulation namely weak opportunistic strong manipulation say extension partial order viable exists manipulative vote extension propose following notions manipulation manipulators incomplete information votes voters eak anipulation manipulators seek vote way makes preferred candidate win least one extension partial votes pportunistic anipulation manipulators seek vote way makes preferred candidate win every viable extension partial votes nonmanipulators trong anipulation manipulators seek vote way makes preferred candidate win every extension partial votes consider several scenarios traditional manipulation problems easy instance borda single manipulator many corresponding manipulative questions propose turn computationally intractable hardness results often hold even little information missing words even instances close complete information setting results show impact paucity information computational complexity manipulation crucially depends notion manipulation consideration overall conclusion computational hardness continues valid obstruction manipulation context realistic model keywords voting manipulation incomplete information algorithm computational complexity introduction many real life related applications agents often need agree upon common decision although different preferences available alternatives natural tool used situations voting classic examples use voting rules context multiagent systems include clarke tax collaborative filtering similarity search etc typical voting scenario set candidates set voters reporting rankings candidates called preferences votes voting rule selects one candidate winner voters provide votes set votes set candidates along voting rule called election central issue voting possibility manipulation many voting rules turns even single vote cast differently alter outcome particular voter manipulates election misrepresenting preference obtains outcome prefers honest outcome cornerstone impossibility result gibbard satterthwaite show every unanimous voting rule three candidates manipulable refer excellent introduction various strategic issues computational social choice theory considering voting rules indeed susceptible manipulation natural seek ways elections protected manipulations works bartholdi approach problem perspective computational intractability exploit possibility voting rules despite vulnerable manipulation theory may hard manipulate practice indeed manipulator faced following decision problem given collection votes distinguished candidate exist vote tallied makes win fixed voting rule manipulation problem subsequently generalized problem oalitional manipulation conitzer one manipulators collude together try make distinguished candidate win election manipulation problem fortunately turns several settings established success approach demonstrating computational barrier manipulation however despite set demonstrate hardness manipulation initial results contrary indicating many voting rules fact easy manipulate moreover even multiple manipulators involved popular voting rules like plurality veto kapproval bucklin fallback continue easy manipulate know computational intractability may provide strong barrier even rules coalitional manipulation problem turns cases possibility manipulation much serious concern motivation problem formulation work propose extend argument computational intractability address cases approach appears fail note incarnations manipulation problem studied far complete information setting manipulators complete knowledge preferences truthful voters assumptions indeed best possible computationally negative results note reflective typical scenarios indeed concerns regarding privacy information cases sheer volume information would significant hurdles manipulators obtain complete information motivated consider manipulation problem natural partial mation setting particular model partial information manipulators votes partial orders set candidates partial order set candidates called partial vote results show several voting rules easy manipulate complete information setting become intractable manipulators know partial votes indeed many voting rules show even ordering small number pairs candidates missing profile manipulation becomes intractable problem results therefore strengthen view manipulation may practical limit information manipulators disposal votes voters introduce three new computational problems natural way extend question manipulation partial information setting problems input set partial votes corresponding votes set manipulators preferred candidate task eak anipulation problem determine way cast manipulators votes wins election least one extension partial votes hand trong anipulation problem would like know way casting manipulators votes wins election every extension partial votes also introduce problem pportunistic anipulation intermediate notion manipulation let call extension partial profile viable possible manipulators vote way manipulators desired candidate wins extension words viable extension standard oalitional anipulation problem opportunistic manipulation possible manipulators cast vote makes win election viable extensions note trong anipulation also pportunistic anipulation may true reverse direction particularly extreme example consider partial profile viable extensions would trong anipulation vacuous pportunistic anipulation pportunistic anipulation problem allows explore relaxed notion manipulation one manipulators obliged successful extensions possible successful note goal trong anipulation successful extensions therefore interesting instances ones extensions viable easy see instance trong anipulation also instance ppor tunistic anipulation eak anipulation beyond remark three problems questions different goals neither render redundant refer reader figure simple example distinguishing scenarios problems generalize oalitional anipulation hence computational intractability result oalitional anipulation immediately yields corresponding intractability result eak anipulation trong anipulation pportunistic nipulation setting example known oalitional anipula tion problem intractable maximin voting rule least two manipulators hence eak anipulation trong anipulation pportunistic anipulation problems intractable maximin voting rule least two manipulators figure example partial profile consider plurality voting rule one manipulator favorite candidate manipulator simply place top vote make win extension favorite candidate vote makes win extension finally favorite candidate vote places top manipulator make win viable extension extension related work notion manipulation partial information considered conitzer focus whether exists dominating manipulation show problem many common voting rules given partial votes dominating manipulation vote manipulator cast makes winner least preferable sometimes preferable winner manipulator votes truthfully dominating manipulation problem eak anipulation ppor tunistic anipulation trong anipulation problems seem apparent connection example dominating manipulation problem common voting rules except plurality veto whereas trong anipulation problem easy cases see table however results establish fact indeed possible make manipulation intractable restricting amount information manipulators possess votes voters elkind study manipulation voting rule uncertainty however work voting rule fixed known manipulators two closely related problems extensively studied context incomplete votes ossible inner ecessary inner ossible inner problem given set partial votes candidate question whether exists extension wins ecessary inner problem question whether winner every extension following work number special cases variants ossible inner problem studied literature flavor eak anipulation problem clearly similar ossible inner however emphasize subtle distinctions two problems elaborate comparison made next section contribution primary contribution work propose study three natural realistic generalizations computational problem manipulation incomplete information setting summarize complexity results work table results provide following interesting insights impact lack information computational difficulty manipulation note number undetermined pairs candidates per vote small constants hardness results observe computational problem manipulation plurality veto voting rules remains polynomial time solvable even lack information irrespective notion manipulation consideration proposition theorem observation note plurality veto voting rule also remain vulnerable notion dominating manipulation impact absence information computational complexity manipulation dynamic bucklin borda maximin voting rules eak anipulation pportunistic anipulation problems computationally intractable theorem theorem bucklin theorem borda observation theorem maximin observation theorem voting rules whereas trong anipulation problem remains computationally tractable theorem table shows interesting behavior fallback voting rule fallback voting rule voting rule among voting rules study eak anipulation problem theorem pportunistic anipulation trong anipulation problems polynomial time solvable theorem observation pportunistic anipulation problem solved fallback voting rule simply making manipulators vote desired candidate results show absence information makes three notions manipulations intractable voting rule every rational eak anipulation problem observation every pportunistic anipulation trong anipulation problems theorem results see table show whether lack information makes manipulation problems harder crucially depends notion manipulation applicable situation consideration three notions manipulations view natural extension manipulation incomplete information setting tries capture different behaviors manipulators example eak anipulation problem maybe applicable optimistic manipulator whereas pessimistic manipulator trong anipulation problem may make sense organization paper define problems introduce basic terminology next section present hardness results section section present polynomially solvable algorithms finally conclude future directions research section plurality veto bucklin fallback borda maximin table summary results denotes number manipulators results white follow immediately literature observation results voting rule hold every rational eak anipulation problem every pportunistic anipulation trong anipulation problems preliminaries section begin providing technical definitions notations need subsequent sections formulate problems capture notions manipulation votes given partial orders finally draw comparisons related problems already studied literature computational social choice theory notations definitions let set voters set candidates specified explicitly denote total number voters total number candidates respectively voter vote preference candidates linear order example two candidates means voter prefers denote set linear orders hence denotes set preference profile map called voting rule preference profile say wins uniquely write whenever say candidate wins mean candidate wins uniquely simplicity restrict unique winner case paper proofs easily extended case general setting election votes partial orders candidates partial order relation reflexive antisymmetric transitive partial vote extended possibly one linear vote depending fix order unspecified pairs candidates example election set candidates valid partial vote partial vote extended three linear votes namely paper often define partial vote like mean partial vote obtained removing order among pair candidates also whenever specify order among set candidates describing complete vote correct whichever way fix order among give examples common voting rules positional scoring rules vector naturally defines voting rule candidate gets score vote placed ith position score candidate sum scores receives votes winners candidates maximum score scoring rules remain unchanged multiply every constant add constant hence assume without loss generality score vector exists call normalized score vector get borda voting rule else voting rule get known voting rule else plurality veto bucklin simplified bucklin let minimum integer least one candidate gets majority within top positions votes winners simplified bucklin voting rule candidates votes within top positions winners bucklin voting rule candidates appearing within top positions votes highest number times fallback simplified fallback voting rules voter ranks subset candidates disapproves rest candidates fallback simplified fallback voting rules apply bucklin simplified bucklin voting rules respectively define winners integer least one candidate gets votes fallback simplified fallback voting rules output candidates approvals winners assume simplicity number candidates partial vote approves known maximin two candidates let respectively number voters prefer respectively election get restricting votes called pairwise election maximin score candidate winners candidates maximum maximin score score candidate candidate number candidates defeats pairwise election plus times number candidates ties pairwise elections winners candidates maximum score problem definitions formally define three problems consider work namely eak anipu lation pportunistic anipulation trong anipulation let fixed voting rule first introduce eak anipulation problem definition eak anipulation given set partial votes set candidates positive integer denoting number manipulators candidate exist votes exists extension define pportunistic anipulation problem first introduce notion opportunistic voting profile voting rule particular candidate definition voting profile let number manipulators set partial votes profile called voting extension exists profile words profile respect partial profile put together truthful votes extension wins extension viable begin ready define pportunistic anipulation problem definition pportunistic anipulation given set partial votes set candidates positive integer denoting number manipulators candidate exist profile finally define trong anipulation problem definition trong anipulation given set partial votes set candidates positive integer denoting number manipulators candidate exist votes every extension use denote instances eak anipulation pportunistic anipulation trong anipulation denotes profile partial votes denotes number manipulators denotes desired winner sake completeness provide definitions oalitional anipulation ossible inner problems definition oalitional anipulation given set complete votes set candidates positive integer denoting number manipulators candidate exist votes definition ossible inner given set partial votes candidate exist extension partial votes linear votes comparison possible winner coalitional manipulation fixed voting rule eak anipulation problem manipulators reduces ossible inner problem achieved simply using set truthful votes introducing empty votes summarize observation observation eak anipulation problem reduces ossible inner problem every voting rule proof let instance eak anipulation let set consisting many copies partial votes clearly eak anipulation instance equivalent ossible inner instance however whether ossible inner problem reduces eak anipulation problem clear since eak anipulation problem instance must exist least one manipulator ossible inner instance may empty vote technical point view difference eak anipulation ossible inner problems may look marginal however believe eak anipulation problem natural generalization oalitional anipulation problem partial information setting thus worth studying similarly easy show oalitional anipulation problem manipulators reduces eak anipulation pportunistic anipulation trong anipulation problems manipulators since former special case latter ones observation oalitional anipulation problem manipulators reduces eak anipulation pportunistic anipulation trong anipulation problems manipulators voting rules positive integers proof follows fact every instance oalitional anipulation problem also equivalent instance eak anipulation pportunistic anipulation trong anipulation problems finally note oalitional anipulation problem manipulators reduced eak anipulation problem one manipulator introducing empty votes votes used witness good extension forward direction reverse direction given extension manipulator successful extension used manipulator votes argument leads following observation observation oalitional anipulation problem manipulators reduces eak anipulation problem one manipulator every voting rule every positive integer proof let instance oalitional anipulation let set consisting many copies partial vote others clearly eak anipulation instance equivalent oalitional anipulation instance observation used derive hardness eak anipulation even one manipulator whenever hardness oalitional anipulation known fixed number manipulators instance case voting rules borda maximin copeland however determining complexity eak anipulation one manipulator requires work voting rules oalitional anipulation polynomially solvable number manipulators plurality bucklin hardness results section present hardness results reductions sible inner problem reductions section xact ets problem also referred problem defined follows definition exact cover given set collection subsets exist use refer complement say instance instance instance rest section organized according problems addressed weak manipulation begin recall oalitional anipulation problem borda maximin voting rules every rational two manipulators therefore follows observation eak anipulation problem borda maximin voting rules every rational even one manipulator voting rules reduce corresponding ossible ner problems natural start voting profile main challenge undoing advantage favorite candidate receives manipulator vote reverse direction begin proving eak anipulation problem voting rule even one manipulator undetermined pairs per vote theorem eak anipulation problem voting rule even one manipulator constant even number undetermined pairs vote proof simplicity presentation prove theorem reduce ossible inner problem even number undetermined pairs vote let set partial votes ossible inner instance let set candidates goal check extension makes win developing instance eak anipulation need reverse advantage candidate obtains vote manipulator notice manipulator increase score one therefore construction artificially increase score candidates one despite manipulator vote win new election possible winner ossible inner instance end introduce many dummy candidates complete votes others every extend given partial votes ossible inner instance force dummy candidates preferred least rest defining every corresponding partial vote follows ensures dummy candidates receive score modified partial votes corresponding partial votes ossible inner instance notice since number undetermined pairs number undetermined pairs also let denote constructed eak anipulation instance claim two instances equivalent forward direction suppose possible winner respect let extension wins easy see manipulator make win extension placing first two positions vote note partial score zero indeed consider extension obtained mimicking extension common partial votes notice since exactly set incomparable pairs extension score strictly greater scores candidates since scores candidates exactly one scores dummy candidates score one reverse direction notice manipulator puts candidates top two positions without loss generality suppose manipulator vote others makes win election extension consider extension obtained restricting notice score candidate extension one less scores therefore candidate wins election well concluding proof proof imitated constant values reducing ossible inner problem introducing dummy candidates use lemma subsequent proofs used lemma let set candidates normalized score vector length given exists voting profile score candidates less moreover number votes poly note number votes used lemma polynomial polynomial every indeed case proofs use lemma next show problem voting rule theorem eak anipulation problem voting rule even one manipulator constant proof reduce ossible inner problem voting rule known let set partial votes ossible inner problem instance let set candidates goal check extension makes win respect assume without loss generality position fixed partial votes fix position high possible every vote introduce many dummy candidates role first dummy candidates ensure manipulator forced place bottom positions vote original candidates get score additional vote manipulator natural way achieving ensure dummy candidates score extension note know score since position fixed partial votes would force manipulator place candidates last positions indeed anything else cause candidates tie even extension makes win end begin placing dummy candidates top positions partial votes formally modify every partial vote follows others every point know scores every using lemma add complete votes final score score every score strictly score relative score every candidate remains completes description construction denote augmented set partial votes argue correctness forward direction extension votes makes win repeat extension vote manipulator puts candidate position candidates arbitrary fashion formally let manipulator vote construction wins election particular setup reverse direction consider vote manipulator extension wins note manipulator vote necessarily places candidates bottom positions indeed win election construction extend partial vote mimicking extension corresponding partial vote simply project extension original set candidates let denote proposed extension claim wins election given indeed suppose let candidate whose score least score extension note scores extension exactly scores except constant offset importantly scores offset amount implies score least score well contradiction hence two instances equivalent next prove reduction eak anipulation problem bucklin simplified bucklin voting rules even one manipulator undetermined pairs per vote theorem eak anipulation problem bucklin simplified bucklin fallback simplified fallback voting rules even one manipulator number undetermined pairs vote proof reduce problem eak anipulation simplified bucklin let instance subset size three construct eak anipulation instance based follows candidate set first introduce following partial votes correspondence sets family follows notice number undetermined pairs every vote introduce following additional complete votes copies others copies others copies others total number voters including manipulator show equivalence two instances forward direction suppose exact set cover let vote manipulator others consider following extension hand claim unique simplified bucklin winner profile notice simplified bucklin score extension since appears top positions votes corresponding set cover votes complete profile one vote manipulator candidate appears top positions times thus get majority top positions making simplified bucklin score least hence unique simplified bucklin winner profile similarly candidate appears times top positions argued remaining candidates reverse direction suppose eak anipulation instance may assume without loss generality manipulator vote others since simplified bucklin score candidates least let extension unique winner profile every candidate ranked within top positions votes win must hold least votes votes candidates also within top positions candidate within top positions unique winner hence corresponding votes form exact set cover reduction also works bucklin voting rule specifically argument forward direction exactly simplified bucklin argument reverse direction follows every candidate ranked within top positions votes never placed within top positions vote hence win must hold least votes votes candidates also within top positions notice never gets placed within top positions vote candidate within top positions gets majority within top positions thus win result fallback simplified fallback voting rules follow corresponding results bucklin simplified bucklin voting rules respectively since every bucklin simplified bucklin election also fallback simplified fallback election respectively strong manipulation know oalitional anipulation problem borda maximin voting rules every rational two manipulators thus follows observation trong anipulation borda maximin voting rules every rational least two manipulators case one manipulator trong anipulation turns solvable voting rules however show problem every single manipulator even number undetermined pairs vote bounded constant achieved careful reduction following lemma used lemma function even exists profile defeats margin moreover even following intractability result trong anipulation problem rule one manipulator undetermined pairs per vote theorem trong anipulation voting rule every even one manipulator number undetermined pairs vote proof reduce trong anipulation rule let instance assume without loss generality even integer replicate set construct corresponding eak anipulation instance follows candidate set partial votes notice number undetermined pairs every vote add set complete votes even poly using lemma achieve following margin victories pairwise elections figure shows weighted majority graph resulting election mod every mentioned figure weighted majority graph reduced instance theorem weight edges shown figure simplicity show edges among one manipulator tries make winner notice number votes trong anipulation instance including manipulator vote odd since even integers therefore never zero every every extension manipulators vote consequently particular value play role reduction hence assume without loss generality zero simply use term copeland instead show instance instance trong anipu lation instance instance trong anipulation instance instance exist vote manipulator makes unique winner every extension partial votes assume without loss generality manipulator puts first position last position vote assume instance instance suppose renaming forms exact set cover claim following extension makes copeland extension summarize pairwise margins rest candidates profile table candidates copeland score table direction notice copeland score least since defeats every candidate every extension also notice copeland score since loses every extension hence way unique winner defeats candidates defeats requires least extensions claim sets remaining extensions forms exact set cover indeed otherwise candidate covered notice votes making opportunistic manipulation reductions pportunistic anipulation start note hardness results hold even one manipulator overall approach following engineer set partial votes way manipulator forced vote limited number ways hope making favorite candidate win vote demonstrate viable extension vote fails make candidate winner leading instance pportunistic anipulation extensions rely existence exact cover hand show exact set cover viable extension thereby leading instance vacuously instance pportunistic anipulation first result pportunistic anipulation shows pportunistic anipulation problem voting rule constant even number manipulators one number undetermined pairs vote theorem pportunistic anipulation problem voting rule constant even number manipulators one number undetermined pairs vote proof reduce pportunistic anipulation rule let instance construct corresponding pportunistic anipulation instance voting rule follows begin introducing candidate every element universe along dummy candidates denoted special candidates formally candidate set every set universe define following total order candidate set denote using define partial vote follows denote set partial votes remark number undetermined pairs partial vote invoke lemma allows achieve scores candidates using polynomially many additional votes using add set complete votes poly ensure following scores denote score candidate set votes reduced instance reasoning score configuration apparent argue equivalence first argue instance words exact cover instance pportunistic anipulation turns follow fact viable extensions show next viable extension implies existence exact set cover end first observe partial votes constructed way gets additional score extension assuming manipulator approves without loss generality final score extension going viable extension every candidate pushed top positions least observe whenever happens forced top positions since behind score votes pushed place votes every lose one point votes must correspond exact cover therefore exact cover viable extension showing one direction reduction hand suppose instance exact cover let forms exact cover use exact cover come two viable extensions require manipulator vote different ways make win therefore single manipulative vote accounts extensions leading instance pportunistic anipulation first consider completion partial votes notice completion accounted along votes score tied scores score one less score therefore candidates manipulator afford approve candidates however consider extension identical except first vote changed hand way unique winner manipulator approves therefore clear way manipulator provide consolidated vote profiles therefore instance pportunistic anipulation next move voting rule show pportunistic anipulation problem every constant even number manipulators one number undetermined pairs vote theorem pportunistic anipulation problem voting rule every constant even number manipulators one number undetermined pairs vote proof reduce pportunistic anipulation rule let instance construct corresponding pportunistic anipulation instance voting rule follows candidate set every define follows using define partial vote every denote set partial votes note number undetermined pairs partial vote using lemma add set complete votes poly ensure following denote score candidate set votes one manipulator tries make winner show instance instance pportunistic anipulation instance instance forward direction let assume instance instance suppose renaming forms exact set cover let assume manipulator vote disapproves every candidate since otherwise never win uniquely show disapprove vote suppose disapprove consider following extension following scores hence win votes however vote disapproves makes unique winner votes hence vote similarly show manipulator vote disapprove vote hence exist vote pportunistic anipulation instance instance reverse direction show instance instance exist vote manipulator extension unique winner votes thereby proving pportunistic anipulation instance vacuously thus every vote notice must least votes corresponding gets pushed bottom positions since however vote placed within top many position thus exactly since notice must least one candidate covered sets corresponding votes instance instance hence win election uniquely irrespective manipulator vote thus every vote pportunistic anipulation instance instance show next similar intractability result borda voting rule undetermined pairs per vote theorem pportunistic anipulation problem borda voting rule even number manipulators one number undetermined pairs every vote proof reduce pportunistic anipulation borda rule let instance without loss generality assume divisible add three new elements set construct corresponding pportunistic anipulation instance borda voting rule follows candidate set every define follows using define partial vote every denote set partial votes note number undetermined pairs partial vote using lemma add set complete votes poly ensure following denote borda score candidate set votes one manipulator tries make winner show instance instance pportunistic anipulation instance instance notice assume without loss generality manipulator places first position second position candidate position every last position since otherwise never win uniquely irrespective extension manipulator vote looks like forward direction let assume instance instance suppose renaming forms exact set cover let manipulator vote argue vote case manipulator vote argued similarly consider following extension following borda scores hence win uniquely votes however unique winner votes hence exist vote pportunistic anipulation instance instance reverse direction show instance instance exist vote manipulator extension unique winner votes thereby proving pportunistic anipulation instance vacuously thus every vote notice score must decrease least win uniquely however every vote score decreases least one extension least one must placed top position vote however candidates placed top positions votes many times ensuring lose election also even manipulator places candidate position every win uniquely score every must decrease least one hence altogether exactly votes denoted set extension placed second position however since instance instance corresponding votes form set cover let element covered corresponding votes notice score decrease extension thus win uniquely irrespective manipulator vote thus every vote thus pportunistic anipulation instance instance thus every vote pportunistic anipulation instance instance maximin voting rule show intractability pportunistic anipulation one manipulator even number undetermined pairs every vote theorem pportunistic anipulation problem maximin voting rule even number manipulators one number undetermined pairs every vote proof reduce pportunistic anipulation maximin rule let instance construct corresponding pportunistic anipulation instance maximin voting rule follows candidate set every define follows using define partial vote every denote set partial votes note number undetermined pairs partial vote define another partial vote follows others using lemma add set complete votes poly ensure following pairwise margins notice pairwise margins among include partial vote figure shows weighted majority graph resulting election every defined one manipulator tries make winner show instance instance pportunistic anipulation instance instance notice assume without loss generality manipulator vote prefers every candidate every forward direction let assume instance instance suppose renaming forms exact set cover notice manipulator vote must prefer either show manipulator vote prefers vote two cases symmetrical consider following extension others votes maximin score hence unique maximn winner figure weighted majority graph reduced instance theorem solid line dashed line represent pairwise margins respectively weight edges shown figure within simplicity show edges among however manipulator vote makes unique maximin winner hence vote reverse direction show instance instance exist vote manipulator extension unique winner votes thereby proving pportunistic anipulation instance vacuously thus every vote consider extension notice win uniquely must least votes call set votes however every vote votes win uniquely hence also win must least one vote possible votes however sets corresponding votes form set cover since instance instance hence must exist every vote thus win uniquely irrespective vote manipulator thus every vote pportunistic anipulation instance instance next result proves pportunistic anipulation problem voting rule every even one manipulator undetermined pairs per vote theorem pportunistic anipulation problem voting rule every even number manipulators one number undetermined pairs vote proof reduce pportunistic anipulation voting rule let instance construct corresponding pportunistic anipulation instance voting rule follows candidate set every define follows others using define partial vote every denote set partial votes note number undetermined pairs partial vote define another partial vote follows others using lemma add set complete votes poly ensure following pairwise margins notice pairwise margins among include partial vote figure shows weighted majority graph resulting election least many every defined one manipulator tries make winner show instance instance pportunistic anipulation instance instance since number voters odd play role reduction thus simply omit notice assume without loss generality manipulator vote prefers every candidate forward direction let assume instance instance suppose renaming forms exact set cover suppose manipulator vote order show vote symmetrically show manipulator vote ordering order consider following extension others others others votes copeland score defeating defeating defeating many defeating defeating defeating defeating every defeating defeating figure weighted majority graph reduced instance theorem solid line dashed line represent pairwise margins respectively weight edges shown figure within weight unlabeled edges simplicity show edges among defeating hence copeland score however manipulator vote makes win uniquely hence vote thus pportunistic anipulation instance instance reverse direction show instance instance exist vote manipulator extension unique winner votes thereby proving pportunistic anipulation instance vacuously thus every vote consider extension notice win uniquely must defeat thus preferred least one vote call votes however every vote preferred thus must defeat win uniquely since instance instance must candidate covered sets corresponding votes thus preferred every vote hence win uniquely irrespective vote manipulator thus every vote pportunistic anipulation instance instance bucklin simplified bucklin voting rules show intractability pportunis tic anipulation problem undetermined pairs per vote one manipulator theorem pportunistic anipulation problem bucklin simplified bucklin voting rules even number manipulators one number undetermined pairs vote proof reduce pportunistic anipulation bucklin simplified bucklin voting rules let instance assume without loss generality divisible introduce three elements set containing even integer duplicate set construct corresponding pportunistic anipulation instance bucklin simplified bucklin voting rules follows candidate set every define follows others using define partial vote every denote set partial votes note number undetermined pairs partial vote introduce following additional complete votes copies others copies others copies others copies others copies others copies others one others one manipulator tries make winner show instance instance pportunistic anipulation instance instance total number voters pportunistic anipulation instance notice within top positions votes appears times appear times appears times appears times every candidate appears times every candidate appears times also every candidate appears times within top positions votes hence bucklin simplified bucklin voting rules assume without loss generality manipulator puts every candidate exactly one forward direction let assume instance instance suppose renaming forms exact set cover suppose manipulator vote puts every candidate within top positions show case manipulator vote puts every candidate within top positions symmetrical consider following extension others others others bucklin simplified bucklin voting rules votes however wins uniquely votes hence vote thus pportunistic anipulation instance instance reverse direction show instance instance exist vote manipulator extension unique winner votes thereby proving pportunistic anipulation instance vacuously thus every vote consider extension notice win uniquely every candidate must pushed top positions least one vote call set votes notice however every vote least one appears within top many positions since manipulator put least one within top positions appear times votes must thus win uniquely however exists candidate covered corresponding votes notice gets majority within top positions votes never get majority within top positions votes hence win uniquely irrespective vote manipulator thus every vote pportunistic anipulation instance instance polynomial time algorithms turn polynomial time cases depicted table section organized three parts one problem considered weak manipulation since ossible inner problem plurality veto voting rules follows observation eak anipulation problem plurality veto voting rules number manipulators proposition eak anipulation problem plurality veto voting rules number manipulators proof ossible inner problem plurality veto voting rules hence result follows observation strong manipulation discuss algorithms trong anipulation problem common flavor algorithms following try devise extension adversarial possible favorite candidate make win extension roughly speaking strategy work extensions well situation improves however challenging come extension globally dominant others sense described instead consider every potential nemesis might win instead build profiles good possible bad possible profile leads constraints much manipulators afford favor terms positions among manipulative votes safe typically show determine whether exists set votes respects constraints either using greedy strategy appropriate reduction flow problem note overall spirit similar approaches commonly used solving ecessary inner problem see differences details begin voting rules theorem trong anipulation problem voting rules number manipulators proof time concentrate votes candidate calculate maximum possible value smax snm snm nonmanipulators votes snm score candidate receives votes done checking possible score combinations get vote choosing one maximizes vote fix position top position manipulators votes check possible place candidates manipulators votes final value smax negative solved easily reducing max flow problem polynomial time solvable prove trong anipulation problem scoring rules one manipulator theorem trong anipulation problem scoring rule one manipulator proof candidate calculate smax using technique described proof theorem put top position manipulator vote candidate placed positions manipulator vote makes smax negative using construct bipartite graph left right edge iff candidate placed manipulator vote according criteria solve problem finding existence perfect matching graph next result proves trong anipulation problem bucklin simplified bucklin fallback simplified fallback voting rules theorem trong anipulation problem bucklin simplified bucklin fallback simplified fallback voting rules number manipulators proof let instance trong anipulation simplified bucklin let denote total number candidates instance recall manipulators cast votes ensure candidate wins every possible extension use denote set manipulating votes construct begin without loss generality manipulators place top position votes organize positioning remaining candidates across votes manipulators ensure necessary winner profile end would like develop system constraints indicating overall number times free place candidate among top positions profile particular let fix let maximum number votes appear top positions first step compute necessary conditions use denote set complete votes construct based given partial votes intuitively votes represent worst possible extensions point view pitted votes engineered ensure manipulators make win elections strongly manipulate favor formally exists voting profile manipulators wins election wins every extension profile describe profile construction based following case analysis goal ensure extent possible position top positions incorporate among top positions let either incomparable add complete vote obtained placing highest possible position lowest possible position extending remaining vote arbitrarily let least candidates preferred add complete vote obtained placing lowest possible position extending remaining vote arbitrarily let forced within top positions add complete vote obtained first placing highest possible position followed placing lowest possible position extending remaining vote arbitrarily remaining votes notice whenever top positions also top positions let denote set votes let number votes consider two cases let number times placed top positions profile let number times placed top positions profile let formulate requirement candidate majority top positions majority top positions note requirement holds strong manipulation possible therefore strongly manipulate favor must ensure every choice able negate conditions derive first condition simply translates second condition amounts requiring first least votes appears top positions note gap majority filled using votes push forward however votes contribute equally top positions respectively therefore difference must less difference summarizing following conditions collectively denote sufficient defeat extension manipulator point view provides set constraints satisfied place remaining candidates across votes whenever manipulators place candidates among top positions freely already majority hand manipulators must respect least one following constraints extending votes manipulator respecting constraints concluding impossible achieved natural greedy strategy construct manipulators votes moving positionally left right position consider manipulator populate vote position available candidate output profile process terminates completing votes otherwise say argue proof correctness suppose algorithm returns implies exists choice voting profile manipulators conditions satisfied indeed exists voting profile violated least one conditions greedy algorithm would discovered therefore matter manipulators cast vote exists extension defeated particular votes extension given choose votes among votes extend placing top positions extending rest profile arbitrary extend remaining votes positioning outside top positions clearly extension fails achieve majority top positions achieve majority top positions hand algorithm returns consider voting profile manipulators claim wins every extension suppose contrary exists extension candidate simplified bucklin score simplified bucklin score extension therefore exists attains majority top positions fails attain majority top positions however note already impossible extension profile design constraints construction number votes appears top positions greater number times appears top positions extension similarly leads desired contradiction bucklin voting rule following modifications algorithm make proof correctness bucklin voting rule similar proof correctness simplified bucklin voting rule fallback simplified fallback voting rules consider number candidates voter approves computing output every every since assume without loss generality manipulator approves candidate proof correctness along similar lines proof correctness simplified bucklin voting rule next show trong anipulation problem maximin voting rule solvable one manipulator theorem trong anipulation problem maximin voting rules one manipulator proof time concentrate votes using algorithm maximin compute pairs computed polynomial time place top position manipulator vote increase one place candidate second position candidate already assigned position manipulator vote else correctness argument similar lines classical greedy manipulation algorithm opportunistic manipulation plurality fallback simplified fallback voting rules turns voting profile manipulators approve voting profile therefore easy devise manipulative vote observation pportunistic anipulation problem plurality fallback voting rules number manipulators veto voting rule however intricate argument needed requires building system constraints reduction suitable instance maximum flow problem network show polynomial time tractability pportunistic anipulation theorem pportunistic anipulation problem veto voting rule constant number manipulators proof let input instance pportunistic anipulation may assume without loss generality manipulators approve view voting profile manipulators tuple many manipulators disapprove denote set tuples polynomial since constant tuple exists another tuple extension following properties denote veto score candidate every candidate define two quantities follows every define every define every define guess value given value check two conditions reducing max flow problem instance follows source vertex sink vertex every call set vertices vertex every vote call set vertices add edge capacity one add edge capacity one vertex vertex candidate corresponding vertex placed last position extension partial vote corresponding vertex add edge vertex capacity voter corresponding vertex also set demand every vertex total amount flow coming vertex must least voter corresponding vertex clearly three conditions met feasible amount flow flow graph since possible values iterate possible pairs tuples possible values find voting profile exists one conclusion revisited many settings complexity barrier manipulation studied problem incomplete information setting results present fresh perspective use computational complexity barrier manipulation particularly cases thought traditional manipulation problem polynomially solvable resurrect argument computational hardness relax model complete information propose incomplete information setting realistic many hardness results work even limited incompleteness information work likely starting point explorations begin leave open problem completely establishing complexity strong opportunistic weak manipulations scoring rules fundamental forms manipulation control exist voting destructive manipulation control adding candidates would interesting investigate complexity problems partial information setting another exciting direction study average case complexity opposed worst case results pursued studies already carried setting complete information studying problems propose averagecase model would reveal insights robustness incomplete information setting captured model involving partial orders results showed impact paucity information computational complexity manipulation crucially depends notion manipulation consideration also argued different notions manipulation may applicable different situations maybe based optimistic pessimistic manipulators one important direction future research run extensive experimentations real synthetic data know people manipulate absence complete information acknowledgement palash dey wishes gratefully acknowledge support google india providing special fellowship carrying doctoral work neeldhara misra acknowledges support inspire faculty scheme dst india project references yoram bachrach nadja betzler piotr faliszewski probabilistic possible winner determination international conference artificial intelligence aaai volume pages felix brandt vincent conitzer ulle endriss lang ariel procaccia handbook computational social choice nadja betzler britta dorn towards dichotomy finding possible winners elections based scoring rules mathematical foundations computer science mfcs pages springer dorothea baumeister piotr faliszewski lang rothe campaigns lazy voters truncated ballots international conference autonomous agents multiagent systems aamas valencia spain june volumes pages john bartholdi iii james orlin single transferable vote resists strategic voting soc choice john bartholdi iii tovey trick computational difficulty manipulating election soc choice nadja betzler rolf niedermeier gerhard woeginger unweighted coalitional manipulation borda rule ijcai volume pages dorothea baumeister magnus roos rothe computational complexity two variants possible winner problem international conference autonomous agents multiagent systems aamas pages dorothea baumeister magnus roos rothe lena schend lirong xia possible winner problem uncertain weights ecai pages steven brams remzi sanver voting systems combine approval preference mathematics preference choice order pages springer yann chevaleyre lang nicolas maudet monnot possible winners new candidates added case scoring rules proc international conference artificial intelligence aaai vincent conitzer tuomas sandholm lang elections candidates hard manipulate acm vincent conitzer toby walsh lirong xia dominating manipulations voting partial information 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achieving time accuracy dec lirong xue princeton university abstract propose simple approach given distributed computing resources nearly achieve accuracy prediction matching improving faster prediction time approach consists aggregating denoised predictors small number distributed subsamples show theoretically experimentally small subsample sizes suffice attain similar performance without sacrificing computational efficiency introduction neighbor classification regression achieve significantly better prediction accuracy practitioners often default achieve much faster prediction scales better large sample size fact much commercial tools nearest neighbor search remain optimized rather biasing practice towards unfortunately statistically inconsistent prediction accuracy plateaus early sample size increases keeps improving longer choices work consider access small number distributed computing units ask whether better tradeoffs achieved harnessing parallelism prediction time simple idea bagging multiple predictors computed distributed subsamples however tends require large number subsamples number computing units often constrained practice fact infinite number subsamples assumed known consistency guarantees bagging approach biau samworth samory kpotufe princeton university particularly interested small numbers distributed subsamples say practical matter hence consider simple variant idea consisting aggregating denoised predictors simple change obtain theoretical guarantees using subsamples individual processing times order better computation time main intuition behind denoising follows increase variance due subsampling hard counter predictors aggregated show problem suitably addressed denoising subsample preprocessing step replacing subsample labels estimates based original data prediction consists aggregating averaging majority predictions denoised subsamples small size interestingly shown theoretically experimentally let subsampling ratio achieving prediction accuracy order improved accuracy vanilla verified experimentally even relatively small number distributed predictors note practice aim minimize number distributed predictors equivalently number computing units usually costly right therefore main focus experiments particular see even single denoised predictor one computer observe significant improvement accuracy vanilla maintaining prediction speed main focus work classification perhaps common form prediction results readily extend regression detailed results related work nearest neighbor prediction methods among oldest enduring data analysis fix hodges cover hart kulkarni posner theoretical performance practical settings still elucidated statistical consistency well known one needs number neighbors vanilla method manuscript review aistats inconsistent either regression classification devroye case regression kpotufe shows convergence rates excess error bayes behave lipschitz regression functions data intrinsic dimension implies rate binary classification via known relations regression classification rates see devroye similar rates recovered cannings much refined parametrization marginal input distribution recent paper moscovich recovers similar rates semisupervised settings classification rates sharpened taking account noise margin mass data away decision boundary done recent work chaudhuri dasgupta obtain faster rates form regression function assumed much faster large characterizing noise margin however rates require large number neighbors growing root sample size large implies much slower prediction time practice exacerbated scarcity optimized tools nearest neighbor search contrast fast commercial tools search readily available building various space partitioning data structures krauthgamer lee clarkson beygelzimer gionis work show classification error proposed approach namely aggregated denoised optimal order plus term subsample size used denoised additional term due subsampling lower order provided words let sampling ratio achieving rate emphasize smaller subsampling ratio faster prediction time rather maintaining prediction time vanilla actually get considerably better prediction time using smaller subsamples time considerably improving prediction accuracy towards finally notice theoretical subsampling ratio best smaller intrinsic dimension data assumed known priori intrinsic dimension smallest structured data ird data unknown manifold sparse data therefore suggests much smaller subsamples hence faster prediction times possible structured data achieving good prediction accuracy mentioned earlier even simpler approach bagging predictors known consistent biau devroye biau samworth however case infinite bag size corresponding infinite number computing units setting assume one subsample per computing unit maintain beat prediction time interestingly first shown biau devroye biau subsampling ratio also tend achieving optimal prediction rates fixed albeit assuming infinite number subsamples contrast show optimal rates par even one denoised subsample suggests verified experimentally denoised subsamples required good prediction accuracy recent work kontorovich weiss theoretical nature considers similar question derives penalized approach shown statistically consistent unlike vanilla approach kontorovich weiss roughly consists finding subsample data whose induced achieves significant margin classes two classes work unfortunately finding subsample prohibitive computable time large data regimes interest contrast training phase involves random subsamples denoising parameter training time akin usual training time finally unlike cited works rates established multiclass classification sake completion depend logarithmically number classes furthermore stated earlier results extend beyond classification regression fact established first obtaining regression rates estimating regression function paper outline section presents theoretical setup prediction approach theoretical results discussed section analysis section experimental evaluations datasets presented section preliminaries distributional assumptions main focus classification although results extend regression henceforth assume given sample conditional distribution fully captured regression function defined manuscript review aistats assume following assumption intrinsic dimension regularity first define ball assume exists integer constant work unknown procedure however understood previous work see kpotufe performance methods depends intrinsic see performance approach interest would also depends unknown particular argued kpotufe low manifolds sparse data would think structured data note assumption also imposes regularity namely ensuring sufficient mass locally nns point arbitrarily far assumption smoothness function use following version tsybakov noise condition audibert tsybakov adapted multiclass setting assumption tsybakov noise condition let denote largest element exists classification procedure classifier interested classification error err well known error mimimized bayes classifier argmaxl therefore estimated classifier interested excess error err err first recall following basic nearest neighbor estimators definition prediction given let denote indices nearest neighbors sample assume simplicity ties resolved classifier defined via regression estimate classifier obtained argmax finally let denote distance nearest neighbor formally describe approach considered work definition denoised consider random subsample without replacement size let denote nearest neighbor denoised estimate given defined fixed estimator corresponds sample prelabeled resulting estimator denote subnn simplicity defined follows definition subnn let denote denoised estimators defined independent subsamples size sets indices corresponding subsample picked independently although indices set picked replacement subnn estimate majority label clear subnn estimate computed parallel machines final step namely computation majority vote takes negligible time thus view prediction time complexity query average time machines takes compute subsample time complexity gets better furthermore show even relatively small increasing variability let get small attaining excess error par verified experimentally overview results main theoretical result theorem concerns statistical performance subnn main technicality involves characterizing effect subsampling denoising performance interestingly rate depend number subsamples due averaging effect taking majority vote accross submodels discussed detail section see proof discussion lemma particular rate bounded terms bad event unlikely random submodel therefore unlikely happen majority manuscript review aistats theorem let let denote dimension balls probability least exists choice estimate satisfies err err constants depending first term function size original sample recovers recent optimal bounds classification chaudhuri dasgupta note however result chaudhuri dasgupta concerns binary classification consider general setting multiclass matching lower bounds established earlier audibert tsybakov second term function subsample size characterizes additional error vanilla due subsampling due using prediction time discussed earlier introduction first term dominates recover rates whenever subsampling ratio goes remarkable suggests smaller subsample sizes sufficient good accuracy large sample regimes motivating present work see later supported experiments mentioned earlier similar vanishing subsampling ratios shown bagged biau devroye biau samworth assuming infinite number subsamples contrast result holds number subsamples improvements supported experiments varying number subsamples along varying subsampling ratios main technicalities insights establishing theorem discussed section proof details relegated appendix analysis overview proof theorem combines statements propositions main technicality involved establishing proposition brings together effect noise margin smoothness overall error due denoising subsample overview supporting results next subsection followed proof theorem supporting results theorem relies first establishing rate convergence regression estimate used denoising subsamples rates exist literature various assumptions see kpotufe require rate holds uniformly given proposition established particular setting takes discrete multiclass values multivariate functions proof follows standard techniques adapted particular aim given appendix supplementary material proposition uniform regression error let let denote regression estimate definition choice ncd probability least simultaneously ncd function statement obtained first remarking structural assumptions namely sizable mass everywhere locally nearest neighbor distances uniformly bounded nearest neighbor distances control bias estimator variance behaves like uniform bound distances given lemma follows standard insights lemma uniform bound distances definition let denote distance nearest neighbor sample probability least following holds sup max subnn convergence ultimately interested particular regression estimates induced subsampling denoised estimates subsample viewed regression estimate evaluated nearest neighbor first step relate error bound distances lemma comes handy since viewed introducing additional bias bias turn controlled distance manuscript review aistats query subsample lemma distance order introducing bias order given smoothness thus combining two results yields following regression error denoised estimates proposition uniform convergence denoised regression let let denote regression estimate definition let denote subsample without replacement define denoised estimate following holds choice probability least simultaneously ncd mcd function proof define two parts decomposition lemma uniform convergence aggregate regresi sion given independent subsamples define regression estimate evaluated nearest neighbor suppose exists max let denote subnn estimate using subsamples probability least randomness following holds simultaneously remark notice statement probability error goes depend number submodels averaging effect majority vote intuition suppose bad event whether happens submodel suppose likelihood happening majority models markov inequality use type intuition proof however sequence related bad events using fact submodels estimates independent conditioned last inequality follows probability proposition proof result isnobtained appropriately boundo ing indicator bound second term inequality notice viewed samples therefore bounded using lemma therefore smoothness condition assumption probability least simultaneously let denote denoised classifier sample short submodel first notice majority vote label least submodels predict words ncd mcd mcd combining yields statement next consider aggregate regression error discrepancy coordinates given labels bounded terms error attainable individual denoised regression estimates bounded proposition therefore fix let manuscript review aistats bound follows suppose labels also definition know maximum entry therefore words thus bound obtain finally use fact event combine fact inequality get sup sup sup proof since classifier excess error classii fier written thus assumption proposition statement probability least yields trivial bound excess error want refine bound let largest entry vector define fixed point refine bound excess error namely separately considering following exhaustive conditions case excess error follows words larger equals case excess error refined however total mass tsybakov noise condition assumption combining conditions probability least excess error satisfies err err final statement obtained integrating sides inequality randomness conditionally independent subsamples next proposition states excess error subnn estimate bounded terms aggregate regression error considered lemma particular proposition serves account effect noise margin parameter towards obtaining faster rates terms smoothness combining results section yield main theorem whose proof given next proof theorem main result follows easily propositions given proposition suppose exists probability least randomi ness subsamples simultaneously proof fix note conditions lemma verified proposition namely probability regression errors submodels bounded probability least excess classification error estimate satisfies err err manuscript review aistats table datasets used evaluating subnn name miniboone twitterbuzz letterbng yearpredmsd winequality train test dimension classes regression regression constant depending next conditions proposition obtained lemma setting follows probability least err err given convex function conclude applying jensen inequality viewing average two terms experiments experimental setup data standardized along coordinate fitting subnn view subsample size number subsamples exogenous parameters determined practical constraints given application domain namely smaller yields faster prediction driven prediction time requirement larger improves prediction error constrained available computing units however much experiments concern sensitivity subnn yield clear insights tradeoffs choices thus fixed choice choose search done two stages first best value minimizdlog ing validation error picked final choice made refined linear range fitting also chosen two stages description particle identification roe buzz social media kawala english alphabet document classification mitchell release year songs quality wine cortez versions subnn subsampling ratio used number subsamples regression datasets error mse classification use error prediction time reported subnn methods maximum time subsamples plus aggregation time reflecting effective prediction time settings motivating work results support theoretical insights namely subnn achieve accuracy close matching time achieving fast prediction time par better sensitivity expected better times achievable smaller subsample sizes better prediction accuracy achievable subsamples tend reduce variability illustrated instance figure vary number subsamples interestingly figure miniboone dataset larger subsampling ratio yields best accuracy number subsamples gap essentially disappears enough subsamples used thus following prescription choosing small values work generally well large values improve accuracy hand subsampling ratios yield good tradeoffs accross datasets table describes datasets used experiments use fast search python datasets perform direct search due highdimensionality sparsity explained earlier main focus classification however theoretical insights previous sections extend regression substantiated section code python found https benefits denoising figure compare subnn pure bagging models suggested theory see bagging approach indeed require considerably subsamples significantly improve error vanilla contrast accuracy subnn quickly tends particular twitterbuzz subsamples sufficient statistically close gap even small subsampling ratio could due hidden beneficial structural aspects data cases experiments highlights benefits simple denoising step variance reduction technique supported std repetitions shown figure results main experimental results described table showing relative errors error method divided vanilla relative prediction time prediction time divided conclusion propose procedure theoretical guarantees easy implement distributed computing resources achieves good tradeoffs nearest neighbor methods manuscript review aistats table ratios error rates prediction times corresponding errors times data miniboone twitterbuzz letterbng yearpredmsd winequality relative error subnn subnn subnn subnn subnn knn time subnn subnn subnn knn error rate miniboone prediction time miniboone prediction error relative time subnn subnn number subsamples number subsamples figure comparing effect subsampling ratios prediction time performance subnn shown subnn estimates using subsampling ratios error rate subnn knn subnn knn error rate miniboone subnn miniboone subnn number subsamples number subsamples subnn knn error rate twitterbuzz subnn subnn knn number subsamples twitterbuzz subnn error rate number subsamples figure bagged compared subnn using subsampling ratios left right manuscript review aistats references audibert alexandre tsybakov fast learning rates classifiers annals statistics yearpredictionmsd data https yearpredictionmsd set beygelzimer kakade langford cover trees nearest neighbors international conference machine learning icml biau luc devroye layered nearest neighbour estimate bagged nearest neighbour estimate random forest method regression classification journal multivariate analysis biau arnaud guyader rate convergence bagged nearest neighbor estimate journal machine learning research feb timothy cannings thomas berrett richard samworth local nearest neighbour classification applications learning arxiv preprint kamalika chaudhuri sanjoy dasgupta rates convergence nearest neighbor classification advances neural information processing systems pages kenneth clarkson searching metric space dimensions methods learning vision theory practice paulo cortez wine quality data set https thomas cover peter hart nearest neighbor pattern classification ieee transactions information theory devroye gyorfi lugosi probabilistic theory pattern recognition springer luc devroye laszlo gyorfi adam krzyzak lugosi strong universal consistency nearest neighbor regression function estimates annals statistics pages evelyn fix joseph hodges discriminatory discrimination consistency properties technical report dtic document kawala buzz social media data set https aristides gionis piotr indyk rajeev motwani similarity search high dimensions via hashing vldb volume pages aryeh kontorovich roi weiss bayes consistent classifier aistats kpotufe regression adapts local intrinsic dimension advances neural information processing systems nips krauthgamer lee navigating nets simple algorithms proximity search symposium discrete algorithms soda sanjeev kulkarni steven posner rates convergence nearest neighbor estimation arbitrary sampling ieee transactions information theory tom mitchell twenty newsgroups data set https newsgroups open open machine learning platform https amit moscovich ariel jaffe boaz nadler minimaxoptimal regression unknown manifolds arxiv preprint byron roe miniboone particle identification data set https richard samworth optimal weighted nearest neighbour classifiers annals statistics manuscript review aistats proof main results section show proof proposition uniform bound knn regression error proof done decomposing regression error bias variance lemma bound separately lemma lemma proved first decomposition following define rkd therefore probability least following holds simultaneously max viewing expectation conditioning sample following variance bias decomposition error let inequality conclude proof using lemma uniform bound get uniform bound bias start introducing known result relative bound lemma lemma use give uniform bound distance query point nearest neighborhood data use lemma bound bias variance separately lemma proposition concluded combing two bounds lemma bias let dimension class balls probability least randomness choice following inequality holds simultaneously lemma relative bound vapnik let set subsets finite dimension drawn sample empirical probpn ability measure defined define let probability least randomness following holds simultaneously proof first fixed sample assumption smoothness using result prove lemma bound proof lemma lemma let probability least randomness closed ball max max assumption intrinsic dimension follows lemma high probability least following holds simultaneously lemma variance let dimension balls probability least randomness following inequality holds simultaneously manuscript review aistats proof consider value fix consider randomness conditioned use hoeffding inequality independent terms summation following holds probability least randomness apply analysis combine union bound following inequality holds probability least randomness consider variations given fixed left hand side inequality seen function subset covered ball sauer lemma number subsets covered ball bounded many different variations inequality varies combine variations union bound let following happens probability least randomness sup inequality holds fixed sample right hand side depend continue hold drawn thus conclude proof combining bias variance bound uniform knn regression error proof proposition apply lemma bias variance inequality probability least aboved minimized depend plug value statement obtain ncd depend solely additional tables plots section present supplemental plots tables table shows experiment table reports average prediction time subsamples rather maximum prediction times plus aggregation time comparing two tables one see differences average maximum times small words prediction time rather stable subsamples expected times mostly controlled subsample size computing resource figure presents experiments figure additional dataset twitterbuzz compares error prediction time subnn models function number subsamples used see subnn yields error rates similar knn even small number subsamples expected best prediction times achieved smaller subsample ratio manuscript review aistats table ratios error rates average prediction times corresponding errors times data miniboone twitterbuzz letterbng yearpredmsd winequality relative error subnn subnn twitterbuzz prediction time subnn subnn subnn knn time subnn subnn subnn knn error rate twitterbuzz prediction error relative average time subnn subnn number subsamples number subsamples figure comparing effect subsampling ratio number models twitterbuzz find subnn predictors reach error similar even using subsamples expected subnn subsampling ratio results best prediction times

complete intersections quadrics weak lefschetz property feb alzati abstract consider graded artinian complete intersection algebras generated homogeneous forms degree show general multiplication linear form injective prove weak lefschetz property holds algebra previously known introduction weak lefschetz property short wlp graded algebra presented quotient xmp homogeneous ideal asserts general linear form multiplication map maximal rank conjectured char complete intersection algebra generated regular sequence wlp also conjectured algebra strong lefschetz property analogous maximal rank property replaced significant case conjectures case artinian considered challenging problems despite affirmative answers see monomial case see article examine case artinian algebra generated degree first main result present article easy geometrical proof generalization called injectivity lemma proposition precisely show injectivity general multiplication map result stated corollary result also used remark obtain new simple proof wlp already covered examine case prove wlp introducing geometrical methods seem new existing literature content theorem section main result paper despite proposing limited progress toward wlp conjecture hope methods present article shed light geometrical aspects general problem inspire investigations plan article following section state general results artinian gorenstein algebras common knowledge commutative algebraists algebraic geometers prerequisites paper since chosen provide essentially exposition results section study interesting stratifications date february mathematics subject classification primary secondary key words phrases weak lefschetz property artinian algebra complete intersection work done within framework national project geometry algebraic varieties prin cofin miur alzati projective space associated generating piece ideal results immediately used give prof injectivity lemma mentioned section introduce general geometrical method approaching wlp conjecture study projectivization variety pairs introduce differential methods purpose final section consider case wlp conjecture complete intersections equal degrees give solution already mentioned general setup notations let homogeneous polynomials common zeros equal degree regular sequence set denote artinian quotient ring set observe since given space denote projective space vector subspaces element denote associated point definition said weak lefschetz property wlp fixed general linear form multiplication map maximal rank note property holds trivially since cases either definition say fails wlp surjectivity degree one dim dim multiplication surjective linear form analogous way one defines failure wlp injectivity general say wlp degree general linear form multiplication map maximal rank collect well known facts following proposition proposition following facts hold injective injective gorenstein consequence enough check wlp case injectivity resp case surjectivity algebra type socle wlp holds general linear form multiplication map injective macaulay inverse system appropriate also mention point dual sub module respect derivation action indeed action linear form defined traspose multiplication map turns derivation indeed duality generally identifies space homogeneous linear differential operators degree constant coefficients acting conversely exchanging roles clear ann usually indicated known macaulay inverse system associated inverse system gorenstein algebra generated single element equal socle degree generated derivatives order case basic certainly well known result useful following complete intersections quadrics weak lefschetz property proposition ideal independent degree ideal contain power linear form case dimension proof well known hfi shows moreover ann respect duality pairing fact explains dimension formula remark observe homogeneous polynomials define hypersurfaces associate derivative defined action therefore means hypersurface cone vertex point remark study inverse system association study wlp algebra main theme article would interesting fully understand connections result approach cited article stratifications section collect easy results homogeneous part ideal degree denote generally denote proposition ideal generated forms degree finite set dim generally one dim min proof assume dim consider irreducible curve contained set consider local analytic parametrization curve general point form impossible otherwise could completed basis vanish contradiction similarly assume dim would local analytic parametrization surface general point form contained would give produces contradiction similar reasons since three elements vanish general case let assume dim let write bijective local analytic parametrization family general point alzati one may expand zij pij find relations pij zij observe forms independent construction vanish codimension contradiction since complete intersection immediate consequence proposition significantly extends injectivity lemma proposition proved paper case corollary injectivity lemma let artinian algebra generated regular sequence forms degree general multiplication map injective proof otherwise general exists note factorization unique hence dim dim contradicts second dimension statement proposition remark observed consequence result case wlp holds indeed cases one remark proposition actually gives precise information corollary dimension non lefschetz locus locus moltiplication map injective showing locus dimension refer reader recent article many deep results context case since quadrics classified rank projective transformations give precise alternative form proposition useful later cover case proposition set projectivized set quadrics rank let ideal generated quadrics dim proof use induction case covered already proposition assume dim let general point dimensional irreducible subvariety closed set hence inductive hypothesis must therefore write since set quadratic forms rank single orbit action find local analytical parametrization given component dim parameters lir moreover construction assume independent computing derivatives find complete intersections quadrics weak lefschetz property independent elements vanish since base must obtain contradiction remark results section although simple strongly depend complete intersection hypothesis general extended case gorenstein algebras even algebras presented quadrics indeed infinite series counterexamples result corollary gorenstein case provided see example corollary differential lemmas set assuming wlp hold algebra general multiplication map injective consider subvariety unique component uniqueness comes fact fibers projections variety linear spaces variety endowed two projections respectively general define vector spaces means set following notations general general dim dim dim dim also set dim dim consider general point system parameters centered denote point neighborhood understanding rational functions defined finally denote lemma notations one particular one moreover following relations hold iii one particular one alzati proof first assertions lemma clear fact surjectivity tangent maps general point applying relation fixed base generating set immediately see multiplying relation using obtain since see choice socle therefore find proves next setting derivations last relation get compute using completes proof iii prove induction assume true prove base case starting relation let assume inductive hypothesis derivation respect obtain inductive step immediately proved multiplying last relation using complete intersection algebras presented quadrics recall following simple formula hilbert function artinian algebra presented quadrics fact let artinian algebra obtained complete intersection quadrics one dim indeed one compute dim koszul resolution fact resolution gives result dim seen directly considering special case case basis given classes mod squarefree monomials exponents whose number exactly following technical lemma useful later lemma let artinian algebra general linearly independent dimhz moreover one dimhz unless following properties hold dim incidence variety dim exist irreducible components denoting two projections complete intersections quadrics weak lefschetz property one dim dim general dimhz one components proof one wiu wiu wiu note dimhz wiu degree piece ideal generated therefore get general bound dimhz sufficient show dim wiu clear generated complete intersection therefore independent element contain linear space possibility obtaining dimhz dim wiu note setting wiu projections introduced note also one hence one dim wiu dim contained pencil rank hence reducible quadrics form either case given pencil fixed hyperplane component therefore contained since latter generated complete intersection case impossible assume general one dim fibres dimension least union form irreducible subvariety general element rank well known quadric spaces contained moreover proposition know dim hence dim dim therefore dim dim since hyperplanes containing fixed see general contain contradiction left case general one dim fibres dimension least union form irreducible subvariety following properties proposition know dim dim consequence since general rank contains spaces one dim hand contained hyperplanes one must dim dim therefore preceding inequalities actually equalities particular dim dim applying arguments every irreducible component one dim irreducible component moreover dim therefore irreducible component also maximal possible dimension dim application case last section apply results obtained far prove wlp holds algebras presented quadrics assume even simple case appears covered existing literature dimensions alzati results stated proposition need examine general multiplication map present case one note letting vary build exact sequence sheaves defined fiberwise assuming wlp hold let general element consider multiplication map notations section following result lemma dim coker dim proof map dual map perfect pairings defined multiplication hence coker proves statement also following formula relating vector space spanned derivatives respect parameters dim introduced previous section lemma arbitrary one dimhq particular dimhq proof embedded tangent space image map defined given hence dimension equal dimhq lemma tangent space dimension statement follows introduce one last preliminary result dimension place paper turns useful consider inverse system mentioned section lemma one dim proof assume dim consider generated since remark cubic cone vertex space assuming dim suitable coordinate system defined degree three homogeneous polynomial two variables easy see vector space generated partial derivatives always contain square linear form obtaining contradiction proposition spaces connected following way lemma one ker proof clear definition lemma finally prove following theorem wlp holds proof assume wlp hold view lemma two cases complete intersections quadrics weak lefschetz property case lemma dim ker dim dim dim coker since ker dim lemma dim note also second part lemma applies case planes varying form irreducible family dimension dim second part lemma know family hence case dim let consider varieties dim note one necessarily codim indeed dim dim dim hence impossible shown hence general line intersect restriction exact sequence gives exact sequence vector bundles line symmetric identifying means multiplication pairing one write sheaf map dualizing exact sequence one immediately sees degree calculation gives deg impossible case hzi lemma ker since general hence apply result corollary dim moreover lemma dim ker dim dim finally lemma dimhq easy conclude dim consider subspace dim claim proof claim recalling claim equivalent assert first using fact space linear forms vanishing one point see dimension dim dim dim dim dim means impossible lemma implies hence since socle generated degree one finds impossible general since general generate alzati references boji migliore nagel locus gondim lefschetz properties artinian gorenstein algebras presented quadrics proc amer math soc harima migliore nagel watanabe weak strong lefschetz properties artinian algebra mezzetti ottaviani laplace equations weak lefschetz property canadian journal mathematics migliore nagel gorenstein algebras presented quadrics collect math stanley weyl groups hard lefschetz theorem sperner property siam algebraic discrete methods watanabe dilworth number artinian rings finite posets rank function commutative algebra combinatorics advanced studies pure math vol kinokuniya north holland amsterdam alberto alzati dipartimento matematica milano via saldini milano italy address riccardo dipartimento scienza alta tecnologia dell insubria via valleggio como italy address

recurrent neural network training dark knowledge transfer may zhiyuan dong zhiyong center speech language technologies cslt riit tsinghua university tsinghua national laboratory information science technology chengdu institute computer applications chinese academy sciences tangzy zhangzy corresponding author abstract recurrent neural networks rnns particularly long memory lstm gained much attention automatic speech recognition asr although successful stories reported training rnns remains highly challenging especially limited training data recent research found model used teacher train child models using predictions generated teacher model supervision knowledge transfer learning employed train simple neural nets complex one final performance reach level infeasible obtain regular training paper employ knowledge transfer learning approach train rnns precisely lstm using deep neural network dnn model teacher different existing research knowledge transfer learning since teacher dnn assumed weaker child rnn however experiments asr task showed works fairly well without applying tricks learning scheme approach train rnns successfully even limited training data index recurrent neural network long shortterm memory knowledge transfer learning automatic speech recognition introduction deep learning gained significant success wide range applications example automatic speech recognition asr powerful deep learning model reported effective asr recurrent neural network rnn obvious advantage rnns compared conventional deep neural networks dnns rnns model temporal properties thus suitable modeling speech signals simple training method rnns backpropagation time algorithm approach work supported national natural science foundation china grant mestdc phd foundation project paper also supported huilan sinovoice however rather inefficient due two main reasons twists objective function caused high nonlinearity vanishing explosion gradients backpropagation order address difficulties mainly second modified architecture called long memory lstm proposed successfully applied asr echo state network esn architecture proposed weights learned training problem odd gradients exist recently special variant optimization approach successfully applied learn rnns random initialization particular problem approach computation demanding another recent study shows carefully designed momentum setting significantly improve rnn training limited computation reach performance method although methods address difficulties rnn training extent either tricky momentum method less optimal esn method particularly limited data rnn training remains difficult paper focuses lstm structure presents simple yet powerful training algorithm based knowledge transfer algorithm largely motivated recently proposed logit matching dark knowledge distiller basic idea knowledge transfer approach model involves rich knowledge target task used guide training models current research focuses learning simple models terms structure powerful yet complex model ensemble models based idea model compression asr idea employed train small dnn models large complex one paper conduct opposite study employs simple dnn model train complex rnn different existing research tries distill knowledge teacher model treat teacher model regularization training process child model smoothed step supervised training located good starting point fact leads new training approach easy perform extended model architecture employ idea address difficulties rnn training experiments asr task database verified proposed method significantly improve rnn training reset paper organized follows section briefly discusses related works section presents method section presents experiments paper concluded section related prior work study directly motivated work dark knowledge distillation important aspect distinguishes work others existing methods focus distilling knowledge complex model use improve simple models whereas study uses simple models teach complex models teacher model work fact knows much sufficient provide rough guide important train complex models rnns present study another related work knowledge transfer dnns rnns proposed however employs knowledge transfer train dnns rnns still follows conventional idea described different rnn training knowledge transfer dark knowledge distiller idea dnn model used teacher guide training models proposed several authors almost time basic assumption teacher model encodes rich knowledge task hand knowledge distilled boost child model often simpler learn many details without teacher guide ways distill knowledge logit matching approach proposed teaches child model encouraging logits activations softmax close teacher model terms norm dark knowledge distiller model proposed encourages posterior probabilities softmax output child model close teacher model terms cross entropy transfer learning applied learn simple models approach performance complex model large model ensemble example learning small dnn large dnn dnn complex rnn focus dark knowledge distiller approach showed better performance experiments basically dnn model plays role teacher generates posterior probabilities training samples new targets training models posterior probabilities called soft targets since class identities deterministic original hard targets make targets softer temperature applied scale logits softmax formulated ezi index output units introduction allows information distilled example training sample hard target involve rank information second third class soft targets rank information second third class reflected additionally large applied target even softer allows classes prominent training note additional rank information classes available original target distilled teacher model additionally larger boosts information classes time reduces information target classes large soft target falls back uniform distribution informative therefore controls knowledge distilled teacher model hence needs set appropriately according task hand dark knowledge complex model training dark knowledge form soft targets used boosting simple models also training complex models argue training soft targets offers least two advantages provides information model training makes training reliable two advantages particularly important training complex models especially training data limited firstly soft targets offer probabilistic class labels definite hard targets one hand matches real situation uncertainty always exists classification tasks example speech recognition often difficult identify phone class frame due effect hand uncertainty involves rich less discriminative information within single example example uncertainty phone classes indicates phones similar easy get confused making use information form soft targets posterior probabilities helps improve statistical strength phones collaborative way therefore particularly helpful phones little training data secondly soft targets blur decision boundary classes offers smooth training smoothness associated soft targets noticed states soft targets result less variance gradient training samples easily verified looking gradients backpropagated logit layer logit target output child model training accumulated argument confused conclusion found also applied child net large equal logit matching assumption equivalence large compared magnitude logit values infinitely large fact large gradient approach zero knowledge distilled teacher model variance given expectation conducted training data assume identical soft hard targets reasonable teacher model well trained data variance given const const constant term assume child model well learn teacher model gradient variance approaches zero soft targets impossible hard targets even training converged reduced gradient variance highly desirable training deep complex models rnns argue mitigate risk gradient vanishing explosion well known hinder rnn training leading reliable training regularization view known including soft hard targets improves performance appropriate setting weight factor balance relative contributions formulated regularized training problem objective function given pij yij represents parameters model cost associated hard soft targets respectively weight factor additionally tij pij hard soft targets sample class respectively note objective function conventional supervised training plays role regularization effect regularization term force model training child model mimic teacher model way knowledge transfer study dnn model used teacher model regularize training rnn regularization rnn training looks optima produce similar targets dnn risk largely reduced view instead training model soft hard targets altogether first train reasonable model soft targets refine model hard targets way transfer learning plays role conventional supervised training plays role rationale soft targets results reliable training used conduct model initialization however since information involved soft targets less discriminative refinement hard targets tends helpful informally interpreted teaching model less important discriminative information firstly model strong enough discriminative information learned leads new strategy based dark knowledge transfer conventional approaches based either restricted boltzmann machine rbm simple models trained stacked construct complex models dark knowledge functions different way makes complex model trainable using less discriminative information soft targets model structure change approach possesses several advantages totally supervised model whole instead layer layer tends fast used complex models layer structure clear rnn model focus paper view related curriculum training method discussed training samples easy learn firstly selected train model difficult ones selected later model fairly strong dark knowledge soft targets regarded easy samples hard targets difficult samples interestingly regularization view view closely related essentially regularization places model location parameter space good local minima easily reached relationship regularization discussed context dnn training experiments verify proposed method use train rnn acoustic models asr task known difficult note rnns mention section indeed lstms experiments conducted database noisy conditions data profile largely standard utterances model training utterances development utterances testing kaldi toolkit used conduct model training performance evaluation process largely follows recipe dnn training specifically training starts constructing system based gaussian mixture models gmm standard mfcc features plus first second order derivatives dnn system trained alignment provided gmm system feature used dnn system fbanks symmetric window applied concatenate neighboring frames lda transform used reduce feature dimension forms dnn input dnn architecture involves hidden layers layer consists units output layer composed units equal total number gaussian mixtures gmm system cross entropy used training criterion stochastic gradient descendent sgd algorithm employed perform training dark knowledge transfer learning trained dnn model used teacher model generate soft targets rnn training rnn architecture involves layers lstms cells per layer unidirectional lstm recurrent projection layer one discarded input features fbanks output units correspond gaussian mixtures dnn rnn trained streams stream contains continuous frames momentum empirically set starting learning rate set default experimental results reported table performance evaluated terms two criteria frame accuracy word error rate wer related training criterion cross entropy wer important speech recognition table fas reported training set cross validation set wer reported test set table rnn baseline trained hard targets trained dark knowledge transfer temperature set respectively dark knowledge transfer model soft targets employed three ways soft way soft targets used rnn training way soft hard targets used together soft targets play role regularization gradients soft scaled pretrain way soft targets hard targets used sequentially soft targets play role weight factor regularization approach empirically set targets dnn soft reg pretrain soft reg pretrain hard hard soft soft hard soft hard soft soft hard soft hard wer table results different models training methods observed rnn baseline beat dnn baseline terms wer although much effort devoted calibrate training process including various trials different learning rates momentum values consistent results published kaldi recipe note mean rnns inferior dnns results clear rnn model leads better quality terms training objective unfortunately advantage propagated wer test set additionally results shown interpreted rnns suitable asr terms wer fact several researchers reported better wers rnns results say database rnn basic training method generalize well terms wer although works well terms training criterion problem largely solved dark knowledge transfer learning demonstrated results systems seen soft targets rnn system obtains equal even better performance comparison dnn baseline means knowledge embedded dnn model transferred rnn model knowledge arranged better form within rnn structure paying attention results seen knowledge transfer learning improve accuracy training set leads better close fas set compared dnn rnn baseline indicates transfer learning soft targets sacrifices performance training set little leads better generalization set additionally advantage wer indicates generalization improved sense data sets also sense evaluation metrics combining soft hard targets either way regularization performance terms wer improved confirms hypothesis knowledge transfer learning play roles regularization note cases results training set lower rnn baseline confirms advantage knowledge transform learning resides improving generalizability resultant model comparing two dark knowledge rnn systems different temperatures see leads little worse fas training set slightly better wers confirms higher temperature generates smoother direction leads better generalization conclusion proposed novel rnn training method based dark knowledge transfer learning experimental results asr task demonstrated knowledge learned simple models effectively used guide training complex models knowledge used either regularization approaches lead models generalizable desired property complex models future work involves applying technique complex models difficult train conventional approaches example deep rnns knowledge transfer heterogeneous models investigation well probabilistic models neural models references deng deep learning methods applications foundations trends signal processing vol online available http graves mohamed hinton speech recognition deep recurrent neural networks proceedings ieee international conference acoustics speech signal processing icassp ieee sutskever martens dahl hinton importance initialization momentum deep learning proceedings international conference machine learning caruana deep nets really need deep advances neural information processing systems hinton vinyals dean distilling knowledge neural network nips deep learning workshop graves jaitly towards speech recognition recurrent neural networks proceedings international conference machine learning bucilu caruana model compression proceedings acm sigkdd international conference knowledge discovery data mining acm sak senior beaufays long memory recurrent neural network architectures large scale acoustic modeling proceedings annual conference international speech communication association interspeech zhao huang gong learning dnn criteria proceedings annual conference international speech communication association interspeech rumelhart hinton williams learning representations errors nature vol online available http chan lane transferring knowledge rnn dnn arxiv preprint bengio simard frasconi 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research vol povey ghoshal boulianne burget glembek goel hannemann motlicek qian schwarz silovsky stemmer vesely kaldi speech recognition toolkit ieee workshop automatic speech recognition understanding ieee signal processing society ieee catalog

implementation distributed coherent quantum observer feb ian petersen elanor huntington paper considers problem implementing previously proposed distributed direct coupling quantum observer closed linear quantum system modifying form previously proposed observer paper proposes possible experimental implementation observer plant system using parametric amplifier chain optical cavities coupled together via optical interconnections shown distributed observer converges consensus time averaged sense output element observer estimates specified output quantum plant ntroduction paper build results providing possible experimental implementation direct coupled distributed quantum observer number papers recently considered problem constructing coherent quantum observer quantum system see coherent quantum observer problem quantum plant coupled quantum observer also quantum system quantum observer constructed physically realizable quantum system system variables quantum observer converge suitable sense system variables quantum plant papers considered problem constructing direct coupling quantum observer given quantum system papers quantum plant consideration linear quantum system recent years considerable interest modeling feedback control linear quantum systems see linear quantum systems commonly arise area quantum optics see addition papers considered problem providing possible experimental implementation direct coupled observer described case quantum plant single quantum harmonic oscillator quantum observer single quantum harmonic oscillator case show possible experimental implementation augmented quantum plant work supported air force office scientific research afosr material based research sponsored air force research laboratory agreement number government authorized reproduce distribute reprints governmental purposes notwithstanding copyright notation thereon views conclusions contained herein authors interpreted necessarily representing official policies endorsements either expressed implied air force research laboratory government work also supported australian research council arc ian petersen elanor huntington research school engineering australian national university canberra act australia email quantum observer system may constructed using nondegenerate parametric amplifier ndpa coupled beamsplitter suitable choice amplifier beamsplitter parameters see description ndpa paper consider issue whether similar experimental implementation may provided distributed direct coupled quantum observer proposed paper proposes direct coupled distributed quantum observer constructed via direct connection many quantum harmonic oscillators chain illustrated figure shown quantum network constructed output direct coupled distributed quantum observer converges plant output interest time averaged sense form time averaged quantum consensus quantum networks consideration however experimental implementation approach extended straightforward way direct coupled distributed quantum observer feasible extend ndpa used allow multiple direct couplings multiple observer elements required theory hence paper modify theory develop new direct coupled distributed observer direct coupling plant first element observer couplings different elements observer via optical field couplings illustrated figure also elements observer except first one implemented passive optical cavities active element augmented plant observer system single ndpa used implement plant first observer element features mean proposed direct coupling observer much easier implement experimentally observer proposed distributed quantum observer quantum plant fig zon distributed quantum observer establish distributed quantum observer proposed paper similar properties distributed quantum observer proposed output distributed observer converges plant output quantum plant observer wna observer observer fig uantum inear ystems distributed quantum observer problem consideration quantum plant distributed quantum observer linear quantum systems see also quantum mechanical behavior linear quantum system described terms system observables operators underlying infinite dimensional complex hilbert space commutator two scalar operators defined also vector operators commutator scalar operator vector operators commutator adjoint matrix operators denotes operator adjoint dynamics closed linear quantum systems consideration described differential equations form ynb zon distributed quantum observer proposed paper interest time averaged sense however important difference observer proposed observer proposed paper output observer element corresponded quadrature whereas paper different quadratures used define outputs phase rotation move observer element element along chain observers observer real matrix vector system observables see assumed even number number modes quantum system initial system variables assumed satisfy commutation relations real matrix components case single quantum harmonic oscillator choose position operator momentum operator commutation relations general matrix assumed form diag denotes real matrix system dynamics determined system hamiltonian operator underlying hilbert space linear quantum systems consideration system hamiltonian quadratic form real symmetric matrix corresponding matrix given defined see case system variables satisfy commutation relations times system physically realizable see remark note hamiltonian preserved time system indeed since symmetric iii irect oupling istributed oherent uantum bservers proposed direct coupling coherent quantum observer quantum plant single quantum harmonic oscillator linear quantum system form described differential equation denotes vector system variables estimated observer assumed quantum plant corresponds plant hamiltonian plant position operator plant momentum operator sequel assume describe linear quantum system form correspond distributed quantum observer see also system described differential equation form observer output distributed observer estimate vector rno also vector system variables see assume distributed observer order even number number elements distributed quantum observer also assume plant variables commute observer variables assume distributed quantum observer chain structure coupled quantum plant shown figure furthermore write mplementation istributed uantum bserver consider distributed quantum observer chain structure coupled quantum plant shown figure distributed quantum observer direct coupling quantum plant first quantum observer direct coupling determined coupling hamiltonian defines coupling quantum plant first element distributed quantum observer however contrast field coupling first quantum observer quantum observers chain observers motivation structure would much easier implement experimentally structure proposed indeed subsystem consisting quantum plant first quantum observer implemented using ndpa beamsplitter similar way described see also details ndpas beamsplitters illustrated figure zon zoi coi xoi note coi augmented quantum linear system consisting quantum plant distributed quantum observer quantum system form described equations form xon con formally define notion direct coupled linear quantum observer definition distributed linear quantum observer said achieve consensus convergence quantum plant corresponding augmented linear quantum system lim ndpa fig ndpa coupled beamsplitter representing quantum plant first quantum observer beamsplitter also remaining quantum observers distributed quantum observer implemented simple cavities shown figure wia yib ith cavity yia wna wib ynb nth cavity fig optical cavity implementation remaining quantum observers distributed quantum observer proposed quantum optical implementation distributed quantum observer simpler however dynamics somewhat different distributed quantum observer proposed proceed analyze dynamics indeed using results write quantum stochastic differential equations qsdes describing observer system shown figure dxp vector position momentum operators quantum plant vector position momentum operators first quantum observer parameters depend parameters beamsplitter ndpa parameters define coupling hamiltonian matrix defined follows addition parameters beamsplitter ndpa need chosen described order obtain qsdes required form qsdes describing ith quantum observer follows xoi dxoi jxoi dwia dwib dyia xoi dwib dyib xoi dwia vector position momentum xoi operators ith quantum observer see parameters relating reflectivity partially reflecting mirrors make cavity qsdes describing quantum observer follows dxon jxon xon dwn dyn xon dwn vector position momenwhere xon tum operators quantum observer parameter relating reflectivity partially reflecting mirror cavity addition equations also following equations describe interconnections observers figure wib order describe augmented system consisting quantum plant quantum observer combine equations indeed starting observer dyn xon therefore dyn xon xon dyn using hence dyn xon follows using dwn xon dyn xon using obtain equation dxon jxon consider observer indeed follows dxo jxo dyn jxo xon jxo xon using using follows using hence using follows therefore substituting obtain dxo jxo xon continuing process obtain following qsdes variables xoi dxoi jxoi observe plant equation dxp construct suitable distributed quantum observer assume finally obtain implies quantity choice matrix means different quadratures used outputs elements distributed quantum observer phase rotation move observer element element along chain observers order construct suitable values quantities require satisfies dzp since matrix therefore combine equations write form indeed let xon write ensure quantity satisfy differential equation combined fact used establishing condition distributed quantum observer require xto xton kxon show candidate distributed quantum observer leads satisfaction condition first note defined satisfy show lim follow satisfied order establish first note write also define complex scalars straightforward verify xoi assume xon show symmetric matrix positivedefinite lemma matrix positive definite proof order establish lemma let satisfied denotes complex conjugate transpose vector follows real symmetric matrix complex hermitian matrix prove first substitute equations definition obtain write thus furthermore null space given span fact implies order show suppose vector follows since must contained null space null space therefore must form however hence null space thus conclude matrix positive definite hence matrix positive definite completes proof lemma verify condition satisfied distributed quantum observer consideration proof follows along similar lines corresponding proof given recall remark quantity remains constant time linear system however therefore follows kxe hence since therefore follows dtk kkro hence lim dtk dtk lim lim dtkkxe implies lim hence follows lim also implies lim therefore condition satisfied thus established following theorem theorem consider quantum plant form distributed direct coupled quantum observer defined equations achieves consensus convergence quantum plant eferences petersen time averaged consensus direct coupled distributed coherent quantum observer proceedings american control conference chicago july vladimirov petersen coherent quantum filtering physically realizable linear quantum plants proceedings european control conference zurich switzerland july miao espinosa petersen ugrinovskii james coherent quantum observers quantum systems australian control conference perth australia november miao james petersen coherent observers linear quantum stochastic systems automatica vol petersen direct coupling coherent quantum observer proceedings ieee systems control antibes france october also available arxiv direct coupling coherent quantum observer single qubit finite level quantum system proceedings australian control conference canberra australia november also arxiv time averaged consensus direct coupled coherent quantum observer network single qubit finite level quantum system proceedings asian control conference kota kinabalu malaysia may james nurdin petersen control linear quantum stochastic systems ieee transactions automatic control vol nurdin james petersen coherent quantum lqg control automatica vol shaiju petersen frequency domain condition physical realizability linear quantum systems ieee transactions automatic control vol gardiner zoller quantum noise berlin springer bachor ralph guide experiments quantum optics weinheim germany petersen huntington possible implementation direct coupling coherent quantum observer proceedings australian control conference gold coast australia november petersen huntington implementation direct coupling coherent quantum observer including observer measurements proceedings american control conference boston july gough james series product application quantum feedforward feedback networks ieee transactions automatic control vol zhang james direct indirect couplings coherent feedback control linear quantum systems ieee transactions automatic control vol

valuation theory generalized ifs attractors fractals mar jan dobrowolski kuhlmann abstract using valuation rings valued fields examples discuss ways notions topological ifs attractor fractal space generalized cover general settings given functions set associate iterated function system ifs denoted view function power set defined one basic approaches calling space fractal ask iterated function system functions system satisfy certain additional forms contracting definition compact metric space called fractal system functions contracting distinct absence metric one find ways encoding meant contracting banakh nowak give topological analogue common definition fractal uses iterated function systems detailed continuation approach see definition compact topological space called fractal iterated function system consisting continuous functions following shrinking condition satisfied every open covering every sequence fik date november mathematics subject classification primary secondary jan dobrowolski kuhlmann clearly suffices check finite coverings fix basis open sets suffices check finite coverings consisting basic sets every finite covering refined covering topology induced valuation field value group one take collection ultrametric balls basis note ultrametric triangle law set closed nonempty intersections reason works restrict subring except values elements form linearly ordered subset example take prime denote finite field elements consists elements consider laurent series ring valuation defined define function understood element therefore iterated function system satisfies ultrametric ball respect valuation form divides integer call radius ball empty sum understood given finite open covering consisting ultrametric balls take maximum radii balls covering covering refined covering form every since functions continuous topology induced ultrametric argument given general case discrete valuation rings see ultrametric balls fractal sense definition obvious generalization previous example example work situation last example fix integer every set therefore iterated function system satisfies stj every generalize observations discrete valuation rings general presented power series form particular mixed characteristic take discrete valuation ring maximal ideal choose uniformizing parameter value smallest positive element value set choose system representatives residue field every define function therefore jan dobrowolski kuhlmann every shows contracting hence continuous topology induced ultrametric finite finitely many functions obtain proposition every discrete valuation ring finite residue field equipped canonical ultrametric fractal definitions given note topological space existencesof continuous ifs satisfying conditions definition implies following example seen conditions imply hausdorff definition could also considered quasicompact spaces example let equipped topology open sets cofinite sets define system consists continuous functions satisfies conditions following definition seems weakest reasonable generalization definition possibly infinite function systems definition let topological space set continuous mappings satisfying finite open covering natural number image contained say topological attractor cardinal number say topological attractor set continuous functions satisfying cardinality normal spaces property implies bound weight minimal cardinality basis topology proposition suppose normal space proof choose system functions cardinality satisfying attractor claim natural number proof claim proceed induction suppose holds every get continuity thus obtain completes proof claim define clearly show basis take open subset since normal choose open sets let condition covering define since disjoint check get remains take show take open neighbourhood claim meets image contained contained therefore meets done proposition applies particular compact spaces known normal particular obtain every topological ifsattractor countable basis thus urysohn metrization theorem get corollary every topological metrizable jan dobrowolski kuhlmann condition satisfied natural examples metric shrinking condition satisfied liml diam example let baire space homeomorphic considered valuation topology field cardinality define follows satisfied witnessed covering thus want consider another topological shrinking condition allowed choose basis covering sets taken however make possible cover way whole space assumed compact allow one covering sets fixed basis leads following definition definition family functions topological space satisfies basis every finite open covering containing one set every sequence fik every space attractor set constant functions covered images say weak attractor attractor set functions satisfying cardinality smaller say attractor attractor finite set functions satisfying clearly remark compact space attractor topological ifs attractor following example seen attractor imply compactness example let considered discrete topology define attractor attractor proof choose basis consisting singletons consider covering form sufficient take max example let let example satisfies weak attractor generally cardinal number space attractor set functions cardinality weak attractor holds example cardinals countable cofinality unboundedly many cardinals proof define choose standard basis write choose open covering form axn put max sequence image either contained one sets axn disjoint thus contained proposition suppose densely ordered abelian group associated absolute value consider collection functions suppose sequence positive elements converges every sequence diam fik satisfies consider order topology proof choose basis order topology consisting open intervals consider covering form intervals contained one sets covering choose every sequence diam min subset sets covering otherwise choice would distance hence could belong intervals case would would mean set diameter smaller contained one sets covering done corollary weak attractor proof take continuous bijection lipschitz constant define integer clearly family satisfies assumptions proposition obtain satisfies course attractor family bigger cardinality fractal space compact space locally compact one ask whether locally fractal whether every element contained fractal subspace example consider laurent series field jan dobrowolski kuhlmann valuation defined arbitrary integer every function homeomorphism topology induced valuation hand see union increasing chain mutually homeomorphic fractal spaces however wish show locally fractal stronger sense idea write union collection mutually homeomorphic fractal subspaces extend functions used suitable way work simultaneously subspaces end observe two function homeomorphism note since finitely many elements modulo write finite union form example extend functions used setting every therefore hand every arbitrary discretely valued fields valuation ring valuation ideal proceed follows choose uniformizing parameter system representatives residue field set snd every unique element every define function every obtain therefore hand every take shows contracting hence continuous topology induced ultrametric define definition locally compact metric space locally fractal union collection mutually homeomorphic subspaces system functions every fractal restrictions functions definition locally compact topological space locally fractal union collection mutually homeomorphic subspaces system functions every topologically fractal restrictions functions note require functions continuous contracting satisfy indeed functions constructed property proved proposition every discretely valued field finite residue field locally fractal definitions references banakh nowak peano continuum ifs attractor proc amer math soc banakh novosad nowak strobin contractive function systems attractors metrization topological methods nonlinear analysis jan dobrowolski kuhlmann faculty mathematics physical sciences university leeds leeds address institute mathematics wielkopolska szczecin poland address fvk

optimal weighted methods aug albert giovanni august abstract consider problem reconstructing unknown bounded function defined domain noiseless noisy samples points measure reconstruction error norm given probability measure given linear space dim study general terms weighted approximations spaces based independent random samples well known approximations inaccurate unstable close even noiseless case recent results shown interest using weighted least squares reducing number samples needed achieve accuracy comparable best approximation compared standard least squares studied contribution present paper twofold theoretical perspective establish results expectation probability weighted least squares general approximation spaces results show optimal choice sampling measure weight depends space measure stability optimal accuracy achieved mild condition scales linearly additional logarithmic factor contrast present analysis covers cases function approximants unbounded might occur instance relevant case gaussian measure numerical perspective propose sampling method allows one generate independent identically distributed samples optimal measure method becomes interest multivariate setting generally tensor product type illustrate particular examples approximation spaces polynomial type domain allowed unbounded high even infinite dimensional motivated certain applications parametric stochastic pdes ams classification numbers keywords multivariate approximation weighted least squares error analysis convergence rates random matrices conditional sampling polynomial approximation introduction let borel set consider problem estimating unknown function pointwise data either noiseless noisy observations points numerous applications interest prior information either established assumed function information may take various forms research supported institut universitaire france erc adv project bread upmc univ paris cnrs umr laboratoire lions place jussieu paris france email cohen sorbonne upmc univ paris cnrs umr laboratoire lions place jussieu paris france email migliorati sorbonne regularity properties sense belongs given smoothness class decay sparsity expansion given basis iii approximability prescribed error given spaces note often related one another sometimes equivalent since many smoothness classes characterized prescribed approximation rates using certain spaces truncated expansions certain bases paper uses third type prior information taking therefore view well approximated space functions defined everywhere dim work following mild assumption exists assumption holds example contains constant functions typically space comes family nested spaces increasing dimension algebraic trigonometric polynomials piecewise polynomial functions hierarchy meshes interested measuring error norm kvk given probability measure denote associated inner product one typical strategy pick estimate space dim ideal estimator given orthogonal projection onto namely argmin general estimator computable finite number observations best approximation error min thus serves benchmark numerical method based finite sample subsequent analysis make significant use arbitrary orthonormal basis space also introduce notation min meant respect observe probability measure weighted method consists defining estimator argmin weights given noiseless case also writes argmin vkn discrete seminorm defined kvkn seminorm associated product expand solution vector solution normal equations matrix entries hlj data vector given system always least one solution unique nonsingular singular may define unique minimal norm solution note nonsingular proper norm space data noisefree may also write orthogonal projection onto norm practice estimator easily computable important functions explicit expressions evaluated point system assembled let note computing estimator solving requires basis space necessarily orthonormal yet since subsequent analysis estimator makes use orthonormal basis simply assume type subsequent analysis sometimes work assumption known uniform bound introduce truncation operator sign min study truncated weighted approximation defined note view pointwise sense therefore truncation operator aims avoiding unstabilities may occur matrix paper use randomly chosen points corresponding weights distributed way resulting random matrix concentrates towards identity increases therefore bound known alternative strategy consists setting zero estimator deviates identity given value spectral norm recall matrices norm defined precisely introduce conditioned approximation defined otherwise choice threshold distance spectral norm related subsequent analysis however value could replaced real number minor changes formulation results note cond well known much close weighted methods may become unstable inaccurate sampling distributions example space algebraic polynomials degree estimator coincides lagrange polynomial interpolation highly unstable inaccurate particular equispaced points question want address general terms therefore given space measure best choose samples weights order ensure error comparable close possible address question case randomly chosen precisely draw independently according certain probabiity measure defined natural prescription success method kvkn approaches kvk tends therefore one first obvious choice use sample according measure plan evaluate error use equal weights using equal weights weighted estimator becomes standard leastsquares estimator particular case strategy analyzed introduction function diagonal integral kernel projector function depends strictly positive due assumption reciprocal function characterized min called christoffel function particular case space algebraic polynomials total degree see obviously function satisfies define kkm recall following results standard method weights sampling measure chosen theorem condition satisfied following hold matrix satisfies tail bound satisfies uniform bound truncated estimator satisfies noiseless case iii truncated nontruncated estimators satisfy noiseless case probability larger second item result shows optimal accuracy met expectation additional term order polynomial decay ensured additional term made negligible taking strictly larger amounts taking small enough condition imposes minimal number samples ensure stability accuracy standard least squares since implies fulfillment condition requires least order however simple examples show restriction severe example uniform probability measure case one choice legendre polynomials proper normalization klj therefore condition imposes least order examples multivariate setting discussed show many relevant approximation spaces probability measures behaviour superlinear leading demanding regime terms needed number samples case multivariate downward closed polynomial spaces precise upper bounds proven measures associated jacobi polynomials addition note theory cover simple situations algebraic polynomials unbounded domains example equipped gaussian measure since orthonormal polynomials unbounded thus main results present paper show limitations overcome using proper weighted leastsquares method thus return general form discrete norm used definition weighted estimator use sampling measure generally differs positive function defined everywhere consider weighted method weights given choice norm kvkn approaches kvk increases particular case corresponds standard method analyzed theorem note changing sampling measure commonly used strategy reducing variance monte carlo methods referred importance sampling denoting orthonormal basis introduce function depends well kkm note since wlj orthonormal basis wvm find thus prove paper following generalization theorem theorem condition satisfied following hold matrix satisfies tail bound satisfies uniform bound truncated weighted estimator satisfies noiseless case iii nontruncated weighted estimators satisfy noiseless case probability larger conditioned weighted estimator satisfies noiseless case let mention quantity considered similar stability approximation results formulated slightly different form see particular theorem therein specific framework total degree polynomial spaces interest theorem leads natural way optimal sampling strategy weighted method simply take choice one readily checks probability measure since addition particular choice wkm therefore thus obtain following result consequence theorem shows choice allows obtain estimates truncated weighted estimator minimal condition least order corollary condition satisfied conclusions iii theorem hold weighted least squares choice given one interests optimal sampling strategy applies polynomial approximation unbounded domains covered theorem particular equipped gaussian measure case relevant target functions often nonuniformly bounded therefore results items iii theorem apply result item conditioned estimator remains valid since require uniform boundedness let remark results independent dimension domain however raising unavoidable effect restricting classes functions best approximation error prescribed decay due curse dimensionality note optimal pair described depends raises difficulty properly choosing samples settings choice fixed adaptive methods certain particular cases known admit limits globally equivalent limits one typical example given univariate polynomial spaces jacobi weight lebesgue measure case pluripotential equilibrium measure see one fixed constants thus case corollary also holds choice condition lnnn development sampling strategies cases varying values without asymptotic equivalences object current investigation closely related weighted strategy recently proposed analyzed polynomial framework authors propose use renormalized christoffel function definition weights however sampling fixed pluripotential equilibrium measure due fact differs main estimate obtained see therein simple form direct comparison theorem particular involves extra term vanish even one intrinsic difficulty using optimal pair described effective sample generation particular multivariate framework since measure generally tensor product type one possible approach use markov chain monte carlo methods algorithm explored methods samples mutually correlated asymptotically distributed according desired sampling measure one contribution present paper propose straightforward effective sampling strategy generating arbitrary finite number independent samples identically distributed according strategy requires tensor product structure spaces spanned tensor product bases multivariate polynomial spaces case generally tensor product type rest paper organized follows proof theorem given concise form since follows lines original results standard least squares devote analog results case samples affected additive noise proving estimates robust condition proposed method sampling optimal measure discussed illustrate effectiveness numerical examples proof theorem proof structurally similar theorem given items item iii therefore sketch observe copies rank random matrix random variable distributed according one obviously invoke chernov bound obtain almost surely exp taking observing yields item may thus take proof item first consider event case write kpm used orthogonal thus solution system since follows event simply write follows second term used fact thus summing obtain therefore obtain proof item iii place event property also means hgv expressed norm equivalence write pnm ukn vkn used pythagorean identity fact dominated since arbitrary obtain finally item proven similar way item writing event kuk conclude way noisy case similar way analyze case observations affected additive noise practical situations noise may come different sources discretization error evaluated numerical code measurement error first one may viewed perturbation deterministic funtion observe second one typically modelled stochastic fluctuation observe independent realizations centered random variable necessarily assume independent however typically assume noise centered also assume uniformly bounded conditional variance sup note may also consider consider noncentered noise amounts adding two contributions following result shows estimates theorem robust presence additive noise theorem condition satisfied following hold noise model satisfies uniform bound truncated weighted estimator satisfies conditioned weighted estimator satisfies cases proof first consider event case write use decomposition proof theorem stands solution problem noise data therefore kpm solution since follows compared proof theorem need estimate expectation third term right side simply write hwlk note first second expectations respect joint density third one respect density summing using condition obtain log rest proceed item proof theorem using event kuk remark note standard method corresponding case know noise term thus takes stardard form seen example theorem theorem note case condition implies term bounded log conclusions theorem include estimate probability similar item iii theorem obtain estimate case bounded noise assume bounded random variable equivalently assuming bounded random variable use noise model bounded noise model following result theorem condition satisfied following hold noise model nontruncated weighted estimator satisfies probability larger proof similar proof iii theorem place event use norm equivalence write kpm first two terms already appeared noiseless case treated way new term corresponds weighted approximation noise vector satisfies kpm leads random sampling analysis previous sections prescribes use optimal sampling measure defined drawing samples weighted method section discuss numerical methods generating independent random samples according measure specific relevant multivariate setting make assumption cartesian product univariate real domains product measure measure defined assume form nonnegative continuous function therefore particular absolutely continuous respect lebesgue measure consider following general setting choose univariate basis orthonormal define tensorized basis orthonormal consider general subspaces form span set thus may rename proper ordering chosen example lexicographical sense given set interest introduce max max measure thus given discuss sampling method generating independent random samples identically distributed according multivariate density note density product structure despite product density exist many methods sampling multivariate densities contrast markov chain monte carlo methods mentioned introduction method next propose exploits particular structure multivariate density order generate independent samples straightforward manner sampling univariate densities given vector coordinates introduce notation dxa dxi following mainly use particular sets may written xaq using notation associate joint density marginal density first variables namely xaq xaq since orthonormal basis obtain xaq xaq xaq xaq xaq therefore marginal density written simple form xaq xaq sequential conditional sampling based previous notation remarks propose algorithm generates samples xkd independent identically distributed realizations density multivariate case coordinates arbitrarily reordered start first coordinate sample points univariate density coincides marginal calculated univariate case algorithm terminates multivariate case iterating consider qth coordinate sample points xnq following way given values calculated previous steps sample point xkq univariate density expression side continuous assumption ensures denominator strictly positive possible choice also ensures marginal strictly positive point density satisfies densities marginals defined evaluated points respectively using simplifying term one obtains side side equation well defined defined points vanishes nonetheless finite limits point xaq limits equal expression according technical terminology side equation conditional density given respect density continuous extension xaq conditional density densities defined concisely rewritten nonnegative weights defined since density convex combination densities note orthonormal basis explicit expressions evaluated point holds univariate densities particular polynomial case standards univariate densities uniform chebyshev gaussian orthonormal polynomials expressions explicitely computable example recursion formulas algorithm summarize sampling method sequentially samples univariate densities generate independent samples multivariate density univariate case algorithm run innermost loop samples multivariate case algorithm runs also innermost loop conditionally samples also algorithm therefore relies accurate sampling methods relevant univariate densities algorithm sequential conditional sampling input output sample xkj sample xkq end xkd end close section discussing two possible methods sampling densities rejection sampling inversion transform sampling methods equally apply univariate density therefore present arbitrarily chosen rejection sampling applying method one needs find suitable univariate density whose support contains support suitable real supp density easier sample efficient pseudorandom number generators sampling available value smallest possible sampling one point using sample point sample standard uniform check case accept realization otherwise reject restart sampling beginning average acceptance occurs every trials therefore given sampling one point requires average evaluations function amounts evaluating times terms subset terms depending coefficients depend terms already evaluated sampling previous coordinates thus use sampling univariate densities overall computational cost algorithm sampling points average proportional basis functions form bounded orthonormal system immediate simple choice parameters algorithm max choice quantify precisely average computational cost sampling points dimension chebyshev polynomials whose norms satisfy obtain bound legendre polynomials whose norms satisfy crude estimate general jacobi polynomials similar upper bounds derived dependence bounds linear inversion transform sampling let cumulative distribution function associated univariate density following using method make assumption vanishes finite number times assumption fulfilled many relevant situations density associated jacobi hermite polynomials orthonormal together assumption ensures function continuous strictly increasing hence bijection unique inverse continuous strictly increasing sampling using therefore performed follows sample independent realizations identically distributed according standard uniform obtain independent samples computing equivalent find unique solution executed elementary numerical methods bisection method newton method alternative using methods one build interpolant operator approximate interpolant constructed example piecewise linear interpolation data tqsq tqsq suitable points tqsq methods interpolation method require evaluating function pointwise general evaluations computed using standard univariate quadrature formulas orthogonal polynomials explicit expression primitive used directly evaluating function finally discuss overall computational cost algorithm sampling points using sampling univariate densities bisection method overall cost amounts maximum number iterations locating zero desired tolerance computational cost iteration interpolation overall cost amounts evaluations interpolant addition cost building interpolants depend examples numerical illustrations section presents numerical performances weighted method compared standard method three relevant situations either uniform measure chebyshev measure gaussian measure one three cases choose weighted method prescribed analysis corollary standard least squares choose tests focus condition number gramian matrix quantifies stability linear system stability weighted standard estimators meaningful quantity therefore probability cond value three threshold related parameter previous analysis probability larger corollary condition gramian matrix weighted least squares satisfies therefore probability larger standard least squares theorem gramian matrix satisfies probability larger condition numerical tests probability approximated empirical probability obtained counting many times event cond occurs repeating random sampling one hundred times examples presented section confine multivariate approximation spaces polynomial type one natural assumption case require set downward closed satisfies means polynomial space spanned monomials orthonormal basis provided taking sequence univariate orthonormal polynomials univariate multivariate forthcoming examples random samples measure generated using algorithm univariate densities sampled using inversion transform sampling method inverse cumulative distribution function approximated using interpolation technique univariate examples univariate case let index set span report fig probability approximated empirical probability gramian matrix weighted method different combinations values tested three choices measure uniform gaussian chebyshev results show perceivable differences among performances weighted least squares three different measures three cases enough obtain empirical probability equal one cond confirms condition choice ensures since demands larger number samples fig shows probability gramian matrix standard least squares uniform measure condition enough empirical probability larger uniform measure gaussian measure chebyshev measure figure weighted least squares cond left uniform measure center gaussian measure right chebyshev measure uniform measure gaussian measure chebyshev measure figure standard least squares cond left uniform measure center gaussian measure right chebyshev measure gaussian measure stability requires large number evaluations roughly linearly proportional exp univariate chebyshev measure proven standard least squares stable minimal condition weighted least squares accordance theory numerical results obtained case weighted standard least squares indistinguishable see fig fig multivariate examples afterwards present numerical tests multivariate setting tests based previous section approximating probability empirical probability dimension larger one many possible ways enrich polynomial space number different downward closed sets whose cardinality equals gets large already moderate values therefore present numerical results chosen sequence polynomial spaces downward closed dim starting set contains null tests fig fig obtained using sequence increasingly embedded polynomial spaces weighted standard least squares three choices measures choice allows establish fair comparison two methods among different measures without additional variability arising modifications polynomial space uniform measure gaussian measure chebyshev measure figure weighted least squares cond left uniform measure center gaussian measure right chebyshev measure uniform measure gaussian measure chebyshev measure figure standard least squares cond left uniform measure center gaussian measure right chebyshev measure report results obtained tests dimension results fig confirm weighted least squares always yield empirical probability equal one cond provided log condition ensures choice implies thus verifying corollary results show significant differences among three choices measure straight line slope three cases uniform chebyshev gaussian separates two regimes corresponding empirical probabilities equal zero one compared univariate case fig results fig exhibit sharper transition two extreme regimes overall lower variability transition regime results standard least squares shown fig case uniform measure fig stability ensured demanding condition needed stability weighted least squares fig much less strict condition required standard least squares univariate case scales like phenomena already observed described similar results uniform measure obtained chebyshev measure fig standard least squares achieve stability using evaluations weighted least squares fig case gaussian measure drastically differs uniform chebyshev cases results fig clearly indicate large number evaluations compared required achieve stability standard least squares let mention analogous results presented figs weighted least squares obtained also dimensions many sequences increasingly embedded polynomial spaces next tables report results selected values choose satisfy condition report table empirical probabilities approximate calculated one hundred repetitions table provides multiple comparisons weighted least squares versus standard least squares three choices measure uniform gaussian chebyshev varying method weighted weighted weighted uniform gaussian chebyshev standard standard standard uniform gaussian chebyshev table cond weighted least squares versus standard least squares uniform versus gaussian versus chebyshev method weighted weighted weighted uniform gaussian chebyshev standard standard standard uniform gaussian chebyshev table average cond weighted least squares versus standard least squares uniform versus gaussian versus chebyshev table empirical probabilities related results weighted least squares equal one confirm theory since chosen values probability larger value computed using estimate proof theorem contrast weighted least squares whose empirical probability equal one independently empirical probability standard least squares depend chosen measure extent dimension well uniform measure empirical probability approximates equals zero equals equals one gaussian case standard least squares always feature null empirical probabilities chebyshev measure condition number standard least squares always lower three tested value addition results table information needed assessing severe lack stability obtaining null empirical probabilities aim table also report average value cond obtained averaging condition number repetitions used estimate empirical probabilities table information table complementary table one hand point stability robustness weighted least squares showing tamed condition number measure dimension hand provide insights stability issues standard least squares dependence standard least squares uniform measure average condition number reduces dimension increases agreement conclusion drawn table gramian matrix standard least squares gaussian measure tested values standard least squares chebyshev measure averaged condition number slightly larger one weighted least squares worth remarking results standard least squares fig table table sensitive chosen sequence polynomial spaces testing different sequences might produce different results however necessarily obey estimates proven theorem uniform chebyshev measures satisfy condition many examples standard least squares extensively discussed previous works also situations satisfy condition therefore theorem apply general satisfy exist multivariate polynomial spaces dimension gramian matrix standard least squares uniform chebyshev measures satisfy examples spaces discussed using spaces would yield null empirical probabilities table standard least squares uniform chebyshev measures weighted least squares satisfy condition sequence polynomial spaces yields empirical probabilities close one according corollary indeed robustness respect choices polynomial space dimension represents one main advantages weighted approach references chardon cohen daudet sampling reconstruction solutions helmholtz equation sampl theory signal image chkifa cohen migliorati nobile tempone discrete least squares polynomial approximation random evaluations application parametric stochastic elliptic pdes cohen davenport leviatan stability accuracy least squares approximations found comput doostan hampton coherence motivated sampling convergence analysis least squares polynomial chaos regression comput methods appl mech jakeman narayan zhou christoffel function weighted least squares algorithm collocation approximations preprint migliorati multivariate inequalities polynomials associated downward closed sets approx theory migliorati nobile von schwerin tempone analysis discrete projection polynomial spaces random evaluations found comput migliorati nobile tempone convergence estimates probability expectation discrete least squares noisy evaluations random points multivar analysis saff totik logarithmic potentials external fields springer nevai freud orthogonal polynomials christoffel functions case study approx theory nevai totik extremum problem unit circle annals mathematics tropp user friendly tail bounds sums random matrices found comput

construct deep recurrent neural networks razvan caglar kyunghyun yoshua apr informatique recherche pascanur gulcehrc department information computer science aalto university school science abstract paper explore different ways extend recurrent neural network rnn deep rnn start arguing concept depth rnn clear feedforward neural networks carefully analyzing understanding architecture rnn however find three points rnn may made deeper function transition function based observation propose two novel architectures deep rnn orthogonal earlier attempt stacking multiple recurrent layers build deep rnn schmidhuber hihi bengio provide alternative interpretation deep rnns using novel framework based neural operators proposed deep rnns empirically evaluated tasks polyphonic music prediction language modeling experimental result supports claim proposed deep rnns benefit depth outperform conventional shallow rnns introduction recurrent neural networks rnn see rumelhart recently become popular choice modeling sequences rnns successfully used various task language modeling see graves pascanu mikolov sutskever learning word embeddings see mikolov online handwritten recognition graves speech recognition graves work explore deep extensions basic rnn depth feedforward models lead expressive models pascanu believe hold recurrent models claim unlike case feedforward neural networks depth rnn ambiguous one sense consider existence composition several nonlinear computational layers neural network deep rnns already deep since rnn expressed composition multiple nonlinear layers unfolded time schmidhuber hihi bengio earlier proposed another way building deep rnn stacking multiple recurrent hidden states top approach potentially allows hidden state level operate different timescale see hermans schrauwen nonetheless notice aspects model may still considered shallow instance transition two consecutive hidden states single level shallow viewed implications kind transitions model represent discussed section based observation paper investigate possible approaches extending rnn deep rnn begin studying parts rnn may considered shallow shallow part propose alternative deeper design leads number deeper variants rnn proposed deeper variants empirically evaluated two sequence modeling tasks layout paper follows section briefly introduce concept rnn section explore different concepts depth rnns particular section propose two novel variants deep rnns evaluate empirically section two tasks polyphonic music prediction language modeling finally discuss shortcomings advantages proposed models section recurrent neural networks recurrent neural network rnn neural network simulates dynamical system input output hidden state notation subscript represents time dynamical system defined state transition function output function respectively function parameterized set parameters given set training sequences xtn ytn parameters rnn estimated minimizing following cost function predefined divergence measure euclidean distance conventional recurrent neural networks conventional rnn constructed defining transition function output function respectively transition input output matrices nonlinear functions usual use saturating nonlinear function logistic sigmoid function hyperbolic tangent function illustration rnn fig parameters conventional rnn estimated instance stochastic gradient descent sgd algorithm gradient cost function computed backpropagation time rumelhart deep recurrent neural networks deep recurrent neural networks deep learning built around hypothesis deep hierarchical model exponentially efficient representing functions shallow one bengio number recent theoretical results support hypothesis see roux bengio delalleau bengio pascanu instance shown delalleau bengio deep network may require exponentially less units represent function compared shallow network furthermore wealth empirical evidences supporting hypothesis see goodfellow hinton findings make suspect argument apply recurrent neural networks depth recurrent neural network figure conventional recurrent neural network unfolded time depth defined case feedforward neural networks multiple nonlinear layers input output unfortunately definition apply trivially recurrent neural network rnn temporal structure instance rnn unfolded time fig deep computational path input time output time crosses several nonlinear layers close analysis computation carried rnn see fig time step individually however shows certain transitions deep results linear projection followed nonlinearity clear functions shallow sense exists intermediate nonlinear hidden layer consider different types depth rnn considering transitions separately may make transition deeper one intermediate nonlinear layers two consecutive hidden states time function made deeper described previously plugging multiple intermediate nonlinear layers hidden state output choices different implication deep function model exploit structure input making function deep previous work shown representations deep networks tend better disentangle underlying factors variation original input goodfellow glorot flatten manifolds near data concentrate bengio hypothesize representations make easier learn temporal structure successive time steps relationship abstract features generally expressed easily instance illustrated recent work mikolov showing word embeddings neural language models tend related temporal neighbors simple algebraic relationships type relationship adding vector holding different regions space allowing form analogical reasoning approach making function deeper line standard practice replacing input extracted features order improve performance machine learning model see bengio recently chen deng reported better speech recognition performance could achieved employing strategy although jointly train deep function together parameters rnn deep function deep function useful disentangle factors variations hidden state making easier predict output allows hidden state model compact may result model able summarize history previous inputs efficiently let denote rnn deep function deep output rnn instead feedforward intermediate layers hidden state output proposed replace output layer conditional rnn stacked rnn figure illustrations four different recurrent neural networks rnn conventional rnn deep transition rnn shortcut connections deep transition deep output dot rnn stacked rnn erative model restricted boltzmann machines neural autoregressive distribution estimator larochelle murray paper consider feedforward intermediate layers deep transition third knob play depth transition state transition consecutive hidden states effectively adds new input summary previous inputs represented hidden state previous work rnns generally limited architecture shallow operation affine transformation followed nonlinearity instead argue procedure constructing new summary hidden state combination previous one new input highly nonlinear nonlinear transition could allow instance hidden state rnn rapidly adapt quickly changing modes input still preserving useful summary past may impossible modeled function family generalized linear models however highly nonlinear transition modeled mlp one hidden layers universal approximator property see hornik rnn deep transition called deep transition rnn throughout remainder paper model shown fig approach deep transition however introduces potential problem introduction deep transition increases number nonlinear steps gradient traverse propagated back time might become difficult train model capture longterm dependencies bengio one possible way address difficulty introduce shortcut connections see raiko deep transition added shortcut connections provide shorter paths skipping intermediate layers gradient propagated back time refer rnn deep transition shortcut connections see fig furthermore call rnn deep function deep transition deep output deep transition rnn see fig illustration consider shortcut connections well hidden hidden transition call resulting model dot approach similar deep transition proposed recently pinheiro collobert context parsing static scene introduced recurrent convolutional neural network rcnn understood recurrent network whose transition consecutive hidden states input hidden state modeled convolutional neural network rcnn shown speed scene parsing obtained result stanford background sift flow datasets dieter proposed deep transitions gaussian process models earlier valpola karhunen used deep neural network model state transition nonlinear dynamical model stack hidden states rnn may extended deeper yet another way stacking multiple recurrent hidden layers top schmidhuber hihi bengio jaeger graves call model stacked rnn srnn distinguish proposed variants goal model encourage recurrent level operate different timescale noticed srnn extend conventional shallow rnn different aspects look recurrent level srnn separately easy see transition consecutive hidden states still shallow argued limits family functions represent example structure data sufficiently complex incorporating new input frame summary seen might arbitrarily complex function case would like model function something universal approximator properties mlp model rely higher layers higher layers feed back lower layer hand srnn deal multiple time scales input sequence obvious feature srnn however orthogonal sense possible features srnn stacking multiple levels build stacked explore paper formal descriptions deep rnns give formal description deep transition recurrent neural network dtrnn deep output rnn well stacked rnn implemented deep transition rnn noticed state transition equation dynamical system simulated rnns restriction form hence propose use multilayer perceptron approximate instead case implement intermediate layers nonlinear function weight matrix layer rnn multilayered transition function deep transition rnn illustration building rnn deep state transition function shown fig illustration state transition function implemented neural network single intermediate layer formulation allows rnn learn highly nonlinear transition consecutive hidden states deep output rnn similarly use multilayer perceptron intermediate layers model output function nonlinear function weight matrix layer rnn implementing kind multilayered output function deep output recurrent neural network fig draws deep output deep transition rnn implemented using deep transition deep output single intermediate layer stacked rnn stacked rnn schmidhuber hihi bengio multiple levels transition functions defined hidden state level time state computed using instead hidden states levels recursively computed bottom level hidden state computed output obtained using usual formulation alternatively one may use hidden states compute output hermans schrauwen hidden state level may also made depend input well graves considered approaches using shortcut connections discussed earlier illustration stacked rnn fig another perspective neural operators section briefly introduce novel approach already discussed deep transition deep output recurrent neural networks rnn may built call approach based building rnn set predefined neural operators operatorbased framework framework one first defines set operators implemented multilayer perceptron mlp instance plus operator may defined function receiving two vectors returning summary may constrain dimensionality identical additionally define another operator predicts likely output symbol given summary possible define many operators paper stick two operators sufficient express proposed types rnns figure view rnn framework plus predict operators respectively clear see plus operator predict operator correspond transition function output function eqs thus step rnn thought performing plus operator update hidden state given input predict operator compute output bht see fig illustration rnn understood framework operator parameterized mlp one hidden layers hence neural operator since simply expect operation linear respect input vector using mlp implement operators proposed deep transition deep output rnn naturally arises framework provides insight constructed rnn regularized instance one may regularize model plus operator commutative however paper explore approach note different mikolov learned embeddings words happened suitable algebraic operators framework proposed rather geared toward learning operators directly experiments train four types rnns described paper number benchmark datasets evaluate performance benchmark dataset try task predicting next symbol task predicting next symbol equivalent task modeling distribution sequence sequence decompose term side replaced single timestep rnn setting rnn predicts probability next symbol sequence given previous symbols train rnn maximizing try task modeling joint distribution three different tasks polyphonic music prediction language modeling test rnns task polyphonic music prediction using three datasets nottingham jsb chorales musedata task characterlevel language modeling use penn treebank corpus marcus model descriptions compare conventional recurrent neural network rnn deep transition rnn shortcut connections transition mlp deep rnn shortcut connections hidden hidden transition mlp dot stacked rnn srnn see fig illustrations models notthingam music jsb chorales musedata language units parameters units parameters units parameters units parameters units parameters rnn dot srnn layers table sizes trained models provide number hidden units well total number parameters two numbers provided number units mean size hidden state intermediate layer respectively dot three numbers size hidden state intermediate layer consecutive hidden states intermediate layer hidden state output layer srnn number corresponds size hidden state level size model chosen limited set minimize validation error polyphonic music task see table final models case language modeling tasks chose size models tasks respectively cases use logistic sigmoid function nonlinearity hidden unit language modeling used rectified linear units glorot intermediate layers output function gave lower validation error training use stochastic gradient descent sgd employ strategy clipping gradient proposed pascanu training stops validation cost stops decreasing polyphonic music prediction nottingham musedata datasets compute gradient step subsequences steps use subsequences steps jsb chorales reset hidden state subsequence unless subsequence belongs different song previous subsequence cutoff threshold gradients set hyperparameter learning rate tuned manually dataset set hyperparameter nottingham musedata jsb chroales correspond two epochs single epoch third epoch respectively weights connections pair hidden layers sparse nonzero incoming connections per unit see sutskever weight matrix rescaled unit largest singular value pascanu weights connections input layer hidden state well hidden state output layer initialized randomly white gaussian distribution standard deviation fixed respectively case deep output functions dot weights connections hidden state intermediate layer sampled initially white gaussian distribution standard deviation cases biases initialized regularize models add white gaussian noise standard deviation weight parameter every time gradient computed graves language modeling used strategy initializing parameters case language modeling modeling standard deviations white gaussian distributions weights weights used respectively hyperparameters modeling case dot sample weights hidden state rectifier intermediate layer output function white gaussian distribution standard deviation using rectifier units language modeling fix biases language modeling learning rate starts initial value halved time validation cost decrease significantly mikolov use regularization modeling modeling use strategy adding weight noise polyphonic music prediction tasks polyphonic music prediction language modeling stacked rnn dot initialized weights conventional rnn similar pretraining feedforward neural network see hinton salakhutdinov use ten times smaller learning rate parameter pretrained either rnn notthingam jsb chorales musedata rnn dot srnn dot table performances four types rnns polyphonic music prediction numbers represent negative test sequences obtained results using dot units deep transition maxout units deep output function dropout gulcehre result analysis polyphonic music prediction test set data presented first four columns tab able observe cases one proposed deep rnns outperformed conventional shallow rnn though suitability deep rnn depended data trained best results obtained notthingam jsb chorales close use update following learning rate max indicate tively learning rate starts decreasing quickly learning rate decreases experiment set coincide time validation error starts increasing first time worse result obtained rnns trained technique fast dropout respectively bayer order quickly investigate whether proposed deeper variants rnns may also benefit recent advances feedforward neural networks use activation method dropout built another set dot recently proposed units gulcehre deep transition maxout units goodfellow deep output function furthermore used method dropout hinton instead weight noise training similarly previously trained models searched size models well learning hyperparameters minimize validation performance however pretrain models results obtained dot maxout units trained dropout shown last column tab every music dataset performance model significantly better achieved models well best results reported recurrent neural networks bayer suggests proposed variants deep rnns also benefit activations using dropout like feedforward neural networks reported results details experiment gulcehre however acknowledge results datasets obtained using rnn combined conditional generative model restricted boltzmann machines neural autoregressive distribution estimator larochelle murray output rnn dot srnn table performances four types rnns tasks language modeling numbers represent perplexity computed test sequence respectively modeling tasks results obtained shallow rnns results obtained rnns term memory units language modeling tab see perplexities test set achieved four models clearly see deep rnns dot srnn outperform conventional shallow rnn significantly tasks dot outperformed models suggests important highly nonlinear mapping hidden state output case language modeling results dot srnn modeling surpassed previous best performance achieved rnn long memory lstm units graves well shallow rnn larger hidden state mikolov even used dynamic results report without dynamic evaluation modeling results obtained using optimization method specific type rnn architecture called mrnn mikolov regularization technique called adaptive weight noise graves result however better performance achieved conventional shallow rnns without advanced note trivial use activation functions conventional rnns may cause explosion activations hidden states however perfectly safe use activation functions intermediate layers deep rnn deep transition reported mikolov using mrnn optimization technique reported mikolov using dynamic evaluation reported graves using dynamic evaluation weight noise dynamic evaluation refers approach parameters model updated data predicted regularization methods mikolov reported best performance using rnn trained learning algorithm martens sutskever discussion paper explored novel approach building deep recurrent neural network rnn considered structure rnn timestep revealed relationship consecutive hidden states hidden state output shallow based observation proposed two alternative designs deep rnn make shallow relationships modeled deep neural networks furthermore proposed make use shortcut connections deep rnns alleviate problem difficult learning potentially introduced increasing depth empirically evaluated proposed designs conventional rnn single hidden layer another approach building deep rnn stacked rnn graves task polyphonic music prediction language modeling experiments revealed rnn proposed deep transition deep output dot rnn outperformed conventional rnn stacked rnn task language modeling achieving result task language modeling polyphonic music prediction different deeper variant rnn achieved best performance dataset importantly however cases conventional shallow rnn able outperform deeper variants results strongly support claim rnn benefits deeper architecture like feedforward neural networks observation clear winner task polyphonic music prediction suggests proposed deep rnns distinct characteristic makes less suitable certain types datasets suspect future possible design train yet another deeper variant rnn combines proposed models together robust characteristics datasets instance stacked may constructed combining srnn quick additional experiment trained dot constructed using nonsaturating nonlinear activation functions trained method dropout able improve performance deep recurrent neural networks polyphonic music prediction tasks significantly suggests important investigate possibility applying recent advances feedforward neural networks novel activation functions method dropout recurrent neural networks well however leave future research one practical issue ran experiments difficulty training deep rnns able train conventional rnn well easily trivial train dot stacked rnn paper proposed use shortcut connections well pretrain either conventional rnn however believe learning may become even problematic size depth model increase future important investigate root causes difficulty explore potential solutions find recently introduced approaches advanced regularization methods pascanu advanced optimization algorithms see pascanu bengio martens promising candidates acknowledgments would like thank developers theano bergstra bastien also thank justin bayer insightful comments paper would like thank nserc compute canada calcul providing computational resources razvan pascanu supported deepmind fellowship kyunghyun cho supported fics finnish doctoral programme computational sciences academy finland finnish centre excellence computational inference research coin references bastien lamblin pascanu bergstra goodfellow bergeron bouchard bengio theano new features speed improvements deep learning unsupervised feature learning nips workshop bayer osendorfer korhammer chen urban van der smagt fast dropout applicability recurrent networks bengio learning deep architectures found trends mach bengio simard frasconi learning dependencies gradient descent difficult ieee transactions neural networks bengio mesnil dauphin rifai better mixing via deep representations icml bergstra breuleux bastien lamblin pascanu desjardins turian wardefarley bengio theano cpu gpu math expression compiler proceedings python 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significantly improving lossy compression scientific data sets based multidimensional prediction quantization jun dingwen tao sheng zizhong chen franck university california riverside usa chen argonne national laboratory usa cappello university illinois usa hpc applications producing extremely large amounts data data storage analysis becoming challenging scientific research work design new lossy compression algorithm scientific data key contribution significantly improving prediction hitting rate prediction accuracy data point based nearby data values along multiple dimensions derive series multilayer prediction formulas unified formula context data compression one serious challenge data prediction performed based preceding decompressed values compression order guarantee error bounds may degrade prediction accuracy turn explore best layer prediction considering impact compression errors prediction accuracy moreover propose adaptive quantization encoder improve prediction hitting rate considerably data size reduced significantly performing variablelength encoding uneven distribution produced quantization encoder evaluate new compressor production scientific data sets compare many compressors gzip fpzip zfp isabela experiments show compressor best class especially regard compression factors bitrates compression errors including rmse nrmse psnr solution better solution increase compression factor reduction normalized root mean squared error average reasonable error bounds bitrates ntroduction one challenging issues performing scientific simulations running parallel applications today vast amount data store disks transmit networks process postanalysis accelerated cosmology code hacc example generate data single simulation yet system mira supercomputer argonne leadership computing facility file system storage single user request total storage capacity simulation climate research also deals large volume data simulation postanalysis indicated nearly data produced community earth system model coupled model intercomparison project cmip introduced postprocessing data submitted earth system grid estimates raw data requirements project exceed data compression offers attractive solution largescale simulations experiments enables significant reduction data size keeping critical information available preserve discovery opportunities analysis accuracy lossless compression preserves information however suffers limited compression factor general far less demand scientific experiments simulations therefore lossy compression error controls fulfill user needs terms data accuracy execution demand key challenge designing efficient errorcontrolled lossy compressor scientific research applications large diversity scientific data many existing lossy compressors isabela try predict data using method spline interpolation method effectiveness compressors highly relies smoothness data local regions however simulation data often exhibits fairly sharp spiky data changes small data regions may significantly lower prediction accuracy compressor eventually degrade compression quality numarck ssem adopt quantization step terms distribution data quantile mitigate dependence smoothness data however unable strictly control compression errors based bounds zfp uses optimized orthogonal data transform strongly rely data smoothness either however requires alignment step might respect user error bound data value range huge shown later paper optimized transform coefficients highly dependent compression data modified users work propose novel lossy compression algorithm deal irregular data spiky changes effectively still strictly respecting error bounds specifically critical contributions threefold propose multidimensional prediction model significantly improve prediction hitting rate prediction accuracy data point based nearby data values multiple dimensions unlike previous work focuses prediction extending prediction multiple dimensions challenging prediction requires solving complicated surface equation system involving many variables become intractable especially number data points used prediction relatively high however since data used prediction must preceding decompressed values order strictly control compression errors prediction accuracy degraded significantly many data points selected prediction paper derive generic formula multidimensional prediction model also optimize number data points used prediction analysis realworld data cases design adaptive quantization encoding model order optimize compression quality optimization challenging need design adaptive solution based careful observation masses experiments encoding tailored reimplemented suit variable numbers quantization intervals implement new compression algorithm namely release source code bsd license comprehensively evaluate new compression method using multiple production scientific data sets across multiple domains climate simulation scientific research hurricane simulation compare compressor five compressors gzip fpzip zfp isabela experiments show compressor best class especially regard compression factors compression errors including rmse nrmse psnr three tested data sets solution better solution nearly increase compression factor reduction normalized root mean squared error average rest paper organized follows section formulate lossy compression issue describe novel compression method section iii optimized multidimensional prediction model bestlayer analysis section adaptive quantization encoding model section evaluate compression quality using multiple production scientific data sets section discuss use compressor parallel data sets perform evaluation supercomputer section vii discuss related work section viii conclude paper summary present future work roblem etrics escription paper focus mainly design implementation lossy compression algorithm scientific data sets given error bounds computing hpc applications applications generate multiple snapshots contain many variables variable specific data type example multidimensional array string data since major type scientific data focus lossy compression research compress multidimensional data sets within reasonable error bounds also want achieve better compression performance measured following metrics pointwise compression error original reconstructed data sets example absolute error relative error average compression error original reconstructed data sets example rmse nrmse psnr correlation original reconstructed data sets compression factor compression decompression speed describe metrics detail let first define necessary notations let original multidimensional data set floatingpoint scalar let reconstructed data set recovered decompression process also denote range xmax xmin discuss metrics may use measuring performance compression method metric data point let eabsi eabsi absolute error let ereli eabsi ereli relative error compression algorithm one set either one bound bounds absolute error relative error depending compression accuracy requirement compression errors guaranteed within error bounds expressed formula ebabs ebrel ebabs absolute error bound ebrel relative error bound metric evaluate average error compression first use popular root mean squared error rmse rmse eabsi diversity variables adopt normalized rmse nrmse rmse nrmse peak ratio psnr another commonly used average error metric evaluating lossy compression method especially visualization calculated following psnr rmse psnr measures size rmse relative peak size signal logically lower value means less error higher value psnr represents less error metric evaluate correlation original reconstructed data sets adopt pearson correlation coefficient cov cov covariance coefficient measurement linear dependence two variables giving note unlike pointwise relative error compared data value relative error compared value range total positive linear correlation apax profiler suggests correlation coefficient original reconstructed data five nines better metric evaluate size reduce result compression use compression factor ilesize forig ilesize fcomp already processed points including colors predicted point first layer second layer third layer ilesizebit fcomp ilesizebit file size bits data size represents amortized storage cost value data set bits per value compression less bits per value compression also mathematical relationship lower means higher compression factor metric evaluate speed compression compare throughput bytes per second based execution time compression decompression compressors iii rediction odel based utidimensional cientific data ets sections iii present novel compression algorithm high level compression process involves three steps predict every data value proposed multilayer prediction model adopt quantization encoder adaptive number intervals perform encoding technique based uneven distributed quantization codes section first present new multilayer prediction model designed multidimensional scientific data sets give solution choosing best layer multilayer prediction model illustrate prediction model works using data sets example prediction model multidimensional scientific data sets consider data set uniform grid size size second dimension size first dimension give data point global coordinate compression algorithm process data point point low dimension high dimension assume coordinates current processing data point processed data points shown figure figure also shows definition layer around processing data point denote data subset since data subset contains layer first one nth one call data build prediction model data sets using symmetric processed data points data subset predict data fourth layer figure example data set showing processed processing data data different layers prediction model first let define surface called prediction surface maximum order follows surface coefficients construct linear system equations using coordinates values data points solve system coefficients consequently build prediction surface however problem every linear system solution also means every set data able surface time fortunately demonstrate linear system constructed data solved explicit solution also demonstrate expressed linear combination data values let give following theorem proof theorem data determine surface shown equation value equals binomial coefficient data value proof transform coordinate data point new coordinate using new coordinates data values construct linear system equations let denote follows coefficient one term containing also equations contains table ormulas layer prediction two dimensional data sets either smaller theory finite differences also polynomial degree less either therefore contains transform current coordinate previous one reversely namely thus theorem know value prediction surface expressed linear combination data values hence use value predicted value words build prediction model using data values follows call prediction model using data subset prediction model consequently proposed model called multilayer prediction model also derive generic formula multilayer prediction model dimensional data sets space limitations give formula follows dimensional size data set represents used prediction model note lerenzo predictor special case multidimensional prediction model prediction formula analysis best layer multilayer prediction model subsection developed general prediction model multidimensional data sets based model need answer another critical question many layers use prediction model compression process words want find best equation exist best use twodimensional data sets explain know better result accurate data prediction accurate prediction bring better compression performance including improvements compression factor compression error speed one hand accurate prediction achieved increasing number layers bring useful information along multiple dimensions hand also note data distance bring uncorrelated information noise prediction means many layers degrade accuracy prediction therefore infer exist best number layers prediction model get best multilayer prediction model data set first need get prediction formulas different layers substituting forth generic formula prediction model shown equation formulas shown table introduce term called prediction hitting rate proportion predictable data whole data set define data point predictable data difference original value predicted value larger error bound denote prediction hitting rate number predictable data points size data set climate simulation atm data sets example hitting rates calculated table based prediction methods described second column shows prediction hitting rate using original data values orig denoted case prediction orig decomp quantization design quantization shown figure first calculate predicted value using multilayer prediction model proposed preceding section call predicted value predicted value represented red dot fig expand values predicted value scaling error bound linearly call values predicted values represented orange dots fig distance two adjacent predicted values equals twice error bound note predicted value also expanded one error bound directions form interval length twice error bound ensure intervals overlapped real value data point falls certain interval mark predictable data use corresponding predicted value interval represent real value compression case difference real value predicted value always lower error bound however real value fall interval mark data point unpredictable data since intervals use codes encode intervals since predictable data real value predicted value error bound predicted value error bound error bound predicted value error bound predicted value aeqve daptive rror controlled uantization variable length ncoding section present adaptive quantization encoding model namely aeqve optimize compression quality first introduce quantization method completely different traditional one second using logic subsection develop adaptive solution optimize number intervals errorcontrolled quantization third show fairly uneven distribution produced quantization encoder finally reduce data size significantly using variablelength encoding technique quantization codes predicted value error bound accurate layers performing prediction original data values however order guarantee compression error absolute relative falls error bounds compression algorithm must use preceding decompressed data values instead original data values therefore last column table shows hitting rate prediction using decomp preceding decompressed data values denoted case prediction become best one compression algorithm atm data sets since best layer different scientific data sets may different best layers thus give users option set value layers compression process default value compressor prediction model based original decompressed data values atm data sets quan code table rediction hitting rate using different layers figure design quantization based linear scaling error bound encoded code corresponding interval since unpredictable data encoded another code need bits encode codes example use codes encode predictable data use code encode unpredictable data process quantization encoding note proposed quantization totally different traditional quantization technique vector quantization used previous lossy compression ssem numarck two properties uniformity vector quantization method nonuniform whereas quantization uniform specifically vector quantization concentratedly data locates shorter quantization interval length quantization intervals fixed twice error bound therefore vector quantization compression error controlled every data point especially points intervals length longer twice error bound thus call quantization method quantization next question many quantization intervals use quantization leave question subsection first introduce technique adopt quantization figure shows example distribution quantization codes produced quantization encoder uses quantization intervals represent predictable data figure see distribution quantization codes uneven degree nonuniformity distribution depends accuracy previous prediction information coding theory strategy called encoding used compress nonuniform distribution source encoding common symbols generally represented using fewer bits less common symbols uneven distribution employ encoding reduce data size significantly note encoding process lossless data compression specifically use popular rate quantization code encoding strategy huffman coding describe huffman coding algorithm detail note huffman coding algorithm implemented lossless compressors market deal source byte byte hence total number symbols higher case however limit greater hence larger quantization codes need compressed using huffman coding thus compression implement highly efficient huffman coding algorithm handle source number quantization codes adaptive scheme number quantization intervals subsection proposed compression algorithm encodes predictable data corresponding quantization code uses encoding reduce data size question remaining many quantization intervals use use bit code encode data point unpredictable data stored reduction analysis however even binaryrepresentation analysis reduce data size certain extent storing unpredictable data point much overhead storing quantization codes therefore select value number quantization intervals small possible provide sufficient prediction hitting rate note rate depends error bound shown figure error bound low ebrel compression close lossless achieving high prediction hitting rate difficult hence focus research reasonable range error bounds ebrel introduce adaptive scheme number quantization intervals used compression algorithm figure shows prediction hitting rate different relative error bounds using different numbers quantization intervals atm data sets hurricane data sets indicates prediction hitting rate suddenly descend certain error bound relatively low value example using quantization intervals prediction hitting rate drop ebrel thus consider quantization intervals cover relative error bound higher however different numbers quantization intervals different capabilities cover different error bounds generally quantization intervals cover lower error bounds baker point ebrel enough climate research simulation data sets atm data sets thus based fig atm data sets using intervals error bound figure distribution produced quantization encoder atm data sets relative error bound relative error bound quantization intervals intervals rate quantization code intervals error bound figure prediction hitting rate decreasing error bounds using different quantization intervals atm data sets hurricane data sets intervals good choices ebrel ebrel respectively hurricane data sets suggest using intervals ebrel intervals ebrel compression algorithm user determine number quantization intervals setting value quantization intervals however unable achieve good prediction hitting rate smaller error bounds compression algorithm suggest user increases number quantization intervals contrast user reduce number quantization intervals reduction results prediction hitting rate smaller practice sometimes user requirement compression accuracy stable therefore user tune good value number quantization intervals get optimized compression factors following compression algorithm figure outlines proposed lossy compression algorithm note input data ddimensional array size size lowest dimension size highest dimension processing data line algorithm needs compute coefficients based equation prediction method line processing data line first algorithm computes predicted value current processing data point using prediction method line next algorithm computes difference original predicted data value encodes data point using quantization codes line data point unpredictable algorithm adopts binaryrepresentation analysis line proposed reduce storage lastly algorithm computes records decompressed value future prediction line processing data point line algorithm compress quantization codes using encoding technique line count number predictable data points line prediction hitting rate lower threshold algorithm suggest user increases quantization interval number line table iii escription data sets used empirical performance evaluation atm aps hurricane figure proposed lossy compression algorithm using prediction aeqve model computation complexity step shown figure note lines since depend number layers used prediction rather data size although line analysis operations lines hence increasing prediction hitting rate result faster compression significantly since adopt huffman coding algorithm encoding total number symbols quantization intervals line theoretical complexity log therefore overall complexity mpirical erformance valuation section evaluate compression algorithm namely various data sets atm data sets climate simulations aps data sets scientific research hurricane data sets hurricane simulation shown table iii also compare compression algorithm losseless gzip fpzip lossy compressors zfp isabela based metrics mentioned section iii conducted experiments single core imac ghz intel core processors mhz ram data source climate simulation instrument hurricane simulation dimension size data size file number compression factor first evaluated compression algorithm based compression factor figure compares compression factors five compression methods gzip fpzip zfp isabela reasonable relative error bounds namely respectively specifically ran different compressors using absolute error bounds computed based listed ratios global data value range checked compression results figure indicates best compression factor within reasonable error bounds example ebrel atm data sets average compression factor higher zfp higher higher isabela higher fpzip higher gzip aps data sets average compression factor higher zfp higher higher isabela higher fpzip higher gzip hurricane data sets average compression factor higher zfp higher higher isabela higher fpzip higher gzip note isabela deal low error bounds thus plot compression factors fails note zfp might respect error bound alignment value range huge example variable cdnumc atm data sets value range compression error data point value using zfp ebabs value range huge maximum compression error zfp much lower input error bound whereas maximum compression errors lossy compression methods including exactly input error bound means zfp overconservative regard user accuracy requirement table shows maximum compression errors zfp different error bounds fair comparison also evaluated setting input error bound maximum compression error zfp make maximum compression errors zfp comparison compression factors shown figure example maximum compression error average compression factor higher zfp atm data sets maximum compression error average compression factor higher zfp hurricane data sets note zfp designed fixed whereas including isabela designed fixed maximum compression error thus fair comparison plot curve table omparison earson correlation coefficient using various lossy compressors different maximum compression errors maximum erel atm zfp maximum erel compression fpzip gzip fpzip gzip compression factor compression relaave error bound figure comparison compression factors maximum compression error using zfp atm hurricane data sets compression fpzip gzip rela error bound figure comparison compression factors using different lossy compression methods atm aps hurricane data sets different error bounds table aximum compression errors normalized value range using zfp different user set value range based error bounds rela error bound compression factor compression factor compression relaave error bound compression rela error bound ebrel compression factor compression factor hurricane zfp atm zfp hurricane zfp lossy compressors compare distortion quality rate rate means use peak ratio psnr measure distortion quality psnr calculated equation decibel generally speaking curve higher bits per value compressed storage higher quality higher psnr reconstructed data decompression figure shows curves different lossy compressors three scientific data sets figure indicates lossy compression algorithm best curve data sets atm aps specifically equals atm data sets psnr higher zfp improvement psnr represents increase accuracy reduction rmse times also accuracy compressor times times isabela aps data sets psnr higher zfp improvement psnr represents increase accuracy times also accuracy compressor times times isabela hurricane data sets curves illustrate low psnr close zfp cases bitrate higher psnr better zfp specifically psnr higher times accuracy zfp higher times accuracy note test show cases lower three data sets means compression factors higher mentioned section lossless compressors provide compression factor reasonable assume users interested lossy compression provides compression factor higher pearson correlation next evaluated compression algorithm based pearson correlation coefficient table ompression decompression speeds using zfp different value range based relative error bounds ebrel atm zfp comp decomp comp decomp aps zfp comp decomp comp decomp psnr compression rate psnr compression rate psnr rate figure using different lossy compression methods atm aps hurricane data sets original decompressed data table shows pearson correlation coefficients using different lossy compression methods different maximum compression errors space limitations compare zfp since previous evaluations outperform isabela significantly note use maximum compression error zfp input error bound make sure three lossy compressors maximum compression error table know three compressors five nines better coefficients marked bold lower relative error bounds atm data sets lower relative error bounds hurricane data sets results mean accuracy pearson correlation decompressed data similar zfp speed let evaluate compression decompression speed compressor evaluate compression decompression speed different lossy compressors different error bound megabytes per hurricane zfp comp decomp comp decomp second first compare overall speed isabela atm aps data sets average compressor faster faster isabela hurricane data sets average faster faster isabela due space limitations show specific values isabela compare speed zfp table shows compression decompression speed zfp illustrates average compression slower zfp decompression slower zfp compression optimized performance primary objective reach high compression factors therefore plan optimize compression different architectures data sets future autocorrelation compression error finally analyze autocorrelation compression errors since applications require compression errors uncorrelated evaluate autocorrelation compression errors two typical variables atm data sets freqsh snowhlnd compression factors freqsh snowhlnd using ebrel thus extent freqsh represent relatively data sets snowhlnd represent relatively data sets figure shows first autocorrelation coefficients zfp compression errors two variables illustrates freqsh maximum autocorrelation coefficient much lower zfp however snowhlnd maximum autocorrelation coefficient higher zfp also evaluate autocorrelation zfp aps hurricane data sets observe generally autocorrelation lower zfp relatively data sets whereas zfp autocorrelation lower relatively data sets therefore plan improve autocorrelation compression errors relatively data sets future effect compression error autocorrelation application specific lossy compressor users might need understand effect using one compressor iscussion section first discuss parallel use compressor data sets perform empirical performance evaluation full atm data sets using cores nodes node two intel xeon processors memory processor cores blues cluster argonne compression zfp compression table vii trong scalability parallel compression using different number processes lues zfp figure autocorrelation analysis first coefficients compression errors increasing delays using lossy compressor zfp variable freqsh variable snowhlnd atm data sets compression wri compressed data wri ini data number processes number nodes comp speed speedup parallel efficiency table viii trong scalability parallel decompression using different number processes lues number processes number nodes decomp speed speedup parallel efficiency number processes decompression reading compressed data reading ini data number processes figure comparison time compressed data time initial data blues parallel compression classified two categories compression compression compressor easily used compressor embedded parallel application process fraction data held memory compression mpi program script used load data multiple processes run compression separately atm data sets shown table iii example total files aps data sets files users load files multiple processes run compressor parallel without communications present strong scalability parallel compression decompression without data time table vii viii different scales ranging processes blues cluster experiments set ebrel compression number processes increased two stages first stage launch one process per node increase ber nodes maximum number request second stage run parallel compression nodes changing number processes per node measure time without time use maximum time among processes test experiment five times use average time calculate speeds speedup parallel efficiency shown tables two tables illustrates parallel efficiency compressor stay nearly processes demonstrates linear speedup number processors however parallel efficiency decreased total number processes greater two processes per node performance degradation due node internal limitations note speeds singe process table vii viii different ones table since run sequential parallel compression two different platforms figure compares time compressed data time initial data bar represents sum time compressed data initial data normalize sum plot dash line ease comparison illustrates time writing reading initial data much longer time writing reading compressed data plus time compression decompression blues number processors demonstrates compressor effectively reduce total time dealing atm data sets also note relative time spent increase number processors inevitable bottleneck bandwidth data simultaneously many processes contract linear speedup number processors means performance gains greater increasing scale vii elated ork scientific data compression algorithms fall two categories losseless compression lossy compression popular lossless compression algorithms include gzip fpzip however mainly limitation lossless compressors fairly low compression factor general order improve compression factor several lossy data compression algorithms proposed recent years isabela performs data compression interpolation sorting data series isabela use extra storage record original index data point loss location information data series thus suffers low compression factor especially large numbers data points lossy compressors using vector quantization numarck ssem guarantee compression error within bound limitation compression factor demonstrated difference numarck ssem numarck uses vector quantization differences adjacent two iterations data whereas ssem uses vector quantization high frequency data wavelet transform zfp lossy compressor using alignment orthogonal block transform encoding however might respect error bound data value range huge viii onclusion uture ork paper propose novel lossy compression algorithm evaluate compression algorithm using multiple production scientific data sets across multiple domains compare five compressors based series metrics implemented released compressor bsd license key contributions listed derive generic model multidimensional prediction optimize number data points used prediction achieve significant improvement prediction hitting rate design adaptive quantization encoding model aeqve deal effectively irregular data spiky changes average compression factor compared compressor reasonable error bounds average compression error reduction atm aps hurricane data sets encourage users evaluate lossy compressor compare existing compressors scientific data sets future work plan optimize compression different architectures data sets also improve autocorrelation compression data sets relatively high compression factors acknowledgments research supported exascale computing project ecp project number collaborative effort two doe organizations office science national nuclear security administration responsible planning preparation capable exascale ecosystem including software applications hardware advanced system engineering early testbed platforms support nation exascale computing imperative submitted manuscript created uchicago argonne llc operator argonne national laboratory argonne argonne department energy office science laboratory operated contract eferences simulation hurricane national center atmospheric research http online austin advanced photon source synchrotron radiation news baker dennis levy nychka mickelson edwards vertenstein wegener methodology evaluating impact data compression climate simulation data hpdc pages bernholdt bharathi brown chanchio chen chervenak cinquini drach foster fox earth system grid supporting next generation climate modeling research proceedings ieee brenner scott mathematical theory finite element methods volume springer science business media chen son hendrix agrawal liao choudhary numarck machine learning algorithm resiliency checkpointing pages community earth simulation model cesm http online deutsch gzip file format specification version cappello fast lossy hpc data compression ipdps pages gleckler durack stouffer johnson forest global ocean heat uptake doubles recent decades nature climate change ibarria lindstrom rossignac szymczak compression decompression large ndimensional scalar fields computer graphics forum volume pages wiley online library lakshminarasimhan shah ethier chang klasky latham ross samatova isabela effective situ compression scientific data concurrency computation practice experience lindstrom compressed arrays tvcg lindstrom isenburg fast efficient compression data tvcg ratanaworabhan burtscher fast lossless compression scientific data dcc pages sasaki sato endo matsuoka exploration lossy compression ipdps pages wegener universal numerical encoder profiler reduces computing memory wall software fpga soc implementations dcc page ziv lempel universal algorithm sequential data compression ieee transactions information theory

dec change point detection autoregressive models moment assumptions fumiya akashi holger dette waseda university bochum department applied mathematics mathematik tokyo japan bochum germany yan liu waseda university department applied mathematics tokyo japan abstract paper consider problem detecting change parameters autoregressive process moments innovation process necessarily exist empirical likelihood ratio test existence change point proposed asymptotic properties studied contrast work change point tests using empirical likelihood assume knowledge location change point particular prove maximizer empirical likelihood consistent estimator parameters autoregressive model case change point derive limiting distribution corresponding test statistic null hypothesis also establish consistency new test nice feature method consists fact resulting test asymptotically distribution free require estimate long run variance asymptotic properties test investigated means small simulation study demonstrates good finite sample properties proposed method keywords phrases empirical likelihood change point analysis infinite variance autoregressive processes ams subject classification introduction problem detecting structural breaks time series studied long time since seminal work page proposed sequential scheme identifying changes mean sequence independent random variables numerous authors worked problem large part literature concentrates cusum tests nonparametric design see aue recent review important references authors make distributional assumptions construct tests structural breaks example gombay suggested likelihood ratio procedure test change mean extensions method found monograph reference therein important problem context detection changes parameters autoregressive process refer work andrews bai davis lee berkes among others proposed likelihood ratio tests practice however distribution random variables rarely known misspecification may result invalid analysis using likelihood ratio methods one seminal method treat likelihood ratio empirically investigated owen qin lawless general context extended chuang chan estimate test parameters autoregressive model change point analysis empirical likelihood approach viewed compromise completely parametric likelihood ratio nonparametric cusum method baragona used concept construct test changepoints showed case location break points known limiting distribution corresponding test statistic distribution ciuperca salloum considered change point problem model independent data without assuming knowledge location derived extreme value distribution limit distribution empirical likelihood ratio test statistic findings similar spirit meanwhile classical results considered likelihood ratio test purpose present paper investigate empirical likelihood test change parameters autoregressive process infinite variance precisely assume existence moments work motivated fact many fields electrical engineering hydrology finance physical systems one often observes data see nolan samoradnitsky taqqu among many others deal data many authors developed methods example chen constructed robust test linear hypothesis parameters based least absolute deviation ling pan proposed least absolute estimators parametric time series models infinite variance innovation process show asymptotic normality estimators however limit distribution statistics usually contains unknown probability density innovation process difficult estimate example ling pan used kernel density estimators purpose choice corresponding bandwidth clear often depends users circumvent problems type context change point analysis combine paper quantile regression empirical likelihood methods remarkable feature asymptotic distribution proposed test statistic involve unknown quantities model even consider autoregressive models infinite variance innovation process would also like emphasize nonparametric cusum tests proposed bai detecting structural breaks parameters autoregressive process assume existence variance innovations however alternative method proposed cusum tests based quantile regression cently considered xiao zhou among others remaining part paper organized follows section introduce model testing problem empirical likelihood ratio test statistic main results given section derive limit distribution proposed test statistic prove consistency finite sample properties proposed test investigated section means simulation study also compare test proposed paper cusum test using quantile regression see empirical likelihood based test suggested competitive cusum test using quantile regression innovation process gaussian performs remarkably better cusum test innovation process heavy tails moreover new test robust respect even process nearly unit root process finally rigorous proofs results relegated section change point tests using empirical likelihood throughout paper following notations symbols used set integers real numbers denoted respectively sequence random vectors denote convergence probability law random vector respectively transpose matrix denoted frobenius norm denote zero vector zero matrix identity matrix respectively consider autoregressive model order model defined assume innovation process sequence independent identically distributed random variables vanishing median let observed stretch model denotes true parameter paper focuses posteriori type change point problem parameters process precisely consider model vector unknown time point change testing problem change point autoregressive process formulated following hypotheses note neither assume knowledge change point null hypothesis true true value null hypothesis holds testing problem construct empirical likelihood ratio elr test precise let denote indicator function median zero moment condition holds null hypothesis measurable function independent motivated moment conditions first introduce moment function function function positive weight function choose weight function arbitrarily provided assumption section holds particular use corresponds case see also section note null hypothesis let vector unit cube empirical likelihood change point change point defined sup subsets cube defined yjp note unconstrained maximum represented sup hence logarithm empirical likelihood ratio elr statistic given log sup kvi log log log yjp obtained lagrange multiplier method multipliers satisfy yjp yjp finally define test statistic change point problem since maximum elr given sup one may define elr test statistic max fixed constants note consider maximum estimated accurately small large values see theorem section details asymptotic properties weighted version statistic investigated following section remark approach presented naturally extended general regression models precise suppose inf denotes conditional assume moment condition still holds null hypothesis define remark method also extended develop change point analysis based generalized empirical likelihood gel gel test statistic change point problem defined sup sup yjp concave twice differentiable function defined open interval real line contains point typical examples choice given log using lagrangian multipliers easy see choice log yields empirical likelihood method discussed far class associated called family see cressie read main results section state main results throughout paper let denote distribution function probability density function respectively impose following assumptions assumption int parameter space compact set nonempty interior iii median zero distribution function continuous differentiable point positive derivative assumption assumption matrix positive definite assumption exists constant let sign sequence strong mixing mixing coefficients satisfy maximum estimator defined sup consistency corresponding rate convergence statistic given following theorem theorem suppose assumptions hold define null hypothesis seen theorem accurate small result hold addition elr statistic computable small reason consider following discussion trimmed ratio test statistic defined max given weight function takes significant large value enough reason reject null hypothesis change point also need assumption control remainder terms stochastic expansion assumption additional assumption limit distribution test statistic derived following theorem theorem suppose assumptions hold null hypothesis change point sup vector independent brownian motions matrix defined denotes square root nonnegative definite matrix test hypotheses easily obtained rejecting null hypothesis whenever distribution random variable defined side equation using appropriate estimate matrix theorem suppose assumptions alternative hold theorem shows power test approaches fixed alternative words test consistent finite sample properties section illustrate finite sample properties elr test hypothesis means small simulation study purpose consider model coefficient satisfies calculation elr statistic use functions throughout section following ling chosen sample trimming parameters definition statistic chosen critical value obtained empirical quantile samples max independent standard brownian motions note case matrix given figures display rejection probabilities elr test hypothesis nominal level chosen horizontal vertical axes show respectively values rejection rate hypothesis point fixed sample sizes given distribution innovation process standard normal distribution figure degrees freedom figure cauchy distribution figure also consider two values parameter definition change point observe small sample sizes test slightly conservative approximation nominal level improves increasing sample size alternatives rejected reasonable probabilities power larger case comparison different distributions figures shows power lower standard normal distributed innovations error process cauchy distribution yields largest rejection probabilities simulations show similar picture results omitted sake brevity figure simulated rejection probabilities elr test model normal distributed innovations rejection rate rejection rate figure simulated rejection probabilities elr test model innovations rejection rate rejection rate figure simulated rejection probabilities elr test model cauchy distributed innovations rejection rate rejection rate second part section compare new test defined cusum test uses quantile regression test statistic median defined sup sup norm median regressor matrix given figures display rejection probabilities test based statistic hypothesis nominal level chosen horizontal vertical axes show respectively values rejection rate hypothesis point fixed distribution innovation process standard normal distribution figure degree freedom figure cauchy distribution figure sample sizes given case consider two different locations change point corresponding values observe tests derived three statistics corresponding weight function corresponding weight function slightly conservative approximation nominal level improves increasing sample size see figure value approximation usually accurate next compare power different tests different distributions innovations case gaussian innovations tests shows similar behavior see figure case elr test based unweighted statistic shows better performance tests based moreover gaussian innovations three tests show remarkable robustness figure display corresponding results innovations differences approximation nominal level negligible observe substantial differences power three tests independently sample size hand tests based elr statistics yield larger rejection probabilities test see right part figure figure interestingly unweighted test based shows better performance test based cases tests robust respect finally figure display rejection probabilities three tests cauchy distributed innovations observe differences approximation nominal level hand differences power tests based elr quantile regression remarkable cases elr tests based substantially power test based elr test based unweighted statistic shows better performance elr test based superiority less pronounced case clearly visible finally contrast test based elr tests based robust cauchy distributed innovations clearly detect change parameters cases figure simulated rejection probabilities various change point tests based statistics defined respectively model given model normal distributed innovations rejection rate rejection rate rejection rate rejection rate iii rejection rate rejection rate figure simulated rejection probabilities various change point tests based statistics defined respectively model given model innovations rejection rate rejection rate rejection rate rejection rate iii rejection rate rejection rate figure simulated rejection probabilities various change point tests based statistics defined respectively model given model cauchy distributed innovations rejection rate rejection rate rejection rate rejection rate iii rejection rate rejection rate proofs section gives rigorous proofs results paper follows denote generic positive constant varies different places probability approaching one abbreviated moreover use following notations throughout section log log proof theorem start proving several auxiliary results required proof theorem lemma suppose assumption holds let sup max sup also max proof let kgi assumption choose finite define obtain consequently maxi inequality sup max sup implies sup similarly follows sup max therefore completes proof lemma lemma suppose assumptions hold exists sequence denote arg max arg max exist moreover proof show statement corresponding statement follows similar arguments since closed set follows arg max exists note concave function lemma follows continuously twice differentiable respect taylor expansion exists point line joining note definition implies lemma uniformly respect furthermore ergodicity implies random variable converges probability hence minimum eigenvalue bounded away obtain dividing sides get hence int lemma concavity convexity follows exists results also imply similar arguments show corresponding results next let consider estimator theorem recall minimizer sup sup let define arg max arg max lemma suppose assumptions hold null hypothesis change point proof define follows implies inequality similar arguments used constant proof lemma hand following inequality inf sup sup sup sup applying lemma yields sup sup get finally implies establishing assertion lemma proof proof theorem lemma follows triangular inequality uniform law large numbers sup since unique zero function must bounded away zero outside neighborhood therefore must inside neighborhood therefore next show lemma central limit theorem implies yields moreover similar arguments given newey mcfadden page differentiability estimate show hence side order completes proof theorem proof theorem first show well approximated function near optima using similar reasoning parente smith purpose let define furthermore hereafter redefine arg min sup arg max arg max lemma suppose assumptions hold proof sufficient show iii proof note taylor expansion leads line joining points observing definition yields estimate since theorem take lemma obtain recalling first term replaced becomes moreover second term order hence get similarly combining estimates yields statement proof first show note function smooth first order conditions interior global maximum conditions satisfied point rewritten matrix form notations system equivalent consequently order respectively therefore arguments given proof follows relationship fact saddle points functions respectively imply hand thus lead finally prove iii similar arguments consequently implies iii proof proof theorem lemma follows sup denotes integer part real number shown lemma max sup second assumption lemma phillips follows vector standard brownian motion hence device continuous mapping theorem lead max sup sup proof theorem proof without loss generality suppose implies exist neighborhood neighborhood alternative follows note ytp ytp uniform law large numbers outside neighborhood sufficiently large consider instead instead find approximated time however since sufficiently large completes proof theorem acknowledgements authors would like thank martina stein typed manuscript considerable technical expertise work authors supported jsps young scientists waseda university grant special research projects deutsche forschungsgemeinschaft sfb statistik nichtlinearer dynamischer prozesse teilprojekt references andrews tests parameter instability structural change unknown change point econometrica journal econometric society aue structural breaks time series journal time series analysis bai partial sums residuals autoregressive moving average models journal time series analysis bai convergence sequential empirical processes residuals arma models annals statistics baragona battaglia cucina empirical likelihood break detection time series electronic journal statistics berkes ling schauer testing structural change model threshold model journal time series analysis chen ying zhang zhao analysis least absolute deviation biometrika chuang chan empirical likelihood autoregressive models applications unstable time series statistica sinica ciuperca salloum empirical likelihood test posteriori nonlinear model metrika cressie read multinomial tests journal royal statistical society series methodological limit theorems analysis john wiley davis huang yao testing change parameter values order autoregressive model annals statistics gombay asymptotic distributions maximum likelihood tests change mean biometrika lee cusum test parameter change time series models scandinavian journal statistics ling least absolute deviation estimation infinite variance autoregressive models journal royal statistical society series statistical methodology newey mcfadden large sample estimation hypothesis testing handbook econometrics nolan stable distributions models heavy tailed data boston birkhauser progress chapter online owen empirical likelihood ratio confidence intervals single functional biometrika page continuous inspection schemes biometrika page control charts warning lines biometrika pan wang yao weighted least absolute deviations estimation arma models infinite variance econometric theory parente smith gel methods nonsmooth moment indicators econometric theory phillips time series regression unit root econometrica qin lawless empirical likelihood general estimating equations annals statistics testing structural change regression quantiles journal econometrics samoradnitsky taqqu stable random processes stochastic models infinite variance volume crc press xiao testing parameter stability quantile regression models statistics probability letters zhou wang tang sequential change point detection linear quantile regression models statistics probability letters

feedback generation performance problems introductory programming assignments sumit gulwani ivan microsoft research usa wien austria sep sumitg radicek florian zuleger wien austria zuleger abstract general terms providing feedback programming assignments manually tedious error prone task paper motivate address problem generating feedback performance aspects introductory programming assignments studied large number functionally correct student solutions introductory programming assignments observed different algorithmic strategies varying levels efficiency solving given problem different strategies merit different feedback algorithmic strategy implemented countless different ways relevant reporting feedback student program propose programming language extension allows teacher define algorithmic strategy specifying certain key values occur execution implementation describe dynamic analysis based approach test whether student program matches teacher specification experimental results illustrate effectiveness specification language dynamic analysis one benchmarks consisting functionally correct implementations programming problems identified strategies able describe using specification language minutes inspecting around implementations dynamic analysis correctly matched implementation corresponding specification thereby automatically producing intended feedback algorithms languages performance categories subject descriptors software engineering testing debugging artificial intelligence automatic analysis algorithms second third author supported vienna science technology fund wwtf grant permission make digital hard copies part work personal classroom use granted without fee provided copies made distributed profit commercial advantage copies bear notice full citation first page copyrights components work owned others author must honored abstracting credit permitted copy otherwise republish post servers redistribute lists requires prior specific permission fee request permissions permissions november hong kong china copyright held publication rights licensed acm acm http keywords education moocs performance analysis trace specification dynamic analysis introduction providing feedback programming assignments tedious task human teacher even standard classroom setting rise massive open online courses moocs much larger number students challenge even pressing hence need introduce automation around task immediate feedback generation automation also enable new pedagogical benefits allowing resubmission opportunity students submit imperfect solutions providing immediate diagnosis class performance teacher adapt instruction accordingly recent research around automation feedback generation programming problems focused guiding students functionally correct programs either providing counterexamples generated using test input generation tools generating repairs however aspects program especially performance also important studied several programming sessions students submitted solutions introductory programming problems platform programming session student submits solution specified programming problem receives counterexample based feedback upon submitting functionally incorrect attempt generated using pex student may inspect counterexample submit revised attempt process repeated student submits functionally correct attempt gives studied different problems observed different programming sessions led functionally correct solutions however unfortunately average around functionally correct solutions different kinds performance problems paper present methodology semiautomatically generating appropriate performance related feedback functionally correct solutions study made two observations form basis feedback generation methodology different algorithmic strategies varying levels efficiency solving given problem rithmic strategies capture global insight solution programming problem also defining key performance characteristics solution different strategies merit different feedback algorithmic strategy implemented countless different ways differences originate local implementation choices relevant reporting feedback student program order provide meaningful feedback student important identify algorithmic strategy employed student program profiling based approach measures running time program use static bound analysis techniques sufficient purpose different algorithmic strategies necessitate different feedback may computational complexity also simple pattern matching based approach sufficient algorithmic strategy syntactically different implementations key insight algorithmic strategy employed program identified observing values computed execution program allow teacher specify algorithmic strategy simply annotating source code level certain key values computed sample program implements corresponding algorithm strategy using new language construct called observe fortunately number different algorithmic strategies introductory programming problems often small per problem experiments easily enumerated teacher iterative process examining student program match existing algorithmic strategy refer step iterative process inspection step propose novel dynamic analysis decides whether student program also referred implementation matches algorithm strategy specified teacher form annotated program also referred specification dynamic analysis executes student implementation teacher specification check whether key values computed specification also occur corresponding traces generated implementation implemented proposed framework evaluated approach programming problems attempted several hundreds students new problems hosted part programming course attempted students course experimental results show manual teacher effort required specify various algorithmic strategies small fraction overall task system automates particular mooc style benchmark functionally correct implementations programming problems specified strategies minutes inspecting implementations standard classroom style benchmark functionally correct implementations programming problems specified strategies minutes inspecting implementations methodology specifying matching algorithmic strategies expressive precise particular able specify strategies using specification language dynamic analysis correctly matched implementation intended strategy paper makes following contributions observe different algorithmic strategies used functionally correct attempts introductory programming assignments strategies merit different performance related feedback describe new language construct called observe specifying algorithmic strategy describe dynamic analysis based approach test whether student implementation matches teacher specification experimental results illustrate effectiveness specification language dynamic analysis overview section motivate problem definition various aspects solution means various examples motivation fig shows running examples programs show sample implementations anagram problem involves testing whether two input strings permuted become equal platform programs examples inefficient implementations quadratic asymptotic complexity efficient solution example first collect array number occurrences character strings compare leading linear asymptotic complexity algorithmic strategies implementations identify three different algorithmic strategies implementations iterate one input strings character string count occurrences character strings counting strategy implementations sort input strings check equal sorting strategy implementations iterate one input strings remove corresponding characters string removing strategy implementation details algorithmic strategy several implementations case counting strategy implementation calls manually implemented method countchar count number characters string lines implementation uses special construct lines implementation uses library function split task lines case sorting strategy implementation employs binary insertion sort implementation employs bubble sort implementation uses library call lines also observe different ways removing character string implementations desired feedback three identified strategies requires separate feedback independent underlying implementation details help student understand fix performance issues first strategy implementations possible feedback might calculate number characters string preprocessing phase instead iteration main loop second strategy might instead sorting input strings compare number character occurrences string third strategy use efficient remove characters specifying algorithmic strategies key values key insight different implementations employ algorithmic strategy generate key values execution input bool puzzle string string return false else return bool puzzle string string return false foreach var item item item return false return true bool puzzle string string bool puzzle string string return false var var char char int return int char bool puzzle string string char return false temp foreach char int index int index return false index return false return true return true puzzle string string string bool puzzle string string int char int char int char return false observe int int observefun split observe int break observe return false return true observefun split observe counting specification bool puzzle string string return false char taux int bool compareletterstring string string char var boolean exists false var int return taux exists true taux break exists false return false return true custom data equality cde void puzzle string string bool puzzle string string string int new int new int return false cover tochararray char cover tochararray char cover int hash new int int int int hash observe foreach char int hash int int foreach char observe hash int else int int observe hash return false return true cover bool puzzle string string return false foreach char countchars countchars return false return true int countchars string char int number foreach char return number efficient specification int binarysearch list char char int low high low high int mid high low low mid high mid else mid low mid else return mid return low char sort string var res new list char foreach var var pos binarysearch res pos return bool puzzle string string return sort sort insertion bool puzzle string string return ispermutation bool ispermutation string string return true return false int index index return false index return ispermutation puzzle string string char observe sorting specification puzzle string string string int return int int observe compareletterstring removing specification bool puzzle string string return false int int int int int int int int int return false return true bool puzzle string string return false string int char bool found false int char else char else char found true break found return false return true computation figure running example implementations specifications anagram assignment example underlined expressions implementations produce key value sequence input strings aba baa framework allows teacher describe algorithmic strategy simply annotating certain expressions sample implementation using special language statement observe framework decides whether student implementation matches teacher specification comparing execution traces common input say implementation matches specification execution trace subsequence execution trace every observed expression expression generated values call matching criterion trace embedding notion trace embedding establishes fairly strong connection specification implementation basically programs produce values corresponding locations order notion trace embedding adaptation notion simulation relation dynamic analysis choice minor differences implementations strategy keyvalues differ example implementation uses library function obtain number characters implementations explicitly count explicitly iterating string moreover counted values incremented one compared thus yields different related trace split split split split split split input strings aba baa address variations implementation details include choice construct specification language fixed execution thus choice merely syntactic sugar succinctly represent multiple similar specifications variables specifications specifications denote specifications counting strategy used implementations sorting strategy used removing strategy used respectively teacher observes characters iterated lines results counting characters lines use library function split lines also teacher uses boolean variables line choose string main loop iterates input strings symmetric anagram problem choose manual library function implementations also decides observed counted values lines teacher observes one input strings sorting allows implementations convert input string uppercase line sort string reverse order line notice enough observe one sorted input case input strings anagrams sorted strings teacher observes string removed characters chooses string iterated line direction iteration line direction remove candidate searched line specifications implementations expression opbin opun statement skip else observe observefun figure syntax language section introduce imperative programming language supports standard constructs writing implementations novel constructs writing specifications language syntax language stated fig discuss features language expressions data value value data domain set contains values language integers characters arrays hashsets variable belongs finite set variables var expression either data value variable operator applied variables array access opbin represents set binary operators opun set unary operators point syntax ensures programs three address code operators applied variables arbitrary expressions motivation choice three address code enables observe expression program observing variables point program automatically translated three address code assigning subexpression new variable example statement translated code follows enables observe subexpression observing statements statements allow build simple imperative programs assignments variables array elements skip statement composition statements looping branching constructs also allow library function calls denoted library function name set library functions two special observe constructs available teacher student discuss observe statements assume statement associated unique program location write functions space reasons define functions could easily extend language recursive functions fact allow recursive functions implementation semantics assume standard imperative semantics execute programs written language assume usual semantics two observe statements semantic meaning skip statement computation domain extend data domain special symbol use represent data value define computation domain val associated language val assume data domain equipped equality relation iff type comparison equals method returns true denote set relations val define default equality relation edef follows edef iff edef iff edef computation trace computation trace finite set programming locations loc finite sequence pairs loc val use notation denote set computation traces loc given loc loc denote sequence obtain deleting pairs val loc strings define value representations implementations regardless used characters equal call function loc comparison function define every statement observe observefun statements left set default value edef choice assume teacher use finite set boolean variables var available student choice allows teacher specify variations implementations discussed variables similar input variables sense assigned program executed note results different program behaviors given input student implementation matching following describe computation trace generated student implementation given input computation trace initialized empty sequence implementation executed according semantics execution append pairs every assignment statement append denote current value point add complete array trace assignment array variable library function call append denote resulting trace jqk construction computation trace achieved instrumenting implementation appropriate manner teacher specification teacher uses observe observefun specifying key values wants observe execution specification defining equality relation computation domain usual rectangular brackets enclose optional arguments following describe computation trace generated specification given input computation trace initialized empty sequence specification executed according semantics execution append pairs observe observefun statements observe append denote current value observefun append ith argument specified left denote resulting trace custom data equality possibility specifying equality relation location useful teacher point practice teacher specify equality function val val true false teacher use define equality similar computation values show usage examples fig examples implement removing strategy discussed almost identical ways difference lines respectively implementations use different characters denote character removed string specification teacher uses equality function compareletterstring defined cde compares letters two section define means implementation partially match fully match specification describe corresponding matching algorithms teacher determine specification definition matching applied case partial matching speak inefficient specifications case full matching efficient specifications trace embedding start discussing problem trace embedding use building block matching algorithms subsequence call partial full matching criterion let val val val val val val two computation traces set locations loc let comparison function defined say subsequence written indices val val case full additionally require length refer val val equality check identity relation val obtain usual definition subsequence since deciding subsequence central operation paper state complexity decision problem easy see deciding subsequence requires equality checks basically one iteration sufficient mapping function let loc loc two disjoint sets locations call injective function loc loc mapping function lift function applying every location set val val val val given comparison function matching criterion computation traces say embedded iff write refer embedding witness executing program set assignments gives rise set traces one assignment say set traces embedded iff definition trace embedding trace embedding problem deciding given sets traces comparison function matching criterion witness mapping function complexity clearly trace embedding assuming equality checks done polynomial time first guess mapping function loc loc check cheap discussed however turns trace embedding npcomplete even singleton set singleton computation domain val full matching criterion theorem trace embedding assuming equality checks done polynomial time proof order show reduce permutation pattern trace embedding first formally define permutation pattern let positive integers let permutation let permutation say occurs injective function monotone subsequence permutation pattern problem deciding whether occurs give reduction permutation pattern trace embedding construct two traces singleton computation domain val sets locations loc loc set identity function val every loc val singleton ignore values rest proof set every occurs exactly twice partial full matching criteria equivalent ignore difference show occurs iff injective function loc loc establish equivalence two observations first every occurs exactly twice iff second iff loc loc monotone algorithm fig shows algorithm embed trace embedding problem straightforward algorithmic solution trace embedding problem simply test possible mapping functions however exponential number mapping functions cardinality loc loc exponential blowup seems unavoidable combinatorial search space responsible hardness core element algorithm narrows space possible mapping functions effectively observe exists trace embedding restricted locations formally algorithm uses insight create bipartite graph loc loc potential mapping pairs lines pair locations potential mapping pair iff exists trace embedding restricted locations described key idea finding embedding witness construct maximum bipartite matching maximum bipartite matching edge connecting every program embed loc loc loc loc loc loc break maximumbipartitematching found true found false break found true return true return false figure algorithm trace embedding problem location loc distinct location loc thus gives rise injective function point injective function need embedding witness observing single location pair time ignores order locations thus maximum bipartite matching algorithm checks lines indeed embedding witness key strength algorithm reduces search space possible embedding witnesses experimental evidence shows approach significantly reduces number possible matchings enables efficient algorithm practice discussed partial matching define notion partial matching also referred simply matching used check whether implementation involves least inefficiency issues underlie given inefficient specification definition partial matching let specification observed locations loc let comparison function specified let implementation whose assignment statements labeled loc implementation partially matches specification set inputs exists mapping function loc loc assignment variables input values jqk partial fig describes algorithm testing implementation partially matches given specification given set input valuations lines implementation executed input values line algorithm iterates assignments variables specification lines specification executed inputs sets traces available line calls subroutine embed returns true iff exists trace embedding witness example give example demonstrates notion programs contains example applications algorithms embed matches fig state two implementations one specification programs represent simplified versions transformed three adress code function inlining matches specification implementation inputs loc observed locations comparison function specified matching criterion loc assignment locations jqk variables assignments embed loc loc return true return false figure matching algorithm puzzle puzzle split split puzzle observe observe observefun split observe observefun split figure implementations spec fig note every assignment observe statement line denote line program location argument left locations specification thus edef specification locations algorithm matches runs three programs input values aab aba program obtain following computation trace similarly program obtain split aab split aba specification obtain two traces depending choice variable split split algorithm matches calls embed check trace embedding algorithm embed first constructs potential graph contains edge two locations specification implementation show values implementation obtain following graph notice shows values locations implementation however one maximal matching also embedding witness thus implementation matches specification implementation true obtain graph construct maximal matching however false obtain also embedding witness thus implementation matches specification full matching define notion full matching used match implementations efficient specifications require every loop every library function call implementation corresponding loop library function call matching specification order need helper definitions observed loop iterations extend construction implementation trace defined statement additionally append element trace whenever loop body entered call loop iteration let embedding witness say observes loop iterations iff every two loop iterations exists pair val words require two iterations loop exists observed location observed library function calls say observes library function calls iff every val val definition full matching let specification observed locations loc let comparison function specified let implementation whose assignment statements labeled loc implementation fully matches specification set inputs exists mapping function loc loc assignment variables input valuations jqk full observes loop iterations library function calls note procedure embed fig easily check line whether current mapping observes loop iterations library function calls tedious teacher exactly specify possible loop iterations library function calls used different efficient implementations add two additional constructs language simplify specification task cover statement extend two cover statements cover cover first statement statement observefun except allow embedding witness map location implementation enables teacher specify function may appear implementation second statement allows map location appears times appearance current value specified variable thus cover enables teacher cover loop iterations example present examples efficient implementations specification anagram problem fig teacher observes computed values lines uses choice line choose implementations count number characters string decrement one number another also teacher allows two library function calls two loops iterations defined cover statements lines extensions section discuss useful extensions core material presented extensions part implementation discuss separately make presentation easier follow mapping according definition trace embedding embedding witness maps one implementation location specification location constructs mapping however possible student splits computation value multiple locations example implementation stated fig student removes character string across three different locations lines depending location removed character string requires map single location specification multiple locations implementation reason extend notion trace embedding mappings loc easy extend procedure embed fig setting potential graph also helpful enumerate every possible mapping however costly unnecessary search arbitrary mappings use heuristics consider mappings example one heuristics implementation checks variable assigned different branches example three locations assignment variable although mappings may seem powerful point teacher always write specification succinct implementation student described mappings provide enough expressivity teacher behaviour trace embedding requires equal values order specification implementation traces however implementation use library function behaviour values returned random generator iteration order set data structure library functions eliminate fixing one particular behaviour fix values returned random generator iteration order set program instrumentation fixes impact functionally correct programs rely behaviour allow apply matching techniques implementation experiments describe implementation present experimental evaluation framework details experiments found website experimental setup implementation algorithm matches fig analyzes programs implementations specifications used microsoft roslyn compiler framework instrumenting every record value program execution data used preexisting problems mentioned anagram problem students asked test two strings could permuted become equal issorted problem students asked test input array sorted caesar problem students asked apply caesar cipher input string chosen specific problems high number student attempts diversity algorithmic strategies problem explicitly stated many problems platform students guess problem failing examples also created new course platform programming problems problems assigned homework students second year undergraduate course created course understand performance related problems students make opposed regular users might previous programming experience encouraged students write efficient implementations giving points performance efficiency mere functional correctness omit description problems descriptions available original course page methodology following describe methodology envision technique paper used teacher maintains set efficient inefficient specifications new student implementation checked available specifications implementation matches specification associated feedback automatically provided student otherwise teacher notified new unmatched implementation teacher studies implementation identifies one following reasons failure match existing specification implementation uses new strategy seen case teacher creates new specification existing specification strategy used implementation specific capture implementation case teacher refines existing specification overall process repeated unmatched implementation new specification teacher creates new specification using following steps copy code unmatched implementation annotate certain values function calls observe statements iii remove unnecessary code needed specification implementation identify input values dynamic analysis matching associate feedback specification specification refinement refine specification teacher identifies one following reasons implementation match implementation differs details specified specification time required specifications required inspection steps matched implementations matched implementations time min required inspection steps time required specifications longestequal doublechar longestword runlength vigenere basetobase catdog minimaldelete commonelement required inspection steps matched implementations inspection steps time min inspection steps matched implementations inspection steps matched implementations matched implementations anagram issorted caesar time required specifications tableaggsum intersection reverselist sortingstrings minutesbetween maxsum median digitpermutation coins time min figure number inspection steps time required completely specify assignments problem name correct implement inefficient implement anagram issorted caesar doublechar longestequal longestword runlength vigenere basetobase catdog minimaldelete commonelement tableaggsum intersection reverselist sortingstrings minutesbetween maxsum median digitpermutation coins performance avg max table list assignments experimental results specification observes values appear implementation iii implementation uses different data representation case teacher adds new nondeterministic choice necessary observes new values function calls case teacher observes less values case iii teacher creates refines custom input values dynamic analysis approach requires teacher associate input values specifications input values cause corresponding implementations exhibit behavior otherwise inefficient implementation might behave similar efficient implementation reason match specification efficient implementation implies trivial inputs avoided example two strings unequal lengths constitute trivial input counting strategy since three implementations fig exit immediately similarly providing sorted input sorting strategy meaningless remark easy teacher understands various strategies provide good input values granularity feedback want point granularity feedback depends teacher example programming problem sorting input value inefficient strategy teacher might want distinguish different sorting algorithms require different feedback however programming problem students asked implement sorting algorithm makes sense provide different feedback different sorting algorithms evaluation report results problems discussed table results manual code study first observe large number students managed write functionally correct implementation problems umn correct implementations shows succeeds guiding students towards correct solution second observation problems large fraction implementations inefficient column inefficient implementations especially anagram problem shows although students manage achieve functional correctness efficiency still issue recall homework students explicitly asked given extra points efficiency also observe except two problems least one inefficient algorithmic strategy problems several inefficient algorithmic strategies column results highly motivate need tool find inefficient implementations also provide meaningful feedback fix problem precision expressiveness programming assignment used described methodology wrote specification algorithmic strategy efficient inefficient manually verified specification matches implementations strategy hence providing desired feedback implementations shows approach precise expressive enough capture algorithmic strategy ignoring low level implementation details teacher effort provide manual feedback students teacher would every implementation look performance characteristics approach teacher take look representative implementations column report total number inspection steps required fully specify one programming problem number implementations teacher would provide feedback implementations problems teacher would around implementations provide full feedback fig shows number matched implementations inspection step well time took specifications measured time takes seeing unmatched implementation matching specification column report largest ratio specification average matched implementation terms lines code observe half cases largest specification size smaller average matched implementation furthermore number input values need provided teacher across problems column one problem issorted one set input values used specifications also one third specifications need variables largest number used one specification column overall approach requires considerably less teacher effort providing manual feedback performance plan integrate framework mooc platform performance web applications critical implementation consists two parts first part execution implementation specification usually small programs relatively small inputs obtaining execution traces cases neglectable terms performance second part embed algorithm discussed challenge consists finding embedding witness observed variables specification obi served variables implementation possible injective mapping functions sortingstrings problem gives possible mapping functions however algorithm reduces huge search space constructing bipartite graph potential mappings pairs report number mapping functions tool explore sortingstrings different mapping functions explored values report maximal number across specifications last column state total execution time required decide one implementation matches specification average maximal note time includes execution programs exploration assignments boolean variables finding embedding witness tool runs cases half second per implementation results show tool fast enough used interactive teaching environment threats validity unsoundness method unsound general since uses dynamic analysis explores possible inputs however observe unsoundness large scale experiments one desires provable soundness embedding witness could used guess simulation relation formally verified techniques otherwise student suspects incorrect feedback always bring attention teacher program size evaluated approach introductory programming assignments although questions applicability larger programs might raised goal analyze arbitrary programs rather develop framework help teachers teach introductory programming providing performance feedback currently manual task difficulty specification language although perform case study instructors report experiences using proposed language would also like point writing specifications investment could performed experienced personnel related work automated feedback lot work area generating automated feedback programming assignments work classified along three dimensions aspects feedback provided functional correctness performance characteristics modularity nature feedback counterexamples bug localization repair suggestions whether static dynamic analysis used ihantola present survey various systems developed automated grading programming assignments majority efforts focussed checking functional correctness often done examining behavior program set test inputs test inputs manually written automatically generated little work testing properties assyst system uses simple form tracing counting execution steps gather performance measurements system counts number evaluations done used coarse complexity analysis authors conclude better error messages important area improvement community built tutors aim bug localization comparing source code student teacher programs laura converts teacher student program graph based representation compares heuristically applying program transformations reporting mismatches potential bugs talus matches student attempt collection teacher algorithms first tries recognize algorithm used tentatively replaces expressions student attempt recognized algorithm generating correction feedback contrast perform trace comparison instead source code comparison provides robustness syntactic variations striewe goedicke proposed localizing bugs trace comparisons suggested creating full traces program behavior running test cases make program behavior visible students also suggested automatically comparing student trace sample solution generating directed feedback however implementation reported also compare student trace teacher trace look similarities opposed differences recently shown automated techniques also provide repair based feedback functional correctness singh sat solving based technology successfully generate feedback corrections around incorrect solutions mit introductory programming course seconds average test inputs provide guidance given solution incorrect bug localization techniques provide guidance error might repairs provide guidance fix incorrect solution also provide repair suggestions manually associated various teacher tions performance based aspects furthermore suggestions necessarily restricted small fixes performance analysis programming languages software engineering communities explored various kinds techniques generate performance related feedback programs symbolic execution based techniques used identifying related issues speed project investigated use static analysis techniques estimating symbolic computational complexity programs goldsmith used dynamic analysis techniques empirical computational complexity toddler tool reports specific pattern computations repetitive similar patterns cachetor tool reports memoization opportunities identifying operations generate identical values contrast interested identifying whether performance issue also identifying root cause generating repair suggestions references http making programs efficient http microsoft roslyn ctp http pex fun http adam laurent laura system debug student programs artif bose buss lubiw pattern matching permutations inf process burnim jalbert stergiou looper lightweight detection infinite loops runtime ase pages goldsmith aiken wilkerson measuring empirical computational complexity fse gulwani learning stem education appear commun acm gulwani mehra chilimbi speed precise efficient static estimation program computational complexity popl pages gulwani zuleger problem pldi pages gupta henzinger majumdar rybalchenko proving popl pages ihantola ahoniemi karavirta review recent systems automatic assessment programming assignments proceedings koli calling international conference computing education research koli calling pages new york usa acm jackson usher grading student programs using assyst sigcse pages masters brief guide understanding moocs internet journal medical education milner algebraic definition simulation programs technical report stanford usa murray automatic program debugging intelligent tutoring systems computational intelligence nguyen cachetor detecting cacheable data remove bloat proceedings joint meeting foundations software engineering pages new york usa acm nistor song marinov toddler detecting performance problems via similar patterns proceedings international conference software engineering icse pages piscataway usa ieee press saikkonen malmi korhonen fully automatic assessment programming exercises proceedings annual conference innovation technology computer science education iticse pages new york usa acm singh gulwani automated feedback generation introductory programming assignments pldi pages striewe goedicke using run time traces automated programming tutoring iticse pages striewe goedicke trace alignment automated tutoring caa tillmann halleux box test generation tap pages tillmann halleux xie gulwani bishop teaching learning programming software engineering via interactive gaming icse uno algorithms enumerating perfect maximum maximal matchings bipartite graphs isaac pages zuleger gulwani sinn veith bound analysis imperative programs abstraction sas pages

nov bayesian identification selecting influenza mitigation strategies pieter timothy diederik jelena kristof philippe ann artificial intelligence lab department computer science vrije universiteit brussel brussels belgium leuven university leuven department microbiology immunology rega institute medical research clinical epidemiological virology leuven belgium november abstract pandemic influenza epidemic potential kill millions people various preventive measures exist vaccination school closures deciding strategies lead effective efficient use remains challenging end epidemiological models essential assist decision makers determining best strategy curve epidemic spread however models computationally intensive therefore pivotal identify optimal strategy using minimal amount model evaluations additionally epidemiological modeling experiments need planned computational budget needs specified priori consequently present new sampling method optimize evaluation preventive strategies using fixed budget identification algorithms use epidemiological modeling theory derive knowledge reward distribution exploit using bayesian identification algorithms thompson sampling bayesgap evaluate algorithms realistic experimental setting demonstrate possible identify optimal strategy using limited number model evaluations times faster compared uniform sampling method predominant technique used epidemiological decision making literature finally contribute evaluate statistic thompson sampling inform decision makers confidence arm recommendation introduction influenza virus responsible deaths half million people year addition seasonal influenza epidemics cause significant economic burden transmission primarily local newly emerging variant may spread pandemic proportions naive fully susceptible host population pandemic influenza occurs less frequently seasonal influenza outcome respect morbidity mortality much severe potentially killing millions people worldwide consequently essential study mitigation strategies control influenza pandemics influenza different preventive measures exist vaccination social measures school closures travel restrictions antiviral drugs however efficiency strategies greatly depends availability preventive compounds well characteristics targeted epidemic furthermore governments typically limited resources implement measures therefore remains challenging formulate public health strategies make effective efficient use preventive measures within existing resource constraints epidemiological models compartment models models essential study effects preventive measures silico models usually associated greater model complexity computational cost compartment models allow accurate evaluation preventive strategies capitalize advantages make feasible employ models essential use available computational resources efficiently possible literature set possible preventive strategies typically evaluated simulating strategies equal number times however approach inefficient identify optimal preventive strategy large proportion computational resources used explore strategies furthermore consensus required number model evaluations per strategy currently lacking moreover show paper number depends hardness evaluation problem reason propose combine epidemiological models bandits preliminary study potential bandits explored regret minimization setting using default strategies however work recognize epidemiological modeling experiments need planned computational budget needs specified priori within constraint aim minimize number required model evaluations determine promising preventive strategy therefore present novel approach formulating evaluation preventive strategies identification problem using fixed budget model evaluations running model computationally intensive minutes hours depending complexity model minimizing number required model evaluations reduces total time required evaluate given set preventive strategies renders use models attainable studies would otherwise computationally feasible additionally reducing number model evaluations also free computational resources studies already use models capacitating researchers explore different model scenarios considering wider range scenarios increases confidence overall utility preventive strategies model arm reward distribution corresponds epidemic size distribution epidemiological model employ epidemiological modeling theory derive distribution approximately gaussian exploit knowledge using bayesian identification algorithms paper contribute novel method evaluate preventive strategies identification problem method enables decision makers obtain recommendations reduced number model evaluations supports decision process providing confidence recommendation statistic section employ concepts epidemiological model theory adapt bayesian identification algorithms incorporate knowledge section evaluate algorithms experimental setting aim find best vaccine allocation strategy realistic simulation environment models seattle social network repeat experiment wide range basic reproduction numbers number infections average generated one single infection typically used influenza literature obtained experimental results show approach able identify best preventive strategy times faster compared uniform sampling predominant technique used epidemiological decision making literature furthermore contribute section evaluate section statistic inform decision makers confidence particular recommendation background pandemic influenza vaccine production primary preventive strategy mitigate seasonal influenza produce vaccine prior epidemic anticipating virus strains expected circulate vaccine pool used inoculate population start epidemic seasonal influenza may restricted susceptible population due vaccination immunity newly emerging strain become pandemic spreading rapidly among naive human hosts worldwide possible stockpile vaccines prepare seasonal influenza case new variants influenza viruses vaccine specifically tailored virus source pandemic therefore appropriate vaccine produced responsible virus needs identified hence vaccine available limited supply beginning pandemic addition production problems result vaccine shortages number vaccine doses limited imperative identify optimal vaccine allocation strategy modeling influenza long tradition use models study influenza epidemics allows accurate evaluation preventive strategies model driver many high impact research efforts flute flute implements contact model population divided communities households population organized hierarchy social mixing groups contact intensity inversely proportional size group closer contact members household colleagues additionally flute implements individual disease progression model associates different disease stages different levels infectiousness support evaluation preventive strategies flute implements simulation therapeutic interventions vaccines antiviral compounds interventions school closure case isolation household quarantine bandits identification bandit game concerns bandit slot machine levers arm returns reward pulled represents sample reward distribution common use bandit game pull sequence arms cumulative regret minimized fulfill goal player needs carefully balance exploitation choose arms highest expected reward exploration explore arms potentially identify even promising arms paper objective recommend best arm arm highest average reward fixed number arm pulls referred fixed budget identification problem instance problem given budget objective minimize simple regret average reward recommended arm time simple regret inversely proportional probability recommending correct arm related work work recognize computational budget needs specified priori meet realities associated high performance computational infrastructure reason consider fixed budget identification setting contrast techniques attempt identify best arm predefined confidence racing strategies strategies exploit confidence bound arms means recently fixed confidence identification algorithms algorithms exist rank select bandit arms bestarm identification best approached using adaptive sampling methods ones study paper moreover use identification methods clears way interesting future work respect evaluating preventive strategies considering multiple objectives see section methods formulate evaluation preventive strategies bandit game aim identifying best arm using fixed budget model evaluations presented method generic respect type epidemic modeled pathogen contact network preventive strategies method evaluated context pandemic influenza next section preventive bandits definition stochastic epidemiological model defined terms model configuration used evaluate preventive strategy evaluating model results sample model outcome distribution outcome note model configuration describes complete model environment aspects inherent model flute mixing model options modeler provide population statistics vaccine properties result model evaluation referred model outcome prevalence proportion symptomatic individuals morbidity mortality societal cost objective find optimal preventive strategy set alternative strategies particular configuration stochastic epidemiological model corresponds studied epidemic definition preventive bandit arms pulling arm corresponds evaluating running simulation epidemiological model preventive bandit thus bandit preventive strategies arms reward distributions corresponding outcome distribution stochastic epidemiological model parameters reward distribution known parameters epidemiological model intractable determine optimal reward analytically epidemiological model hence must learn outcome distribution via interaction epidemiological model outcome distribution previously defined reward distribution associated preventive bandit arm corresponds outcome distribution epidemiological model evaluated pulling arm therefore employing insights epidemiological modeling theory allows specify prior knowledge reward distribution well known disease outbreak two possible outcomes either able spread beyond local context becomes fully established epidemic fades stochastic epidemiological models reflect reality hence epidemic size distribution bimodal evaluating preventive strategies objective determine preventive strategy suitable mitigate established epidemic practice observe act established epidemics epidemics faded simulation would bias evaluation consequently necessary focus mode distribution associated established epidemic therefore censor discard epidemic sizes correspond faded epidemic size distribution remains one corresponds established epidemic approximately gaussian study consider scaled epidemic size distribution proportion symptomatic infections hence assume bimodality full size distribution approximately gaussian size distribution established epidemic verified experimentally assumptions hold reward distributions observed experiments see section censor size distribution use threshold represents number infectious individuals required ensure outbreak fade low probability epidemic threshold heterogeneous host populations population significant variance among individual transmission rates case influenza epidemics number secondary infections accurately modeled using negative binomial offspring distribution basic reproductive number dispersion parameter specifies extent heterogeneity probability epidemic extinction pext computed solving probability generating function pgf offspring distribution epidemic individuals targeted preventive measures vaccination use case obtain following pgf popc popc popc signifies random proportion controlled individuals pext compute threshold limit probability extinction cutoff identification fixed budget objective identify best preventive strategy strategy minimizes expected outcome set preventive strategies particular configuration using fixed budget model evaluations successive rejects first algorithm solve identification fixed budget setting bandit successive rejects operates phases end phase arm lowest average reward discarded thus end phase one arm survives arm recommended phase arm still available played times log log successive rejects serves useful baseline however support incorporate prior knowledge bayesian identification algorithms able take account knowledge defining appropriate prior posterior arms reward distribution show prior knowledge increase identification accuracy additionally time arm recommended posteriors contain valuable information used formulate variety statistics helpful assist decision makers consider two bayesian algorithms bayesgap thompson sampling thompson sampling derive statistic based posteriors inform decision makers confidence arm recommendation probability success established previous section arm preventive bandit reward distribution approximately gaussian unknown mean variance make method generic type preventive bandit problem assume uninformative jeffreys prior honda takemura demonstrate prior leads following posterior nth pull tnk reward mean sum squares tnk standard student degrees freedom bayesgap bayesian algorithm algorithm requires arm upper bound lower bound defined posterior time step using bounds gap quantity max defined arm represents upper bound simple regret defined section step algorithm arm minimizes gap quantity compared arm maximizes upper bound arm highest confidence diameter pulled reward results pull observed used update posterior budget consumed arm argmin recommended arm minimizes simple regret bound times order use bayesgap preventive bandit setting contribute bounds given posteriors equation define mean standard deviation posterior arm time step exploration coefficient amount exploration feasible given particular bandit game proportional available budget inversely proportional game complexity complexity modeled taking account game hardness variance rewards following hoffman define hardness quantity hardness max max considering budget hardness generalized reward variance arms define theorem supplementary formally proves using bounds results probability simple regret asymptotically reaches exponential lower bound presented hoffman supplementary information available end manuscript unknown order compute quantities need estimated firstly estimate upper bound estimating follows max hoffman secondly need measure variance representative reward distribution arms end arms initialized observe sample variance compute average bounds depend standard deviation posterior arm posterior needs initialized times ensure defined initialization also ensures proper posteriors thompson sampling reformulation thompson sampling algorithm used context thompson sampling operates directly arms posterior mean time step thompson sampling obtains one sample arm posterior arm highest sample pulled reward subsequently used update arm posterior approach proven highly successful minimize cumulative regret balances identify best arm adapt thompson sampling minimize simple regret thompson sampling increases amount exploration end exploration probability needs specified time step one sample obtained arm posterior arm atop highest sample pulled probability probability repeat sampling posteriors find arm highest posterior sample atop arm found pulled observed reward used update posterior pulled arm available budget consumed arm highest average reward recommended thompson sampling requires samples arms posteriors use posteriors equation avoid improper posteriors arm needs initialized times specified previous subsection reward distribution censored observe reward consider update arm value exceeds threshold receive sample mode epidemic represents established epidemic probability success probability arm recommendation correct presents useful confidence statistic support policy makers decisions thompson sampling recommends arm highest average reward probability success max random variable represents mean recommended arm assume arm reward distributions independent probability computed using recommended arm posterior probability density function arms cumulative density function max integral computed analytically estimate using gaussian quadrature important notice aiming generality made conservative assumptions reward distributions approximated gaussian uninformative jeffreys prior used assumptions imply derived probability success actual recommendation success experiments composed performed experiment context pandemic influenza analyze mitigation strategy vaccinate population limited number vaccine doses available details rationale behind scenario section experiment extend simulation environment presented accommodate realistic setting evaluate vaccine allocation contrast consider large realistic social network city seattle wide range values consider scenario pandemic emerging particular geographical region vaccines becomes available albeit limited number doses number vaccine doses limited imperative identify optimal vaccine allocation strategy experiment explore allocation vaccines five different age groups children children young adults older adults elderly presented chao consider experiment wide range values influenza model configuration epidemiological model used experiments flute stochastic model experiment consider population seattle united states includes individuals population realistic respect number individuals community structure provides adequate setting validation vaccine strategies first day simulated epidemic random individuals seeded infection epidemic simulated days time infections seeded thus new infections established run time simulation result mixing infectious susceptible individuals assume immunity towards circulating virus variant choose number vaccine doses allocate approximately population size experiment explore efficacy different vaccine allocation strategies consider one vaccine variant available simulation environment flute allows vaccine efficacy configured levels efficacy protect infection individual susceptible esus efficacy avoid infected individual becoming infectious einf efficacy avoid infected individual becoming symptomatic esym experiment choose esus einf esym influenza vaccine becomes fully effective certain period upon administration effectiveness increases gradually period experiment assume vaccine effectiveness build exponentially period weeks perform experiment set values within range steps range considered representative epidemic potential influenza pandemics refer set values note setting described subsection conjunction particular value corresponds model configuration definition computational complexity flute simulations depends size susceptible population proportion population becomes infected population seattle simulation run time minutes median minutes hardware details supplementary information section formulating vaccine allocation strategies consider age groups vaccine doses allocated children years old children years old young adults years old older adults years old elderly years old allocation scheme encoded boolean position tuple corresponds respective age group boolean value particular position tuple denotes whether vaccines allocated respective age group vaccine allocated particular age group done proportional size population part age group decide best vaccine allocation strategy enumerate possible combinations tuple tuple encoded binary number different allocation strategies represented integers influenza preventive bandit influenza preventive bandit exactly arms arm associated allocation strategy integer encoding given model configuration definition arm pulled flute invoked vaccine allocation strategy definition associated arm flute finishes outputs proportion population experienced symptomatic infection reward computed outcome distributions establish proxy ground truth concerning outcome distributions considered preventive strategies strategies evaluated times values use ground truth reference validate correctness recommendations obtained throughout experiments presents interesting evaluation problem demonstrate visualize outcome distribution figure figure outcome distributions values shown section supplementary information firstly observe different values distances top arms means differ additionally outcome distribution variances vary set values differences produce distinct levels evaluation hardness see section demonstrate setting usefulness benchmark evaluate preventive strategies secondly expect outcome distribution bimodal however probability sample mode outcome distribution represents epidemic decreases increases expectation confirmed inspect figure shows bimodal distribution figure shows unimodal outcome distribution samples established epidemic obtained analysis identified best vaccine allocation strategy allocate vaccine school children strategy values identification experiment assess performance different identification algorithms successive rejects bayesgap thompson sampling run epidemic size vaccine allocation strategy figure violin plot depicts density outcome distribution epidemic size vaccine allocation strategies epidemic size vaccine allocation strategy figure violin plot depicts density outcome distribution epidemic size vaccine allocation strategies algorithm budgets range evaluation performed influenza bandit game defined earlier budget run algorithms times report recommendation success rate previous section optimal vaccine allocation strategy identified vaccine allocation strategy thus consider recommendation correct equals vaccine allocation strategy evaluate algorithm performance respect respect uniform sampling current art evaluate preventive strategies uniform sampling method pulls arm step given budget index sampled uniform distribution consider different levels hardness obtain insight effect unestablished outcome distribution perform analysis value bayesian identification algorithms prior specifications detailed section bayesgap requires upper lower bound defined terms used posteriors experiments use upper bound lower bound established section toptwo thompson sampling requires parameter modulates amount exploration important identification algorithms differentiate top two arms choose limit thompson sampling explore top two arms uniformly censor reward distribution based threshold defined section threshold depends basic reproductive number dispersion parameter defined explicitly experimental settings dispersion parameter choose conservative choice according literature define probability cutoff figure figure show recommendation success rate identification algorithms respectively results values visualized section supplementary information results different values clearly indicate selection identification algorithms significantly outperform uniform sampling method experiment consider different values uniform sampling method requires double amount evaluations achieve similar recommendation performance harder problems setting recommendation uncertainty remains considerable even consuming times budget required thompson sampling identification algorithms require initialization phase order output recommendation successive rejects needs pull arm least thompson sampling bayesgap need pull arm respectively times details section reason algorithms performance evaluated initialization phase bayesgap performance par successive rejects except hardest setting studied comparison thompson sampling consistently outperforms successive rejects pulls initialization phase thompson sampling needs initialize arm posterior pulls double amount uniform sampling successive rejects however experiments clearly show none algorithms reach acceptable recommendation rate using less pulls thereby alleviating concerns using posterior success rate ttts uni budget figure figure present results experiment curve represents rate successful arm recommendations range budgets curve shown considered algorithms bayesgap legend successive rejects legend thompson sampling legend ttts uniform sampling legend uni section derived statistic express probability success concerning recommendation made thompson sampling analyzed probability thompson sampling recommendations obtained experiment described provide insights statistic used support policy makers show values thompson sampling recommendations figure figures values section supplementary information figure indicates closely follows recommendation correctness success rate ttts uni budget figure figure present results experiment curve represents rate successful arm recommendations range budgets curve shown considered algorithms bayesgap legend successive rejects legend thompson sampling legend ttts uniform sampling legend uni uncertainty inversely proportional size available budget additionally figure figures values section supplementary information confirm underestimates recommendation correctness observations indicate potential serve conservative statistic inform policy makers confidence particular recommendation thus used define meaningful cutoffs guide policy makers interpretation recommendation preventive strategies conclusion formulate objective select best preventive strategy individualbased model fixed budget identification problem experiment set evaluate setting context pandemic influenza assess best arm recommendation performance preventive bandit report success rate independent bandit runs probability success success failure budget figure thompson sampling run times budget experiment recommendations computed values shown scatter plot point color reflects correctness recommendation see legend demonstrate possible efficiently identify optimal preventive strategy using limited number model evaluations even large number preventive strategies consider compared uniform sampling method able recommend best preventive strategy reducing number required model evaluations times additionally show using bayesian identification algorithms statistics defined support policy makers decisions confident method potential used decision support tool mitigating epidemics enable use models studies would otherwise computationally prohibitive allow researchers explore wider variety model scenarios identify two particular directions future work firstly method evaluated context pandemic influenza important stress used evaluate preventive strategies infectious diseases since recently dengue vaccine available optimal allocation vaccine remains important research topic recognize dengue epidemics interesting use case secondly paper empirical success rate estimated probability succes figure thompson sampling run times budget experiment recommendations computed values binned steps per bin thus set bernoulli trials show empirical success rate blue scatter confidence interval blue confidence bounds orange reference line denotes perfect correlation empirical success rate estimated probability success preventive bandits learn respect single model outcome proportion symptomatic infections however many pathogens interesting incorporate multiple objectives morbidity mortality cost future aim use bandits contrast current preventive bandits approach plan learn coverage set containing optimal strategy every possible preference profile decision makers might statement respect reproducibility research manuscript accepted source code used experiments made publicly available acknowledgments pieter libin supported phd grant fwo fonds wetenschappelijk onderzoek vlaanderen vub research council timothy verstraeten supported phd grant fwo fonds wetenschappelijk onderzoek vlaanderen vub research council kristof theys supported postdoctoral grant fwo fonds wetenschappelijk onderzoek vlaanderen diederik roijers supported postdoctoral grant fwo fonds wetenschappelijk onderzoek vlaanderen computational resources services used work provided hercules foundation flemish government department krediet aan navorsers theys supplementary information introduction supplementary information provide proof bayesgap simple regret bound section furthermore provide additional figures omitted main manuscript figures outcome epidemic size distributions section figures experimental success rates section figures probabilities success values per budget section figures binned distribution values section finally section describe computational resources used execute simulations bayesgap simple regret bound posteriors lemma consider jeffrey prior parameters gaussian reward distributions posterior mean arm following nonstandardized pull tnk number pulls arm sample mean sum squares proof lemma presented proved honda lemma consider random variable variance probability within radius mean written normalizing constant standard proof consider random variable probability greater probability greater lower bound integral probability density function starting lower bound integral introduce factor greater considered values take note following derivative use result analytically solve integral finally solve primitive infinity next apply union bound obtain lower bound probability magnitude smaller finally consider lemma consider bandit problem budget arms let upper lower bounds hold times arms probability finally let monotonically decreasing function bound simple regret proof first define event every mean bounded associated bounds time step probability deviating single bound time definition applying union bound obtain probability regret equal probability event occuring proven theorem consider gaussian bandit problem budget unknown variance let generalization variance arms respectively upper lower bounds arm time simple regret bounded min min note min bound decreases exponentially simik lar problem setting presented intuitively result makes sense known variances gaussian used describe posterior means indeed number pulls approaches infinity converge gaussians proof according lemma posterior average reward distribution scaling factor therefore arm time variance equals nkn scaling factor described lemma denote variance rewards per arm define upper bound expression specified lemma next compute inverse generalize variance representative approximating hardness problem hardness defined obtain follows finally conditions lemma function satisfied simple regret bound obtained using lemma probability true mean bounds given lemma main paper choose mean variances obtained initialization phase vaccine allocation strategy vaccine allocation strategy outcome epidemic size distributions outcome distributions vaccine allocation strategy outcome distributions vaccine allocation strategy outcome distributions outcome distributions outcome distributions vaccine allocation strategy vaccine allocation strategy outcome distributions epidemic size epidemic size epidemic size epidemic size epidemic size epidemic size bandit run success rates success rate success rate ttts uni budget ttts uni budget success rate ttts uni budget bandit run results budget ttts uni budget bandit run results bandit run results ttts uni success rate success rate bandit run results success rate ttts uni bandit run results budget bandit run results values thompson sampling probability success probability success success failure budget success failure budget probability success probability success budget values success failure success failure budget budget values values probability success probability success values success failure success failure values budget values binned distribution values thompson sampling empirical success rate empirical success rate estimated probability succes estimated probability succes binned distribution empirical success rate empirical success rate binned distribution estimated probability succes binned distribution estimated probability succes binned distribution empirical success rate empirical success rate estimated probability succes binned distribution estimated probability succes binned distribution computational resources simulations run high performance cluster hpc hpc used ivy bridge nodes specifically nodes two ivy bridge xeon cpus ghz level cache ram infrastructure allowed run flute simulations per node references abul abbas andrew lichtman shiv pillai cellular molecular immunology elsevier health sciences milton abramowitz irene stegun handbook mathematical functions formulas graphs mathematical tables volume courier corporation shipra agrawal navin goyal analysis thompson sampling bandit problem conference learning theory pages maira aguiar nico stollenwerk dengvaxia efficacy dependency serostatus closer look recent data clinical infectious diseases wenchi chiu david goldsman lim lee kwokleung tsui beate sander david fisman azhar nizam reactive strategies containing developing outbreaks pandemic influenza bmc public health audibert bubeck best arm identification bandits 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convex regularization methods spatial point processes intensity estimation achmad mar laboratory jean kuntzmann department probability statistics grenoble alpes france department mathematics uqam canada march abstract paper deals feature selection procedures spatial point processes intensity estimation consider regularized versions estimating equations based campbell theorem derived two classical functions poisson likelihood logistic regression likelihood provide general conditions spatial point processes penalty functions ensure consistency sparsity asymptotic normality discuss numerical implementation assess finite sample properties simulation study finally application tropical forestry datasets illustrates use proposed methods introduction spatial point pattern data arise many contexts interest lies describing distribution event space examples include locations trees forest gold deposits mapped geological survey stars cluster star animal sightings locations specific cells retina road accidents see waagepetersen illian baddeley interest methods analyzing spatial point pattern data rapidly expanding accross many fields science notably ecology epidemiology biology geosciences astronomy econometrics one main interests analyzing spatial point pattern data estimate intensity characterizes probability point event occurs infinitesimal ball around given location practice intensity often assumed parametric function measured covariates waagepetersen guan loh waagepetersen waagepetersen waagepetersen guan guan shen coeurjolly paper assume intensity function parameterized vector specification exp spatial covariates measured location real parameter intensity function many variables covariates selection becomes inevitable variable selection regression number purposes provide regularization good estimation obtain good prediction identify clearly important variables fan mazumder identifying set relevant features list many features general combinatorially hard computationally intensive context convex relaxation techniques lasso tibshirani effectively used variable selection parameter estimation simultaneously lasso procedure aims minimizing log likelihood function model interest penalty shrinks coefficients towards zero also set coefficients exactly zero context variable selection lasso often thought convex surrogate selection problem log penalty penalizes number nonzero coefficients model since lasso suboptimal model selection cases fan zou zhang huang many regularization methods developped motivating beyond regime aggressive penalties bridges gap scad fan zhang recently several works implementing variable selection spatial point processes order reduce variance inflation overfitting bias underfitting thurman zhu focused using adaptive lasso select variables inhomogeneous poisson point processes study later extended clustered spatial point processes thurman established asymptotic properties estimates terms consistency sparsity normality distribution also compared results employing adaptive lasso scad adaptive elastic net simulation study application using regularized weighted unweighted estimating equations derived poisson likelihood yue loh considered modelling spatial point data poisson pairwise interaction point processes cluster models incorporated lasso adaptive lasso elastic net regularization methods generalized linear model framework fitting point models note study yue loh also used estimating equation derived poisson likelihood however yue loh provide theoretical study detail although application many penalty functions employed regularization methods spatial point processes intensity estimation theoretical study still restricted specific penalty functions paper propose regularized versions estimating equations based campbell formula derived poisson logistic regression likelihoods estimate intensity spatial point processes consider convex penalty functions provide general conditions penalty function ensure oracle property central limit theorem thus extend work thurman obtain theoretical results general penalty functions less restrictive assumptions asymptotic covariance matrix see remark logistic regression method proposed baddeley easy implement poisson likelihood method less biased since require deterministic numerical approximation prove estimates obtained regularizing logistic regression likelihood also satisfy asymptotic properties see remark procedure straightforward implement since need combine spatstat package two packages glmnet ncvreg remainder paper organized follows section gives backgrounds spatial point processes section describes standard parameter estimation methods regularization regularization methods developed section section develops numerical details induced methods introduced sections asymptotic properties following work fan generalized linear models presented section section investigates properties proposed method simulation study followed application tropical forestry datasets section finished conclusion discussion section proofs main results postponed appendices spatial point processes let spatial point process let compact set lebesgue measure play role observation domain view locally finite random subset random number points almost surely finite whenever bounded region suppose denotes realization observed within bounded region represent locations observed points number points note random empty point pattern background material spatial point processes see example waagepetersen moments first properties point process described intensity measure factorial moment measure properties point process indicate spatial distribution events domain interest intensity measure given intensity measure written nonnegative function called intensity function constant said homogeneous stationary intensity otherwise said inhomogeneous may interpret probability occurence point infinitesimally small ball centre volume properties point process indicate spatial coincidence events domain interest factorial moment measure given summation sign means sum runs pairwise different points indicator function factorial moment measure written dudv nonnegative function called product density intuitively dudv probability observing pair points occuring jointly two infinitesimally small balls centres volume fore detail description moment measures order see appendix waagepetersen suppose intensity function product density campbell theorem see waagepetersen states function dudv order study whether point process deviates independence poisson point process often consider pair correlation function given exist convention poisson point process section example resp indicates pair points likely resp less likely occur locations poisson point process intensity function spirit define order intensity function see waagepetersen details depends point process said reweighted stationary modelling intensity function discuss spatial point process models specified deterministic random intensity function particularly consider two important model classes namely poisson cox processes poisson point processes serve tractable model class interaction complete spatial randomness cox processes form major classes clustering aggregation conciseness focus two later classes models could also presented determinantal point processes lavancier constitute interesting class repulsive point patterns explicit moments investigated sake brevity paper focus models intensity function given poisson point process point process poisson point process intensity function assumed locally integrable following conditions satisfied oisson conditionally points joint density proportional poisson point process intensity function also called modulated poisson point process waagepetersen waagepetersen particular poisson point processes cox processes cox process natural extension poisson point process obtained considering intensity function poisson point process realization random field suppose nonnegative random field conditional distribution given poisson point process intensity function said cox process driven see waagepetersen several types cox processes consider two types cox processes point process log gaussian cox process point processes let stationary poisson process mother process intensity given let independent poisson processes offspring processes intensity function exp probability density function determining distribution offspring points around mother points parameterized special case inhomogeneous point process mothers offspring point process cox process driven exp waagepetersen coeurjolly verify intensity function indeed exp one example point process thomas process exp density conditionally parent event location children events normally distributed around smaller values correspond tighter clusters smaller values correspond fewer number parents parameter vector referred interaction parameter modulates spatial interaction dependence among events log gaussian cox process suppose log gaussian random field given point process follows poisson process said log gaussian cox process driven waagepetersen random intensity function written log stationary gaussian random field covariance function depends parameter waagepetersen coeurjolly intensity function log gaussian cox process indeed given exp one example correlation function exponential form waagepetersen guan exp constitutes interaction parameter vector variance correlation scale parameter parametric intensity estimation one standard ways fit models data maximizing likelihood model data maximum likelihood method feasible parametric poisson point process models section computationally intensive markov chain monte carlo mcmc methods needed otherwise waagepetersen mcmc methods yet straightforward implement estimating equations based campbell theorem developed see waagepetersen waagepetersen waagepetersen guan shen baddeley review estimating equations derived poisson likelihood section logistic regression likelihood section maximum likelihood estimation inhomogeneous poisson point process intensity function parameterized likelihood function exp function log omitted constant term form reduces intensity function exp rathbun cressie showed maximum likelihood estimator consistent asymptotically normal asymptotically efficient sample region goes poisson likelihood let true parameter vector applying campbell theorem score function gradient vector denoted exp exp exp exp exp score function poisson appears unbiased estimating equation even though poisson point process estimator maximizing referred poisson estimator properties poisson estimator carefully studied schoenberg showed poisson estimator still consistent class point process models asymptotic normality fixed observation domain obtained waagepetersen guan loh established asymptotic normality increasing domain assumption suitable mixing point processes regarding parameter see section waagepetersen guan studied procedure estimate proved certain mixing conditions parameter estimates enjoy properties consistency asymptotic normality weighted poisson likelihood although estimating equation approach derived poisson likelihood simpler faster implement maximum likelihood estimation potentially produces less efficient estimate maximum likelihood waagepetersen guan shen information interaction events ignored regain lack efficiency guan shen proposed weighted poisson function given log weight surface regarding see larger weight makes observations infinitesimal region influent campbell theorem still unbiased estimating equation addition guan shen proved conditions parameter estimates consistent asymptotically normal guan shen showed weight surface minimizes trace asymptotic matrix estimates maximizing result efficient estimates poisson estimator particular proposed weight surface pair correlation function poisson point process note hence reduces maximum likelihood estimation general point processes weight surface depends intensity function pair correlation function thus incorporates information inhomogeneity dependence spatial point processes clustering present hence weight decreases weight surface achieved setting get estimate substituted given poisson estimates alternatively also computed nonparametrically kernel method furthermore guan shen suggessted approximate ripley estimated guan extended study guan shen considered complex estimating equations specifically replaced function derivative respect procedure results slightly efficient estimate one obtained however computational cost important since combine estimating equations penalization methods see section considered extension logistic regression likelihood although estimating equations discussed section unbiased methods general produce unbiased estimator practical implementations waagepetersen baddeley proposed another estimating function indeed close score poisson able obtain less biased estimator poisson estimates addition proposed estimating equation fact derivative logistic regression likelihood following baddeley define weighted logistic regression loglikelihood function log log nonnegative function role well explanation name logistic method explained section note score unbiased estimating equation waagepetersen showed asymptotic normality poisson certain clustered point processes estimator obtained similar procedure furthermore methodology results studied baddeley considering spatial gibbs point processes determine optimal weight surface logistic method follow guan shen minimized trace asymptotic covariance matrix estimates obtain weight surface defined estimated section regularization techniques section discusses convex regularization methods spatial point process intensity estimation methodology regularization techniques introduced alternatives stepwise selection variable selection parameter estimation general regularization method attempts maximize penalized function function number observations nonnegative penalty function parameterized real number let either weighted poisson function weighted logistic regression function similar way define penalized weighted function given volume observation domain plays role number observations setting nonnegative tuning parameter corresponding penalty function described details next section penalty functions regularization methods say penalty function nonnegative function examples penalty function norm norm elastic net scad first second derivatives functions given table noticed differentiable resp scad resp penalty table first second derivatives several penalty functions penalty elastic net scad first penalization technique improve ordinary least squares ridge regression hoerl kennard works minimizing residual sum squares subject bound norm coefficients continuous shrinkage method ridge regression achieves better prediction ridge also extended fit generalized linear models however ridge reduce model complexity since always keeps predictors model introduced method called lasso tibshirani employs penalty obtain variable selection parameter estimation simultaneously despite lasso enjoys attractive statistical properties limitations senses fan zou hastie zou zhang huang zhang making huge possibilities develop methods scenario high correlations among predictors zou hastie proposed elastic net technique convex combination penalties method particularly useful number predictors much larger number observations since select eliminate strongly correlated predictors together lasso procedure suffers nonnegligible bias satisfy oracle property asymptotically fan fan zhang among others introduced penalties get around drawbacks idea bridge gap trying keep unbiased estimates nonzero coefficients shrinking less important variables exactly zero rationale behind penalties scad also understood considering first derivative see table start applying similar rate penalization lasso continuously relax penalization rate penalization drops zero however employing penalties regression analysis main challenge often minimization possible objective function penalty longer dominated convexity likelihood function issue carefully studied fan proposed local quadratic approximation lqa zou proposed local linear approximation lla yields objective function optimized using least angle regression lars algorithm efron finally breheny huang mazumder investigated application coordinate descent algorithm penalties table details regularization methods method ridge lasso enet aenet scad enet aenet respectively stand elastic net adaptive lasso adaptive elastic net worth emphasizing allow direction different regularization parameter elastic net penalty functions extended adaptive lasso zou adaptive elastic net zou zhang table details regularization methods considered study numerical methods present numerical aspects section nonregularized estimation two approaches consider weighted poisson regression explained section logistic regression reviewed section penalized estimation procedure done employing coordinate descent algorithm section separate use convex penalties section weighted poisson regression berman turner developed numerical quadrature method approximate maximum likelihood estimation inhomogeneous poisson point process approximated likelihood finite sum analytical form weighted likelihood generalized linear model poisson response method extended gibbs point processes baddeley turner suppose approximate integral term riemann sum approximation points consisting data points dummy points quadrature weights implement method domain firstly partitioned rectangular pixels equal area denoted one dummy point placed center pixel let indicator whether point event point process dummy point without loss generality let observed events dummy points thus poisson function approximated rewritten log equation corresponds quasi poisson function maximizing equivalent fitting weighted poisson generalized linear model performed using standard statistical software similarly approximate weighted poisson function using numerical quadrature method log value weight surface point estimate obtained suggested guan shen similarity beetween allows compute estimates using software generalized linear model well fact particular exploited ppm function spatstat package baddeley turner baddeley option mpl make presentation becomes general number dummy points denoted next sections logistic regression perform well approximation often requires quite large number dummy points hence fitting generalized linear models computationally intensive especially dealing quite large number points unbiased estimating equations approximated using deterministic numerical approximation section always produce unbiased estimator achieve unbiased estimator estimate log log dummy point process independent intensity function form related estimating equation defined baddeley besides consider form since apply campbell theorem last term obtain log log exactly last term addition conditional weighted likelihood function bernoulli trials exp log exp log precisely weighted logistic regression offset term log thus parameter estimates straightforwardly obtained using standard software generalized linear models approach fact provided spatstat package calling ppm function option logi baddeley spatstat dummy point process generates points average poisson binomial stratified binomial point process baddeley suggested choose number points furthermore determine option considered starting point approach see baddeley details coordinate descent algorithm lars algorithm efron remarkably efficient method computing entire path lasso solutions linear models computational cost order order least squares fit coordinate descent algorithm friedman appears competitive algorithm computing regularization paths costs operations therefore adopt cyclical coordinate descent methods work really fast large datasets take advantage sparsity coordinate descent algorithms optimize target function respect single parameter time iteratively cycling parameters convergence criterion reached detail convex penalty functions next two sections present coordinate descent algorithm fitting generalized weighted poisson regression similar approach used fit penalized weighted logistic regression convex penalty functions since given concave function parameters newtonraphson algorithm used maximize penalized function done using iteratively reweighted least squares irls method current estimate parameters construct quadratic approximation weighted poisson function using taylor expansion constant working response values weights exp exp exp regularized poisson linear model works firstly identifying decreasing sequence starting minimum value entire vector value outer loop created compute secondly coordinate descent method applied solve penalized weighted least squares problem minp minp coordinate descent method explained follows suppose estimate method consists partially optimizing respect min friedman provided form update penalized regression using several penalties nonnegative garrote breiman lasso elastic net fused lasso tibshirani group lasso yuan lin berhu penalty owen wang instance update elastic net embraces ridge lasso regularization setting respectively zij zil fitted value excluding contribution covariate zij operator value sign update repeated convergence coordinate descent algorithm several convex penalties implemented package glmnet friedman set implement ridge lasso set apply elastic net regularization adaptive lasso follow zou take replace initial estimate say ols ridge positive tuning parameter avoid computational evaluation choosing follow zou section wasserman roeder also considered choose ridge ridge estimates obtained ridge regression implementing adaptive elastic net follows along similar lines penalty functions breheny huang investigated application coordinate descent algorithm fit penalized generalized linear model using scad penalty mazumder also studied coordinatewise optimization algorithm linear models considering general penalties mazumder concluded known current estimate univariate penalized least squares function convex ensure procedure converges stationary point mazumder found turns case scad penalty satisfied bridge power penalty cases breheny huang derived solution coordinate descent algorithm scad generalized linear models cases implemented ncvreg package let vector containing estimates wish partially optimize respect define zij update scad maxj maxj definition update operator given selection regularization tuning parameter worth noticing coordinate descent procedures computation procedures computing penalized likelihood estimates rely tuning parameter choice also becoming important task estimation using large value tends smaller variance larger biases whereas estimation using small value leads zero biases larger variance biases variances yields optimal choice fan select reasonable identify range values extending maximum value penalized coefficients zero friedman breheny huang select value optimizes criterion fixing path select tuning parameter minimizes wqbic weighted version bic criterion defined wqbic log number selected covariates nonzero regression coefficients observation volume represents sample size linear regression models wang proposed bictype criterion choosing bic log log number observations degree freedom criterion consistent meaning selects correct model probability approaching large samples set candidate models contains true model findings line study zhang criterion presented general way called generalized information criterion gic criterion wqbic specific form gic proposed zhang selection scad another task fix scad following fan breheny huang respectively avoid complexities asymptotic theory section present asymptotic results regularized weighted poisson likelihood estimator considering point process observed sequence observation domain expands regularization parameters indexed asymptotic results also hold regularized unweighted poisson likelihood estimator sake conciseness present asymptotic results regularized logistic regression estimate results similar main difference lying conditions matrices different expression see remark notation conditions recall classical definition strong mixing coefficients adapted spatial point processes politis define sup generated minimal distance sets denotes class borel sets let denote vector true coefficient values vector nonzero coefficients vector zero coefficients define matrices zdn zdn dvdu consider following conditions required derive asymptotic results denotes origin every convex compact contains interior assume intensity function specification given open convex bounded set covariates weight function satisfy sup sup exists integer product density exists satisfies strong mixing coefficients assume exists exists positive definite matrix sufficiently large exists positive definite matrix sufficiently large penalty function nonnegative continuously differentiable derivative assumed lipschitz function furthermore given assume exists sufficiently large thrice continuously differentiable ball centered radius assume third derivative uniformly bounded condition define sequences max inf inf max sequences detailed table different methods considered paper play central role results even discussed later section specify right require table details sequences given regularization method method ridge max lasso enet aenet max max min max min max scad main results state main results proofs relegated appendices first show theorem penalized weighted poisson likelihood estimator converges probability exhibits rate convergence theorem assume conditions hold let given exists local maximizer implies penalized weighted poisson likelihood estimator consistent furthermore demonstrate theorem consistent estimator ensures sparsity estimate correctly set zero probability tending asymptotically normal theorem assume conditions hold consistent local maximiz ers theorem satisfy sparsity asymptotic normality resp corner resp consequence asymptotic covariance matrix note inverse square matrix remark lasso adaptive lasso penalties since since conditions asymptotically negligible respect remark theorems remain true regularized weighted logistic regression likelihood estimates extend condition replacing expression matrices adding remark want highlight main theoretical differences work thurman first methodology results available logistic regression likelihood second consider general penalty function thurman considered adaptive lasso method third assume thurman positive definite matrix instead assume sharper condition assuming either smallest eigenvalue positive definite matrix makes proofs little bit technical discussion conditions adopt conditions based paper coeurjolly condition assumption contains interior made without loss generality instead interior point condition could modified ball centre radius contained sufficiently large condition quite standard conditions matrices bounded see coeurjolly combination conditions used establish central limit theo rem using general central limit theorem triangular arrays nonstationary random fields obtained extension bolthausen later extended nonstationary random fields guyon pointed coeurjolly condition spatial average assumption like establishing asymptotic normality ordinary least square estimators linear models condition also useful make sure matrix invertible conditions ensure matrix invertible sufficiently large conditions discussed details several models coeurjolly satisfied large class intensity functions large class models including poisson cox processes discussed section condition controls higher order terms taylor expansion penalty function roughly speaking ask penalty function least lipschitz thrice differentiable neighborhood true parameter vector condition looks technical however obviously satisfied ridge lasso elastic net adaptive versions according choice satisfied scad equal theorem requires conditions simultaneously requiring assumptions corresponding penalized weighted poisson likelihood estimators possess oracle property perform well weighted poisson likelihood estimator estimates knowing fact ridge regularization method preventing applying theorem penalty lasso elastic net constant lasso two conditions satisfied simultaneously different adaptive versions compromise found adjusting well two penalties scad adjusted regularization methods considered paper condition implied condition simulation study conduct simulation study three different scenarios described section compare estimates regularized poisson likelihood regularized weighted poisson likelihood wpl also want explore behaviour estimates using different regularization methods empirical findings presented section furthermore compare section estimates regularized weighted logistic likelihood ones regularized weighted poisson likelihood simulation setting quite similar waagepetersen thurman spatial domain center scale pixel images elevation gradient elevation contained bei datasets spatstat library core team use two true covariates addition create three different scenarios define extra covariates scenario generate eighteen pixel images covariates standard gaussian white noise denote define covariates vector regression coefficients set zero scenario first generate eighteen pixel images covariates scenario second transform together multicollinearity third define precisely except preserve correlation regression coefficients set zero scenario consider complex situation center scale soil nutrients covariates obtained study tropical forest barro colorado island bci central panama see condit hubbell use extra covariates together keep structure covariance matrix preserve complexity situation setting regression coefficients set zero different maps covariates obtained scenarios depicted appendix except high correlation extra covariates obtained scenario tend constant value figure completely different ones obtained scenario figure mean number points domain chosen set true intensity function represents relatively large effect elevation reflects relatively small effect gradient selected realization points average furthermore erode regularly domain intensity function mean number points new domain becomes erosion used observe convergence procedure observation domain expands consider default number dummy points poisson likelihood denoted suggested spatstat package number points scenarios simulate spatial point patterns thomas point process using rthomas function spatstat package also consider two different parameters different levels spatial interaction let four combinations fit intensity simulated point pattern realizations also fit oracle model uses two true covariates models fitted using modified internal function spatstat baddeley glmnet friedman ncvreg breheny huang modification ncvreg package required include penalized weighted poisson logistic likelihood methods simulation results better understand behaviour thomas processes designed study figure shows plot four realizations using different smaller value tighter clusters since fewer parents considering realizations observed mean number points replications standard deviation resp resp mean number points standard deviation resp resp figure realizations thomas process row row column column tables present selection properties estimates using penalized penalized wpl methods similarly van geer indices consider true positive rate tpr false positive rate fpr positive predictive value ppv tpr corresponds ratio selected true covariates number true covariates fpr table empirical selection properties tpr fpr ppv based replications thomas processes domain different values three different scenarios different penalty functions considered well two estimating equations regularized poisson likelihood regularized weighted poisson likelihood wpl method regularized regularized wpl regularized regularized wpl tpr fpr ppv tpr fpr ppv tpr fpr ppv tpr fpr ppv scenario ridge lasso enet aenet scad scenario ridge enet lasso aenet scad scenario ridge lasso enet aenet scad approximate value corresponds ratio selected noisy covariates number noisy covariates tpr explains model correctly select finally fpr investigates model uncorrectly select among scenarios scenario ppv corresponds ratio table empirical selection properties tpr fpr ppv based replications thomas processes domain different values three different scenarios different penalty functions considered well two estimating equations regularized poisson likelihood regularized weighted poisson likelihood wpl method regularized regularized wpl regularized regularized wpl tpr fpr ppv tpr fpr ppv tpr fpr ppv tpr fpr ppv scenario ridge lasso enet aenet scad scenario ridge lasso enet aenet scad scenario ridge lasso enet aenet scad approximate value selected true covariates total number selected covariates model ppv describes model approximate oracle model terms selection therefore want find methods tpr ppv close fpr close generally penalized penalized wpl methods best selection properties obtained larger value shows weaker spatial dependence clustered one indicated smaller value seems difficult select true covariates increases table table tpr tends improve model select frequently ridge lasso elastic net regularization methods satisfy theorems firstly emphasized covariates always selected ridge rates never changed whatever method used penalized lasso elastic net regularization shown tend quite large value fpr meaning wrongly keep noisy covariates frequently penalized wpl applied gain smaller fpr suffer smaller tpr time smaller tpr actually comes unselection smaller coefficient apply adaptive lasso adaptive elastic net scad achieve better performance especially fpr closer zero automatically improves ppv adaptive elastic net resp elastic net slightly larger fpr adaptive lasso resp lasso among regularization methods considered paper adaptive lasso seems outperform ones considering scenarios observe best selection properties penalized combined adaptive lasso design getting complex scenario applying penalized suffers much larger fpr indicating method may able overcome complicated situation however use penalized wpl properties seem stable different designs simulation study one advantage considering penalized wpl remove almost extra covariates worth noticing may suffer smaller tpr apply penalized wpl lose less informative covariates tables faced complex situation would recommend use penalized wpl method adaptive lasso penalty focus selection properties otherwise use penalized combined adaptive lasso penalty preferable tables give prediction properties estimates terms biases standard deviations square root mean squared errors rmse criterions define rmse bias respectively empirical mean variance estimates scenarios scenario general properties improve larger value due weaker spatial dependence larger sample size oracle model model contains wpl estimates efficient estimates particularly clustered case agreeing findings guan shen table empirical prediction properties bias rmse based replications thomas processes domain different values three different scenarios different penalty functions considered well two estimating equations regularized poisson likelihood regularized weighted poisson likelihood wpl method regularized regularized wpl regularized regularized wpl bias rmse bias rmse bias rmse bias rmse scenario oracle ridge lasso enet aenet scad scenario oracle ridge lasso enet aenet scad scenario oracle ridge lasso enet aenet scad table empirical prediction properties bias rmse based replications thomas processes domain different values three different scenarios different penalty functions considered well two estimating equations regularized poisson likelihood regularized weighted poisson likelihood wpl method regularized regularized wpl regularized regularized wpl bias rmse bias rmse bias rmse bias rmse scenario oracle ridge lasso enet aenet scad scenario oracle ridge lasso enet aenet scad scenario oracle ridge lasso enet aenet scad regularization methods applied bias increases general especially consider penalized wpl method regularized wpl larger bias since method select much frequently furthermore weighted method seems introduce extra bias even though regularization considered oracle model low clustered process using penalized wpl similar penalized may weaker dependence represented larger making weight surface closer however larger rmse obtained penalized wpl observe clustered process obtain smaller using penalized wpl explains cases mainly scenario rmse gets smaller ridge method bias closest oracle model largest among regularization methods adaptive lasso method best performance terms prediction considering scenarios obtain best properties apply penalized adaptive lasso penalty design getting much complex scenario use penalized adaptive lasso doubled even quadrupled due overselection many unimportant covariates particular clustered process better properties even obtained applying regularized wpl combined adaptive lasso tables focus prediction properties would recommend apply penalized wpl combined adaptive lasso penalty observed point pattern clustered covariates complex stucture covariance matrix otherwise use penalized combined adaptive lasso penalty favorable recommendations terms prediction support recommend terms selection logistic regression concern compare estimates penalized weighted logistic likelihood penalized weighted poisson likelihood different number dummy points remind number dummy points comes discretize integral terms following ease presentation use term poisson estimates resp logistic estimates parameter estimates obtained using regularized poisson likelihood resp regularized logistic regression likelihood consider three different numbers dummy points denoted different numbers dummy points want observe properties three different situations number points following note choice default poisson likelihood spatstat corresponds case baddeley showed datasets large number points structured point processes logistic likelihood method clearly preferable requires smaller number dummy points perform quickly efficiently want investigate similar comparison methods regularized table empirical selection properties tpr fpr ppv based replications thomas processes domain two different scenarios three different numbers dummy points different estimating equations considered regularized weighted poisson weighted logistic regression likelihoods employing adaptive lasso regularization method scenario method unweighted scenario weighted unweighted weighted tpr fpr ppv tpr fpr ppv tpr fpr ppv tpr fpr ppv poisson logistic approximate value repeat results scenarios use selection prediction indices examined section consider adaptive lasso method table presents selection properties poisson logistic likelihoods adaptive lasso regularization unweighted versions procedure regularized logistic method outperforms regularized poisson method number dummy points much smaller number points methods tend similar performances consider weighted versions regularized logistic poisson likelihoods results change much regularized poisson likelihood method slightly outperforms regularized logistic likelihood method addition scenario considers complex situation methods tend select noisy covariates much frequently empirical biases standard deviation square root mean squared errors presented table include empirical results standard poisson logistic estimates regularization considered let first consider unweighted methods regularization logistic method clearly smaller bias especially explains situations rmse smaller however weighted methods although logistic method smaller bias general produces much larger leading larger rmse cases compare weighted unweighted methods logistic estimates general fail reduce also larger bias adaptive lasso regularization considered combined unweighted methods preserve bias general simultaneously improve hence improve rmse logistic likelihood method table empirical prediction properties bias rmse based replications thomas processes domain two different scenarios three different numbers dummy points different estimating equations considered regularized weighted poisson weighted logistic regression likelihoods employing adaptive lasso regularization method scenario method unweighted scenario weighted unweighted weighted bias rmse bias rmse bias rmse bias rmse regularization poisson logistic adaptive lasso poisson logistic slightly outperforms poisson likelihood method weighted methods considered obtain smaller larger bias weighted versions poisson logistic likelihoods results change much weighted poisson method slightly outperforms weighted logistic method tables number dummy points chosen would recommend apply poisson likelihood method number dummy points chosen logistic likelihood method favorable recommendations regarding whether weighted unweighted methods follow ones section application forestry datasets region tropical moist forest barro colorado island bci central panama censuses carried woody stems least diameter breast height identified tagged mapped resulting maps individual trees species see condit hubbell interest know high number different tree species continues coexist profiting different habitats determined topography soil properties see waagepetersen waagepetersen guan particular selection covariates among topological attributes soil minerals well estimation coefficients becoming concern figure maps locations bpl trees top left elevation top right slope bottom left concentration phosporus bottom right particularly interested analyzing locations beilschmiedia pendula lauraceae bpl tree stems model intensity bpl trees loglinear function two topological attributes soil properties covariates figure contains maps locations bpl trees elevation slope concentration phosporus bpl trees seem appear greater abundance areas high elevation steep slope low concentration phosporus covariates maps depicted figure apply regularized weighted poisson logistic likelihoods combined adaptive lasso regularization select estimate parameters since deal datasets large number points set default number dummy points poisson likelihood spatstat package number dummy points chosen larger number points perform quickly efficiently worth emphasizing center scale covariates observe one largest effect intensity results presented table covariates poisson likelihood logistic method selected covariates using unweighted methods covariates poisson logistic methods selected using weighted versions unweighted methods tend overfit model overselecting unimportant covariates weighted methods tend keep uninformative covariates poisson logistic estimates similar selection estimation results first table barro colorado island data analysis parameter estimates regression coefficients beilschmiedia pendula lauraceae trees applying regularized weighted poisson logistic regression likelihoods adaptive lasso regularization unweighted method weighted method poisson estimates logistic estimates poisson estimates logistic estimates elev slope cov find differences estimation unweighted weighted methods especially slope manganese weighted methods approximately two times larger estimators second may loose nonzero covariates apply weighted methods even though covariates relatively small coefficient boron high correlation many covariates particularly selected possibly boron selected may nonnegligible coefficient unweighted methods chosen model may explain weighted methods introduce extra biases however since situation appears quite close scenario simulation study weighted methods favorable terms selection prediction application face computational problem nevertheless model species trees much points default value lead numerical problems case logistic likelihood would good alternative results suggest bpl trees favor live areas higher elevation slope result different findings waagepetersen guan loh concluded based standard error estimation bpl trees really prefer either high low altitudes however conclusion analysis guan shen thurman bpl trees prefer live higher altitudes higher levels manganese lower levels phosporus zinc concentrations soil associated higher appearance bpl trees conclusion discussion develop regularized versions estimating equations based campbell theorem derived poisson logistic likelihoods procedure able estimate intensity function spatial point processes intensity function many covariates form furthermore procedure also generally easy implement since need combine spatstat package glmnet ncvreg packages study asymptotic properties regularized weighted poisson logistic estimates terms consistency sparsity normality distribution find among regularization methods considered paper adaptive lasso adaptive elastic net scad methods satisfy theorems carry scenarios simulation study observe selection prediction properties estimates compare penalized poisson likelihood penalized weighted poisson likelihood wpl different penalty functions results deal covariates complex covariance matrix point pattern looks quite clustered recommend apply penalized wpl combined adaptive lasso regularization otherwise regularized adaptive lasso preferable careful investigation choose tuning parameters may needed improve selection properties note bias increases quite significantly regularized wpl applied penalized wpl considered procedure may needed improve prediction properties use penalized wpl combined adaptive lasso chose covariates use selected covariates obtain estimates inference procedure investigated paper also compare estimates obtained poisson logistic likelihoods number dummy points chosen either similar larger number points recommend use poisson likelihood method nevertheless number dummy points chosen smaller number points logistic method favorable work would consist studying situation number covariates much larger sample size situation coordinate descent algorithm used paper may cause numerical troubles dantzig selector procedure introduced candes tao might good alternative implementaion linear models generalized linear els results linear programming would interesting bring approach spatial point process setting acknowledgements thank thurman kindly shared code used simulation study thurman breheny kindly provided code used ncvreg package also thank drouilhet technical help research coeurjolly funded project persyvact research funded project bci soils data sets collected analyzed dalling john harms stallard yavitt support nsf oise stri soils initiative ctfs assistance segre trani datasets available center tropical forest science website http references adrian baddeley rolf turner practical maximum pseudolikelihood spatial point patterns 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estimates nonconcave penalized likelihood models annals statistics hui zou hao helen zhang adaptive diverging number parameters annals statistics auxiliary lemma following result used proof theorems throughout proofs notation random vector sequence real numbers means kxn kxn way vector squared matrix notation mean kvn kmn lemma conditions following convergence holds distribution moreover proof let first note using campbell theorems var proof follows coeurjolly let unit box centered define set exp zero mean condition sup sup kyi combine conditions apply theorem central limit theorem triangular arrays random fields obtain also implies second result deduced condition particular implies proof theorem proof result following ones notation stands generic constant may vary line line particular constant independent proof let remind reader estimate defined maximum function given open convex bounded set kkk sufficiently large assume valid following prove theorem follow main argument fan aim proving given exists sufficiently large sup equation imply probability least exists local maximum ball kkk therefore local maximizer decompose since infinitely continuously differentiable using taylor expansion exists tdn since convex bounded since uniformly bounded conditions exists nonnegative constant kan tdn let smallest eigenvalue squared matrix condition lim inf lim inf hence kkk regarding term since penalty function nonnegative since sufficiently large twice continuously differentiable every tdn therefore using taylor expansion exist sign definition condition deduce exists san inequality since sufficiently large kkk sdn return sufficiently large sup sdn since choosing large enough exists sufficiently large sup given proof theorem prove theorem provide lemma follows lemma assume conditions condition hold probability tending satisfying constant max proof sufficient show probability tending satisfying small first note obtain second conditions exists third let sequence given condition since assumption particular sufficiently large therefore sufficiently large sign since proves proceed similarly prove proof focus proof theorem since theorem proved lemma need prove theorem asymptotic normality shown theorem consistent local maximizer shown exists estimator theorem consistent local maximizer regarded function satisfies exists sign sign sign sign sign decompose sequence defined condition condition following taylor expansion derived term exists sign latter equation ensues theorem condition theorem implies regarding term since lipschitz function exists theorem deduce let resp first components resp corner resp let also matrix containing finally let vector vector matrix sign sign rewrite sides definition given obtain using deduce premultiplying sides kmn kmn kmn kmn condition implies exists positive definite matrix sufficiently large hence kmn conditions theorem theorem theorem finally since assumption deduce kmn kmn kmn kmn therefore theorem slutsky theorem deduce rewritten particular given map covariates figure maps covariates designed scenario first two top left images elevation slope covariates generated standard gaussian white noise transformed get multicollinearity figure maps covariates used scenario application left right elevation slope aluminium boron calcium row copper iron potassium magnesium manganese row phosporus zinc nitrogen nitrigen mineralisation row

wireless network design control systems survey aug pangun park sinem coleri ergen carlo fischione chenyang karl henrik johansson networked control systems wncs composed spatially distributed sensors actuators controllers communicating wireless networks instead conventional wired connections due main benefits reduction deployment maintenance costs large flexibility possible enhancement safety wncs becoming fundamental infrastructure technology critical control systems automotive electrical systems avionics control systems building management systems industrial automation systems main challenge wncs jointly design communication control systems considering tight interaction improve control performance network lifetime survey make exhaustive review literature wireless network design optimization wncs first discuss call critical interactive variables including sampling period message delay message dropout network energy consumption mutual effects communication control variables motivate joint tuning discuss effect controllable wireless network parameters layers communication protocols probability distribution interactive variables also review current wireless network standardization wncs corresponding methodology adapting network parameters moreover discuss analysis design control systems taking account effect interactive variables control system performance finally present wireless network design optimization wncs highlighting tradeoff achievable performance complexity various approaches conclude survey highlighting major research issues identifying future research directions index networked control systems wireless sensor actuator networks joint design delay reliability sampling rate network lifetime optimization ntroduction recent advances wireless networking sensing computing control revolutionizing control systems interact information physical processes systems cps internet things iot tactile internet wireless networked control systems wncs sensor nodes attached physical plant sample transmit measurements controller wireless channel controllers compute control commands park department radio information communications engineering chungnam national university korea email pgpark coleri ergen department electrical electronics engineering koc university istanbul turkey email sergen department computer science engineering washington university louis louis usa fischione johansson access linnaeus center electrical engineering royal institute technology stockholm sweden carlofi kallej park coleri ergen contributed equally work based sensor data forwarded actuators order influence dynamics physical plant particular wncs strongly related cps tactile internet since emerging techniques deal control physical systems networks strong technology push behind wncs rise embedded computing wireless networks advanced control cloud computing well pull emerging applications automotive avionics building management industrial automation example wncs play key role industry ease installation maintenance large flexibility increased safety make wncs fundamental infrastructure technology control systems wncs applications backed several international organizations wireless avionics alliance zigbee alliance alliance international society automation highway addressable remote transducer communication foundation wireless industrial networking alliance wncs require novel design mechanisms address interaction control wireless systems maximum overall system performance efficiency conventional control system design based assumption instantaneous delivery sensor data control commands extremely high reliabilities usage wireless networks data transmission introduces delay message error probability times transmission failures deadline misses may result degradation control system performance even serious economic losses reduced human safety hence control system design needs include mechanisms tolerate message loss delay hand wireless network design needs consider strict delay reliability constraints control systems data transmissions sufficiently reliable deterministic latency order seconds even milliseconds depending time constraints closedloop system furthermore removing cables data communication sensors actuators motivates removal power supply nodes achieve full flexibility limited stored battery harvested energy components brings additional limitation energy consumption wireless network interaction wireless networks control systems illustrated example wncs connects sensors attached plant controller via wireless networking protocol ieee fig shows control cost wncs using ieee protocol message loss probability sfrag replacements iii wireless networked control systems critical interactive system variables sampling period network delay message dropout network energy consumption allowable control cost network constraints wireless network sampling period control cost various sampling periods message loss probabilities standardization wireless network parameters control system analysis design sampling sampling message delay vii wireless network design techniques control systems interactive design joint design sfrag replacements allowable control cost network constraints fig main section structure relations message loss probability control cost various message delays message loss probabilities fig control cost wncs using ieee protocol various sampling periods message delays message loss probabilities different sampling periods message delays message loss probabilities quadratic control cost defined sum deviations plant state desired setpoint magnitude control input maximum allowable control cost set transparent region indicates maximum allowable control cost network requirements feasible instance control cost would minimized message loss delay point infeasible since requirements met ieee protocol control cost generally increases message loss probability message delay sampling period increase since short sampling periods increase traffic load message loss probability message delay closer critical values system unstable hence area shape feasible region significantly depends network performance determining optimal parameters minimum network cost achieving feasibility trivial complex interdependence control communication systems recently network lpwan wan lora narrowband iot iot developed enable iot connections even though related works wncs applicable control applications smart grid smart transportation remote healthcare survey focuses wireless control systems based wireless personal area networks lowpan radios applications recent excellent surveys exist wireless networks particularly industrial automation specifically discusses general requirements representative protocols wireless sensor networks wsns industrial applications compares popular industrial wsn standards terms architecture design mainly elaborates scheduling algorithms protocols wirelesshart networks experimentation joint design approaches industrial automation focused wirelesshart networks control applications article provides comprehensive survey design space wireless networks control systems potential synergy interaction control communication designs specifically survey touches importance interactions recent advanced works ncs wsn well different approaches wireless network design optimization various wncs applications goal survey unveil address requirements challenges associated wireless network design wncs present review recent advances novel design approaches optimizations algorithms protocols effectively developing wncs section structure relations illustrated fig section introduces inspiring applications wncs automotive electronics avionics building automation industrial automation section iii describes wncs multiple plants remotely controlled wireless network section presents critical interactive variables communication control systems including sampling period message delay message dropout energy consumption section introduces basic wireless network standardization key network parameters various protocol layers useful tune distribution critical interactive variables section provides overview recent control design methods incorporating interactive variables section vii presents various optimization techniques wireless networks integrating control systems classify design approaches two categories based degree integration interactive designs joint designs interactive design wireless network parameters tuned satisfy given requirements control system joint design wireless network control system parameters jointly optimized considering tradeoff performances section viii describes three experimental testbeds wncs conclude article highlighting promising research directions section otivating pplications section explores inspiring applications wncs wireless network wireless networks recently proposed goal reducing manufacturing maintenance cost large amount wiring harnesses within vehicles wiring harnesses used transmission data power delivery within current vehicle architecture may parts weigh much contain wiring eliminating wires would additionally potential improve fuel efficiency greenhouse gas emission spur innovation providing open architecture accommodate new systems applications wireless network consists central control unit battery electronic control units wireless sensors wireless actuators wireless sensor nodes send data corresponding electronic control unit scavenging energy either one electronic control units energy scavenging devices attached directly actuators receive commands corresponding electronic control unit power electronic control units energy scavenging device reason incorporating energy scavenging envisioned architecture eliminate lifetime limitation fixed storage batteries applications exploit wireless architecture fall one three categories powertrain chassis body powertrain applications use automotive sensors engine transmission onboard diagnostics control vehicle energy use driveability performance chassis applications control vehicle handling safety steering suspension braking stability elements vehicle body applications include sensors mainly used vehicle occupant needs occupant safety security comfort convenience information first wireless network applications tire pressure monitoring system tpms intelligent tire tpms based wireless transmission tire pressure data sensors vehicle body currently integrated new cars europe intelligent tire based placement wireless sensors inside tire transfer accelerometer data coordination nodes body car goal improving performance active safety systems since accelerometer data generated much higher rate pressure data batteries placed within tire intelligent tire contains ultralow power wireless communication system powered energy scavenging technology commercialized pirelli wireless avionics wireless avionics waic tremendous potential improve aircraft performance flight operations reduction overall weight maintenance costs enhancement safety currently cable harness provides connection sensors corresponding control units sample process sensor information among multiple control units backbone network safetycritical flight control due high demands safety efficiency modern aircraft relies large wired sensor actuator networks consist devices wiring harness usually represents aircraft weight instance wiring harness airbus weights waic alliance considers wireless sensors avionics located various locations within outside aircraft sensors used monitor health aircraft structure smoke sensors ice detectors critical systems engine sensors landing gear sensors sensor information communicated central onboard entity potential waic applications categorized two broad classes according application data rate requirements low high data rate applications data rates less respectively world radio conference international telecommunication union voted grant frequency band ghz waic systems allow replacement heavy wiring used aircraft waic alliance dedicating efforts performance analysis assigned frequency band design wireless networks avionics control systems space shuttles international space stations already using commercially available wireless solutions ewb microtau ultrawis invocon building automation wireless network based building automation provides significant savings installation cost allowing large retrofit market addressed well new constructions building automation aims achieve optimal level occupant comfort minimizing energy usage control systems integrative component fans pumps equipment dampers thermostats modern building control systems require wide variety sensing capabilities order control temperature pressure humidity flow rates european environment agency shows electricity water consumption buildings total resource consumptions respectively world survey reports early adopters five continents interested new technologies help better manage energy consumption willing pay energy management equipment could save energy bill smart energy home applications example energy management systems using wsns intelligent building ventilation control described underfloor air distribution indoor climate regulation process set injection fresh airflow floor exhaust located ceiling level considered system composed ventilated rooms fans plenums wireless sensors underfloor air distribution systems reduce energy consumption buildings improving thermal comfort ventilation efficiency indoor air quality using wsns industrial automation wireless sensor actuator network wsan effective smart infrastructure process control factory automation emerson process management estimates wsns enable cost savings compared deployment cost wired field devices industrial automation domain industrial process control product processed continuous manner oil gas chemicals factory automation discrete manufacturing instead products processed discrete steps individual elements cars drugs food industrial wireless sensors typically report state fuse heating ventilation vibration levels pumps since discrete product factory automation requires sophisticated operations robot belt conveyors high speed sampling rates requirements often stricter process automation furthermore many industrial automation applications might future require networks hundreds sensors actuators communicating access points according technavio wsn solutions industrial control applications one major emerging industrial trends many wireless networking standards proposed industrial processes wirelesshart abb emerson siemens isa honeywell industrial wireless solutions also commercially available deployed tropos abb smart wireless emerson iii ireless etworked ontrol ystems fig depicts generalized diagram wncs multiple plants remotely controlled actuators sensors plant sensors actuators plant wireless networks controller controller fig overview considered ncs setup multiple plants controlled multiple controllers wireless network closes loop sensor controller controller actuator network includes nodes attached plant controller also relay nodes wireless network wireless network includes sensors actuators attached plants controllers relay nodes plant physical system controlled inputs outputs plant continuoustime signals outputs plant sampled periodic aperiodic intervals wireless sensors packet associated state plant transmitted controller wireless network controller receives measurements computes control command control commands sent actuator attached plant hence system contains component since channels use wireless network general wncs fig also called feedback ncs system scenario quite general applies interconnection plant controller control systems objective feedback control system ensure system desirable dynamic steadystate response characteristics able efficiently attenuate disturbances handle network delays loss generally system satisfy various design objectives stability fast smooth responses setpoint changes elimination errors avoidance excessive control actions satisfactory degree robustness process variations model uncertainty particular stability control system extremely important requirement ncs design methods consider subsets requirements synthesize estimator controller subsection briefly introduce fundamental aspects modeling stability control cost controller estimator design ncss ncs modeling ncss modeled using three main approaches namely approach sampleddata approach approach dependent controller plant approach considers controllers plant model representation leads often uncertain system uncertainties appear matrix exponential form due discretization typically approach applied ncs linear plants controllers since case exact models derived secondly approach considers discretetime controllers model describes ncs dynamics without exploiting form discretization equations used model dynamics approach able deal simultaneously delays sampling intervals finally approach designs continuoustime controller stabilize plant model controller needs approximated representation suitable computer implementation whereas typical wncs consider controller discuss details analysis design wncs deal network effects section stability stability base requirement controller design briefly describe two fundamental notions stability namely stability internal stability stability ability system produce bounded output bounded input internal stability system ability return equilibrium perturbation linear systems two notions closely related nonlinear system stability concerns forced response system bounded input system defined bibo stable every bounded input system results bounded output bounded input output bounded system said unstable internal stability based magnitude system response steady state response unbounded system said unstable system said asymptotically stable response initial conditions decays zero asymptotically steady state system defined exponentially stable system response addition decays exponentially towards zero faster convergence often means better performance fact many ncs researches analyze exponential stability conditions furthermore response due initial conditions remains bounded decay zero system said marginally stable hence system asymptotically stable marginally stable linear system asymptotically stable bibo stable however bibo stability generally imply internal stability internal stability stronger sense bibo stability hide unstable internal behaviors appear output control cost besides stability guarantees typically certain control performance desired closedloop performance control system quantified control cost function plant state control inputs general regulation control goal keep state error setpoint close zero minimizing control actions hence control cost often consists two terms namely deviations plant state desired setpoint magnitude control input common controller design approach via linear quadratic control formulation linear systems quadratic cost function quadratic control cost defined sum quadratic functions state deviation control effort formulation optimal control policy minimizes cost function explicitly computed riccati equation controller design controller ensure system desirable dynamic steady state response characteristics ncs network delay loss may degrade control performance even destabilize system surveys present controller design ncss historical review see survey briefly describe three representative controllers namely pid controller linear quadratic regulator lqr control model predictive control mpc pid control almost century old remained widely used controller process control today one main reasons controller widely used designed without precise knowledge plant model pid controller calculates error value difference desired setpoint measured plant state control signal sum three terms pterm proportional error proportional integral error proportional derivative error controller parameters proportional gain integral time derivative time integral proportional derivative part interpreted control actions based past present future plant state several parameter tuning methods pid controllers exist historically pid tuning methods require trial error process order achieve desired stability control performance linear quadratic problem one fundamental optimal control problems objective minimize quadratic cost function subject plant dynamics described set linear differential equations quadratic cost sum plant state cost final state cost control input cost optimal controller linear feedback controller lqr algorithm basically automated way find controller furthermore lqr important subproblem general linear quadratic gaussian lqg problem lqg problem deals uncertain linear systems disturbed additive gaussian noise lqr problem assumes noise full state observation lqg problem considers input measurement noise partial state observation finally mpc solves optimal linear quadratic control problems receding horizon hence optimization problem similar controller design problem lqr solved moving horizon order handle model uncertainties contrast controllers pid lqr controller compute current control action function current plant state using information plant past predictive controllers compute control based systems predicted future behaviour mpc tries optimize system behaviour receding horizon fashion takes control commands sensing measurements estimate current message generated sampling period future state plant based control system model control command optimized get desired plant plant state based quadratic cost practice often time message delay hard constraints imposed state control input psfrag replacements compared pid lqr control mpc framework maximum allowable control cost controller efficiently handles constraints moreover mpc handle network constraints disordered missing measurements control commands message dropout packet loss message arrivals message delay appear ncs setting packet loss probability estimator design due network uncertainties plant sampling period actuator state estimation crucial significant research field fig timing diagram control wireless ncss estimator used predict plant state network sampling period message delay message dropouts using partially received plant measurements moreover estimator typically compensates measurement noise network energy harvesting techniques hand may rely delays packet losses predicted state sometimes natural sources solar indoor lighting vibrational used calculation control command kalman filter thermal inductive magnetic resonant coupling one popular approaches obtain estimated radio frequency efficient usage energy harvesting plant states ncs modified kalman filters may attain infinite lifetime sensor actuator nodes posed deal different models network delay situations actuations need powered sepaloss state estimation problem often rately significant amount energy required formulated probabilistically modeling uncertainties actuation commands opening valve curring sensor controller however approach ritical nteractive ystem variables measurement packets proposed critical system variables creating interactions lqg control kalman filter used estimate wncs control communication systems sampling state plant output optimal state estimator period message delay message dropout fig illustrates optimal state feedback controller combined timing diagram control wireless lqg problem controller linear feedback controller lqr optimal lqg estimator controller network sampling period message delay message designed separately communication protocol dropouts distinguish messages control application supports acknowledgement packet transmission layer packets communication layer control channels system generates messages sensor samples sharp contrast separation principle estimator channel control commands controller hold acknowledgement channel control system generally supported hence underlying network operation determines sampling period communication protocols critical design overall estimator controller convert message packet format transmit packet destination since wireless channel lossy transmitter may multiple packet retransmissions wireless networks associated one message depending communication vast majority control applications protocol packet transmissions message fail traffic wireless network consists sensor due bursty channel message considered data sensor nodes towards one controllers lost controller either sits backbone reachable fig message delay time delay via one backbone access points therefore data message generated control system sensor flows sensor nodes controllers necessarily controller received destination hence symmetric wncs particular asymmetrical link cost message delay successfully received message depends unidirectional routes common part number packet retransmissions furthermore since sensor traffic furthermore multiple sensors attached routing path network congestion affects message single plant may independently transmit measurements delay message arrivals possibly disordered shown controller process automation fig environments multicast may used deliver data multiple design wireless network multiple protocol nodes may functionally similar delivery layers determines probability distribution message delay alerts multiple nodes automation control room message dropout variables together samwireless sensors actuators control environments pling period influence stability ncs powered battery energy scavenging power cable energy consumption network fig presents battery storage provides fixed amount energy requires dependences critical system variables since replacement energy consumed therefore efficient wncs design requires understanding interplay usage energy vital achieving high network lifetime communication control discuss effect psfrag replacements aximum allowable control cost network constraints message delay packet loss probability sampling period communication system aspect control system aspect sampling period energy consumption message delay message dropout message discard message loss packet delay packet loss transmission shadow fading medium access multipath fading queueing doppler shift interference fig complex interactions critical system variables arrows represent explicit relationships system variables control communication system performance sampling period control system aspect signals plant need sampled transmitted wireless network important note choice sampling related desired properties system response reference signals influence disturbances network traffic computational load two methods sample continuoustime signals wncs sampling sampling next sampling instant occurs elapse fixed time interval regardless plant state periodic sampling widely used digital control systems due simple analysis design systems based experience simulations common rule selection sampling period make sure range desired natural frequency system sampling period implies typically sampling samples per period dominating mode system traditional digital control system based wired connections smaller sampling period chosen better performance achieved control system however wireless networks decrease sampling period increases network traffic turn increases message loss probability message delay therefore decrease sampling period eventually degrades control performance illustrated fig recently control schemes control systems proposed sensing actuating performed system needs attention hence traffic pattern selftriggered control systems asynchronous rather periodic control execution control tasks determined occurrence event rather elapse fixed time period control events triggered stability control performance lost control significantly reduce traffic load network minor control performance degradation since traffic generated signal changes specified amount however since trigger conditions depend instantaneous state plant state required monitored control proposed prevent monitoring control estimation next event time instant made online detection plant disturbances corresponding control actions generated selftriggered control combination control therefore often desirable communication system aspect choice timetriggered sampling control system determines pattern message generation wireless network sampling results regular periodic message generation predetermined rate random medium access mechanism used increase network load results worse performance critical interactive system variables message delay message dropout energy consumption increase control system performance higher sampling rates therefore hold due network effects hand predetermined nature packet transmissions sampling allows explicit scheduling sensor node transmissions hand reducing message loss delay caused random medium access scheduled access mechanism predetermine transmission time components additional nodes minimal effect transmission existing nodes transmission periodically transmitting nodes distributed uniformly time rather allocated immediately arrive additional nodes may allocated without causing jitter periodic allocation optimal choice medium access control mechanism trivial control overall performance control systems significantly depends plant dynamics number control loops random access mechanism good alternative large number slow dynamical plants share wireless network case scheduled access mechanism may result significant delay triggering event transmission assigned slot due large number control loops however time slots utilized since traffic load low slow plants hand scheduled access mechanism performs well small number fast plants controlled control algorithm random access generally degrades reliability delay performance high traffic load fast plants packet losses random access scheme control increases traffic load may eventually incur stability problems possible prediction control alleviates high network load problem sampling random message generation nature eventtriggered sampling predicting evolution triggering threshold crossings plant state prediction allows explicit scheduling sensor node transmissions eliminating high message delays losses random medium access existing works control assume message dropouts message disorders occur assumption practical packets messages transmitted wireless network dealing message dropouts message disorders control schemes challenging wireless network control system message delay control system aspect mainly two kinds message delays ncss delay delay illustrated fig controller delay represents time interval instant physical plant sampled instant controller receives sampled message actuator delay indicates time duration generation control message controller reception actuator increase delays prevents timely delivery control feedback degrades system performance exemplified fig control theory delays cause phase shifts limit control bandwidth affect stability since delays especially pernicious systems forms modeling prediction essential overcome effects techniques proposed overcome delays use predictive filters including kalman filter practice message delay estimated time stamped data receiving node synchronized wireless network control algorithm compensates measured predicted delay unless large compensation generally impossible delays hence actuator delays critical delays packet delay variation another interesting metric since significantly affects control performance causes possible instability even mean delay small particular heavy tail delay distribution significantly degrades stability system amount degradation depends dynamics process distribution delay variations one way eliminate delay variations use buffer trading delay variation communication system aspect message delay multihop wireless network consists transmission delay access delay queueing delay hop path source destination transmission delay defined time required transmission packet transmission delay depends amount data transmitted destination transmission rate depends transmit power node simultaneously active neighboring nodes transmit power node increases transmission rate increases decreasing transmission delay causing interference simultaneously transmitting nodes increasing delay optimization transmission power rate take account tradeoff medium access delay defined time duration required start actual transmission packet access delay depends choice medium access control mac protocol random access mechanism used delay depends network load mechanism used transmitter receiver random access control protocol network load increases access delay increases due increase either busy sensed channel failed transmissions receiver decoding capability determines number simultaneously active neighboring transmitters decoding technique may based interference avoidance one packet received time cancellation node transmit another packet receiving interference cancellation node may receive multiple packets simultaneously eliminate interference similarly transmitter may capability transmit multiple packets simultaneously execution random access algorithm together parameters also affect message delay hand access used access delay general increases network load increases however effect may minimized designing efficient scheduling algorithms adopting uniform distribution transmissions via exploiting periodic transmission control similar random access advanced capability nodes may decrease access delay moreover packet losses channel may require retransmissions necessitating repetition medium access transmission delay time increases message delay illustrated fig queueing delay depends message generation rate nodes amount data relaying multihop routing path message generation forwarding rate nodes kept acceptable level allow packet build queue moreover scheduling algorithms consider multihop forwarding order minimize delay source destination destination may observe disordered messages since packet associated message travels several hops multiple routing paths experiences network congestion message dropout control system aspect generally two main reasons message dropouts namely message discard due control algorithm message loss due wireless network logical hold zoh mechanism one popular simplest approaches discard disordered messages mechanism latest message kept old messages discarded based time stamp messages however alternatives also proposed utilize disordered messages filter bank message considered lost packet transmissions associated message eventually failed effect message dropouts critical message delay since increases updating interval multiple sampling period mainly two types dropouts message dropouts message dropouts controller estimates plant state compensate possible message dropouts channel remind kalman filtering one popular approaches estimate plant state works well significant message loss since control command directly affects plant dropouts critical dropouts many practical ncss several channels whereas controllers collocated actuators heat ventilation control systems ncs literatures often model message dropout stochastic variable based different assumptions maximum consecutive message dropouts particular significant work devoted deriving upper bounds updating interval stability guaranteed upper bounds could used update deadline network discuss detail section bursty message dropout critical control systems since directly affects upper bounds updating interval communication system aspect data packets may lost transmissions due susceptibility wireless channel blockage multipath doppler shift interference obstructions transmitter receiver variation time cause random variations received signal called shadow fading probabilistic distribution shadow fading depends number size material obstructions environment multipath fading mainly caused multipath components transmitted signal reflected diffracted scattered surrounding objects occurs shorter time periods distances shadow fading multipath components arriving receiver cause constructive destructive interference changing rapidly distance doppler shift due relative motion transmitter receiver may cause signal decorrelate time impose lower bound channel error rate furthermore unintentional interference simultaneous transmissions neighboring nodes intentional interference form disturb successful reception packets well network energy consumption truly wireless solution wncs requires removing power cables addition data cables provide full flexibility installation maintenance therefore nodes need rely either battery storage energy harvesting techniques limiting energy consumption wireless network prolongs lifetime nodes enough energy scavenging extracted natural sources inductive magnetic resonant coupling radio frequency infinite lifetime may achieved decreasing sampling period message delay message dropout improves performance control system cost higher energy consumption communication system higher sampling rate greater number packets transmitted channel increases energy consumption nodes moreover decreasing message delay requires increasing transmission rate data capability transceivers comes cost increased energy consumption finally decreasing message dropout requires either increasing transmit power combat fading interference increasing data capabilities translates energy consumption ireless etwork standardization frequently adopted communication standards wncs ieee ieee enhancements particularly wirelesshart ieee based ieee furthermore recent works ietf consider internet protocol version lossy networks routing protocol lossy networks rpl compatible ieee physical layer medium access control data link layer ieee dsss gts allocation wirelesshart ieee phy ieee mac tdma channel hoping channel blacklisting tdma channel hoping channel blacklisting compaction fragmentation management resource allocation performance monitoring ieee phy ieee mac ieee rpl ieee phy ieee phy tsch dsme lldn ieee mac ieee phy tsch ieee ieee dsss ofdm dsss ofdm dcf pcf edca hcca routing source routing graph routing source routing graph routing source routing distance vector routing table comparison wireless standards ieee originally developed lowrate personal area networks pans without concern delay reliability standards wirelesshart ieee built top physical layer ieee additional time division multiple access tdma frequency hopping multiple path features provide delay reliable packet transmission guarantees lowering energy consumption subsection first introduce ieee discuss wirelesshart ieee higher layers ietf activities rpl hand although key intentions ieee family wireless local area network wlan standards provide high throughput continuous network connection several extensions proposed support qos wireless industrial communications particular ieee specification amendment introduces significant enhancements support soft applications subsection describe fundamental operations basic ieee ieee standards summarized table ieee ieee standard defines physical mac layers protocol stack pan consists pan coordinator responsible managing network many associated nodes standard supports star topology associated nodes directly communicate pan coordinator topology nodes communicate neighbouring node still managed pan coordinator physical layer adopts direct sequence spread spectrum based spreading transmitted signal large bandwidth enable greater resistance interference single channel mhz channels mhz channels ghz used transmission data rate kbps ghz band kbps mhz kbps mhz band standard defines two channel access modalities beacon enabled modality uses slotted optional guaranteed time slot gts allocation mechanism simpler unslotted without beacons communication organized temporal windows denoted superframes fig shows superframe structure beacon enabled mode beacon beacon abaseslotduration gts gts gts inactive period min abasesuperframeduration fig superframe structure ieee following focus beacon enabled modality network coordinator periodically sends beacon frames every beacon interval tbi identify pan synchronize nodes communicate coordinator nodes communicate active period called superframe duration tsd enter mode inactive period structure superframe defined two parameters beacon order superframe order determine length superframe active period given tbi abasesuperframeduration tsd abasesuperframeduration respectively abasesuperframeduration number symbols forming superframe equal addition superframe divided equally sized superframe slots length abaseslotduration active period divided contention access period cap optional contention free period cfp composed gtss slotted mechanism used access channel non data frames gts requests cap cfp dedicated bandwidth used data frames fig illustrates date transfer mechanism beacon enabled mode cap cfp following describe data transmission mechanism cap cfp mechanism cap used cap beacon enabled mode time enabled mode cap nodes access network using slotted described fig major difference different channel access modes backoff timer starts beginning next backoff slot beacon enabled mode immediately enabled mode upon request transmission packet following steps pan coordinator device pan coordinator device gts request beacon cap length data macm inbe acknowledgement cap cap delay random unit backoff periods beacon gts descriptor acknowledgement optional data cfp acknowledgement optional perform cca non data packet data packet transgts request transmission mission channel idle fig data transfers beacon enabled mode cap cfp yes min macm axbe algorithms performed channel access variables initialized contention window size denoted initialized slotted backoff exponent called number backoff stages denoted set macm inbe respectively backoff time chosen randomly interval node waits backoff time units backoff period slots backoff timer expires clear channel assessment performed channel free nonbeacon enabled mode packet transmitted channel free beacon enabled mode updated subtracting packet transmitted otherwise second channel assessment performed channel busy variables updated follows min macm axbe algorithm continues step macm axcsm abackof otherwise packet discarded gts allocation cfp coordinator responsible gts allocation determines length cfp superframe request allocation new gts node sends gts request command coordinator coordinator confirms receipt sending ack frame within cap upon receiving gts allocation request coordinator checks whether sufficient resources possible allocates requested gts recall fig illustrates gts allocation mechanism cfp length depends gts requests current available capacity superframe sufficient bandwidth next superframe coordinator determines node list gts allocation based policy coordinator transmits beacon including gts descriptor announce node list gts allocation information note receipt ack gts request command node continues track beacons waits agtsdescpersistencetime superframes node uses dedicated bandwidth transmit packet within cfp wirelesshart wirelesshart released september first wireless communication standard process control applications standard adopts ieee physical layer channels ghz tdma used allow nodes put radio sleep scheduled transmit receive packet better energy efficiency eliminate collisions better reliability slot size tdma fixed yes yes macm axcsm abackof transmission failure fig slotted algorithm ieee beacon enabled mode increase robustness interference harsh industrial environments channel hopping channel blacklisting mechanisms incorporated direct sequence spread spectrum technique adopted ieee standard frequency hopping spread spectrum used alternate channel transmission packet level channel change packet transmission frequency hopping pattern explicitly defined standard needs determined network manager distributed nodes channel blacklisting may also used eliminate channels containing high interference levels network manager performs blacklisting based quality reception different channels network wirelesshart defines two primary routing approaches multihop networks source routing graph routing source routing provides single route flow graph routing provides multiple redundant routes since source routing approach establishes fixed single path source destination link node failure disturbs communication reason source routing mostly used network diagnostics purposes test connection multiple redundant routes graph routing provide significant improvement source routing terms routing reliability routing paths determined network manager based periodic reports received nodes including historical instantaneous quality wireless links standard released september many similar features wirelesshart providing flexibility adaptivity similar wirelesshart standard adopts ieee physical layer channels ghz optional additional usage channel tdma used better energy consumption reliability performance configurable slot size superframe base adopts channel hopping blacklisting anism improve communication robustness similar wirelesshart flexibility standard adopts three channel hopping mechanisms slotted hopping slow hopping hybrid hopping slotted hopping channel varied slot wirelesshart slow hopping node stays channel consecutive time slots number configurable slow hopping facilitates communication nodes imprecise synchronization join process new nodes transmission packets transmissions slow hopping period performed using mechanism decreases delay packets increasing energy consumption due unscheduled transmission reception times hybrid hopping slotted hopping combined slow hopping accommodating slotted hopping periodical messages slow hopping less predictable new messages five predetermined channel hopping patterns standard contrast wirelesshart explicitly define hopping patterns ieee standard released goal introducing new access modes address delay reliability constraints industrial applications ieee defines three major mac modes namely time slotted channel hopping tsch deterministic synchronous multichannel extension dsme low latency deterministic network lldn time slotted channel hopping tsch medium access protocol based ieee standard industrial automation process control main idea tsch combine benefits time slotted access multichannel channel hopping capabilities time slotted access increases network throughput scheduling links meet traffic demands nodes multichannel allows nodes exchange packets time using different channel offsets since tsch based scheduling tdma slot fdma delay deterministically bounded depending timefrequency pattern furthermore packet based frequency hopping supported achieve high robustness interference channel impairments tsch also supports various network topologies including star tree mesh tsch mode exhibits many similarities wirelesshart including slotted access multichannel communication frequency hopping mesh networks fact defines details mac operation respect wirelesshart tsch mode nodes synchronize periodic slotframe consisting number time slots node obtains synchronization channel hopping time slot slotframe information enhanced beacons ebs periodically sent nodes order advertise network slots may dedicated one link shared among links dedicated link defined pairwise assignment directed communication nodes given time slot given channel offset hence link communicating nodes represented pair specifying time slot slotframe channel offset used nodes time slot however tsch standard specify derive appropriate link schedule since collisions may occur shared slots exponential backoff algorithm used retransmit packet case transmission failure avoid repeated collisions differently original ieee algorithm backoff mechanism activated collision experienced rather waiting random backoff time transmission deterministic synchronous multichannel extension dsme designed support stringent timeliness reliability requirements factory automation home automation smart metering smart buildings patient monitoring dsme extends beacon enabled mode ieee standard relying superframe structure consisting caps cfps increasing number gts time slots frequency channels used channel access dsme relies specific structure called consists collection superframes defined ieee beacon transmission interval multiple number without inactive period adopting structure dsme tries support periodic aperiodic traffic even large multihop networks dsme network coordinators periodically transmit used keep nodes synchronized allow new nodes join network distributed beacon gts scheduling algorithms dsme allow quickly react traffic changes network topology specifically dsme allows establish dedicated links two nodes network multihop mesh networks deterministic delay dsme scalable suffer single point failure beacon scheduling slot allocation performed distributed manner major difference tsch relies central entity given large variety options features dsme turns one complex modes ieee standard due major complexity issue dsme still lacks complete implementation moreover current studies dsme limited networks investigate potentialities mesh topologies low latency deterministic network lldn designed low latency applications industrial automation large number devices sense actuate factory production specific location differently tsch dsme lldn designed star topologies number nodes need periodically send data central sink using one channel frequency specifically design target lldn support data transmissions sensor nodes every since former ieee standard fulfill constraint lldn mode defines fine granular deterministic tdma access similarly ieee lldn device obtain exclusive access time slot superframe send data pan coordinator number time slots superframe determines many nodes access channel many nodes need send packets pan coordinator needs equip multiple transceivers allow simultaneous communications different channels lldn short mac frames mac header used accelerate frame processing reduce transmission time moreover node omit address fields header since packets destined pan coordinator compared tsch lldn nodes need wait beginning time slot order start transmitting moreover lldn provides group ack feature hence time slots much shorter one tsch since necessary accommodate waiting times ack frames provides compaction fragmentation mechanism efficiently transport packets ieee frames header compressed removal fields needed always contents inferring addresses link layer addresses moreover fragmentation rules defined multiple ieee frames form one packet allows devices communicate using rpl rpl routing protocol lossy networks llns proposed meet delay reliability high availability requirements critical applications industrial environmental monitoring rpl distance vector source routing protocol operate top link layer mechanism including ieee phy mac rpl adopts destination oriented directed acyclic graphs dodags popular destination nodes act roots directed acyclic graphs directed acyclic graphs structures allow nodes associate multiple parent nodes selection stable set parents node based objective function objective function determines translation routing metrics delay link quality connectivity ranks rank defined integer strictly decreasing downlink direction root rpl left routing metric open implementation integrates upper stack including rpl ieee tsch link layer integration allows achieving industrial performance terms reliability power consumption providing upper stack operation sublayer used manage tsch schedule allocating deallocating resources within schedule monitor performance collect statistics uses either centralized distributed scheduling centralized scheduling entity network collects topology traffic requirements nodes network computes schedule sends schedule nodes network distributed scheduling nodes communicate compute schedule based local topology information labels scheduled cells either hard soft depending dynamic reallocation capability hard cell scheduled centralized entity moved deleted inside tsch schedule entity maintains statistics network performance scheduled cells information used centralized scheduling entity update schedule needed moreover information used objective function rpl hand soft cell typically scheduled distributed scheduling entity cell performs significantly worse cells scheduled neighbor reallocated providing interference avoidance mechanism network distributed scheduling policy called scheduling specifies structure interfaces scheduling outgoing packet queue node fills scheduling negotiates additional time slots corresponding neighbors queue empty negotiates removal time slots ieee basic mac layer uses distributed coordination function dcf simple flexible exponential backoff based optional medium sharing medium sensed idle transmitting node transmits frame otherwise postpones transmission medium sensed free time interval equal sum arbitration interframe spacing aifs random backoff interval dcf experiences random unpredictable backoff delay result periodic ncs packets may miss deadlines due long backoff delay particularly congested network conditions enforce timeliness behavior wlans original mac defines another coordination function called point coordination function pcf available infrastructure mode nodes connected network access point aps send beacon frames regular intervals beacon frames pcf defines two periods contention free period cfp contention period dcf used cfp sends packets give right send packet hence node opportunity transmit frames cfp pcf data exchange based periodically repeated cycle superframe within time slots defined exclusively assigned nodes transmission pcf provide differentiation traffic types thus fulfill deadline requirements control systems furthermore mode optional widely implemented wlan devices ieee extension basic dcf mechanism enhances dcf pcf using new coordination function called hybrid coordination function hcf similar defined legacy mac two methods channel accesses namely enhanced distributed channel access edca hcf controlled channel access hcca within hcf edca hcca define traffic categories support various qos requirements ieee edca provides differentiated access individual traffic known access categories acs mac layer node high priority traffic basically waits little less sends packet node low priority traffic accomplished variation using shorter aifs contention window range higher priority packets considering requirements ncss periodic ncs traffic defined high priority saturation must avoided high priority acs hcca extends pcf supporting parametric traffic comes close actual transmission scheduling pcf hcca enable access support collisionfree transmissions contrast pcf hcca allows cfps initiated almost anytime support qos differentiation coordinator drives data exchanges runtime according specific rules depending qos traffic demands although hcca quite appealing like pcf hcca also widely implemented network equipment hence researches adapt dcf edca mechanisms practical control applications wireless network parameters fulfill control system requirements bandwidth wireless networks needs allocated high priority data sensing actuating specific deadline requirements however existing wireless standards explicitly consider deadline requirements thus lead unpredictable performance wncs wireless network parameters determine probability distribution critical interactive system variables design parameters different layers transmission power rate nodes decoding capability receiver physical layer protocol channel access energy saving mechanism mac layer protocol packet forwarding routing layer physical layer physical layer parameters determine values critical interactive system variables transmit power rate network nodes decoding capability receiver depends ratio sinr receiver sinr criteria sinr obviously ratio signal power total power noise interference sinr criteria determined transmission rate decoding capability receiver increase transmit power transmitter increases sinr receiver however increase transmit power neighboring nodes causes decrease sinr due increase interference optimizing transmit power neighboring nodes therefore critical achieving sinr requirements receivers transmit rate determines sinr threshold receivers transmit rate increases required sinr threshold increases moreover depending decoding capability receiver may multiple sinr criteria instance successive interference cancellation multiple packets received simultaneously based extraction multiple signals received composite signal successive decoding ieee allows adjustment transmit power rate however wirelesshart use fixed power rate operating suboptimal region medium access control mac protocols fall one three categories access access hybrid access protocols access protocol random access protocols used wncs mostly adopt mechanism ieee values parameters determine probability distribution delay message loss probability energy consumption include minimum maximum value backoff exponent denoted macm inbe macm axbe respectively maximum number backoff stages called macm axcsm abackof similarly ieee corresponding parameters ieee mac include ifs time contention window size number tries sense clean channel retransmission limits due missing acks energy consumption shown mostly dominated constant listening channel therefore various energy conservation mechanisms adopting low operation later proposed low operation nodes periodically cycle sleep listening state corresponding durations sleep time listen time respectively low protocols may synchronous asynchronous synchronous protocols listen sleep time neighboring nodes aligned time however requires extra overhead synchronization exchange schedules asynchronous protocols hand transmitting node sends long preamble multiple short preambles guarantee wakeup receiver node parameters sleep time listen time significantly affect delay message loss probability energy consumption network using larger sleep time reduces energy consumption idle listening receiver increasing energy consumption transmitter due transmission longer preamble moreover increase sleep time significantly degrades performance message delay reliability due high contention medium increasing traffic access protocol protocols based assigning time slots possibly variable length frequency bands subset nodes concurrent transmission since nodes know transmit receive packet put radio sleep mode scheduled activity scheduling algorithms classified two categories fixed priority scheduling dynamic priority scheduling fixed priority scheduling flow assigned fixed priority function periodicity parameters including sampling period delay constraint instance rate monotonic deadline monotonic scheduling flows assigned priorities function sampling periods deadlines respectively shorter sampling period deadline higher priority fixed priority scheduling algorithms preferred due simplicity lower scheduling overhead typically since take urgency transmissions account hand dynamic priority scheduling algorithms priority flow changes time depending execution schedule instance earliest deadline first edf scheduling transmission closest deadline given highest priority scheduled next whereas least laxity first algorithm priority assigned based slack time defined amount time left transmission transmission started although dynamic priority scheduling algorithms higher scheduling overhead perform much better due dynamic adjustment priorities time hybrid access protocol hybrid protocols aim combine advantages random access protocols random access eliminates overhead scheduling synchronization whereas scheduled access provides message delay reliability guarantees eliminating collisions ieee already provides hybrid architecture flexible usage depending application requirements network routing network layer routing protocol plays extremely important role achieving high reliability forwarding together energy efficiency large scale wncs aircraft avionics industrial automation various routing protocols proposed achieve energy efficiency traditional wsn applications however deal much harsher noisier environments routing protocol must additionally provide reliable transmissions multipath routing extensively studied wireless networks overcoming wireless errors improving routing reliability previous works focus identifying multiple paths guarantee energy efficiency robustness node failures isa wirelesshart employ simple reliable routing mechanism called graph routing enhance network reliability multiple routing paths using graph routing network manager builds multiple graphs flow graph includes device numbers forwarding list unique graph identification based graphs manager generates corresponding subroutes node transmits every node hence nodes path destination graph information specifies neighbors packets may forwarded example link broken node forwards packet another neighbor corresponding flow increasing interest developing new approaches graph routing different routing costs dependent reliability delay energy consumption rpl employs objective function specify selection routes meeting qos requirements applications various routing metrics proposed objective function compute rank value nodes network rank represents virtual coordinate node distance dodag root respect given metric approaches propose usage single metric including link expected transmission count node remaining energy link delay mac based metrics considering packet losses due contention queue utilization proposes two methods control system analysis design sampling hard sampling period unbounded consecutive message dropout bounded consecutive message dropout sampling control control soft sampling period comparison sampling fig subsection structure section namely simple combination lexical combination combining two routing metrics among hop count expected transmission count remaining energy received signal strength indicator simple combination rank node determined using composition function weighted sum ranks two selected metrics lexical combination node selects neighbor lower value first selected metric equal first metric node selects one lower value second composition metric finally combines set metrics order provide configurable routing decision depending application requirements based fuzzy parameters ontrol ystem nalysis esign section provides brief overview analysis design control systems deal critical interactive system variables resulting wireless network presence imperfect wireless network degrades performance control loop even lead instability therefore important understand interactive system variables influence closedloop performance quantitative manner fig illustrates section structure relations control system analysis two main usages requirement definition network design actual control algorithm design first since control cost depends network performance message loss delay explicit set requirements wireless network design determined meet certain control performance allows optimization network design meet given constraints imposed control system instead improving reliability delay energy efficiency second based control system analysis controller designed guarantee control performance imperfect network operation despite interdependence three critical interactive variables sampling period message delay message dropout discussed section much available literature ncs considers subset variables due high complexity problem since practical wireless network incurs imperfect network performance wncs designers must carefully consider performance feasibility tradeoffs previous studies literature analyze stability control systems considering either wireless channel hybrid system markov jump linear system applied modeling control ncs message dropout message delay hybrid switched system approach refers dynamics isolated discrete switching events mathematically components usually described collection indexed differential difference equations ncs control system modelled continuous dynamics network effects message dropouts message delays modelled discrete dynamics compared switched systems markov jump linear system mode switches governed stochastic process statistically independent state values markov systems may provide less conservative requirements switched systems however network performance must support independent transitions states words technique effective network performance statistically independent modelled simple markov model theoretical approaches used derive network requirements function sampling period message dropout message delay network requirements explicitly related message dropout message delay maximum allowable message dropout probability number consecutive message dropouts message delay furthermore since various analytical tools provide sufficient conditions stability requirements might conservative fact many existing results shown conservative simulation studies finding tighter bounds network area great interest highlight importance sampling mechanism classify ncs analysis design methods sampling sampling sampling ncss classified two categories based relationship sampling period message delay hard sampling period soft sampling period message delay hard sampling period smaller sampling period network discards message successfully transmitted within sampling period tries transmit latest sampled message hard sampling period hand node soft sampling period continues transmit outdated messages even sampling period wireless network design must take account sampling method implemented hard sampling period message dropouts ncss generally modelled stochastic variables without limited number consecutive message dropouts hence classify hard sampling period unbounded consecutive message dropout bounded consecutive message dropout unbounded consecutive message dropout controller collocated actuators markov jump linear system used analyze effect message dropout message dropout modelled bernoulli random process dropout probability bernoulli dropout model system model augmented state special case markov jump linear system matrix theory used show exponential stability ncs dropout probability stability condition interpreted linear matrix inequality useful tool design output feedback controller well requirement derivation maximum allowable probability message dropouts network design however main results hard apply wireless network design since ignore message delay fixed sampling period furthermore link reliability wireless networks follow bernoulli random process since wireless links highly correlated time space practice communication considered without delays channels modelled two switches indicating whether corresponding message dropped switched system used model ncs message dropouts message delay sampling period fixed using switched system theory sufficient conditions exponential stability presented terms nonlinear matrix inequalities proposed methods provide explicit relation message dropout rate stability ncs quantitative relation enables design state feedback controller guaranteeing stability ncs certain message dropout rate network may assign fixed time slot single packet associated message guarantee constant message delay however since allow retransmissions significantly degrade message dropout rate another way achieve constant message delay may buffer received packet sink however degrade control performance higher average delay order apply results wireless network needs monitor message dropout probability adapt operation order meet maximum allowable probability message dropouts results used save network resources preserving stability ncs dropping messages certain rate fact ncs research focuses stability analysis design control algorithm rather explicit derivation network requirements useful wireless network design since joint design controller wireless networks necessitates derivation required message dropout probability message delay achieve desired control cost provides formulation control cost function function sampling period message dropout probability message delay ncs researches use linear quadratic cost function control objective model combines stochastic models message dropout message delay furthermore estimator controller obtained extending results optimal stochastic estimator controller given control cost numerical methods used derive set network requirements imposed sampling period message dropout message delay one major drawbacks high computation complexity quantify control cost order find feasible region network requirements bounded consecutive message dropout ncs literatures assume limited number consecutive message dropouts hard requirements unreasonable wireless networks packet loss probability greater zero point time hence approaches set stochastic constraints maximum allowable number consecutive message dropouts control theory provides deterministic bounds maximum allowable number consecutive message dropouts switched linear system used model ncss constant message delay arbitrary finite message dropout channel message dropout said arbitrary sampling sequence successfully applied actuation arbitrary variable within maximum number consecutive message dropouts based stability criterion switched system linear matrix inequality used analyze sufficient conditions stability maximum allowable bound consecutive message dropouts feedback controllers derived via feasible solution linear matrix inequality characterization stability provided explicit bounds maximum allowable transfer interval mati maximally allowable delay mad derived guarantee control stability ncss considering sampling period message delays message dropouts sampling effect modelled timevarying sampling period receiver mati upper bound transmission interval stability guaranteed network performance exceeds given mati mad stability overall system could guaranteed developed results lead tradeoff curves mati mad tradeoff curves provide effective quantitative information network designer selecting requirements guarantee stability desirable level control performance many control applications wireless industrial automation air transportation systems autonomous vehicular systems set stochastic mati constraint form keeping time interval subsequent state vector reports mati value predefined probability guarantee stability control systems stochastic mati constraint efficient abstraction performance control systems since directly related deadline scheduling network design soft sampling period sometimes reasonable relax strict assumption message delay smaller sampling period works assume eventual successful transmission messages various types deterministic stochastic message delays since packet retransmission corresponding message allowed beyond sampling period one consider packet loss message delay actuating signal updated message delay sampling period delay smaller sampling period delays longer one sampling period may result one none arriving single sampling period makes derivation recursive formulas augmented matrix system harder compared hard sampling period case avoid high computation complexity alternative approach defines slightly different augmented state use stability results switched systems even though stability criterion defines mati mad requirements fundamental limits approach apply wireless networks stability results hold message dropout fixed sampling period constant message delay since augmented matrix consiered function fixed sampling period constant message delay hence mati mad requirements used set fixed sampling period message delay deadline hand ncs uses sampling varying message delay take account message dropout stochastic message delay hence mati mad requirements practical control constraints ones apply wireless network design stochastic optimal controller proposed compensate long message delays channel fixed sampling period stochastic delay assumed bounded known probability density function hence network manager needs provide stochastic delay model analyzing delay measurements ncss assume eventual successful transmission messages approach reasonable mati large enough compared sampling period guarantee eventual successful transmission messages high probability however applicable fast dynamical system small mati requirement explicitly consider message dropouts jointly considers message dropout message delay longer fixed sampling period channel derived stability criteria controller designed mad requirement determined fixed message dropout rate solving set matrix inequalities even though message dropout message delay considered tradeoff performance measures explicitly derived however still possible obtain tradeoff curves using numerical methods network allowed transmit packet associated message within mad network also monitors message dropout rate stability guaranteed message dropout rate lower maximum allowable rate furthermore network may discard outdated messages efficiently utilize network resource long message dropout rate requirement satisfied sampling control reactive since generates sensor measurements control commands plant state deviates certain threshold desired value hand control proactive since computes next sampling actuation instance ahead current time control demonstrated significantly reduce network traffic load motivated advantages systematic design eventbased implementations stabilizing feedback control laws performed control systems consist two elements namely feedback controller computes control command triggering mechanism determines control input updated triggering mechanism directly affects traffic load many proposals triggering rule literature suppose state physical plant available one traditional objectives control maintain condition denotes time instant last control task executed last event time threshold next event time instant defined inf sensor control loop continuously monitors current plant state evaluates triggering condition network traffic generated plant state deviates threshold network design problem particularly challenging wireless network must support randomly generated traffic furthermore eventtriggered control provide high energy efficiency since node must continuously activate sensing part hardware platform control determines next execution time based previously received data triggering rule control basically emulation rule one considers model plant controller compute next triggering time hence predictive sampling based plant models controller rules approach generally conservative approach since based approximate models predicted events explicit allocation network resources based predictions improves performance energy efficiency wireless network however since selftriggered control generate fewer messages message loss message delay might seem critical control comparison sampling one fundamental issues compare performance sampling sampling approaches using various channel access mechanisms fact many control researches show performance improvement since often reduces network utilization however recent works control using random access show control performance limitations case large number control loops considers control system number control loops closed shared communication network research one inspiring works wncs codesign problem control policy network scheduling policy taken account overall target framework minimize sum stationary state variance control loops dirac pulse applied achieve minimum plant state variance control law sampling either depending mac schemes traditional tdma fdma csma schemes intuitively tdma used sampling sampling applied csma based previous work approach also used fdma since sampling minimum event interval performs better one using sampling time interval authors assume mac protocol gains network resource network busy specific delay sensor actuator control command applied plant simulation results show eventtriggered control using csma gives best performance even though main tradeoffs conclusions paper interesting assumptions realistic practice dirac pulse controls unrealistic due capability limit actuators simplicity authors assume contention resolution time csma negligible compared transmission time assumption realistic general wireless channel access schemes ieee ieee furthermore total bandwidth resource fdma assumed scale proportion number plants transmission delay sensor actuator inversely proportional number plants assumptions practical since frequency spectrum limited resource general wireless networks thus studies needed previous works control consider single control loop small number control loops compares control control ncs consisting large number plants pure aloha protocol used control ncss authors show packet losses due collisions drastically reduce performance control packets transmitted whenever control generates event remark instability aloha network well known problem communications turns setup control superior control authors also analyze tradeoff delay loss control slotted aloha show slotted aloha significantly improve control cost state variance respect one pure aloha however timetriggered control still performs better therefore hard generalize performance comparison triggered sampling sampling approaches since really depends network protocol topology vii ireless etwork esign echniques ontrol ystems wireless network control system analysis design standardization sampling wireless network parameters sampling section presents various design optimization techniques wireless networks wncs distinguish interactive design approach joint design approach interactive design approach wireless network parameters tuned satisfy given constraints critical interactive system variables possibly enforced required control system performance joint design approach wireless network control system parameters jointly optimized considering interaction critical system variables fig illustrates section structure related previous sections table summarize characteristics related works table demonstrated whether indications requirements communication control parameters included network design optimization wncs table iii classifies previous design approaches wncs based control communication aspects furthermore table categorizes previous works based wireless standards described section interactive design approach interactive design approach wireless network parameters tuned satisfy given requirements control system interactive design approaches assume control systems sensor samples generated periodically predetermined rates generally assume requirements control systems given form upper bounds message delay message dropout fixed sampling period adoption wireless communication technologies supporting control applications heavily depends ability guarantee bounded service times messages least probabilistic point view aspect particularly important control systems requirement considered much significant performance metrics throughput usually important application areas note performance wireless networks heavily depends message delay message dropout hence mainly discuss mac protocols ieee ieee different analytical techniques provide explicit requirements control systems wireless networks discussed section focus previous research mainly design optimization mac network resource scheduling routing layer limited efforts additionally considering physical layer parameters medium access control research networks classified two groups first group solutions called access includes adaptive mac protocols qos differentiations adapt parameters backoff mechanism retransmissions dependent constraints second group interactive design approach medium access control joint design approach sampling access access access access physical layer extension routing traffic generation control network resource schedule scheduling algorithm robustness enhancement network routing sampling access mixed approach disjoint path routing graph routing controlled flooding vii wireless network design technique control systems routing fig subsection structure section vii related previous sections called access relies contention free scheduling netowrk access random access protocols wncs aim tune parameters mechanism ieee ieee improve delay packet loss probability energy consumption performance adaptive tuning algorithms either adaptation adaptation techniques require network model rather depend local measurements packet delivery characteristics early works ieee propose adaptive algorithms dynamically change value single parameter adaptively determine minimum contention window size denoted macm inbe decrease delay packet loss probability nodes increase overall throughput references extend studies autonomously adjust parameters adapt protocol adapts parameter values goal minimizing energy consumption meeting packet delivery probability based local estimates however adapt tends oscillate two parameters sets results high energy consumption solves oscillation problem triggering adaptation mechanism upon detection change operating conditions furthermore aims optimize parameters based linear decrease depending comparison successfully received packet rate target value minimizing energy consumption parameter optimization mainly use theoretical derivations probability table comparison related works circle plus denotes paper explicitly considers indication column dot denotes paper include indication hence control simulation experiment results include terms sim exp evaluation column mean proposed solution evaluated theoretical analysis simulation realistic experiment respectively requirements loss delay sampling period system parameters communication parameters control cost power rate schedule energy routing control parameters sampling control period algorithm contention scenarios evaluation multihop multihop multihop multihop multihop multihop multihop multihop multihop multihop multihop multihop multihop multihop multihop multihop sim exp sim sim sim sim sim multihop multihop multihop multihop multihop multihop multihop multihop multihop tion delay packet error probability energy consumption markov model per node ieee used capture state node moment time individual markov chains coupled memory introduced fixed duration two slot clear channel assessment proposed markov model used derive analytical formulation throughput energy consumption networks extension work leads derivation reliability delay energy consumption function protocol parameters ieee paper provides analytical models delay reliability energy consumption function parameters considering effects random backoff ieee successful transmissions models used minimize energy consumption given constraints delay reliability hand derives experimental based models using curve fitting techniques validation extensive experiments adaptive algorithm also proposed adjust coefficients models introducing learning phase without explicit information data traffic network topology table iii classification wncs design techniques interactive design access medium access control access physical layer extension network resource schedule scheduling algorithm robustness enhancement disjoint path graph network routing controlled flooding traffic generation control joint design approach sampling sampling table classification wncs design techniques based wireless standards interactive design physical layer contention hybrid schedule wirelesshart routing contention schedule mac parameters considering ieee mac protocol qos differentiation presented soft realtime ncss handles periodic traffic using two specific mechanisms namely backoff mechanism retry limit assignment mechanism backoff algorithm offers bounded backoff delays whereas retry limit assignment mechanism differentiates retry limits periodic traffic terms respective deadline requirements markov chain model established describe proposed mac protocol evaluate performance terms throughput delay reliability critical traffic condition provides experimental measures analysis network better understand statistical distribution delay industrial applications statistical distribution network delay first evaluated experimentally traffic patterns support resemble realistic industrial scenarios varying background traffic experimental results validated means theoretical analysis unsaturated traffic condition quite common condition industrial communication systems performance evaluation shows delays generally bounded traffic industrial wlan light traffic grows higher qos mechanism provided edca used achieve behavior bounded delays selected high priority messages access explicit scheduling transmissions allows meet strict delay reliability constraints nodes giving priority nodes tighter joint design approach sampling sampling straint support soft industrial applications combines number various mechanisms ieee transmission retransmission scheduling seamless channel redundancy basic bandwidth management improve deterministic network performance proposed protocol relies centralized transmission scheduling coordinator according edf strategy furthermore coordinator takes care number retransmissions achieve delay reliability lossy links addition scheduling seamless channel redundancy concurrently transmits copies frame multiple distinct radio channels mechanism appealing systems since improves reliability without affecting timeliness moreover bandwidth manager reallocates unused bandwidth failed data transmission additional attempts data transmissions within deadlines presents design implementation wireless communication protocol called support control systems typically require higher sampling rate tdma data link layer protocol based ieee physical layer provides deterministic timing performance packet delivery since different control applications different communication requirements data delivery provides configurable platform adjust design tradeoffs including sampling rate delay variance reliability middleware proposed uses method top csma assign specific time slots node send traffic pollingbased scheduling using edf policy top mac incorporated feedback mechanism adjust maximum number transmission attempts moreover maximum delay implements communication architecture based standard networking framework rtnet wireless ralink chipset rtnet robust schedule used support strict network scheduling requirements systems performance indicators suchreplacements packet psfrag loss ratio delay experimentally evaluated varying protocol parameters star topology experimental results edf schedule show proper tuning system parameters support fig illustrative example two schedulers robust network performance physical layer extension propose priority optimal assignment scheduling algorithm function sampling ssf periods transmission deadlines provide maximum level edf adaptivity accommodate packet losses timeleast laxity first triggered nodes transmissions nodes adaptivity metric illustrated using following example let assume network consists sensor nodes denoted sensor node packet generation period transmission time sensor respectively packet generation period sensor nodes whereas packet transmission times given respectively figs psfrag replacements show robust schedule time slots uniformly distributed time edf schedule respectively number nodes schedule given fig robust packet losses edf schedule given fig indeed suppose fig comparison maximum delay experienced eventthat data packet sensor first triggered components ssf edf least laxity first optimal scheduling algorithms successfully transmitted fig robust schedule includes enough unallocated intervals retransmission sensor whereas edf schedule furthermore authors present theoretical analysis validation robust scheduler accommodate traffic set experiments experimental analysis shows smaller delay edf schedule shown fig possibility select ieee parameters ensure witness suppose additional packet deterministic behavior applications transmission time generated sensor particular shown good mimo configuration node beginning scheduling frame standard enhances communication reliability triggered packet transmission allocated delay sacrificing network throughput network resource schedule several scheduling algoof robust schedule edf rithms proposed efficiently assign time slot schedule uniform distribution paradigm quantified channel multihop networks order meet strict imizing maximum total active length subframes delay reliability requirements scheduling algorithm scheduling algorithms focus subframe length minimum packet generation period among components total active length meeting common deadline packets generated subframe sum transmission time within sampling period formulates nents allocated subframe proposed smallest period delay minimization packet transmissions shortest subframe first ssf algorithm sensor nodes common access point optimization demonstrated significantly decrease maximum delay problem shown proposed schedulexperienced packet component ing algorithms provide upper bounds packet delivery compared edf schedule shown fig moreover time considering transmission characteristics time diversity form retransmission formulation scheduling algorithms however lost packets included framework proposed take packet losses account introduce novel adaptive framework decreases average number missed procedures provide reliability case packet failures deadlines per unit time defined average number proposes optimal schedule increment strategy based packets successfully transmitted within repetition suitable slot comdelay constraint significantly compared edf schedule mon deadline objective optimization problem since ieee encompasses several enhancements maximize reliability providing endboth phy mac layers wlan analyzes transmission delay guarantees physical network performance indicators service time reliability nodes reorganized logical nodes improved ieee industrial communication systems scheduling flexibility two scheduling algorithms evaluated dedicated scheduling shared scheduling dedicated scheduling packets transmitted scheduled time slots whereas shared scheduling packets share scheduled time slots better reliability proposes faster scheduling algorithm problem introduced algorithm based gradually increasing network model one multiple transmitted packets function given link qualities guarantee reliability scheduling algorithms combined multiple path routing algorithms authors assume bernoulli distribution arrival success packets link moreover consider transmission power rate packet length variable assigning exactly one time slot transmission scheduling algorithms consider variation sampling periods deadlines nodes network fall one two categories fixed priority dynamic priority delay analysis periodic flows sensors actuators wirelesshart network fixed priority scheduling policy performed upper bound delay periodic flows obtained mapping scheduling scheduling exploiting response time analysis scheduling channel contention transmission conflict delay due higher priority flows considered channel contention happens channels assigned higher priority flows transmission slot whereas transmission conflict occurs exists common node transmission higher priority flow study later extended reliable graph routing handle transmission failures retransmissions route diversity similarly probabilisitic delay bounds derived considering channel contention transmission conflicts analyses consider multihop multichannel networks fixed time slots without incorporating transmit power rate adjustment mechanism dynamic priority scheduling periodic flows wirelesshart network shown upon determining necessary condition schedulability optimal scheduling proposed effectively discarding infeasible branches search space moreover faster heuristic least laxity first algorithm developed assigning priorities nodes based criticality transmission laxity defined laxity discarding time slots wasted waiting avoid transmission conflicts lower laxity higher transmission criticality algorithm provide guarantee timely packet delivery provides delay analysis periodic realtime flows sensors actuators edf policy delay bounded considering channel contention transmission conflict delays edf shown outperform fixed priority scheduling terms performance robustness enhancement predetermined nature transmissions allows incorporation ious retransmission mechanisms case packet losses random time instants although explicit scheduling used prevent various types conflict contention still transmission failures may occur due multipath fading external interference harsh unstable environments retransmission mechanisms introduced link layer since schedule known apriori nodes network retransmissions minimized exploiting determinism packet headers recover unknown bytes header moreover various efficient retransmission procedures used minimize number bits retransmissions uses symbol decoding confidence whereas uses received signal strength variations determine parts packet received error retransmitted retransmission mechanisms network layer aim determine best timing quantity shared separate time slots given link quality statistics combines retransmissions realtime scheduling analysis number possible retransmissions packet limited considering corresponding deadline already guaranteed delay bounds packets proposes scheduling algorithm provides delay guarantees periodic flows considering link bursts interference new metric called maximum burst length defined maximum length error burst estimated using empirical data algorithm provides reliability guarantee allocating link one plus corresponding maximum burst length time slots novel algorithm used conjunction scheduling algorithm minimize sum worst case burst lengths links route similarly increases spacing actual transmission first retransmission maximum reliability instead allocating time slots improves retransmission efficiency using limited number shared slots efficiently fast slot competition segmented slot assignment shared resources allocated retransmission due unpredictability fast slot competition introduced embedding one clear channel assessment beginning shared slots reduce rate collision hand segmented slot assignment provides retransmission chances routing hop following hop arrives network routing increasing interest developing efficient multipath routing improve network reliability energy efficiency wireless networks previous works multipath routings classified four categories based underlying key ideas routing metric operation disjoint path routing graph routing controlled flooding routing disjoint path routing previous works focus identifying multiple disjoint paths source destination guarantee routing reliability node link failures since multiple paths may fail independently disjoint paths two types paths relay node common paths common link may common nodes provides braided multipath schemes provide resilience node failures multipath distance vector aomdv multipath extension single path routing protocol distance vector aodv graph routing graph routing isa wirelesshart leads significant improvement single path terms reliability due usage multiple paths since standards explicitly define mechanism build multiple paths possible use existing algorithms disjoint path multiple routing paths node destination formed generating subgraphs containing shortest paths source destination pair link quality estimation integrated generation subgraphs better reliability proposes algorithm construct three types reliable routing graphs namely uplink graph downlink graph broadcast graph different communication purposes uplink graph graph connects nodes upward gateway downlink graph gateway graph send unicast messages node network broadcast graph connects gateway nodes network transmission operational control commands three algorithms proposed build graphs based concepts minimum required number incoming outgoing edges nodes excluding gateway respectively communication schedule constructed based traffic load requirements hop sequence routing paths recently graph routing problem formulated optimization problem objective function maximize network lifetime namely time interval first node exhausts battery given connectivity graph battery capacity nodes optimization problem shown suboptimal algorithm based integer programming greedy heuristic algorithm proposed optimization problem proposed algorithm shows significant improvement network lifetime guaranteeing high reliability graph routing controlled flooding previous approaches disjoint routing graph routing focus build routing paths distribute traffic load network control applications may define stringent requirements routing reliability harsher noisier environments address major reliability issue reliable routing protocol realflow proposed industrial applications realflow controls flooding mechanism improve multipath diversity reducing overhead node transmits received packet corresponding multiple routing paths instead feasible outgoing links furthermore discards duplicated packets outdated packets reduce overhead uplink downlink transmissions packets forwarded according related node lists relay nodes due redundant paths flooding mechanism realflow tolerant network topology changes furthermore since related node lists distributively generated workloads gateway greatly reduced flooding schedule also extended using received signal strength routing even though multipath routings disjoint path graph routing controlled flooding lead significant reliability improvement also increase cost energy consumption routing jointly considers application requirements energy consumption network several energybalanced routing strategies proposed maximize network lifetime meeting strict requirements industrial applications breath proposed ensure desired packet delivery delay probabilities minimizing energy consumption network protocol based randomized routing mac jointly optimized energy efficiency design approach relies constrained optimization problem whereby objective function energy consumption constraints packet reliability delay optimal working point protocol achieved simple algorithm adapts traffic variations channel conditions negligible overhead earq another energy aware routing protocol reliable communications industrial applications earq proactive routing protocol maintains ongoing routing table updated exchange beacon messages among neighboring nodes beacon message contains expected values energy cost residual energy node reliability message delay node gets new path destination broadcast beacon message neighbors node wants send packet destination next hop selections based estimations energy consumption reliability deadlines packet chooses path low reliability source forward redundant packet via paths proposes minimum transmission power cooperative routing algorithm reducing energy consumption single route guaranteeing certain throughput however algorithm ignores residual energy communication load neighboring nodes result unbalanced energy consumption among nodes addition loadbalanced routing algorithm proposed node always chooses based communication load neighboring nodes however algorithm heavy computation complexity communication load high propose routing protocol aiming enhancing performance energy efficiency routing decision based integration velocity information neighbors energy balancing mechanism whereas routing decision based number hops source destination information velocity joint design approach joint design approach wireless network control system parameters jointly optimized considering control cost replacements throughput throughput jreq sampling period fig quadratic control cost control systems throughput wireless networks different sampling periods refer control cost bound using ideal network realistic network respectively tradeoff performances parameters include sampling period control level crossings control control system transmission power rate physical layer access parameters algorithm mac protocol parameters routing paths communication system high complexity problem led different abstractions control communication systems many considering subset parameters sampling joint design approaches control classified three categories based communication layers access access routing traffic generation control access usage protocols joint optimization control communication systems requires modeling probabilistic distribution delay packet loss probability wireless network effect control system general framework optimization sampling period together link layer parameters first proposed objective optimization problem maximize control system performance given delay distribution packet error probability constraints linear quadratic cost function used control performance measure simplified models packet loss delay assumed random access mechanism without considering spatial reuse solution strategy based iterative numerical method due complexity control cost used objective function optimization problem aims minimize error state estimation subject delay packet loss probability induced random access error estimator derived function sampling period delay distribution bernoulli random process packet losses discusses several fundamental tradeoffs wncs ieee networks fig shows quadratic control cost communication throughput different sampling periods figure refer control cost bound using ideal network packet loss delay realistic lossy network ieee respectively due absence packet delays losses control performance using ideal network increases monotonically sampling period increases however using realistic network shorter sampling period minimize control cost higher packet loss probability delay traffic load high addition two curves control cost coincide longer sampling periods meaning sampling period larger sampling period dominant factor control cost compared packet loss probability delay fig consider desired maximum control cost jreq greater minimum value control cost feasible range sampling periods however performance wireless network still heavily affected operating point sampling period let consider two feasible sampling periods choosing throughput network stabilized control cost also stabilized respect small perturbations network operation furthermore longer sampling period leads lower network energy consumption one shorter sampling period based observations adaptation wncs proposed considering constrained optimization problem objective minimize total energy consumption network subject desired control cost variables problem include sampling period mac parameters ieee network manager predicts energy consumption corresponding feasible network requirement optimal network requirements obtained minimize energy consumption network feasible set network requirements proposes interesting approach design wncs decomposing overall concerns two design spaces control layer passive control structure used guarantee stability ncss overall ncs performance optimized adjusting retransmission limits ieee standard control layer authors leverage architecture handle message delay message loss authors consider passive controller produces trajectory plant track define control performance absolute tracking error extensive simulation results convex relationship retransmission limit ieee control performance shown based observation mac parameter controller introduced dynamically adjust retransmission limit track optimal tradeoff packet losses delays thus optimize overall control system performance simulation results show mac adaptation converge proper retransmission limit optimizes performance control system even though proposed approach interesting fundamental tradeoff relationships communication parameters control performance trivial derive practice presents ncs implementation wireless relay networks ieee cooperative mac protocol proposed approach deals problem control perspective basically employs mpc actuator state adaptive ieee mac reduce unbounded packet delay improve tolerance packet loss furthermore cooperative mac protocol used improve control performance enabling reliable timely data transmission harsh wireless channel conditions access novel framework joint optimization proposed encompassing efficient abstraction control system form stochastic mati mad constraints remember mati mad defined maximum allowed time interval subsequent state vector reports maximum allowed packet delay transmission respectively discussed section since hard guarantees satisfied wireless network packet loss probability stochastic mati introduced goal keeping time interval subsequent state vector reports mati value predefined probability guarantee stability control systems novel schedulability constraint form forcing adaptive upper bound sum utilization nodes defined ratio delay sampling periods included guarantee schedulability transmission variable transmission rate sampling period values objective optimization minimize total energy consumption network guaranteeing mati mad requirements control system maximum transmit power schedulability constraints wireless communication system solution specific case quadrature amplitude modulation edf scheduling based reduction resulting programming problem integer programming problem based analysis optimality conditions relaxation reduced problem formulation also extended nondecreasing function power consumption nodes objective modulation scheme scheduling algorithm first exact solution method based analysis optimality conditions smart enumeration techniques introduced two polynomialtime heuristic algorithms adopting intelligent search space reduction smart searching techniques proposed energy saving demonstrated increase network containing nodes studies utility maximization problem subject wireless network capacity delay requirement control system utility function defined ratio system counterpart utility function demonstrated strictly concave function sampling period inversely proportional tracking error induced discretization based assumption plants follow reference trajectories provided controllers wireless network capacity derived adopting slotted time transmission conflict graph vertex represents wireless link edge two vertices corresponding links interfere sampling period used multihop delay bound solution methodology based approach inner loop relaxed problem fixed delay bound independent sampling period solved via dual decomposition outer loop determines optimal delay bounds based sampling period output inner loop proposes mathematical framework modeling analyzing multihop ncss authors present formal syntax semantics dynamics composed system providing explicit translation multihop control networks switched systems proposed method jointly considers control system network topology routing resource scheduling communication error formal models applied analyze robustness ncss data packet exchanged multihop communication network subject disruptions authors consider two communication models namely permanent error model transient error model dependent length communication disruptions authors address robustness multihop ncs case worst case analysis scheduling routing packet losses stochastic case stability analysis node fault probability packet loss probability joint optimization sampling period sensors packet forwarding policy control law computing actuator command addressed multihop wirelesshart network objective optimization problem minimize control cost subject energy delay constraints nodes linear quadratic cost function used control cost similar one solution methodology based separation joint design problem fixed sampling rate transmission scheduling maximizing deadlineconstrained reliability subject total energy budget optimal control packet loss optimal solution transmission scheduling based dynamic programming allows nodes find optimal forwarding policy based statistics outgoing links distributed fashion bounds control loss function derived optimal kalman filter estimator static linear feedback control law joint optimal solution found search sampling period recent researches wncs investigate fault detection fault tolerant issues develops design framework ncss industrial automation applications framework relies integrated design parametrization tdma mac protocols controller fault diagnosis algorithms multilayer system main objective determine data transmission wireless networks reduce traffic load meeting requirements control fault detection identification performance considering distributed control groups hierarchical wncs configuration considered lower layer tightly integrates sensors actuators microprocessors local feedback control loops tdma resource higher layer implements control context resource management tdma mac protocol modeled scheduler whose design parameterization achieved development control fault detection identification algorithms different functional layers similar way investigates fault estimation problem based deterministic model tdma mechanism discrete periodic model control systems integrated periodic information scheduling model without packet collisions adopting linearity state equations fault estimator proposed periodic system model arbitrary sensor inputs fault estimation obtained solving deterministic quadratic minimization problem control systems means recursive calculation however scheduler wireless network consider realistic message delays losses routing traffic generation control optimized control cloc protocol proposed minimizing performance loss multiple control systems cloc designed general wireless sensor actuator network connections multihop mesh network design approach relies constrained maxmin optimization problem objective maximize minimum resource redundancy network constraints stability control systems schedulability communication resources stability condition control system formulated form stochastic mati constraint optimal operation point protocol automatically set terms sampling period slot scheduling routing achieved solving linear programming problem adapts system requirements link conditions performance analysis shows cloc ensures control stability fulfills communication constraints maximizing system performance presents case study wireless process control system integrates control design wireless routing wirelesshart standard network supports two routing strategies namely source routing graph routing remind graph routing wirelesshart standard reduces packet loss path diversity cost additional overhead energy consumption mitigate effect packet loss wncs control design integrates observer based extended kalman filter mpc actuator buffer recent control inputs experimental results show sensing actuation different levels robustness packet loss design approach specifically plant state observer highly effective mitigating effects packet loss sensors controller control performance sensitive packet loss controller actuators despite buffered control inputs based observation paper proposes asymmetric routing configuration sensing actuation source routing sensing graph routing actuation improve control performance addresses sampling period optimization goal minimizing overall control cost ensuring delay constraints multihop wirelesshart network linear quadratic cost function used control performance measure function sampling period optimization problem relies multihop problem formulation delay bound due difficulty resulting optimization problem solution methodologies based subgradient method simulated penalty method greedy heuristic method approximated convex optimization method proposed tradeoff execution time achieved control cost analyzed methods sampling communication system design sampling mostly focused mac layer particular researches focus contentionbased random access since suitable control systems due unpredictability message generation time access tradeoff level threshold crossings control system packet losses communication system analyzed studies eventtriggered control lossy communication information generated sent level crossings plant output packet losses assumed bernoulli distribution independent link dependence stochastic control criterion level crossings message loss probability derived class integrator plants allows generation design guideline assignment levels optimal usage communication resources provides extension considering markov chain model attempted successful transmissions lossy channel particular algorithm used transmit control command controller actuator combining communication model retransmissions analytical model performance theoretical framework proposed analyze tradeoff communication cost control performance used adapt event threshold however proposed markov chain considers packet loss bernoulli process capture contention multiple nodes hand access nodes assigned fixed time slots independent message generation times considered alternative random access control however introduces extra delay triggering event transmission assigned slot analyzes ncs consisting multiple linear control systems multichannel slotted aloha protocol multichannel slotted aloha system considered random access model long term evolution authors separate resource allocation problem multichannel slotted aloha system two problems namely transmission attempt problem channel selection problem given time slot control loop decides locally whether attempt transmission based error thresholds local algorithm used adapt error thresholds based knowledge network resource control loop decides transmit selects one available channels uniform random fashion given plant controller dynamics proposes random access policies address coupling control loops shared wireless channel particular authors derive sufficient mathematical condition random access policy sensor violate stability criterion control loops authors assume packet loss due interference simultaneous transmissions network propose mathematical condition decoupling control loops based condition random access policy proposed adapting physical plant states measured sensors online however still computationally challenging verify condition sampling appproaches use csma protocol share network resource analyzes performance ncss csma protocol access shared network authors present markov model captures joint interactions policy contention resolution mechanism csma proposed markov model basically extends bianchi analysis ieee decoupling interactions multiple systems network investigates data scheduling multiple loop control systems communicating shared lossy network proposed scheduling scheme combines deterministic probabilistic approaches scheduling policy deterministically blocks transmission requests lower errors exceeding predefined thresholds subsequently medium access granted remaining transmission requests probabilistic manner message error modeled homogeneous markov chain analytical uniform performance bounds error variance derived proposed scheduling policy numerical results show performance improvement terms error level respect one periodic random scheduling policies proposes distributed adaptation algorithm control system system adjusts communication parameter control gain meet global control cost stochastic linear system coupled csma model allows close limited number feedback loops every time instant backoff intervals csma assumed exponentially distributed homogeneous backoff exponents furthermore data packets discarded limited number retransmission trials individual cost function defined linear quadratic cost function design objective find optimal control laws optimal eventtriggering threshold minimize control cost design problem formulated average cost markov decision process mdp problem unknown global system eters estimated execution techniques distributed optimization adaptive mdps used develop distributed adapt request rate accommodate global resource constraint particular dual price mechanism forces system adjust thresholds according total transmission rate control mixed approach sampling allows save energy consumption reduce contention delay predicting level crossings future explicitly scheduling corresponding transmissions sensor nodes set sleep mode predicted level crossing proposes new approach ensure stability controlled processes shared ieee network control selftriggered sampler selects next sampling time function current previous measurements measurement time delay estimated disturbance superframe duration transmission scheduling contention free period ieee adapted minimize energy consumption meeting deadlines joint selection sampling time processes protocol parameters scheduling allows address tradeoff system performance network energy consumption however drawback sampling methodology lack robustness uncertainties disturbances due predetermined control communication models explicit scheduling sampling therefore recently extended include additional time slots communication schedule assigned apriori nodes case presence disturbance extra slots used fashion random access used slots due unpredictability transmissions joint optimization framework presented objective function process state cost actuations energy consumption transmit control commands subject communication constraints limited capabilities actuators control requirements control adopted controller dynamically determining next task execution time actuator including command broadcasting changing action sensors assumed perform sampling periodically simulated annealing based algorithm used online optimization optimizes sampling intervals addition authors propose mechanism estimating predicting system states may known exactly due packet losses measurement noise proposes joint design approach control adaptive sampling multiple control loops proposed method computes optimal control signal applied well optimal time wait taking next sample basic idea combine concept sampling mpc cost function penalizes plant state control effort well time interval next sample taken latter considered generate adaptive sampling scheme overall system sampling time increases system state error goes zero multiple loop case authors also present transmission scheduling algorithm avoid conflicts proposes mixed sampling eventtriggered sampling scheme ensure control stability ncss improving energy efficiency ieee wireless networks basic idea mixed approach combine sampling sampling schemes sampling scheme first predicts next activation time eventtriggered sampler controller receives sensing information sampler begins monitor predefined triggering condition computes next sampling instance compared typical sampling sensor continuously check eventtriggered condition since sampling component proposed mixed scheme estimates next sampling priori furthermore compared alone utilization sampling conservativeness reduced since sampling component extends sampling interval coupling sampling unified framework proposed scheme extends inactive period wireless network reduces conservativeness induced selftriggered sampling guarantee high preserving desired control performance viii xperimental estbeds contrast previous surveys wsn testbeds introduce representative wncs testbeds existing wncs research often relies experiments however usually suffers limited size capture delays losses realistic large wireless networks several simulation tools developed investigate ncs research unfortunately simulation tools control systems often lack realistic models wireless networks exhibit complex stochastic behavior environments section describe three wncs testbeds namely simulator wsn testbed building automation testbed industrial process testbed fig wsn testbed bryan hall jolley hall washington university louis fed tossim delays drops packets sends outputs actuators furthermore also possible use experimental wireless traces wsn testbed inputs tossim simulator laboratory washington university louis developed experimental wsn testbed study evaluate wsn protocols system comprises network manager server network protocol stack implementation tinyos telosb nodes node equipped microcontroller radio compatible ieee standard fig shows deployment nodes campus building testbed consists nodes placed throughout several office areas testbed architecture hierarchical nature consisting three different levels deployment sensor nodes microservers desktop class machine lowest tier sensor nodes placed throughout physical environment order take sensor readings perform actuation connected microservers second tier usb infrastructure consisting usb compliant hubs messages exchanged sensor nodes microservers interface directions testbed two nodes connected microserver typically one microserver per room final tier includes dedicated server connects microservers ethernet backbone server machine used host among things database containing information different sensor nodes microservers connected simulator wsn testbed wireless simulator wcps designed provide realistic simulation wncs wcps employs federated architecture integrates simulink simulating physical system dynamics controllers tossim simulating wireless networks simulink commonly used control engineers design study control systems tossim widely used sensor network community simulate wsns based realistic wireless link models wcps provides opensource middleware orchestrate simulations simulink tossim following software architecture wcps sensor data generated simulink fed wsn simulated using tossim tossim returns packet delays losses according behavior network fed controller simulink controller commands building automation testbed heating ventilation air conditioning hvac systems guarantee indoor air quality thermal comfort levels buildings price high energy consumption reduce energy required hvac systems researchers trying efficiently use thermal storage capacities buildings proposing advanced estimation control schemes using wireless sensor nodes example hvac testbed currently comprised second floor electrical engineering building kth campus depicted fig floor houses four laboratories office room lecture hall one storage room boiler room room testbed considered thermal zone set wireless sensors actuators individually controlled wsn testbed implemented wireless actuator wireless sensors lower tank pump tap upper tank fig hvac testbed second floor kth five rooms considered contain sensors actuators used hvac control additional sensors located corridor outside building fig coupled tank system setup diagram code integrated application mathscript zone industrial process testbed fig hvac system architecture users able design experiments labview application remotely connect hvac testbed additionally web browser user download experimental data testbed database tinyos telosb nodes testbed consists wireless sensors measuring indoor outdoor temperature humidity concentrations light intensity occupancy levels events like several rooms note nodes equipped humidity temperature light sensors external sensors sensors using converter channel telosb expansion area furthermore laboratory includes people counter measure occupancy laboratory collection tree protocol used collect sensor measurements multihop networks actuators flow valve heating radiator flow valve air conditioning system air vent fresh air flow constant temperature air vent air exhaust corridor overview testbed architecture shown fig hvac testbed developed labview comprised two separate components experimental application server system database responsible logging data hvac components hand experimental application developed user interacts supervisory control module testbed server connects programmable logic controller component allows sensing computation actuation even though application developed labview matlab control liquid levels tanks flows tanks basic problems process industry liquids need processed chemicals mixed treatment tanks levels tanks must controlled flows tanks must regulated fig depicts experimental apparatus diagram physical system used coupled tank system consists pump water basin two tanks uniform cross sections system simple yet representative testbed dynamics water tanks used practice water lower tank flows water basin pump responsible pumping water basin upper tank flows lower tank holes tanks diameter controller regulates level water upper lower tank sensing water levels performed pressure sensors placed tank process control testbed built multiple control systems quanser coupled tanks wireless network consisting telosb nodes control loops regulating two coupled tank processes tanks collocated sensors actuators communicate wirelessly controller node wireless node interfaces sensors converter order sample sensors tanks actuation implemented converter wireless actuator node connected amplification circuit convert output voltage pump motor pen hallenges uture esearch irections although large number results wsn ncss reported literature still number challenging problems solved presented follows tradeoff joint design joint design communication control layers essential guarantee robustness resilience overall wncs several different approaches wncs design categorized dependent degree interaction increasing interaction may improve control performance risk high complexity design problem thus eventually leading fundamental scalability tractability issues hence critical quantify benefit control performance cost complexity depending design approaches benefit adaptation design parameters significantly depends dynamics control systems researches control communication focus design controller network protocol certain optimization problems fixed sampling period ncs researches propose possible alternatives set sampling periods based stability analysis however consider fundamental tradeoff qos sampling period wireless networks adaptive sampling period might provide control performance improvement results complex stability problem control systems requires adaptation wireless networks adaptation sampling period might needed fast dynamical system hand may increase complexity implementation overhead slow control systems hence critical quantify benefit cost joint design approach control communication systems control system requirement various technical approaches hybrid system markov jump linear system system used analyze stability ncss different network assumptions wireless network designers must carefully consider detailed assumptions ncs using results wireless network design similarly control system designers need consider wireless network imperfections encompassing message dropout message delay framework assumptions control system design affect protocol operation assumptions may infeasible meet overall network instance protocol operation consider sampling period check whether allowed retransmit outdated messages sampling period hand ncs design requires strict bound maximum allowable number consecutive packet losses achieved wireless system packet error probability times numerical methods mostly used derive feasible sets wireless network requirements terms message loss probability delay achieve certain control system performance even though feasible requirements meet control cost may give significantly different network costs energy consumption robustness thus eventually affect overall control systems two ways solve problems first one provide efficient tools quantifying feasible sets corresponding network costs previous researches wncs still lack comparison different network requirements effect network design cost second one provide efficient abstractions control communication systems enabling usage methods instance usage stochastic mati mad constraints control system enables generation efficient solution methodologies joint optimization systems communication system abstraction efficient abstractions communication systems need included achieve benefit joint design reducing complexity wncs interactive joint design approaches mostly focus usage constant transmit power rate physical layer simplify problem however variable transmit power rate already supported network devices integration variability time slots variable transmit power rate demonstrated improve communication energy consumption significantly work extended integrate power rate variability wncs design approaches bernouilli distribution commonly used packet loss model analyze control stability simplicity however wireless links highly correlated time space practice time dependence packet loss distribution significantly affect control system performance due effect consecutive packet losses control system performance packet loss dependencies efficiently integrated interactive joint design approaches network lifetime control systems must continuously operate process without interruptions oil refining chemicals power plants avionics continuous operation requires infrequent maintenance semiannual annual since effects downtime losses may range production inefficiency equipment destruction irreparable financial environmental damages hand energy constraints widely regarded fundamental limitation wireless devices limited lifetime due battery constraint particularly challenging wncs attached main physical process equipment fact battery replacement may require maintenance since may possible replace control process operating recently two major technologies energy harvesting wireless power transfer emerged promising technology address lifetime bottlenecks wireless networks solutions also commercially available deployed abb wisa based wireless power transfer industrial automation enocean based energy harvesting building automation wncs using energy efficient technologies encounters new challenges layers network design well overall joint design approach particular joint design approach must balance control cost network lifetime considering additional constraint arrival energy harvesting timing amount energy harvesting may random generation energy natural sources solar vibration controlled inductive magnetic resonant coupling latency communication recently communication latency requirements attracted much interest research community due many control related applications industrial automation autonomous driving healthcare virtual augmented reality particular tactile internet requires extremely low latency combination high availability reliability security network deliver control physical sensing information remotely diversity techniques previously proposed maximize total data rate users adapted achieve reliability corresponding packet error probability order within latency millisecond less latency requirement may prohibit sole usage time diversity form arq transmitter resends packet case packet losses hybrid arq transmitter sends incremental redundancy rather whole packet assuming processing information available receiver therefore investigated usage space diversity form multiple antennas transmitter receiver transmission multiple base stations user cellular networks schemes however mostly focus reliability single user multiple users interference scenario multiple users meet single deadline nodes extended works consider separate packet generation times individual packet transmission deadlines multiple users high reliability communication previous work wncs investigated time path diversity achieve high reliability low latency communication requirements corresponding applications explained detailed time diversity mechanisms either adopt efficient retransmission mechanisms minimize number bits retransmissions link layer determine best timing quantity time slots given link quality statistics hand path diversity based identification multiple disjoint paths source destination guarantee routing reliability node link failures extension techniques include diversity mechanisms space frequency context ultra low latency communication requires reformulation joint design balancing control cost network lifetime addressing new challenges layers network design networks one major issues large scale smart grid smart transportation industry allow communications sensors actuators using levels recently several lpwan protocols lora sigfox ltem proposed provide low data rate communications battery operated devices use licensed spectrum supported generation partnership project standardization hand lora sigfox rely unlicensed spectrum wireless channel behavior lpwans significantly different behavior wireless channel commonly used wncs standards wirelesshart bluetooth due different fading characteristics spectrum usage thus design physical link layers completely different moreover protocol design needs consider effect interoperation different protocols lpwans overall message delay hence control system engineers must validate feasibility traditional assumptions wireless networks wncs based lpwans furthermore network architecture lpwan must carefully adapt operation order support requirements control message priority large scale control systems onclusions wireless networked control systems fundamental technology control systems many areas including automotive electronics avionics building automation industrial automation article provided tutorial reviewed recent advances wireless network design optimization wireless networked control systems discussed critical interactive variables communication control systems including sampling period message delay message dropout energy consumption discussed effect wireless network parameters protocol layers probability distribution interactive variables moreover reviewed analysis design control systems consider effect various subsets interactive variables control system performance considering degree interactions control communication systems discussed two design approaches interactive design joint design also describe practical testbeds wncs finally highlighted major existing research issues identified possible future research directions analysis tradeoff benefit control performance cost complexity joint design efficient abstractions control communication systems usage joint design inclusion energy harvesting diversity techniques joint design extension joint design wireless networked control systems eferences sztipanovits 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mar notes finitely generated flat modules abolfazl tarizadeh abstract article projectivity finitely generated flat module commutative ring studied exterior powers invariant factors consequently related results endo vasconcelos wiegand projectivity flat modules generalized introduction main purpose present article investigate projectivity finitely generated flat modules commutative ring worthy mention main topic many articles literature years still current interest see note general flat modules necessarily projective see example also see tag another example use place finitely generated paper projectivity finitely generated flat module commutative ring studied exterior powers invariant factors important outcome study major results literature projectivity flat modules directly without using homological methods time vastly generalized particular theorem vastly generalizes theorem theorem generalizes theorem theorem generalizes theorem proposition also generalizes proposition corollary commutative case fact theorem viewed generalization mentioned results main motivation investigate projectivity flat modules essentially originates fact every flat module local ring free article also prove general result theorem result particular implies fact mathematics subject classification key words phrases flat module invariant factor projectivity abolfazl tarizadeh see corollary reading present article reasonable knowledge exterior powers module necessary article rings commutative preliminaries lemma let let annr let multiplicative subset anns proof easy ring map canonically isomorphic also projective resp flat natural number projective resp flat finally shall use facts freely throughout article main results lemma annihilator projective module generated idempotent element proof let let annr let ideal generated elements map clearly consider free basis onto map map exists map identity may write finite number indices implies moreover tag may find element let clearly notes finitely generated flat modules remark flat module scalars vectors involved linear relation peculiar properties precisely let consider linear relation let consider map maps get following exact sequence ker morern ker flatness canonically isomorphic therefore exist natural number also elements applying canonical isomor phism maps pure tensor obtain moreover since light remark following result obtained theorem let local ring let flat let subset image canonical map linearly independent linearly independent proof suppose prove assertion shall use induction argument remark elements hypotheses therefore follows let remark elements since abolfazl tarizadeh follows get let note image canonical map linearly independent linearly independent subset module also linearly independent subset arbitrary scalars therefore induction hypothesis also implies corollary let local ring let flat free particular either finitely generated maximal ideal nilpotent free proof every vector space basis let rxi theorem free clearly finitely generated nakayama lemma nilpotent natural number follows immediate consequence corollary obtain following result plays major role article corollary every flat module local ring free first application corollary obtain lemma annihilator flat module idempotent ideal proof let flat module ring let annr let prime ideal lemma annrp corollary free therefore either whole localization zero ideal since contained thus may choose clearly therefore notes finitely generated flat modules invariant factor denoted defined annihilator exterior power therefore annr lemma invariant factors flat module idempotent ideals proof flat well thus lemma idempotent ideal remark let flat corollary leads function spec defined rankrp called rank map also easy see supp spec rankrp theorem let flat following conditions equivalent invariant factors ideals iii rank map locally constant proof projective lemma principal ideal iii suffices show rank map zariski continuous lemma tag exists clearly remark supp spec supp supp moreover supp since therefore open subset spec iii apply corollary tag following result vastly generalizes theorem theorem let extension rings let flat proof first shall prove anna principal ideal let annb claim let abolfazl tarizadeh prime ideal clearly thus corollary either whole localization zero ideal since contained corollary either whole localization zero ideal corollary isomorphic contradiction therefore follows establishes claim lemma idempotent let clearly exists prime ideal thus corollary therefore extension canonical map zero thus exists element hence contradiction therefore follows element let flat moreover since canonically isomorphic thus proved principal ideal hence theorem lemma let flat let ideal let annr annr proof clearly let prime ideal lemma annrp thus corollary either whole localization null ideal primes since annrp recall nonzero free annr hand lemma annrp thus hence following result generalizes theorem theorem let flat let ideal contained radical jacobson proof first shall prove annr principal ideal lemma also lemma notes finitely generated flat modules principal ideal implies since canonically isomorphic let maximal ideal corollary either whole localization zero ideal since contained thus also contained since hence therefore let let isomorphic flat therefore proved annr principal ideal thus invariant factors ideals theorem ring called refers sakhajev every flat theorem let ring map whose kernel contained radical jacobson flat particular well proof clearly flat ker moreover viewed subring via therefore theorem apply theorem finally assume flat flat hypothesis therefore remark let subset ring polynomial ring modulo denoted ideal generated elements form call pointwise localization respect amongst pointwise localization respect namely interesting properties information please instead consult note wiegand utilizes notation clearly canonical map pair satisfies following universal property pair ring map abolfazl tarizadeh exists unique ring map let prime ideal consider canonical map residue field universal property unique ring map thus induces surjection corresponding spectra particular implies kernel contained using following result vastly generalizes theorem corollary let flat exists subset proof immediate consequence theorem proposition let ideal ring annr proof first assume suppose annr thus exists prime annr therefore corollary element contradiction win conversely let injective map prove assertion suffices show induced map given injective may write hypothesis elements annr follows thus therefore final result following give example flat module projective note finding explicit examples flat modules projective easy one may think first example let infinite direct product copies ring let ideal let notes finitely generated flat modules element exists finite subset consider sequences elements clearly annr thus proposition suppose lemma sequence thus exists finite subset clearly pick kronecker delta particular contradiction therefore references cox pendleton rings certain flat modules projective trans amer math soc endo flat modules commutative rings math soc japan aise johan jong stacks project see http jondrup finitely generated flat modules math scand olivier anneaux absolument plats universels buts samuel commutative tomme puninski rothmaler every finitely generated flat module projective journal algebra vasconcelos finitely generated flat modules trans amer math soc wiegand golobalization theorems locally finitely generated modules pacific journal math vol department mathematics faculty basic sciences university maragheh box maragheh iran address

intphys framework benchmark visual intuitive physics reasoning ronan riochet ecole normale superieure inria mario ynocente castro ycmario mar mathieu bernard ecole normale superieure inria adam lerer facebook research rob fergus facebook research alerer robfergus izard paris descartes cnrs emmanuel dupoux coml research university abstract predict interact physical world experimental evidence shows young infants many animals intuitive grasp objects interact world exploit intuitive physics make predictions future outcomes plan actions months infants able parse visual inputs terms permanent solid spatiotemporally continuous objects months understand notion stability support causality months grasp notions gravity inertia conservation momentum collision months shape constancy tacit knowledge intuitive nonverbal opposed formal knowledge taught physics classes follows developmental path parallel early language acquisition occur quickly spontaneously without explicit training caregivers living organisms intuitive physics latent construct observed measured indirectly effects specific tasks like planning problem solving humans verbal descriptions explanations also revealed measurement surprise reactions magic tricks physically impossible events objects disappearing appearing nowhere passing defying gravity review latent nature intuitive physics raises two difficult challenges vision systems evaluation challenge engineering challenge evaluation challenge formulated follows given artificial vision system define measure quantifies much system understands order reach human performance complex visual tasks artificial systems need incorporate significant amount understanding world terms macroscopic objects movements forces etc inspired work intuitive physics infants propose evaluation framework diagnoses much given system understands physics testing whether tell apart well matched videos possible versus impossible events test requires systems compute physical plausibility score entire video free bias test range specific physical reasoning skills describe first release benchmark dataset aimed learning intuitive physics unsupervised way using videos constructed game engine describe two deep neural network baseline systems trained future frame prediction objective tested possible versus impossible discrimination task analysis results compared human data gives novel insights potentials limitations next frame prediction architectures introduction despite impressive progress machine vision many tasks face recognition object recognition object segmentation etc artificial systems still far understanding complex scenes scene understanding involves segmenting tracking objects across time also representing spatial temporal relationships objects able figure popular applications involving scene understanding proposed evaluation method based physical plausibility judgments itive physics one possible answer would measure intuitive physics applications like visual question answering vqa object tracking action planning see figure however runs two risks dataset bias noisy measure first risk also known clever hans problem real life application datasets often contain inherent statistical biases make sometimes possible achieve good performance minimal involvement solving problem hand second risk overall performance system complicated function performance parts therefore vqa system better performance another one could better understands physics better language model propose framework call physical plausibility test directly evaluates intuitive physics fashion framework inspired research infant animal intuitive physics recasts physical reasoning simple classification problem presented video simple scene question whether depicted event physically possible trick preparing matched sets videos physical violation introduces minimal differences frames possible impossible movies varying nature physical violation one probe different types reasoning laws regarding objects properties laws regarding objects movement interactions given method involves videos events could arise spontaneously nature taken diagnostic test way practical method training physical reasoning systems yet advantage applied variety systems engineered trained task long systems minimal requirement compute global scalar number given scene interpret plausibility score system based probability reconstruction error easily derive score engineering challenge formulated follows construct system incorporates much intuitive physics possible least much infants start already discarded use impossible movies train system grounds practicality another approach using supervised learning high level annotations physical entities laws relations etc would also impractical first system could good physical understanding scene without performing full reconstruction second shown infants possible learn intuitive physics without fed high level tag label fact experience positive physical instances physically possible events additionally infants get useful feedback environment become competent motor control although feedback consists possible events one way address challenge would therefore construct unsupervised weakly supervised system learns laws physics using type data available infants abundant amount observational sensory data limited informative environmental feedback positive instances propose intphys intuitive physics benchmark aims getting first stab engineering evaluation challenges consists synthetic videos constructed python interfaced game engine unrealengine enabling realistic physics precise control training set consists positive cases possible movies seen perspective immobile agent probably difficult task one faced infants explore interact environment however interesting establish far one get simplified inputs easy gather abundant amounts real world video cameras addition enables easier comparison models get training data test set constructed according evaluation framework requires system output plausibility score evaluated ability separate possible impossible movies test set also used standalone diagnostic evaluation systems trained ways real videos interactive training virtual environment see structure paper follows section review related work high level vision evaluation models section detail intuitive physics evaluation framework section present first release intphys benchmark addresses basic component intuitive physics namely object permanence section present two baseline systems trained frame prediction cost dataset analyze results compared human performance related work previous work relevant intuitive physics conducted context particular applications distinguish three broad classes applications depending type data use first class includes tasks interface model ability reason image assessed language task generating caption answering question image second class includes tasks use images videos input predicting future events video tracking objects time third class involves interface applications robotics case systems require vision reasoning control actions predict outcome interface going beyond standard object classification tasks recent work visionlanguage interface focused classifying relations objects requires principle understanding underlying physics distinction hanging supporting two tasks currently receiving lot attention captioning vqa scene captioning consists generating sentence describes image matching image one several captions task requires recognize objects image also understand spatial relations interactions objects visual question answering requires provide verbal answer verbal question image alternatively one required rank several potential answers like image captioning task requires understand spatial relations objects addition understand question extract right information answer datasets use videos input instead static images beyond closed set predefined outputs two tasks raise evaluation difficulties shown vqa systems cheat obtain good performance exploiting statistical biases dataset example question covers ground highly correlated images ground statistical learner would perform well question always answering snow whatever image biases make harder understand models weaknesses strengths clevr dataset authors focus testing visual reasoning ability minimizing questionconditional biases provide representations images questions well detailed annotations describing kind reasoning question requires similar spirit aim propose diagnostic tool visual reasoning systems providing systematically constructed tasks minimizing statistical biases proposed task gets rid language altogether directly taps understanding objects interactions high level vision many research projects vision define tasks aim recovering high level structure low level pixel information one example recovery structure static moving images two tasks proposed tackle temporal dynamics objects videos object tracking future prediction classic formulations object tracking focus matching instance labels video frames given video results collection object instances location every time step problem literature challenges contrary learning systems tracking models use priors regarding objects motion assuming instance objects constant speed occluded appearance small changes time intuitive physics forward modelling prediction seen general task object tracking given image stack images around time task produce predicted image time future recent studies investigated models predicting stability forward modeling dynamics towers blocks proposes model based intuitive physics engine follow supervised approach using convolutional neural networks cnns makes comparison models models improves predictions cnn model providing prediction generative model authors propose different feature learning strategies architecture adversarial training method image gradient difference loss function predict future frames raw videos even though next frame prediction tasks could used learned aspects intuitive physics observed regularities object motion current forward models still struggle predict outcomes beyond frames models use structured representation objects derive predictions authors learn objects dynamics modelling pairwise interactions predicting resulting objects states representation position velocity object intrinsic properties authors combine factored latent object representations object centric dynamic models visual encoders frame parsed set object state representations used input dynamic model authors use visual decoder reconstruct future frames allowing model learn raw though synthetic videos interface studies focused interface potential applications robotics main tasks two kinds predicting outcome action visual environment forward modelling predicting optimal action make order reach desired outcome action planning forward modelling studied authors train model predict outcome interactions robot object conditioned input image state action robot model trained predict resulting image action use forward model train robot execute given action object authors construct indoor scenes physics simulator apply forces various objects scenes train deep neural network predict effects forces objects ground truth simulated physics simulator using properties like mass friction gravity solidity train model predict dynamics object static images studies focused learning control visual inputs authors train deep neural networks coordinate robotic vision action specific task object grasping learning visual control policies reinforcement learning investigated either simulation real world systems use prediction model purpose action planning authors train model predict future frames videos using predictions train playing atari games integrating model dynamic external world agent also done plan novel actions running multiple internal simulations finally robots pushing poking objects investigated model predict image directly rather latent representation latent representation forward model learned jointly inverse model predicts move object desired position latent representation keep information object location able manipulate objects predict dynamics billiard game seems require notions like solidity mass collisions causality even though proposed framework testing intuitive physics involves vision integrating vision action training may help learn notions diagnostic test intuitive physics saw great diversity systems applications rely way physical understanding propose single diagnostic test run provided minor modifications systems captioning vqa systems systems performing reconstruction tracking planning etc engineered hand trained using statistical learning main idea draw work developmental comparative psychology infants animals construct well controlled test avoiding potential statistical biases cheap tricks obtain relatively pure tests measuring different types physical reasoning abilities intuitive physics best described latent body knowledge allow organisms predict events plan actions applicable describe body knowledge may incomplete totally coherent used situations due variations attention memory etc take view rudimentary version newtonian physics far deals solid macroscopic objects existing world intrinsic properties mass shape position velocities illustrate diagnostic test object permanence one basic principle intuitive physics states object continue exist even seen present design features test minimal sets parametric task difficulty evaluation metric show extended wide range intuitive physics reasoning problems minimal sets design important design principle evaluation framework relates organization possible impossible movies extremely well matched sets avoid clever hans problem illustrated figure object permanence constructed matched sets comprising four movies contain initial scene time either one two objects final scene time either one two objects separated potential occlusion screen raised lowered variable amount time maximal height screen completely occludes objects impossible know frame many objects behind occluder four movies constructed combining two possible beginnings two possible endings giving rise two possible two impossible movies importantly across movies possible impossible ones made exact frames factor distinguishing temporal coherence frames design intended make difficult algorithms use cheap tricks distinguish possible impossible movies focusing low level details rather requires models focus higher level temporal dependencies frames parametric manipulation task complexity second design principle block vary stimulus complexity parametric fashion hierarchy intuitive physics problems figure illustration minimal sets design object permanence schematic description static condition one two objects one occluder two possible movies green arrows number objects remains constant despite occlusion two impossible movies red arrows number objects changes goes case object permanence block instance stimulus complexity vary according three dimensions first dimension whether change number objects occurs plain view visible hidden behind occluder occluded change plain view evidently easier detect whereas hidden change requires element short term memory order keep trace object time second dimension complexity object motion tracking immobile object easier object complicated motion third dimension number objects involved scene tests attentional capacity system defined number objects track simultaneously manipulating stimulus complexity important establish limit vision system fail instance humans well known fail number objects track simultaneously greater four physical possibility metrics evaluation metrics depend system ability compute plausibility score given movie test movies structured matched figure positive negative movies oski impki derive two different metrics relative error rate computes score within set requires within set positive movies plausible negative movies osji impji absolute error rate requires globally score positive movies greater score negative movies computed osji impji area roc curve plots true positive rate false positive rate various threshold settings explain design principles presented applied study physical reasoning progressively complicated problems taking advantage behavioral work intuitive studies organize tests levels blocks one corresponding core principle intuitive physics raising particular machine vision challenge typology problems presented table organized two levels level problems deal properties movement characteristics single objects level problems involve interactions objects first level define blocks follows first two related conservation time intrinsic properties objects object permanence already discussed corresponds fact objects continuously exist time pop existence turns computational challenge tracking objects occlusion second block shape constancy describes tendancy rigid objects preserve shape time principle challenging preceding one even rigid objects undergo change appearance due factors illumination distance viewpoint partial occlusion three blocks relate object movement time conservation laws govern movements rigid inanimate macroscopic objects principles map progressively challenging problems trajectory prediction regarding interactions objects level also define blocs increasing order complexity first two test basic principles solidity states two objects occupy physical space causality states object interactions occur physical contact course true gravitational electromagnetic forces still principle contact deeply entrenched human perception forces acting distance would limited practical value many applications another reason leave extensions last three blocks correspond modes interactions objects contact elastic collision support containment blocks raise difficult challenges action planning also correctly describing scene event using language causality support containment intphys benchmark present first release intphys benchmark designed address engineering evaluation challenges intuitive physics vision systems first release focused unsupervised learning tests first block hierarchy problems table list conceptual blocks intuitive physics framework block name object permanence shape constancy continuity energy momentum gravity solidity causality physical principles computational challenge objects objects attributes objects pop existence object tracking objects keep shapes object tracking trajectories objects continuous object trajectories constant kinetic energy momentum object trajectories downwards gravitational field predicting objects trajectories relations interactions two objects occupy space predicting objects mechanical interactions require description mass elastic collisions temporal proximity objects keep mass conservation system momentum planning support containment gravity solidity polygon support solidity continuity event description solving shell ject permanence future releases include blocks table benchmark consists three components training set containing physically possible events involving simple inanimate objects moving interacting virtual environment dev test set containing physically possible physically impossible videos carefully matched tuples described evaluation software describe three components well results humans plausibility judgments test set serve reference algorithms modeling human perception training set training set constructed using unreal engine contains large variety objects interacting one another occluders textures etc see figure examples composed videos possible events around seconds totalling hours videos video delivered stacks raw image pixels totalling uncompressed data also release source code data generation allowing users generate larger training set desired even though spirit intphys unsupervised learning intuitive physics provide additional information may help learner first one depth field image unreasonable given infants stereo vision motion cues could provide approximation information also deliver object instance segmentation masks given information probably available infants provide training set test set pretraining purposes figure examples frames training set dev test sets section describes dev test sets block object permanence design dev test sets follow general structure matched sets described section parametric complexity vary number objects presence absence occluder complexity movement static dynamic dynamic static case objects move dynamic case bounce roll left right right left types events one occluder may present scene objects may sometimes pop existence event disappear suddenly impossible events occur behind occluder present full view otherwise dynamic events illustrated supplementary material figure two occluders present existence objects may change twice example one object may present scene first disappear going behind first occluder later reappearing behind ond occluder dynamic events designed prevent systems detecting inconsistencies merely comparing number objects visible beginning end movie matched sets contain four videos two possible events two impossible events total block test set contains types movies objects occlusions types movements dev set instantiated different renderings scenarios objects positions shapes trajectories resulting movies test set instantiated different renderings scenarios total movies uses different objects textures motions etc objects textures dev test sets present training set purpose dev set released intphys help selection appropriate plausibility score comparison various architectures hyperparameters serve train model parameters done training set dev set kept intentionally small test set statistical power enables fine grained evaluation results across different movie subtypes evaluation software movie model issue scalar plausibility score number together movie fed evaluation software outputs two tables results one absolute score relative score evaluation software provided dev set test set evaluating test set participants invited submit system results see results registered website leaderboard human judgments presented videos test set block human participants using amazon mechanical turk experiment human judgements results detailed supplementary section baseline systems section present two baseline systems attempt learn intuitive physics unsupervised setting using possible movies training set two baselines consist training deep neural networks future frame prediction objective based literature next frame prediction propose two neural network models predicting future frame given set current frames first model cnn structure second conditional generative adversarial network gan similar structure dcgan model architectures investigate two different training procedures first train models predict images prediction span frames second predict images prediction span frames preliminary work predictions pixel level revealed models failed predicting convincing object motions especially small objects rich background reason switched computing predictions higher level using object masks use metadata provided benchmark training set train semantic mask deep neural network dnn dnn uses pretrained imagenet extract features image deconvolution network trained predict semantic mask distinguished three types entities background occluders objects use mask input prediction component predicts future masks based past ones evaluate models benchmark system needs output plausibility score movie compute prediction loss along movie given past frames plausibility score frame derived comparing prediction like use analogy agent running internal simulation visual imagination assimilate greater distance prediction observation lower plausibility subsection detail aggregate scores frames plausibility score whole video models movie models take input two frames predict future frame ftarget prediction span independent model architecture depends triplets ftarget provided training phase two architectures trained either short term prediction task frames future long term prediction task frames intuitively prediction robust prediction allow model grasp dependencies deal long occlusions cnn use pretrained imagenet extract features input frames deconvolution network trained predict semantic mask future frame ftarget conditioned features using loss generative adversarial network second baseline propose conditional generative adversarial network gan takes input predicted semantic masks frames predicts semantic mask future frame ftarget setup discriminator distinguish mask predicted ftarget directly real mask predicted past frames like model combines conditional approach similar structure dcgan test time derive plausibility score computing conditioned discriminator score every conditioned frame novel approach based observation optimal discriminator computes score pdata pdata events pdata therefore long events physical events note strong assumption guarantee generator ever support part distribution corresponding impossible videos models architectures well training procedures samples predicted semantic masks found supplementary material tables figure code made available video plausibility score forward models presented compute plausibility score every frame ftarget conditioned previous frames however temporal positions impossible events given must decide score video given scores conditioned frames impossible event characterized presence one impossible frame conditioned previous frames hence natural approach compute video plausibility score take minimum conditioned frames scores plaus min ftarget plaus ftarget video ftarget frame triplets given training phase results prediction first training procedure prediction task takes input frames predicts note following train two architectures presented prediction task evaluate test set relative classification task cnn encoderdecoder error rate impossible events visible occluded gan error rate visible occluded absolute classification task cnn see impossible events visible occluded gan visible occluded results detailed supplementary material tables observe prediction models show good performances impossible events visible especially relative classifications task however perform poorly impossible events occluded easily explained fact prediction span frames usually lower occlusion time hence models enough memory catch occluded impossible events prediction second training procedure consists prediction task relative classification task cnn error rate impossible events visible occluded gan error rate visible occluded absolute classification task cnn impossible events visible occluded gan visible occluded results detailed supplementary material tables expected models perform better shortterm models occluded impossible events moreover results absolute classification task confirm way challenging relative classification task movies complex others average score quadruplet movies may vary lot results cases one model returns higher plausibility score impossible movie imp easy easy quadruplet possible movie pos complex complex quadruplet aggregated model grasp short dependencies aggregate scores longterm models pagg relative classification task cnn error rate impossible events visible occluded gan error rate visible occluded absolute classification task cnn impossible events visible occluded gan visible occluded results detailed supplementary material tables figure discussion defined general framework measuring intuitive physics artificial systems inspired research conceptual development infants framework system asked return plausibility score video sequence showing physical interaction objects system performance assessed measuring ability discriminate possible impossible videos illustrating several types physical principles addition present intphys benchmark designed test unsupervised learning intuitive physics learning positive examples first release benchmark dedicated object permanence provide human performance proof principle baseline systems humans show generally good performance although attentional limitations start appear using occlusion several objects track simultaneously computational system shows possible obtain chance performance using mask prediction task although occlusion presents particularly strong challenge relative success mask prediction system compared expected systems indicates operating abstract level worth pursuing strategy new blocks benchmark released see table prediction task become difficult progressively reach level scene comprehension achieved humans references agrawal batra parikh analyzing behavior visual question answering models arxiv preprint agrawal nair abbeel malik levine learning poke poking experiential learning intuitive physics corr antol agrawal mitchell batra lawrence 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pattern recognition cvpr ieee conference pages ieee pylyshyn storm tracking multiple independent targets evidence parallel tracking mechanism spatial vision radford metz chintala unsupervised representation learning deep convolutional generative adversarial networks corr real shlens mazzocchi pan vanhoucke large data set object detection video arxiv preprint ren kiros zemel exploring models data image question answering advances neural information processing systems pages rezende eslami mohamed battaglia jaderberg heess unsupervised learning structure images advances neural information processing systems pages rolfs dambacher cavanagh visual adaptation perception causality current biology russakovsky deng krause satheesh huang karpathy khosla bernstein berg imagenet large scale visual recognition challenge international journal computer vision saxe carey perception causality infancy acta psychologica saxena sun learning scene structure single still image ieee transactions pattern analysis machine intelligence spelke kestenbaum simons wein spatiotemporal continuity smoothness motion object identity infancy british journal developmental psychology tapaswi zhu stiefelhagen torralba urtasun fidler movieqa understanding stories movies proceedings ieee conference computer vision pattern recognition pages watters tacchetti weber pascanu battaglia zoran visual interaction networks arxiv june wright yang ganesh sastry robust face recognition via sparse representation ieee transactions pattern analysis machine intelligence song khosla zhang tang xiao shapenets deep representation volumetric shapes proceedings ieee conference computer vision pattern recognition pages carey infants metaphysics case numerical identity cognitive psychology young lai hodosh hockenmaier image descriptions visual denotations new similarity metrics semantic inference event descriptions transactions association computational linguistics park berg berg visual madlibs fill blank description generation question answering proceedings ieee international conference computer vision pages zhang goyal batra parikh yin yang balancing answering binary visual questions proceedings ieee conference computer vision pattern recognition pages zhang zhang freeman tenenbaum comparative evaluation approximate probabilistic simulation deep neural networks accounts human physical scene understanding cogsci zitnick parikh bringing semantics focus using visual abstraction proceedings ieee conference computer vision pattern recognition pages supplementary material framework details ties reversed presumably participants started using heuristics checking number objects beginning end therefore missed intermediate disappearance object results suggest human participants responding according gold standard laws physics due limitations attentional capacity even though number objects track theoretical limit objects performance human observers thus serve reference besides ground truth especially systems intended model human perception models training procedure figure illustration dynamic condition two possible movies green arrows number objects remains constant despite occlusion two impossible movies red arrows number objects changes temporarily goes detailed models see tables models architectures figure samples predicted semantic masks code made available https training procedure human judgement experiment presented videos test set block human participants using amazon mechanical turk participants first presented examples possible scenes training set simple complex told test movies incorrect corrupted showed events could possibly take place real world without specifying participants presented randomly selected videos labeled possible impossible completed task minutes paid response counted error possible movie classified impossible vice versa total persons participated data discarded failed respond correctly easiest condition static one object visible mock sample amt test available http physics experiment average error rates computed across condition number objects visibility remaining participant shown table overall error rate rather low general observers missed violations often scene occluded increase error going static dynamic dynamic dynamic pattern consistently observed occluded condition visible scenario dynamic appeared difficult dynamic probably due fact objects visible dynamic impossible scenarios contain two local discontinuities therefore easier spot one discontinuity present discontinuities occurred behind occluder pattern separate training dataset control overfitting forward predictions models trained using adam cnn use adam default parameters stop training one epoch gan use parameters set generator learning rate discriminator learning rate prediction task train gan epoch longterm prediction task train epochs learning rate decays set set generator discriminator detailed baseline results table average error rate plausibility judgments collected humans using mturk intphys block test set datapoint forced zero inclusion criterion visible occluded type scene obj obj obj avg obj obj obj avg static dynamic violation dynamic violations avg figure results baselines cases impossible event occurs open visible behind occluder occluded represents losses see equation relative performance see equation absolute performance table mask predictor parameters stands input frame first layers pretrained frozen weights reshape reshape upsamplingnearest conv relu upsamplingnearest conv relu upsamplingnearest conv relu sigmoid target mask table cnn forward prediction parameters stands input frames first layers pretrained frozen weights applied frame reshape reshape upsamplingnearest conv relu upsamplingnearest conv relu upsamplingnearest conv relu sigmoid target mask table generator parameters sfconv stands spatial full convolution stands batchnormalization input masks conv relu conv relu noise conv relu unif conv relu conv relu stack input noise sfconv relu sfconv relu sfconv relu sfconv relu sfconv relu sigmoid target mask table discriminator parameters stands history input reshape convolution strides leakyrelu convolution strides leakyrelu convolution strides leakyrelu convolution strides leakyrelu convolution strides leakyrelu layer sigmoid figure output examples semantic mask predictor left right input image ground truth semantic mask predicted semantic mask table detailed relative classification scores cnn prediction span visible occluded type scene obj obj obj total obj obj obj total static dynamic violation dynamic violations total table detailed absolute classification scores cnn prediction span visible occluded type scene obj obj obj total obj obj obj total static dynamic violation dynamic violations total table detailed relative classification scores gan prediction span visible occluded type scene obj obj obj total obj obj obj total static dynamic violation dynamic violations total table detailed absolute classification scores gan prediction span visible occluded type scene obj obj obj total obj obj obj total static dynamic violation dynamic violations total table detailed relative classification scores cnn prediction span visible occluded type scene obj obj obj total obj obj obj total static dynamic violation dynamic violations total table detailed absolute classification scores cnn prediction span visible occluded type scene obj obj obj total obj obj obj total static dynamic violation dynamic violations total table detailed relative classification scores gan prediction span visible occluded type scene obj obj obj total obj obj obj total static dynamic violation dynamic violations total table detailed absolute classification scores gan prediction span visible occluded type scene obj obj obj total obj obj obj total static dynamic violation dynamic violations total table detailed relative classification scores aggregation cnn models prediction spans visible occluded type scene obj obj obj total obj obj obj total static dynamic violation dynamic violations total table detailed absolute classification scores aggregation cnn models prediction spans visible occluded type scene obj obj obj total obj obj obj total static dynamic violation dynamic violations total table detailed relative classification scores aggregation gan models prediction spans visible occluded type scene obj obj obj total obj obj obj total static dynamic violation dynamic violations total table detailed absolute classification scores aggregation gan models prediction spans visible occluded type scene obj obj obj total obj obj obj total static dynamic violation dynamic violations total

published conference paper iclr hen earn onvolutional ilter asy simon carnegie mellon university ssdu jason lee university southern california jasonlee yuandong tian facebook research yuandong feb bstract analyze convergence stochastic gradient descent algorithm learning convolutional filter rectified linear unit relu activation function analysis rely specific form input distribution proofs use definition relu contrast previous works restricted standard gaussian input show stochastic gradient descent random initialization learn convolutional filter polynomial time convergence rate depends smoothness input distribution closeness patches best knowledge first recovery guarantee algorithms convolutional filter input distributions theory also justifies learning rate strategy deep neural networks focus theoretical also present experiments justify theoretical findings ntroduction deep convolutional neural networks cnn achieved performance many applications computer vision krizhevsky natural language processing dauphin reinforcement learning applied classic games like silver despite highly nature objective function simple algorithms like stochastic gradient descent variants often train networks successfully hand success convolutional neural network remains elusive optimization perspective input distribution constrained existing results mostly negative hardness learning neural network blum rivest convolutional filter brutzkus globerson recently shamir showed learning simple fully connected neural network hard specific input distributions negative results suggest order explain empirical success sgd learning neural networks stronger assumptions input distribution needed recently line research tian brutzkus globerson yuan soltanolkotabi zhong assumed input distribution standard gaussian showed stochastic gradient descent able recover neural networks relu activation polynomial time one major issue analysis rely specialized analytic properties gaussian distribution section thus generalized case distributions fall general input distributions new techniques needed paper consider simple architecture convolution layer followed relu activation function average pooling formally let input sample image generate patches size column patch generated known function filter size stride pixels since convolutional filters need focus patches instead input following definitions theorems refer input let distribution max relu activation function published conference paper iclr relu label estimate input figure architecture network considering given input extract patches send shared weight vector outputs sent relu summed yield final label estimation two conditions proposed convergence want data highly correlated concentrated direction aligned ground truth vector see figure graphical illustration architectures used first layer many works computer vision lin milletari address realizable case training data generated unknown teacher parameter input distribution consider loss learn stochastic gradient descent step size may change time random function expectation equals population gradient goal analysis understand conditions optimized stochastic gradient descent setup main contributions follows learnability filters show input patches highly correlated section small gradient descent stochastic gradient descent random initialization recovers filter polynomial furthermore strong correlations imply faster convergence best knowledge first recovery guarantee randomly initialized algorithms learning filters even simplest network input distribution answering open problem tian convergence rate formally establish connection smoothness input distribution convergence rate filter weights recovery smoothness paper defined ratio largest least eigenvalues second moment activation region section show smoother input distribution leads faster convergence gaussian distribution special case leads tightest bound theoretical finding also justifies twostage learning rate strategy proposed szegedy step size allowed change time elated orks recent years theorists tried explain success deep learning different perspectives optimization point view optimizing neural network optimization note since paper focus continuous distribution results conflict previous negative results blum rivest brutzkus globerson whose constructions rely discrete distributions published conference paper iclr problem pioneered class optimization problems satisfy strict saddle property optimized perturbed stochastic gradient descent polynomial time jin motivates research studying landscape neural networks soltanolkotabi kawaguchi choromanska hardt haeffele vidal mei freeman bruna safran shamir zhou feng nguyen hein however results directly applied analyzing convergence methods relu activated neural networks learning theory point view well known training neural network hard worst cases blum rivest livni recently shamir showed either niceness target function input distribution alone sufficient optimization algorithms used practice succeed additional assumptions many works tried design algorithms provably learn neural network polynomial time sample complexity goel zhang sedghi anandkumar janzamin gautier goel klivans however algorithms tailored certain architecture explain stochastic gradient based optimization algorithms work well practice focusing algorithms line research analyzed behavior stochastic gradient descent gaussian input distribution tian showed population gradient descent able find true weight vector random initialization model brutzkus globerson showed population gradient descent recovers true weights convolution filter input polynomial time yuan showed sgd recover true weights resnet model relu activation assumption spectral norm true weights bounded small constant methods use explicit formulas gaussian input enable apply trigonometric inequalities derive convergence gaussian assumption soltanolkotabi shows true weights exactly recovered projected gradient descent enough samples linear time number inputs less dimension weights approaches combine tensor approaches assumptions input distribution zhong proved sufficiently good initialization implemented tensor method gradient descent find true weights fully connected neural network however approach works known input distributions soltanolkotabi used gaussian width definition soltanolkotabi concentrations approach directly extended learning convolutional filter paper adopt different approach relies definition relu show long input distribution satisfies weak smoothness assumptions able find true weights sgd polynomial time using conclusions justify effectiveness large amounts data may eliminate saddle points adaptive learning rates used szegedy etc rganization paper organized follows section analyze simplest model state key observation establish connection smoothness convergence rate section discuss performance stochastic gradient descent learning convolutional filter provide empirical illustrations section conclude section place detailed proofs appendix otations let denote euclidean norm vector matrix use denote largest singular value smallest singular value note positive semidefinite matrix represent largest smallest eigenvalues respectively let denote standard notations hide absolute gradient descent guaranteed converge local minima polynomial time lee published conference paper iclr figure four regions considered analysis illustration defined definition assumption constants assume gradient function uniformly bounded exists condition satisfied long patches noise bounded warm nalyzing ayer euron odel diving convolutional filter first analyze special case equivalent architecture analysis simple case give insights fully general case ease presentation define following two events corresponding second moments indicator function intuitively joint activation region joint activation region see figure graphical illustration simple algebra derive population gradient one key observation write inner product sum two terms lemma observation directly leads following theorem theorem suppose initialization satisfies gradient descent algorithm recovers first assumption input distribution one case assumption fails input distribution supported space degenerated second assumption initialization ensure gradient descent converge gradient undefined general convergence theorem holds wide class input distribution initialization points particular includes theorem tian special case input distribution degenerate holes input space gradient descent may stuck around saddle points believe data needed facilitate optimization procedure also consistent empirical evidence data helpful optimization onvergence ate ayer euron odel previous section showed distribution regular weights initialized appropriately gradient descent recovers true weights converges practice also want know many iterations needed characterize convergence rate need quantitative assumptions note different set assumptions lead different rate one possible choice paper use following quantities published conference paper iclr definition eigenvalue values second moment intersection two half spaces define min max two conditions quantitatively characterize angular smoothness input distribution given angle difference large one direction large probability mass one direction small probability mass meaning input distribution smooth hand close directions similar probability mass means input distribution smooth smoothest input distributions rotationally invariant distributions standard gaussian analogy think lipschitz constant gradient strong convexity parameter optimization literature also allow change angle also observe intersection measure monotonically decreasing next assumption growth note intersection measure also grows angle becomes larger following assume operator norm increases smoothly respect angle intuition long input distribution bounded probability density respect angle operator norm bounded show theorem rotational invariant distribution theorem standard gaussian distribution assumption assume exists maxw ready state convergence rate theorem initialization satisfies denote suppose arcsin step size set kwt note increases decreases choose constant step size theorem implies find solution iterations also suggests direct relation smoothness log distribution convergence rate smooth distribution close small relatively small need fewer iterations hand much larger need iterations verify intuition section able choose step sizes adaptively like using proposed lin xiao may improve computational complexity log justifies use learning rate strategy proposed szegedy beginning need choose learning small small later choose large learning rate angle becomes smaller becomes bigger theorem requires initialization satisfying achieved random initialization constant success probability see section detailed discussion esults earning onvolutional ilter section generalize ideas previous section analyze convolutional filter first given define four events divide input space patch event published conference paper iclr corresponds different activation region induced similar please check figure illustration ease presentation also define average patches region next generalize smoothness conditions analogue definition assumption smoothness defined average patches assumption define max min assume maxw main difference simple network convolution filter two patches may appear different regions given sample may exists patch interaction plays important role convergence stochastic gradient descent assume second moment interaction also grows smoothly respect angle assumption assume exists lcross max lcross first note measure assumption models growth next note lcross represents closeness patches similar joint probability density small implies lcross small extreme setting lcross case events measure ready present result learning convolutional filter gradient descent theorem initialization satisfies kwt satisfies arcsin cross denote choose kwt arcsin kwt theorem suggests initialization satisfies obtain linear convergence rate section give concrete example showing closeness patches implies large small lcross similar theorem step size chosen published conference paper iclr log iterations find solution proof also similar theorem practice never get true population gradient stochastic gradient equation following theorem shows sgd also recovers underlying filter theorem let denote sufficiently small arcsin iterations probability log cross least kwt unlike vanilla gradient descent case convergence rate depends instead randomness sgd need robust initialization choose average ease presentation apparent proof require close proof relies constructing martingale use inequality idea previously used hat distribution easy sgd learn convolutional filter different model also requires lipschitz constant closeness lcross relatively small relatively large natural question input distributions satisfy condition give example show patches close input distribution small probability mass around decision boundary assumption theorem satisfied see figure graphical illustrations theorem denote zavg suppose patches unit norm zavg assume exists cos lcross analogue definition several comments sequel view quantitative measure closeness different patches small means similar decreasing function note bound monotonically recovers definition upper bond lcross represents bound density around decision boundary example small neighborhood around say radius assumption usually satisfied real world examples like images image patches usually close decision boundary example computer vision local image patches often form clusters evenly distributed appearance space therefore use linear classifier separate cluster centers rest clusters near decision boundary probability mass low ower andom nitialization model need initialization convolution filter need stronger initialization cos following theorem condition relaxed norm angle patch independent norm pair independent others published conference paper iclr shows uniformly random initialization constant probability obtain good initialization note theorem hand boost success probability arbitrary close random restarts proof similar tian theorem ball radius uniformly sample probability least apply general initialization theorem convolution filter case choose cos therefore simple algebra following corollary corollary suppose cos uniformly sampled ball center radius cos probability least cos assumption corollary satisfied patches close discussed previous section xperiments section use simulations verify theoretical findings first test smoothness affect convergence rate model described section construct input distribution different definition assumption fix patch unit norm use mixture truncated gaussian distribution model angle around around specifically probability density sampled note definitions probability mass centered around distribution spiky large hand input distribution close rotation invariant distribution small figure verifies prediction fix initialization step size next test closeness patches affect convergence rate convolution setting using model generate unit norm first generate single patch sampled figure shows whose angle variance patches becomes smaller obtain faster convergence rate coincides theorem also test whether sgd learn filter real world data choose mnist data generate labels using two filters one random filter entry sampled standard gaussian distribution figure gabor filter figure figure figure show convergence rates sgd different initializations better initializations give faster rates coincides theory note report relative loss logarithm squared error divided square mean data points instead difference learned filter true filter found sgd often converge exact filter rather filter near zero loss believe data approximately lying low dimensional manifold learned filter true filter equivalent justify conjecture try interpolate learned filter true filter linearly result filter similar low loss figure lastly visualize true filters learned filters figure see similar patterns onclusions uture orks paper provide first recovery guarantee stochastic gradient descent algorithm random initialization learning convolution filter input distribution gaussian analyses used definition relu mild structural assumptions input distribution list future directions one possibility extend result deeper wider architectures even fullyconnected network convergence stochastic gradient descent random initialization known existing results either requires sufficiently good initialization zhong log log published conference paper iclr epochs epochs figure convergence rates sgd different smoothness larger smoother different closeness patches smaller closer learning random filter different initialization mnist data learning gabor filter different initialization mnist data random generated target filters gabor filters figure visualization true learned filters pair left one underlying truth right filter learned sgd published conference paper iclr relies special architecture yuan however believe insights paper helpful understand behaviors algorithms settings another direction consider agnostic setting label equal output neural network lead different dynamics stochastic gradient descent may need analyze robustness optimization procedures problem also related expressiveness neural network raghu underlying function equal bot close neural network believe analysis extend setting acknowledgment authors would like thank hanzhang tengyu yuanzhi jialei wang kai zhong useful discussions eferences avrim blum ronald rivest training neural network advances neural information processing systems alon brutzkus amir globerson globally optimal gradient descent convnet gaussian inputs arxiv preprint anna choromanska mikael henaff michael mathieu ben arous yann lecun loss surfaces multilayer networks artificial intelligence statistics yann dauphin angela fan michael auli david grangier language modeling gated convolutional networks arxiv preprint simon chi jin jason lee michael jordan barnabas poczos aarti singh gradient descent take exponential time escape saddle points arxiv preprint daniel freeman joan bruna topology geometry network optimization arxiv preprint antoine gautier quynh nguyen matthias hein globally optimal training generalized polynomial neural networks nonlinear spectral methods advances neural information processing systems rong furong huang chi jin yang yuan escaping saddle pointsonline stochastic gradient tensor decomposition proceedings conference learning theory surbhi goel adam klivans learning neural networks polynomial time arxiv preprint surbhi goel varun kanade adam klivans justin thaler reliably learning relu polynomial time arxiv preprint benjamin haeffele vidal global optimality tensor factorization deep learning beyond arxiv preprint moritz hardt tengyu identity matters deep learning arxiv preprint kaiming xiangyu zhang shaoqing ren jian sun deep residual learning image recognition proceedings ieee conference computer vision pattern recognition majid janzamin hanie sedghi anima anandkumar beating perils guaranteed training neural networks using tensor methods arxiv preprint chi jin rong praneeth netrapalli sham kakade michael jordan escape saddle points efficiently arxiv preprint published conference paper iclr kenji kawaguchi deep learning without poor local minima advances neural information processing systems alex krizhevsky ilya sutskever geoffrey hinton imagenet classification deep convolutional neural networks advances neural information processing systems jason lee max simchowitz michael jordan benjamin recht gradient descent converges minimizers conference learning theory yuanzhi yang yuan convergence analysis neural networks relu activation arxiv preprint min lin qiang chen shuicheng yan network 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learning international conference machine learning shai ohad shamir shaked shammah weight sharing crucial succesful optimization arxiv preprint ohad shamir hardness learning neural networks arxiv preprint david silver aja huang chris maddison arthur guez laurent sifre george van den driessche julian schrittwieser ioannis antonoglou veda panneershelvam marc lanctot mastering game deep neural networks tree search nature training single sigmoidal neuron hard neural computation mahdi soltanolkotabi learning relus via gradient descent arxiv preprint mahdi soltanolkotabi adel javanmard jason lee theoretical insights optimization landscape shallow neural networks arxiv preprint published conference paper iclr christian szegedy sergey ioffe vincent vanhoucke alexander alemi impact residual connections learning aaai yuandong tian analytical formula population gradient relu network applications convergence critical point analysis arxiv preprint yuchen zhang jason lee martin wainwright michael jordan learning halfspaces neural networks random initialization arxiv preprint yuchen zhang jason lee michael jordan neural networks improperly learnable polynomial time international conference machine learning kai zhong zhao song prateek jain peter bartlett inderjit dhillon recovery guarantees neural networks arxiv preprint pan zhou jiashi feng landscape deep learning algorithms arxiv preprint published conference paper iclr roofs dditional heorems roofs heorem ection lemma terms proof since first term one part second term part second term proof theorem assumption input distribution ensures gradient descent converges following theorem assumption since gradient descent decreases function value converge note critical points lemma suppose converging critical point two cases contradicts eqn without loss generality let assumption know second equation becomes contradicts eqn therefore proof theorem proof relies following simple crucial observation arcsin denote observation recall gradient descent dynamics consider squared distance optimal weight kwt analysis previous section second term smaller kwt published conference paper iclr used assumption angle third term expand kwt kwt kwt kwt kwt kwt kwt kwt kwt therefore summary kwt kwt kwt first inequality assumption step size second monotonically decreasing theorem rotational invariant distribution unit norm rotational invariant input distribution proof theorem without loss generality need focus plane spanned suppose sin cos cos cos sin sin sin cos two eigenvalues sin sin therefore maxw theorem proof note previous theorem integrate angle radius separately multiply together gaussian distribution result follows roofs heorems ection proof theorem proof similar theorem two events use shorthand shorthand denote first note routine algebra write gradient published conference paper iclr first examine inner product gradient published conference paper iclr kwt kwt kwt kwt kwt kwt first inequality used definitions regions second inequality used definition operator norm third inequality used fact kwt fourth inequality used definition lcross fifth inequality used sin next upper bound norm gradient using similar argument kwt kwt kwt kwt therefore using dynamics gradient descent putting two bounds together kwt kwt kwt last step used choice proof theorem consists two parts first show chosen properly big high probability iterates stat neighborhood next conditioning derive rate lemma denote sin given sin number step size log probability least kwt proof lemma let denote sigmaalgebra generated define event consider ict kwt ict kwt ict inequality follows analysis gradient descent together definition define kwt iterations failure probability denote arcsin satisfies published conference paper iclr analysis ict ict last inequality subset therefore may apply inequality need bound difference ict expectation note kwt kwt kwt therefore first inequality used second used third used assumption let bound step iterate goes region exp second inequality used inequality last one used assumption therefore probability least happens derive rate lemma denote sin given sin number iterations failure probability denote arcsin step size satisfies log log published conference paper iclr probability kwt proof lemma use notations proof lemma analysis lemma know ict kwt ict therefore kwt ict bound failure probability kwt kwt kwt ict kwt ict kwt ict first inequality used last assumption second inequality used probability event upper bound superset event third one used lemma union bound fourth one used markov inequality specify derive convergence rate sgd learning convolution filter proof theorem choice straightforward check satisfies conditions lemma proof theorem first prove lower bound published conference paper iclr note unit norm law cosines kop cos therefore cos prove upper bound lcross notice kzi kzj assumption hand let angle therefore using similar arguments show proof theorem use argument tian let rinit initialization radius failure probability lower bounded cos rinit rinit rinit rinit therefore rinit cos maximizes lower bound plugging optimizer using formula volume euclidean ball failure probability lower bounded cos cos used gautschi inequality last step dditional xperimental esults figure show loss linear interpolation learned filter ground truth filter interpolation form winter interpolation ratio note interpolation ratios loss remains low published conference paper iclr log relative loss log relative loss gabor filter random filter interpolation ratio interpolation ratio figure loss linear interpolation learned filter true filter

ieee transactions circuits regular papers optimal tracking performance limitation networked control systems limited bandwidth additive colored white gaussian noise feb guan chen gang feng tao paper studies optimal tracking performance issues linear systems networked control limited bandwidth additive colored white gaussian noise channel tracking performance measured control input energy energy error signal output system reference signal respect brownian motion random process paper focuses two kinds network parameters basic network additive colored white gaussian noise studies tracking performance limitation problem best attainable tracking performance obtained impact limited bandwidth additive colored white gaussian noise communication channel attainable tracking performance revealed shown optimal tracking performance depends nonminimum phase zeros gain frequencies directions unitary vector given plant well limited bandwidth additive colored white gaussian noise communication channel simulation results finally given illustrate theoretical results index control systems bandwidth additive colored white gaussian noise performance limitation ntroduction ore researchers interested networked control systems past decade please see example references therein works focus analysis synthesis networked control systems quantization effects time delays bandwidth constraint data rate constraint data packet dropout spite significant progress studies inspiring challenging issues control performance limitation network environment remain largely open guan chen department control science engineering huazhong university science technology wuhan china gang feng department mechanical biomedical engineering city university hong kong kowloon hong kong sar china tao college electronics information yangtze university jingzhou china corresponding author zhguan guan work supported part national natural science foundation china grants doctoral foundation ministry education china grant grant research grants council hong kong special administrative region china project cityu performance limitations resulting nonminimum phase nmp zeros unstable poles given systems known long time issue attracting growing amount interest control community see example tracking performance achievable via feedback studied respect siso stable systems result extended mimo unstable systems found minimal tracking error depends location system nonminimum phase zeros also input signal may interact zeros angles input zero directions optimal tracking regulation control problems studied objective functions tracking error regulated response defined integral square measures minimized jointly control effort latter measured system input energy optimal tracking control problem studied forward feedback channel disturbances authors investigated regulation performance limitations unstable phase simo systems respectively however mentioned works taken account effects networks would make study optimal performance limitation much challenging networked control systems ubiquitous industry control systems operating network recent years research performance limitation networked control systems attracts attention example authors studied tracking performance siso networked feedback systems modeling quantization error white noise tracking performance mimo systems additive white gaussian noise awgn studied control schemes result generalized noisy channels bandwidth limitation optimal tracking performance measured achievable minimal tracking error however showed optimal tracking problem order attain minimal tracking error control input systems often required infinite energy requirement met general practice thus control input energy systems considered performance index address issue paper ieee transactions circuits regular papers consider optimal tracking problem terms tracking error energy control input energy meanwhile consider communication link bandwidthlimited additive colored gaussian noise acgn channels realistic models communication link paper study optimal tracking performance issues pertaining mimo feedback control systems objective minimize tracking error output reference signals feedback system constraint control input energy optimal tracking performance attained stabilizing compensators structure tracking error defined square error sense reference signals considered brownian motion roughly considered integral standard white noise tracking performance index given weighted sum power tracking error energy system input energy rest paper organized follows problem formulation preliminaries given section section iii main results paper presented results extensive simulation studies discussions shown validate theoretical results section concluding remarks made section associated zero always true unitary vector hand complex number said pole unstable pole equivalent statement unitary vector order facilitate subsequent proof introduce two specific factorization zand allpass factor form uih reliminaries consider class functions analytic lemma found lemma let denote suppose conjugate symmetric lemma consider conjugate symmetric function suppose analytic zero log provided log lemma let defined equality lnz holds proof assume allpass factor lemma pnz let lnz therefore proof completed begin summarizing briefly notations used throughout paper complex number denote complex conjugate expectation operator denoted respectively vector denote conjugate transpose euclidean norm kuk matrix denote conjugate transpose vectors matrices involved sequel assumed compatible dimensions simplicity dimensions omitted let open plane denoted open plane imaginary axis define measurable hilbert space inner product next define subspace functions functions analytic inr analytic orthogonal complement ras analyticin thus use notation denote corresponding norm finally denote class stable proper rational transfer function matrices introduce factorization formula phase systems rational transfer function matrix let right left coprime factorizations given complex number said zero unitary vector called output direction vector unitary vectors obtained factorizing zeros one time matrices together form unitary matrix similarly definition nature likewise allpass factorization allpass factor minimum phase part one particular allpass factor qnof given iii racking performance limitations consider control feedback loop shown plant model rational transfer function matrix channel model acgn channel zeromean stationary white gaussian noise process spectral density note reference signal vector step signal generated passing standard white noise integrator ieee transactions circuits regular papers roughly considered brownian motion process emulate step signal deterministic setting therefore formulation resembles tracking deterministic step signal channel denote spectral density note assumed system reference inputs different channels independent reference input noise uncorrelated denotes fig feedback control bandwidth limited acgn channels optimal performance attainable possible stabilizing controllers inf theorem let uncorrelated white gaussian signals suppose integer proper zero supposed unstable nmp invertible including right invertible left invertible denote nmp zeros assume also zeros distinct define factorize nonminimum phase zeros minimum phase noted controller given compensators communication channel characterized three parameters awgn channel transfer functions channel transfer function modeling bandwidth limitation assumed stable nmp diag distinct nmp zeros channel transfer function colors additive white gaussian noise performance index system defined parameter used weigh relative importance tracking objective plant control input energy constraint transfer function matrices let right left coprime factorizations respectively given satisfy double bezout identity set stabilizing two parameter compensators characterized log res dir dirh dil djl proof ryn rucr rucn ryn rucr rucn autocorrelation functions random processes ucr ucn respectively denote spectral densities respectively according may rewrite performance index kyn outputs response respectively tracking error given tyn tucr ucr tucn ucn tucr tucn ieee transactions circuits regular papers following equation obtained cft one define following matrix function module equal according property matrix norm becomes inf inf diag diag evidently inf inf inf firstly using allpass factorization inf inf obtain inf similar may invoke lemma obtains light lemma one also obtains resi log thus minimum phase part direction vector associated zero unitary column vector whose element remaining elements inf inf furthermore perform factorization given inner matrix function outer according definition inner matrix function resi log secondly nom minimum phase part similar equation perform factorization ieee transactions circuits regular papers addition similar factorize minimum phase part allpass factor formed wih fig feedback control awgn channels hence light lemma rcd dil dir dil factorize nonminimum phase zeros minimum phase pitl noted twoparameter controller given resi log dir since right invertible left invertible inf dir dil dir proof similar proof theorem performance index dil dir dirh dil djl proof thus completed remark network channel brownian motion random process different step signal vector deterministic direction result degraded results literature corollary system siso theorem optimal tracking performance written nzx resi log corollary consider simple channel case assumptions theorem fine ryn rucr rucn ktyn ktucr ktucn let obtain res log addition factorize minimum phase part allpass factor formed ieee transactions circuits regular papers similar proof theorem obtain predetermined input power threshold performance index system stabilizable obtain optimal tracking performance channel snr must satisfy pad pad nom proof thus completed channel noise configuration feedback control system depicted following result immediately obtained corollary consider case suppose channel assumptions described theorem resf log remark consider impact system control input setting expression corollary observed feedback control system compensators tracking target brownian motion performance limitation depends nonminimum phase zeros plant gain frequencies directions unitary vectors follows discuss relationship stabilizability performance limits channel characteristics simplified conditions appropriate simplifications assumptions consider siso system simplified performance kyn relationship stabilizability tracking performance channel ratio snr summarized shown following theorem theorem consider feedback control system suppose scalar transfer function assumptions theorem system stabilizable admissible channel snr satisfies nom optimal tracking performance given nom nom proof using equation similar proof theorem ryn nom rnm nom rnm nom rnm based allpass factorization lemma write nom nom ieee transactions circuits regular papers nom rnm nom rnm nom nom rnm kpow nom kpow residue addition suppose input channel input required satisfy power constraint kukpow predetermined input power level kpow run tun tutn nom nom nom kpow pad pad given equation kpow pad proof completed remark theorem shows system achieve best tracking performance addition stabilization ratio must greater required stabilization imulation studies consider plant lti filters used model finite bandwidth colored noise communication link chosen butterworth filters order kno knom nom result however many cases stabilizability needs considered also system tracking performance case via noting equations equations nom nom residue nom therefore feedback system stabilizable channel snr must satisfy using bezout identity written nom nom stabilizability considered regardless tracking performance clearly minimum phase shows optimal performances plotted different values two observations obtained optimal performance plotted respect bandwidth ieee transactions circuits regular papers first system tracking performance becomes better available bandwidth communication channel decreases secondly noise colored low pass filter decrease cutoff frequency would lead better tracking performance shows reference signal acgn deteriorate tracking performance onclusions fig respect different bandwidth bandwidth respect fig paper investigated best attainable tracking performance networked mimo control systems tracking brownian motion limited bandwidth additive colored white gaussian noise channel derived explicit expressions best performance terms tracking error control input energy shown due existence network best achievable tracking performance adversely affected several factors nonminimum phase zeros directions plant colored additive white gaussian noise basic network parameters bandwidth finally simulation results given illustrate obtained results furthermore one possible future work consider realistic constraints dropout issues much challenging networked control system contains nondeterministic hybrid switching issue tracking performance also deserves study eferences fig respect braslavsky middleton freudenberg feedback stabilization ratio constrained channels ieee transactions automatic control vol tuncel chen optimal tracking additive white gaussian noise channel proceedings american control conference usa zhan guan xiao wang performance limitations tracking linear system measurement noise proceedings ieee conference chinese control conference beijing china rojas braslavsky middleton fundamental limitations control communication channel automatica vol xiao xie feedback stabilization stochastic multiplicative input channels case proceedings international conference control automation robotics vision singapore menon edwards static output feedback stabilisation synchronisation complex networks performance international journal robust nonlinear control vol guan zhan feng optimal tracking performance mimo systems communication constraints international journal robust nonlinear control wiley online library zhang yan yang chen quantized control design impulsive fuzzy networked systems ieee transactions fuzzy systems vol azuma sugie dynamic quantization nonlinear control systems ieee transactions automatic control vol optimal tracking tracking performance constraints quantization proceedings asian control conference china xie optimality logarithmic quantizer stabilization linear systems achieving minimum data rate proceedings ieee conference decision control chinese control conference china ieee transactions circuits regular papers xiao xie stabilization markov jump linear systems using quantized state feedback automatica vol luan shi liu stabilization networked control systems random delays ieee transactions industrial electronics vol wei wang shu filtering networked stochastic systems sector nonlinearity ieee transactions circuits systems express briefs vol liu predictive controller design networked systems communication delays data loss ieee transactions circuits systems express briefs vol rojas braslavsky middleton output feedback stabilisation bandwidth limited signal noise ratio constrained communication channels proceedings american control conference minnesota usa trivellato benvenuto state control networked control systems packet drops limited transmission bandwidth ieee transactions communications vol chen design networked control systems packet dropouts ieee transactions automatic control vol wang liu zhu survey networked control systems delay packet dropout proceedings chinese control decision conference ccdc xie mean square stability kalman filtering markovian packet losses automatics toker chen qiu tracking performance limitations lti multivariable systems ieee transactions circuits systems fundamental theory applications vol chen hara chen best tracking regulation performance control energy constraint ieee transactions automatic control vol bakhtiar hara regulation performance limitations simo linear feedback control systems automatica vol wang guan yuan optimal tracking twochannel disturbance rejection control energy constraint automatica vol morari zafiriou robust process control englewood cliffs chen qiu toker limitations maximal tracking accuracy ieee transactions automatic control vol ding wang guan chen tracking additive white gaussian noise effect iet control theory application vol qiu ren chen fundamental performance limitations estimation problems communications information systems vol zhan guan liao yuan optimal performance tracking stochastic signal disturbance rejection asian journal control doi wang ding guan chen limitations minimum tracking energy siso plants proceedings control decision conference china tuncel chen optimal tracking power allocation additive white noise channel proceedings ieee international conference control automation new zealand francis course control theory ser lecture notes control information science berlin germany guan hill shen hybrid impulsive switching systems application nonlinear control ieee transactions automatic control vol zhang cui liu zhao asynchronous filtering switched linear systems average dwell time ieee transactions circuits systems regular papers vol zhang james necessary sufficient conditions analysis synthesis markov jump linear systems incomplete transition descriptions ieee transactions automatic control vol guan received phd degree automatic control theory applications south china university technology guangzhou china full professor mathematics automatic control jianghan petroleum institute jingzhou china since december full professor department control science engineering executive associate director centre nonlinear complex systems director control information technology huazhong university science technology hust wuhan china since held visiting positions harvard university usa central queensland university australia loughborough university national university singapore university hong kong city university hong kong currently associate editor journal control theory applications international journal nonlinear systems application severs member committee control theory chinese association automation executive committee member also director control theory committee hubei province association automation research interests include complex systems complex networks impulsive hybrid control systems networked control systems systems chen born hunan china graduated mathematics hunan university science technology xiangtan china received degree department mathematics guangxi teachers education university currently working towards degree department control science engineering huazhong university science technology wuhan china research interests include networked control systems complex dynamical networks impulsive hybrid control systems gang feng received degrees automatic control nanjing aeronautical institute china respectively degree electrical engineering university melbourne australia city university hong kong since present chair professor changjiang chair professor nanjing university science technology awarded ministry education china lecturer school electrical engineering university new south wales australia awarded alexander von humboldt fellowship ieee transactions fuzzy systems outstanding paper award current research interests include piecewise linear systems intelligent systems control feng ieee fellow associate editor ieee trans fuzzy systems associate editor ieee trans systems man cybernetics part journal control theory applications conference editorial board ieee control system society tao received degree huazhong university science technology wuhan china also currently associate professor college electronics information yangtze university jingzhou china current research interests include nonlinearity complex network systems complex network theory application complex networks spreading dynamics

comparing powers edge ideals sep mike janssen thomas kamp jason vander woude bstract given nontrivial homogeneous ideal problem great recent interest comparison rth ordinary power mth symbolic power comparison undertaken directly via exploration exponents guarantee subset containment asymptotically via computation resurgence number guarantees recently third quantity symbolic defect introduced symbolic defect minimal number generators required add order get consider various means comparison edge ideal certain graphs describing ideal edge ideal odd cycle description structure yields solutions direct asymptotic containment questions well partial computation sequence symbolic defects ntroduction let algebraically closed field nonzero proper homogeneous ideal recall mth symbolic power ideal last years structure object ongoing study see recent survey one avenue study examination relationship algebraic structure rth ordinary power naive context examine relationship via subset containments fact line inquiry extremely productive straightforward see determining give delicate seminal results established ideals additional information ideal consideration generally leads tighter results see phenomenon led bocci harbourne introduction quantity known resurgence denoted least upper bound set thus recently galetto geramita shin van tuyl introduced new measure difference known symbolic defect since quotient mike janssen thomas kamp jason vander woude finite thus let sdefect denote number minimal generators known symbolic defect symbolic defect sequence sequence sdefect authors study symbolic defect sequences star configurations pnk homogeneous ideals points work considers questions context class edge ideals let simple graph vertex set edge set edge ideal introduced ideal given generated products pairs variables edges authors establish edge ideal bipartite natural question explore relationship bipartite equivalent containing odd cycle thus sought explore relationship cycle vertices continue problem exploring structure symbolic power certain classes graphs focus odd cycle main results work theorem corollary together describe decomposition form ideal able use decomposition resolve conjecture compute theorem establish partial symbolic defect sequence theorem close showing ideas theorem apply complete graphs graphs consist odd cycle plus additional vertex edge remark preparation manuscript concluding summer dao posted preprint particular theorem bears striking resemblance corollary similarities worth noting part evidence interest symbolic powers high also worth noting aims two works distinct complementary aim relevant sections investigate packing property edge ideals directly describe difference ordinary symbolic powers investigating structure set minimal generators use information generators compute invariants related containment acknowledgements work supported dordt college summer undergraduate research program summer three authors wish express deep gratitude dordt college office research scholarship opportunity undertake project ackground results edge ideals important class examples squarefree monomial ideals ideal generated elements form xnan comparing powers edge ideals squarefree monomial minimal primary decomposition form edge ideal variables precisely vertices minimal vertex covers recall given graph vertex cover subset minimal vertex cover vertex cover minimal respect inclusion minimal vertex covers especially useful describe variables needed decompose edge ideal minimal primes see corollary lemma let graph vertices edge ideal minimal vertex covers let monomial prime ideal generated variables prm symbolic powers squarefree monomial ideals specifically edge ideals enjoyed great deal recent interest see linear programming approach used compute invariants related containment question adapt technique lemma edge ideals consideration paper one result use following reduces problem determining whether given monomial problem checking certain linear constraints exponents variables lemma let squarefree monomial ideal minimal primary decomposition remark throughout work exploring questions related ideals related graphs vertex set use interchangeably represent vertices variables specific use clear context see opportunity emphasize close connection graph ideal factoring monomials along odd cycles section introduce main ideas approach studying symbolic powers edge ideals begin defining means writing monomial power edge ideal respect minimal vertex covers graph study factorization describe situation improved follows let let edge ideal odd cycle mike janssen thomas kamp jason vander woude definition let monomial let denote degree two monomial representing jth edge cycle may write large possible observe deg written way call optimal factorization say expressed optimal form addition form called ancillary factor optimal factorization ancillary short observe optimal form representation unique sense different edges may appear factors example may write lemma let optimal factorization also optimal factorization proof let since variable exponent less equal corresponding exponent know divides thus must exist suppose optimal form must exist way expressing sum exponents edge factors greater new expression edge exponent sum must true edge exponent sum greater contradicts premise expressed optimal form thus optimal factorization next lemma describes process critical proof main result intuitively says monomial factored product odd number consecutive edges ancillaries ends path edges monomial written optimal form rewritten product strictly edges lemma let case optimal form proof let notice string adjacent edges ancillaries either end goal rewrite optimal form clarity without loss generality let suppose evenly indexed edge exponents comparing powers edge ideals let note lemma must optimal form expressed optimally however since initially expressed optimal form know could optimal factorization example let cycle vertices consider edge factorization note factorization ancillary show optimal form graphically represent drawing edge creating bold outline ancillary shown using method outlined lemma break red bolded edges back standard notation create new ancillaries every vertex note define new monomial based graphical representation still true merely changing factorization monomial value one see consecutive ancillaries pair new way shown new edges highlighted green bolded second line third possible representation monomial note still true see monomial representation one edge original representation means optimal mike janssen thomas kamp jason vander woude example let cycle vertices consider following edge factorization goal determine whether optimal note equal monomial example except ancillaries create graphical representation shown however impossible remove right combination edges create ancillary every vertex edge exists therefore use lemma conclude optimal fact conclusive way determine whether optimal factorization point despite example without value note nonexistence edge fact would able prove optimal nonexistence least one following useful latter stages proof theorem owers edge ideals structures turn decomposition terms another ideal approach numerous strengths including ability easily compute symbolic defect certain powers well determining additional elements needed generate although primarily focus odd cycles section show underlying principles extended edge ideals types graphs see section definition let set vertices monomial exponent vector define vertex weight usually interested case minimal vertex cover using language vertex weights definition symbolic power edge ideal given lemma becomes minimal vertex covers define sets deg minimal vertex covers deg minimal vertex covers comparing powers edge ideals generate ideals respectively note main work section show edge ideal odd cycle content theorem lemma let graph defined proof suppose write optimal form xrar know given arbitrary minimal vertex cover edge dividing must true thus since know deg means lemma let graph defined ancillaries single ancillary degree proof ancillaries deg thus also means none divisors similar reason furthermore reach conclusion one ancillary exponent deg since odd remainder section let odd cycle size vertices edge ideal minimal vertex cover make following definition describes sum exponents given monomial relative set vertices theorem given defined proof lemma know must show reverse containment let implies show lemma allows consider cases either multiple ancillaries single ancillary least degree given arbitrary monomial let optimal factorization ancillary goal show exists vertex cover weight equal since generating set sufficient claim neither divisors whose vertex weights less generating set construct minimal vertex cover sequence subsets cover induced subgraph vhq sake simplicity let pair consecutive ancillaries let wraparound case let case mike janssen thomas kamp jason vander woude single ancillary degree greater addition let note lemma optimal form show subgraph exists set vertices vhq covers case suppose vhq odd number elements consider claim shown follows xiai intuitively selecting alternating vertices would guarantee edge contributes weight twice edges connect sequentially indexed vertices also ancillaries would increase weight included definition know weight monomial respect set variables equal sum powers variables given monomial case case suppose vhq even number elements note must contain vertices simply would imply vertices two ancillaries adjacent could thus expressed would contradict statement expressed optimal form lemma know satisfying edge product appear current optimal form consider claim see comparing powers edge ideals xiai intuitively reasons given vhq odd number elements since alternating vertices chosen exception however edge product appear including redundant powers weight means hence matter whether vhq odd even number vertices regardless since covers respective set vertices union disjoint subcovers vertex cover addition completely disjoint subgraph cover number edges existed induced subgraph representation two subgraphs contained edges total number edges optimal factorization constructed vertex cover thus therefore corollary given proof theorem states also know simply substitute thus proved use result carry various computations related interplay ordinary symbolic powers close section brief remark proof theorem specifically relies fact cycle odd cycle however focus odd cycle case even cycle bipartite showed case mike janssen thomas kamp jason vander woude pplications deal ontainment uestions given edge ideal odd cycle corollary describes structural relationship given section exploit relationship establish conjecture compute resurgence explore symbolic defect various powers given recall definitions generate ideals respectively deg minimal vertex covers deg minimal vertex covers begin examining lemma given monomial exists proof recall ideal generated although graph many different minimal vertex covers certain type vertex cover guaranteed exist odd cycle type cover includes two adjacent vertices alternating vertices thereafter without loss generality consider suppose two minimal vertex covers include order must true means adding inequalities yields follows contradicts requirement deg hence monomial least one exponent equal element extension lemma given monomial deg divisible proof let element deg suppose divisible means exists moreover since must odd consider minimal vertex covers even use order must true odd means comparing powers edge ideals similarly even combining see contradiction following corollary partially answers conjecture affirmative note restatement theorem corollary let odd cycle size edge ideal proof suppose recall element generating set must degree less however since variables least two would need exponent order element lemma know none variables monomial exponent therefore monomials satisfy conditions means empty thus recent paper galetto geramita shin van tuyl introduced notion symbolic defect denoted sdefect measure difference symbolic power ordinary power number minimal generators corollary thus implies sdefect satisfying corollary let odd cycle size edge ideal sdefect particular proof let proposition states recall ideal generated deg minimal vertex covers know degree monomial must strictly less lemma also know variables exponent least variables see variables exponent least total degree monomial becomes least valid thus every monomial straightforward check therefore thus recall homogenous ideal minimal degree denoted least degree nonzero polynomial particular edge ideal general may conclude converse need hold however next lemma demonstrates lemma let edge ideal odd cycle mike janssen thomas kamp jason vander woude proof forward direction clear converse suppose definition symbolic powers know minimal vertex covers note thus deg observe deg minimal vertex covers completes proof despite providing condition guarantees containments form lemma actually compute delicate computing next adapt lemma linear programming approach compute order make following definition definition fix list minimal vertex covers define minimal vertex cover matrix matrix defined remark note minimum cardinality minimal vertex cover fact minimal vertex covers size seen exist minimal vertex covers size greater covers accounted rows higher minimal vertex cover matrix first seek lower bound using linear programming let consider following linear program minimal vertex cover matrix minimize subject refer alpha program observe value realizes consider following partition let submatrix consisting first rows thus corresponding minimal vertex covers contain exactly vertices matrix consisting remaining rows thus create following comparing powers edge ideals minimize subject lemma value proof claim column vector feasible solution indeed column vector whose entries satisfying constraint case show value make use fundamental theorem linear programming showing existence produces value dual linear program maximize subject specifically let rows exactly see satisfied straightforward check lemma value bounded proof observe obtained possibly introducing additional con straints thus value least value proposition let proof lemma see bounded value enough find element degree claim element note minimal vertex cover hence minimal prime contain one least contains contains one vertices former case latter case see mike janssen thomas kamp jason vander woude thus integer satisfying whence recall given nontrivial homogeneous ideal gence introduced denoted number sup theorem odd cycle size edge ideal proof let suppose order subset must true lemma since know follows thus conclude next goal prove smallest upper bound finding sequence lim first make following claim claim proof claim lemma follows enough show conclude proposition recall odd cycle size let recursively define claim note definition sequence equivalent explicit formula moreover lim finally implies new measure failure contain introduced measure known symbolic defect given number minimal generators recall corollaries comparing powers edge ideals imply sdefect next explore additional terms symbolic defect sequence general approach rely decomposition described corollary parlance work symbolic defect size minimal generating set ideal observe general computing cardinality set may monomials divisible monomials set thus goal determine cardinality subset forms minimal generating set theorem let satisfying sdefect proof stated wish count number minimal generators recall definition everything degree less see consists monomials degree collection distinct monomials degree linearly independent thus minimal generating set consider arbitrary note deg since edge monomials degree see divisible product edge monomials proposition lemma gives thus divisible least edge monomials thus must divisible exactly edge monomials optimal factorization single ancillary exponent lemma must divisible write optimal form odd even observe either case product single variable edge monomials thus monomial product exactly edge monomials thus factored product exactly edge monomials observe product exactly edge monomials deg minimal vertex cover follows fact definition thus mike janssen thomas kamp jason vander woude therefore count monomials suffices count monomials products edge monomials visualize problem counting number ways place edges around cycle assuming place multiple edges two vertices end let number pairs vertices place least one edge ways place edges first choose among choices pairs vertices place edges choose sdefect ways arrange edges thus particular sdefect computation sdefect becomes much complicated additional containment question proof hold graph cycle relies fact path ancillaries disjoint every path true general leads naturally following question question let graph vertices containing odd cycle suppose edge ideal let retain usual definitions respect following example answers question negative example consider graph defined write edges products vertices let observe every minimal vertex cover contains three thus however observe following two theorems certain classes graphs one case question holds case odd cycle one additional vertex connected exactly one vertex cycle see figure example graph constructed theorem let graph consisting vertices edges form cycle remaining edge connects remaining vertex existing vertex cycle let edge ideal let retain usual definitions respect comparing powers edge ideals igure additional vertex edge appended proof without loss generality consider cycle formed newly added edge recall let monomial expressed optimal form recall cycle follow lemma know must show reverse containment let implies show lemma allows consider cases either multiple ancillaries single ancillary least degree note deg else definition construct minimal vertex cover first assume ancillary observe may write true would possible divide monomial must optimal form lemma however case contradicting optimal form thus least one construct follows let let suppose ancillaries cycle ancillaries come set adapting argument theorem may assume either one ancillary exponent least multiple ancillaries use construction proof theorem decompose subgraph define proof theorem provides minimal subcovers mike janssen thomas kamp jason vander woude covers case may let hand let next assume ancillaries least one cycle may write contradicting assumption optimal form use construction theorem decompose cycle subgraphs note vertex observe since ancillary let vertices represented ancillaries wrap around representing proof theorem gives construction minimal vertex subcover required properties construct subgraph follows decompose two induced subgraphs vertices observe may use construction proof theorem build minimal covers containing whose union gives cover given note union required property cases extension also verify answer question positive complete graph thus additional study needed identify precise property question affirmative answer theorem let let denote complete graph let maintain definitions proof let denote edge show lemma must show reverse containment let implies recall lemma allows consider cases either multiple ancillaries single ancillary least degree let monomial optimal form ancillary xnan ancilaries could expressed optimal form xnan guaranteed edge complete thus exactly ancillary must degree least without loss generality let ancillary note case could expressed optimal form xnan comparing powers edge ideals let observe covers thus argument divisor means therefore corollary leads desired result eferences bocci cooper harbourne containment results ideals various configurations points pure appl algebra bocci harbourne comparing powers symbolic powers ideals algebraic cristiano bocci susan cooper elena guardo brian harbourne mike janssen uwe nagel alexandra seceleanu adam van tuyl thanh waldschmidt constant squarefree monomial ideals journal algebraic combinatorics susan cooper robert embree huy andrew hoefel symbolic powers monomial ideals proceedings edinburgh mathematical society dao stefani grifo huneke symbolic powers ideals arxiv august denkert janssen containment problem points reducible conic journal algebra marcin dumnicki tomasz szemberg halszka counterexamples containment journal algebra ein lazarsfeld smith uniform bounds symbolic powers smooth varieties invent galetto geramita shin van tuyl symbolic defect ideal arxiv october tuan hoa tran nam trung regularity symbolic powers twodimensional monomial ideals commut algebra hochster huneke comparison symbolic ordinary powers ideals invent simis vasconcelos villarreal ideal theory graphs journal algebra adam van tuyl beginner guide edge cover ideals pages springer berlin heidelberg berlin heidelberg mike janssen thomas kamp jason vander woude rafael villarreal graphs manuscripta mathematica dec scarlet worthen ellis lesley wilson symbolic powers edge ideals undergraduate math journal athematics tatistics epartment ordt ollege ioux enter usa address athematics tatistics epartment ordt ollege ioux enter usa address thmskmp athematics tatistics epartment ordt ollege ioux enter usa address jsnvndrw

mar noether number cziszter institute mathematics hungarian academy sciences budapest hungary abstract group order prime indecomposable polynomial invariant degree least group cyclic subgroup index isomorphic elementary abelian group order heisenberg group order keywords polynomial invariants degree bounds sequences introduction let finite group field characteristic dividing group order maximal degree minimal generating set ring polynomial invariants known see even observed supv runs base field typically much less algebraically closed base field characteristic zero proved holds cyclic turned group see moreover holds cyclic subgroup index two exception four particular groups small order see theorem recently asymptotic extensions result given goal present article establish following strengthening kind results class theorem finite prime characteristic base field zero greater inequality partially supported national research development innovation office nkfih grants erc holds cyclic subgroup index elementary abelian group heisenberg group order proof theorem reduced study single critical case heisenberg group extraspecial group order exponent odd prime prove following result theorem prime base field characteristic greater paper organised follows section contains technical results sequences abelian groups needed later section reduce proof theorem theorem section explain main invariant theoretic idea behind proof theorem also applicable general setting proof theorem carried full detail section finally section completes argument showing case characteristic preliminaries sequences follow notations terminology usage fixed let abelian group noted additively sequence subset mean multiset elements form free commutative monoid respect concatenation denoted unit element empty sequence distinguished zero element sequence obtained repetition element denoted distinguished product multiplicity element sequence denoted also write indicate say subsequence write sequence case also write length sequence denoted expressed whereas sum sequence convention set say sequence relevance sequences topic due fact abelian group noether number coincides davenport constant defined maximal length sequence containing proper subsequence see chapter value given following formula theorem cpnr pni variant notion kth davenport constant defined maximal length sequence factored concatenation sequences numerical value much less known recent results see shall need fact according theorem following consequence definition also used lemma lemma sequence abelian group length least factors sequences define sequence set partial sums called free next result could also deduced theorem see corollary provide elementary proof reader convenience lemma let prime sequence min proof use induction length claim trivial otherwise consider sequence claim holds either else subgroup containing since two subgroups assumption means lemma theorem sequence prime length free lemma proposition let prime sequence length subsequence length close section technical result motivation relevance become apparent application proof proposition function defined sequence write sequence obtained applying lemma let sequence length sequence proof let denote maximal integer sequences irreducible hence free hence assuming get whence follows contradiction assumption proposition let prime projection onto first component sequence given subsequence length factorisation sequence proof let maximal subsequence assumption subsequence length lemma two cases sequences take lemma find sequence length cases construction consequently hence lemma factorisation sequences finally cases hence lemma reduction theorem theorem main tool kth noether number defined greatest integer invariant degree exists contained ideal generated products least invariants positive degree notion introduced section goal estimating ordinary noether number information composition factors made possible lemma according normal subgroup observed chapter abelian group coincides use applications proof theorem assuming theorem part follows proposition states subgroup cyclic index moreover proposition part follows theorem rest may assume let group order holds normal subgroup lemma claim must cyclic otherwise applying lemma factor group find subgroup get using lemma get contradiction let generates order first case hgi index done case hgi hence acts trivially contains subgroup hence contradiction shows must act well known aut order sylow must order isomorphic therefore must act trivially subgroup isomorphic excluded case remains open factor group acts heisenberg group denoted theorem assumption characteristic base field among heisenberg groups inequality hold remark precise value noether number already known satisfy according theorem theorem states equality holds rest groups order cyclic subgroup index classified burnside see theorem follows abelian either cyclic case isomorphic modular group mpn mpn remark iii dihedral group group generalised quaternion group theorem altogether results imply inequality sharp case remark notion davenport constant originally defined abelian groups section extended finite group conjectural connection noether number generalisation davenport constant see section invariant theoretic lemmas let fix notations related invariant rings vector space field denote coordinate ring say group left action group homomorphism given abbreviate writing setting obtain right action ring polynomial invariants defined ring already known normal subgroup vector space spanned elements form runs set monomials epimorphism defined see chapter trivial definition amounts hom reynolds operator given character set constitutes weight restriction trivial obtained projection map defined analogous formula graded rings denotes vector space degree homogeneous polynomials set maximal ideal ideal generated polynomials positive degree called ideal main object interest since observed section graded factor ring finite dimensional top degree denoted yields upper bound noether number easy argument using reynolds operator well known unchanged extend base field assume throughout paper algebraically closed lemma let finite group normal subgroup abelian let assume proof regarded direct sum decomposition used assumptions means element written sum term belongs ideal whenever quence contains subsequence holds every term right lemma lemma factor group cyclic prime order elements relation proof observe weight sequence free lemma result get replacing respectively observing definition infer must belong residue class modulo ideal belong proves claim heisenberg group heisenberg group defined presentation denotes commutator subgroups normal isomorphic center derived subgroup coincide hci extraspecial particular also isomorphic taking account subgroup structure best upper bound give noether number means following goal section enhance estimate analysing closely invariant rings let algebraically closed field char primitive root unity regarded fixed throughout paper irreducible two types composing group homomorphism hom canonic surjection yields irreducible representations primitive root unity take induced representation inda hvi hvi left basis representation given terms matrices following form identity matrix irreducible mackey criterion see easily seen matrix corresponding adding squares dimensions irreducible get irreducible exist result arbitrary canonic direct sum decomposition consists irreducible representations hci kernel isotypic consisting direct sum isomorphic copies irreducible representation times next recall action extend coordinate ring speaking coordinate ring always tacitly assume variables form dual basis basis used convention section acts right variables rewrite xbi mod xai xci abuse notation identified integers occurring indexes modulo residue classes represent shows action subgroup variable completely determined modulo residue classes exponents call weight variable shall also refer projections notation immediate subtraction multiplication understood implies observation used frequently later variable arbitrarily given always element hbi discussion also shows variable otherwise value determines isotypic monomial hence associate weight obviously monomials variables repetitions allowed form sequence called weight sequence obviously notations section observe monomial sequence finally set definition call two monomialsqu homologous denoted deg deg variables repetitions allowed group elements hbi observe monomial obtained monomial repeated applications homologous sense proposition let monomial deg monomial deg homologous monomial homologous monomial proof use induction degree deg deg done taking suppose claim holds suffices prove given divisor variable deg hbi monomial exists inductive hypothesis already monomial divides hbi divides applying proposition weight sequences obtain factorisation divides two cases similarly take lemma otherwise assumption divisor hbi lemma take factorisation rest construction divides factorisation falls case done need notations decomposition induces isomorphism turn yields monomial factorisation decomposition gives identifications set variable introduced placed jth tensor factor monomial factorisation monomial depends set variables observe finally two monomials homologous deg deg shall also need polarisation operators defined polynomial formula denotes partial derivation respect variable polarisation operations degree preserving deg deg therefore leibniz rule proposition let assume char greater monomial deg proof consider factorisation derived described observe weight sequence hci deg deg lemma done hci hence lemma remains deg must deg say otherwise deg deg would follow take factorisation corresponding direct decomposition proceed induction deg assume first means deg say let arbitrary divisor degree deg let variable construction moreover consequently find proposition monomial done case let deg take divisor indices deg deg monomial homologous consequently proposition monomial exists claim follow proving end observe monomial hence induction hypothesis already holds moreover construction hence finished assumption allowed divide char proof theorem proposition see module generated elements degree equivalently top degree factor ring estimate whence conclude case proposition consider primitive third root unity given proof let variables conforming conventions spanned elements monomial easy argument shows xyz irreducible monomials enumerating ainvariant monomials degree see degree xyz assume follows observe however generators symmetric polynomials hand symmetric polynomial whence contradiction proves upper bound obtained argument similar propositions since many different details preferred give treatment case proposition char proof suppose holds monomial deg otherwise space spanned elements would contained let identify hci let recall factorisation corresponding direct decomposition deg may assume symmetry claim sequence possible mod let integer denoting maximum number sequences factored otherwise lemma applied hci hci get since hand subtracting inequality previous one yields whence claim deg factorisation variable deg factorisation setting enforces deg otherwise lemma contradiction therefore deg deg contradicting assumption sequence result variable dividing whence claim divisor deg monomial monomial let deg hbi induction deg assume already monomial hbi according factorisations variable two cases take one factorisations lemma done otherwise necessarily still however hbi hence lemma obtain factorisation falling case setting done proceed proof proposition sake nthe simplicity rename variables moreover abbreviate deg apply concluding contradiction otherwise deg still deg application may assume divisible monomial mxi falls case hence contradiction finally deg deg application may assume consider relation multiplying get left hand side three monomials occurring fall case right hand side whence follows contradiction completes proof comparing proposition immediately gives corollary char remark would interesting know theorem also extends whole case field whose characteristic divide case result acknowledgements author grateful domokos many valuable comments manuscript paper also thanks anonymous referee many suggestions improve presentation material references berkovich groups prime power order volume gruyter expositions mathematics gruyter berlin new york cziszter domokos groups large noether bound ann institut fourier cziszter domokos noether number groups cyclic subgroup index two journal algebra cziszter domokos generalised davenport constant noether number central european journal mathematics cziszter domokos geroldinger interplay invariant theory multiplicative ideal theory arithmetic combinatorics scott chapman fontana geroldinger olberding eds multiplicative ideal theory factorization theory cziszter domokos noether numbers davenport constants groups order less domokos noether bound polynomial invariants finite groups arch math basel fleischmann noether bound invariant theory finite groups fogarty noethers bound polynomial invariants finite group electron res announc amer math soc freeze schmid remarks generalization davenport constant discrete math issue december geroldinger grynkiewicz large davenport constant groups cyclic index subgroup pure appl algebra geroldinger factorizations algebraic combinatorial analytic theory monographs textbooks pure applied mathematics chapman grynkiewicz large davenport constant general upper bounds pure appl algebra pyber finite groups large noether number almost cyclic neusel smith invariant theory finite groups mathematical surveys monographs providence american mathematical society noether der endlichkeitssatz der invarianten endlicher gruppen math schmid finite groups invariant theory malliavin editor topics invariant theory number lecture notes mathematics pages springer serre representations des groupes finis hermann paris sezer sharpening generalized noether bound invariant theory finite groups algebra

secret sharing shared information nov johannes rauh abstract secret sharing cryptographic discipline goal distribute information secret set participants way specific authorized combinations participants together reconstruct secret thus secret sharing schemes systems variables clearly specified subsets information secret provide perfect model systems information decompositions however following intuition far leads information decomposition negative partial information terms difficult interpret one possible explanation partial information lattice proposed williams beer incomplete extended incorporate terms corresponding higher order redundancy results put bounds information decompositions follow partial information framework hint partial information lattice needs improved introduction williams beer proposed general framework decompose multivariate mutual information target random variable predictor random variables different terms called partial information terms according different ways combinations variables provide unique shared synergistic information williams beer argue decomposition based measure shared information underlying idea information classified according knows true situation question knows easy answer precisely secret sharing part cryptography goal distribute information secret set participants secret reconstructed certain authorized combinations participants join information see beimel survey set authorized combinations called access structure formally secret modelled random variable secret sharing scheme assigns random variable participant way authorized set participants function xik xik conversely authorized xik assumed participants know scheme authorized combination participants reconstruct secret join information secret sharing scheme perfect sets participants know nothing secret xik thus perfect secret sharing scheme clearly specified knows sense perfect secret sharing schemes provide model systems easy write information decomposition one connection secret sharing information decompositions set access structures secret sharing schemes participants mathematics subject classification key words phrases information decomposition partial information lattice shared information secret sharing johannes rauh correspondence partial information terms williams beer correspondence makes possible give another interpretation partial information terms namely partial information term measure similar given system random variables secret sharing scheme given access structure correspondence also allows introduce secret sharing property makes precise intuition information decomposition satisfies property perfect secret sharing scheme single partial information term corresponds access structure lemma states secret sharing property implied williams beer axioms shows secret sharing property plays well together ideas williams beer proposition shows information decomposition satisfies natural generalization property possible prescribe arbitrary nonnegative values partial information terms results suggest perfect secret sharing schemes fit well together ideas williams beer however following intuition far leads inconsistencies theorem shows extending secret sharing property pairs perfect secret sharing schemes leads negative partial information terms authors started build intuition negative partial terms argue may unavoidable information decompositions concluding section collects arguments claims proposes another possible solutions williams beer framework incomplete missing nodes represent higher order redundancy cryptography goal transport information coding theory also keep concealed unauthorized parties initiated many interesting developments information theory example introducing new information measures older ones see example maurer wolf csiszar narayan manuscript focuses another contribution cryptography probabilistic systems distribution information remainder article organized follows section summarizes definitions results secret sharing schemes section introduces different secret sharing properties fix values measure shared information assigns perfect secret sharing schemes combinations thereof main result section pairwise secret sharing property leads negative partial information terms section discusses implications incompatibility result perfect secret sharing schemes consider participants among want distribute information secret way control subsets participants together decrypt secret definition access structure family subsets closed taking supersets elements called authorized sets secret sharing scheme access structure family random variables whenever subsets secret sharing scheme perfect whenever secret sharing shared information condition perfection equivalent see beimel survey secret sharing theorem access structure exists perfect secret sharing scheme access structure entropy secret equals proof perfect secret sharing schemes arbitrary access structures first constructed ito construction entropy secret equals bit combining copies secret sharing scheme gives secret sharing scheme secret bit explained beimel claim distribution secret may perturbed arbitrarily long support distribution remains way possible prescribe entropy secret perfect secret sharing scheme example let independent uniform binary random variables let denotes addition modulo xor operation perfect secret sharing scheme access structure may little surprise integer addition modulo important building block many secret sharing schemes existence perfect secret sharing schemes solved remains problem finding efficient secret sharing schemes sense variables small possible sense small entropy given fixed entropy secret instance example see beimel survey since access structure closed taking supersets uniquely determined elements instance example first three elements belong set property element subset another element collection sets called antichain conversely antichain equals set elements unique access structure antichains natural lattice structure used williams beer order different values shared information organize call partial information lattice lattice also description terms secret sharing definition let antichains exists partial information lattice case depicted figure lemma let access structure let antichain authorized proof statement directly follows definitions johannes rauh information decompositions secret sharing schemes williams beer proposed decompose total mutual information target random variable predictor random variables according different ways combinations variables provide unique shared synergistic information one main ideas base decomposition single measure shared information function takes arguments list random variables first takes special role arrive decomposition variables taken combinations corresponding subsets simplicity xak denoted williams beer proposed list axioms measure satisfy follows axioms suffices consider function case antichain moreover monotone function partial information lattice definition thus natural write value lattice sum local terms corresponding antichains lie lattice terms called partial information terms representation always exists partial information terms uniquely defined using inversion however guaranteed always nonnegative nonnegative called locally positive williams beer also defined function denoted imin satisfies axioms locally positive framework intriguing attracted lot research special issue illustrates function imin critiziced measuring right thing difficulty finding reasonable measure shared information locally positive bertschinger rauh led argue maybe local positivity necessary requirement information decomposition issue discussed section goal section present additional natural properties measure shared information relate secret sharing intuition behind information decompositions perfect secret sharing scheme combination participants knows either nothing everything motivates following definition definition measure shared information secret sharing property access structure perfect secret sharing scheme access structure following holds authorized otherwise lemma secret sharing property implied williams beer axioms proof williams beer axioms imply whenever authorized hand authorized monotonicity axiom implies secret sharing shared information perfect secret sharing schemes lead information decompositions single nonzero partial information term lemma secret sharing property perfect secret sharing scheme access structure otherwise proof suppose let right hand side need show since inversion unique suffices show lemma authorized otherwise claim follows happens several secret sharing schemes involving participants order clear intuition assume secret sharing schemes satisfy following definition definition let access structures combination perfect secret sharing schemes access structures consists random variables perfect secret sharing scheme access structure definition ensures secrets independent sense knowing secrets provides information secrets formally one see secrets probabilistically independent follows example definition two access structures identical replace single random variable obtain smaller combination perfect secret sharing schemes combination perfect secret sharing schemes clear knows namely group participants knows secrets authorized knows nothing remaining secrets motivates following definition definition measure shared information combined secret sharing property combination perfect secret sharing schemes access structures entropy secrets authorized pairwise secret sharing property holds true special case johannes rauh combined secret sharing property implies pairwise secret sharing property pairwise secret sharing property follow williams beer axioms example imin satisfies williams beer axioms pairwise secret sharing property become apparent theorem one ask whether pairwise combined secret sharing properties compatible williams beer axioms question difficult answer since currently two proposed measures shared information satisfy williams beer axioms namely imin minimum mutual informations barrett immi min measures satisfy pairwise secret sharing property proposal function satisfies williams beer axioms arbitrarily many arguments several measures proposed bivariate case notably ired harder bertschinger appendix shows least satisfies combined secret sharing property far combinations perfect secret sharing schemes lead information decompositions nonzero partial information terms lemma assume combined secret sharing property combination perfect secret sharing schemes pairwise different access structures otherwise proof similar proof lemma omitted combined secret sharing property implies combination nonnegative values prescribed partial information values proposition suppose nonnegative number given antichain measure shared information satisfies combined secret sharing property exist random variables corresponding partial measure satisfies antichains proof theorem antichain exists perfect secret sharing scheme combine independent copies perfect secret sharing schemes let runs antichains independent combination perfect secret sharing schemes statement follows lemma unfortunately every random variable decomposed way combination secret sharing schemes however proposition suggests given measure shared information satisfies combined secret sharing property informally interpreted measure quantifies much looks like perfect secret sharing scheme access structure lemma suppose measure shared information satisfies pairwise secret sharing property independent secret sharing shared information language ince lemma says pairwise secret sharing property implies independent identity property pair perfect secret sharing proof let schemes access structures statement follows definition since authorized authorized incompatibility local positivity unfortunately although combined secret sharing property much fits intuition behind axioms williams beer incompatible nonnegative decomposition according partial information lattice theorem let measure shared information satisfies williamsbeer axioms pairwise secret sharing property nonnegative proof xor example already used bertschinger rauh prove incompatibility results properties information decompositions also used let independent binary uniform random variables let let observe situation symmetric particular also independent following values computed assumptions bit since function monotonicity axiom lemma monotonicity moreover bit since bit total entropy system bit bit bit denotes values vanish thus nonnegative note random variables proof theorem form three perfect secret sharing schemes satisfy definition combination perfect secret sharing schemes three secrets independent independent lemma apply remark xor example proof theorem already used bertschinger rauh criticized chicharro panzeri grounds involves random variables stand deterministic functional relation sense chicharro panzeri argue case appropriate use full partial information lattice instead functional relationship used eliminate identify nodes lattice thus monotonicity axiom williams beer implies part partial information lattice axiom also implies xor example similarly excluded lattice analyzing particular example note first argument johannes rauh figure partial information lattice node indexed antichain values bit shared information xor example proof theorem according pairwise secret sharing property given colon formal argument valid joint distributions second argument takes account particular underlying distribution easy work around objection deterministic relationship disappears arbitrarily small stochastic noise added joint distribution precise let independent binary random variables let binary otherwise example proof recovered assuming partial information terms depend continuously joint distribution partial information term still negative small thus assuming continuity conclusion theorem still holds true information decomposition according full partial information lattice considered random variables satisfy functional deterministic constraint remark analyzing proof theorem one sees independent identity axiom lemma main ingredient arrive contradiction property also arises uses xor example bertschinger rauh discussion perfect secret sharing schemes correspond systems random variables clearly specified knows system easy assign intuitive values shared information nodes partial information lattice one may conjecture intuition behind assignment intuition underlies williams beer axioms define partial information lattice moreover following intuition independent combinations perfect secret sharing schemes used tool construct systems random variables prescribable nonnegative values partial information secret sharing shared information unfortunately extension independent combinations perfect secret sharing schemes without problems theorem leads decompositions negative partial information terms mean examples derived intuition williams beer axioms contradict axioms way indication whole idea information decomposition work question posed first paragraph introduction answered affirmatively several ways dilemma first solution assign different values combinations perfect secret sharing schemes solution pursued text would change interpretation information decomposition measuring knows second solution accept negative partial values information decomposition argued negative values information given intuitive interpretation terms confusing misleading information also called local information quantities mutual information log interpretation goes back early days information theory fano sometimes phenomenon called misinformation ince wibral however usual language misinformation refers false incorrect information especially intended trick someone macmillan publishers limited retrieved effect modelled thus word misinformation avoided order mislead reader wrong intuition negative information quantities situation problematic average quantities agent receives sideinformation form value relevant random variable changes strategy prior strategy based prior distribution new strategy based posterior clearly probabilistic setting change strategy lead better worse result single instance average though never hurts never advantageous average ignore mutual information never negative similarly natural expect information quantities difficult imagine correct aspect thereof misleading average situation different incorrect information interpretation negative value much easier conceptually would suspect averaged information quantity may change sign actually conflates different aspects interaction information conflates synergy redundancy williams beer case allowing negative partial values alters interpretation information decomposition point questionable whether word decomposition still appropriate decomposing object parts parts reasonable way example fourier decomposition function fourier components never larger function sense sum squared fourier coefficients equals squared original function another example given positive amount money two investment options may indeed possible invest negative share total amount one two options order increase funds invested second option argue whether true quantities recently ince suggested also write mutual information difference quantities johannes rauh however short selling regulated many countries much stronger rules ordinary trading claim information decomposition negative partial information terms possibly make sense however made clear precisely interpret negative terms important distinguish correct information leads suboptimal decision due unlikely events happening bad luck incorrect information leads decisions based wrong posterior probabilities opposed correct conditional probabilities third solution change underlying lattice structure decomposition first step direction done chicharro panzeri propose decompose mutual information according subsets partial information lattice however also conceivable lattice enlarged williams beer derived partial information lattice axioms together assumption everything expressed terms shared information according knows shared information sometimes equivalently called redundant information may necessary distinguish two information shared several random variables information accessible single random variable redundancy also arise higher orders example infamous xor example proof theorem example pair independent contains two bits total system two bits therefore one bit redundancy however redundancy bit located anywhere specifically contained either thus shared information since redundant bit part shared sense phenomenon corresponds fact random variables pairwise independent without independent kind higher order redundancy place partial information lattice may nodes corresponding higher order redundancy added lattice enlarged way structure inversion changed possible resulting lattice leads nonnegative partial information terms without changing cumulative information values already present original lattice approach succeeds answer question introduction negative simply classifying information according knows shared information work since capture higher order redundancy analysis extensions partial information lattice scope future work acknowledgments thank fero teaching secret sharing schemes grateful guido pradeep banerjee remarks manuscript nils bertschinger jost eckehard olbrich many inspiring discussions topic thank reviewers many comments particular concerning discussion thank organizers participants pid workshop december frankfurt material first presented secret sharing shared information appendix combined secret sharing properties small section discusses defining equation combined secret sharing property case incorporated definition combination perfect secret sharing schemes following lemma implies measure shared information satisfies satisfies recall williams beer axiom implies lemma let combination perfect secret sharing schemes access structures proof suppose secret authorized hand independence remark definition thus next result shows bivariate measure shared information proposed bertschinger satisfies reader referred loc cit definitions elementary properties proposition let combination perfect secret sharing schemes access structures proof given suppose secrets least one authorized secrets neither authorized alone let joint distribution let set alternative joint distributions marginal distributions subsets according need compare elements find definition maximum subscript indicates respect joint distributions conditional entropy evaluated define distribution secrets independent marginally independent independent johannes rauh pair hand function either function thus hand joint distribution secrets functions whence follows solves optimization problem definition suppose secrets authorized secrets authorized one computes whence references williams beer nonnegative decomposition multivariate information beimel schemes survey proceedings third international conference coding cryptology berlin heidelberg maurer wolf intrinsic conditional mutual information perfect secrecy proc ieee isit csiszar narayan secrecy capacities multiple terminals ieee transactions information theory ito saito nishizeki secret sharing scheme realizing general access structure proceedings ieee global telecommunication bertschinger rauh olbrich jost shared information new insights problems decomposing information complex systems proc eccs springer rauh bertschinger olbrich jost reconsidering unique information towards multivariate information decomposition proc ieee isit barrett exploration synergistic redundant information sharing static dynamical gaussian systems corr harder salge polani bivariate measure redundant information phys rev bertschinger rauh olbrich jost quantifying unique information entropy secret sharing shared information ince measuring multivariate redundant information pointwise common change surprisal entropy chicharro panzeri synergy redundancy dual decompositions mutual information gain information loss entropy fano transmission information mit press cambridge wibral lizier priesemann bits brains biologically inspired computing frontiers robotics macmillan publishers limited macmillan dictionary available http retrieved ince partial entropy decomposition decomposing multivariate entropy mutual information via pointwise common surprisal address jrauh max planck institute mathematics sciences leipzig germany

new class permutation trinomials constructed niho exponents oct tao bai yongbo abstract permutation polynomials finite fields interesting subject due important applications areas mathematics engineering paper investigate trinomial xpq finite field odd prime positive integer shown permutation trinomial even property also true general class polynomials nonnegative integer gcd moreover also show permutation trinomials proposed new sense multiplicative equivalent previously known ones similar form index terms finite fields permutation polynomials trinomials niho exponents multiplicative inequivalent ams introduction let denote finite field elements prime power polynomial called permutation polynomial associated mapping permutes permutation polynomials firstly studied hermite finite prime fields dickson arbitrary finite fields wide applications coding theory cryptography combinatorial designs finite field total permutation polynomials obtained lagrange interpolation permutations terms particular interest simple algebraic expressions especially permutation binomials trinomials attracted particular attention recent achievements study permutation polynomials surveyed let prime positive integer niho exponent finite field positive integer satisfying mod nonnegative integer case called normalized niho exponent researches past decades demonstrate niho exponents corresponding author bai xia department mathematics statistics university nationalities wuhan china xia also hubei key laboratory intelligent wireless communications university nationalities wuhan china xia good resources lead desirable objects sequence design coding theory cryptography recently lot permutation trinomials form proposed two integers coefficients restricted helleseth gave rather detailed list known pairs new pairs permutation polynomial permutation trinomials similar form also presented investigated several permutation trinomials form proposed three conjectures later confirmed recently derived series sufficient conditions permute also permutation polynomials constructed niho exponents power arbitrary prime hou characterized necessary sufficient conditions coefficients polynomial bxq permutation mod necessary sufficient conditions permutation polynomial determined positive integer let denote trace function permutation trinomials form obtained niho exponent paper investigate permutation property following trinomial xpq odd prime positive integer easily verified niho exponents show permutation polynomial even however case result may hold furthermore prove property also true general polynomials nonnegative integer gcd addition permutation polynomials presented shown new sense multiplicative equivalent permutation polynomials form remainder paper organized follows section gives preliminaries notation including useful lemmas section give proofs main results section devoted demonstrating permutation trinomials given new section concludes study preliminaries let prime positive integer trace function norm function denoted respectively namely unit circle defined order prove permutation property trinomials constructed niho exponents authors mainly used following lemma proved park lee reproved zieve lemma let prime positive integer assume positive integer fpn integer permutation fpn gcd permutes set root unity polynomials constructed niho exponents always rewritten form determine permutation property polynomials constructed niho exponents lemma main task decide unit circle however sometimes corresponding polynomial permutes leads fractional polynomial high degree still difficult problem general verify permutes another general approach investigating permutation property polynomials constructed niho exponents concentrate subset specifically assume polynomial constructed niho exponents coefficients show permutation permutation respectively permutation polynomial end usually required property key step proof prove permutation idea originated later used paper use idea prove main result following lemma needed sequel proof trivial omitted lemma let prime power denote trace function norm function respectively uniquely determined pair following lemma obtained direct computations lemma let prime power trace function norm function respectively proof give proof lemma expressions given compute illustrate obtain results note implies substituting get desired result lemma let defined following results root even two roots otherwise root even two roots otherwise proof let primitive root roots belong divisible since divisible odd follows desired result note irreducible polynomial solution since degree follows two roots belong rewrite two roots denote two roots even since odd thus even equal odd therefore odd computations follows desired result following lemma special case exercise reader convenience include proof lemma let finite field characteristic permutation polynomial power element proof note permutation polynomial root thus permutation polynomial root latter exactly means power new class permutation trinomials niho exponents section present main results permutation property defined defined first main result given following theorem theorem let trinomial defined permutation polynomial even prove theorem detail mention characterizations three exponents appearing niho exponents since however exponents may niho exponents see sequel permutation property depend condition utilizing lemma obtain condition permute also true addition permutation polynomial form closely related permutation polynomial form next section study relationship theorem previously known permutation trinomials form comparison show permutation polynomial proposed new order prove theorem following preparatory lemma needed lemma let defined even proof assume note xpq equivalent xpq dividing equation get setting rewritten equals note defined conclude satisfies lemma even root thus case odd two roots shows elements similarly obtain conclusion lemma discussions follows desired result proof theorem note proof restricted prove permutation suffices show exactly one root note permutation since gcd hand odd proof lemma elements therefore odd must exist least two distinct elements thus permutation polynomial odd next prove even permutation polynomial exactly one root consider following two cases case lemma derive must belong thus case equivalent obviously latter one root case lemma roots belong next show exactly one root given conditions let defined compute follows xpq xpq xpq xpq lemma expressed terms follows follows prove uniquely determined aforementioned conditions give proof conclusion case proved way thus sequel always assume note gcd thus one knows uniquely determined therefore suffices show also uniquely determined consider following two subcases subcase follows second equation thus also uniquely determined subcase convenience put divided second equation rewritten equivalently claim equal otherwise implies leading contradiction assumption let becomes note equal since equal therefore transformed show fourth power lemma conclude uniquely determined follows uniquely determined thus suffices show fourth power end express terms follows since follows let suppose contrary fourth power note thus rewrite implies denote two root note even two roots belong follows contradicts since therefore fourth power desired result follows discussions subcases show even derive uniquely determined conclusion similarly proved furthermore lemma one conclude set uniquely determined note one satisfies instance since thus even one root cases follows desired result corollary let defined following fractional polynomial permutes even proof note written lemma theorem follows permutes even note permutes equal zero thus written exactly since based lemma corollary obtain second main result gives permutation property defined theorem let nonnegative integer satisfying gcd permutation polynomial even proof rewrite note gcd gcd lemma permutation polynomial permutes latter equivalent permutes since desired result follows corollary remark shown proofs corollary theorem permutation polynomial permutes shows permutation property depend condition however seems difficult verify directly whether condition holds thus paper use different approach investigate permutation property obtain permutation property remark let defined respectively polynomial contains special case taking transformed exactly note xpq always permutation polynomials positive integer theorems may hold magma obtained numerical results table table permutation permutation yes yes yes comparison known related permutation trinomials section compare permutation polynomials proposed theorem previously known ones form straightforward composition two permutation polynomials finite field also permutation polynomial recall definition multiplicative equivalence definition let prime power two permutation polynomials called multiplicative equivalent exists integer gcd next determine whether permutation trinomials given theorem multiplicative equivalent previously known ones form make preparations follows proposition let defined even following results multiplicative equivalent following permutation trinomials iii multiplicative equivalent following permutation trinomials iii proof prove case case result proved way note thus multiplicative equivalent let even gcd gcd extended euclidean algorithm denotes inverse modulo note therefore multiplicative equivalent easily seen pairwise multiplicative equivalent following claims needed claim recall inverse normalized niho exponent exists normalized niho exponent product two normalized niho exponents also normalized niho exponents let permutation polynomial form one invertible say also permutation polynomial form multiplicative equivalent analysis together claim gives following claim claim let permutation polynomial form permutation trinomials form multiplicative equivalent given xdi provided gcd claims together proposition give following claim claim let permutation polynomial theorem permutation trinomial form multiplicative equivalent must one proposition sequel permutation polynomial form denoted tuple according lemma permutation polynomial associated polynomial permutes unit circle permutation polynomial written table known permutation trinomials form fractional polynomial ref mod theorem odd theorem conjecture mod conjecture even theorem even theorem even theorem even theorem fractional polynomial since called fractional polynomial comparison purposes collect known permutation trinomials form list tables respectively best knowledge tables contain permutation trinomials completely note last one table exactly proposition iii hou determined permutation trinomials form bxq according theorem permutation polynomial square equivalent even permutation polynomial equals thus permutation property derived theorem according tables proposition claim conclude following result proposition let permutation polynomial proposed theorem multiplicative equivalent permutation polynomial contained theorem multiplicative equivalent permutation trinomial listed table proposition shows proposed theorem indeed new permutation polynomial proposed theorem multiplicative equivalent known one contained nevertheless method proving permutation property paper different table known permutation trinomials form fractional polynomial ref theorem odd theorem odd theorem odd theorem even theorem even theorem even theorem even theorem even theorem iii odd proposition theorem even proposition theorem even theorem even theorem even theorem even theorem odd even theorem iii theorem denotes exponent canonical factorization conclusion paper construct class permutation trinomials precisely prove xpq permutation trinomial even conclusion also true general polynomials nonnegative integer satisfying gcd moreover prove presented multiplicative equivalent known permutation trinomial form numerical experiments show theorems may hold would nice construction generalized arbitrary finite field namely readers invited determine permutation polynomials form xpq prime positive integer acknowledgment bai xia supported part national natural science foundation china grant grant part natural science foundation hubei province grant bai also supported graduate innovation fund university nationalities references bartoli giulietti permutation polynomials fractional polynomials algebraic curves available online http chen zeng class binary cyclic codes generalized niho exponents finite fields vol ding yuan family skew hadamard difference sets comb theory ser vol ding wang yuan yuan permutation trinomials finite fields even characteristic siam dis vol dobbertin almost perfect nonlinear power functions welch case ieee trans inf theory vol apr dobbertin felke helleseth rosendahl niho type functions via dickson polynomials kloosterman sums ieee trans inf theory vol dobbertin leander canteaut carlet felke gaborit construction bent functions via niho power functions comb theory ser vol gupta sharma new classes permutation trinomials finite fields even characteristic finite fields vol hou determination type permutation trinomials finite fields acta vol hou determination type permutation trinomials finite fields finite fields vol hou permutation polynomials finite survey recent advances finite fields vol hou permutation polynomials form nonexistence result available online http kyureghyan zieve permutation polynomials form available online https permutation polynomials applications coding theory finite fields vol helleseth tang results class permutation polynomials finite fields finite fields vol helleseth several classes permutation trinomials niho exponent cryptogr vol helleseth new permutation trinomials niho exponents finite fields even characteristic available online http two conjectures permutation trinomials finite fields vol lidl niederreiter finite fields encyclopedia mathematics applications vol amsterdam netherlands chen new classes permutation binomials permutation trinomials finite fields finite fields vol new permutation trinomials constructed fractional polynomials available online https note permutation polynomials finite fields available online https mullen panario handbook finite fields boca raton taylor francis park lee permutation polynomials group permutation polynomials bull austral math vol tan tan constructing differentially permutations via switching method ieee trans inf theory vol july zeng class binomial permutation polynomials available online http zeng several classes complete permutation polynomials finite fields vol yuan ding permutation trinomials finite fields vol several classes permutation trinomials niho expeonents available online http xia zeng helleseth open problem distribution niho type function ieee trans inf theory vol zeng tian permutation polynomials trace functions finite fields finite fields vol zeng jiang yue cao weight distribution class cyclic codes finite fields vol zieve permutation polynomials form proc amer math soc vol zha fan results permutation trinomials finite fields even characteristic finite fields vol

dynamic ntegration background nowledge eural nlu ystems dirk weissenborn german research center chris dyer deepmind tkocisky cdyer oct bstract background knowledge required understand natural language neural natural language understanding nlu systems requisite background knowledge indirectly acquired static corpora develop new reading architecture dynamic integration explicit background knowledge nlu models new reading module provides refined word representations nlu architecture processing background knowledge form statements together taskspecific inputs strong performance tasks document question answering dqa recognizing textual entailment rte demonstrate effectiveness flexibility approach analysis shows models learn exploit knowledge selectively semantically appropriate way ntroduction understanding natural language depends crucially background knowledge example knowledge concepts expressed words read lexical knowledge relations hold concepts relational knowledge simple illustration agent needs understand statement king farouk signed abdication entailed king farouk exiled france signing resignation must know among things abdication means resignation king neural natural language understanding nlu systems requisite background knowledge implicitly encoded models parameters background knowledge present learned task supervision also word embeddings distributional information reliably reflects certain kinds useful background knowledge semantic relatedness however acquisition background knowledge static training corpora limiting two reasons first expect background knowledge could important solving nlu task extracted limited amount training data second world changes facts may influence text understood likewise change short building suitably large corpora capture relevant information keeping corpus derived models date changes world would impractical paper develop new architecture dynamically incorporating external background knowledge nlu models rather relying static knowledge implicitly present training data supplementary knowledge retrieved knowledge base assist understanding text inputs since nlu systems must necessarily read understand text inputs approach incorporates background knowledge repurposing reading read text understood together supplementary natural language statements assert facts assertions relevant understanding content nlu systems operate series phases first given text input system must understand call context set relevant supporting assertions retrieved learning retrieve relevant information solving nlu tasks important question nogueira cho narasimhan inter alia work focus learning incorporate retrieved information use simple heuristic retrieval methods identify plausibly relevant background external knowledge base supplementary texts retrieved use word embedding refinement strategy incrementally reads context retrieved assertions starting word embeddings building successively refined embeddings words ultimately reflect relevant supporting assertions input context contextually refined word embeddings serve dynamic memory store newly incorporated knowledge used reading architecture overall architecture illustrated figure although incorporating new kind information nlu pipeline strength approach architecture reading module independent final nlu requirement final architecture use word embeddings carry experiments several different datasets tasks document question answering dqa recognizing textual entailment rte evaluating impact proposed solution basic task architectures sophisticated task architecture rte find embedding refinement strategy quite effective four standard benchmarks show refinement refining embeddings using context additional background information improve performance significantly adding background knowledge helps results competitive setting new recent triviaqa benchmarks remarkable considering simplicity chosen architecture finally provide detailed analysis knowledge used rte system including experiments showing system capable making appropriate counterfactual inferences provided false knowledge xternal nowledge upplementary ext nputs knowledge resources make information could potentially useful improving nlu available variety different formats subject predicate object relational databases structured formats rather tailoring solution particular structured representation assume supplementary information either already exists natural language statements easily recoded natural language contrast mapping unstructured structured representations inverse problem terribly difficult example given triple monkey isa animal construct assertion monkey animal using simple rules finally format means knowledge exists unstructured text form usable system major question remains answered given text understood supplementary knowledge incorporated retrieval contextually relevant information knowledge sources complex research topic likewise crucially dependent format underlying knowledge base several statistical manning recently neural approaches mitra craswell approaches based reinforcement learning nogueira cho work make use simple heuristic almost exhaustively retrieve potentially relevant assertions see rely reading architecture learn extract relevant information next section turn question leverage retrieved supplementary knowledge encoded text nlu system efining ord mbeddings eading order incorporate information retrieved input texts propose compute contextually refined word representations prior processing nlu task hand pass task form word embeddings word embeddings thus serve form memory contains knowledge typical neural nlu systems also contextual information including retrieved background knowledge incremental refinement process encodes input texts followed updates word embedding matrix using encoded input multiple reading steps words first represented standard word type embeddings conceived columns embedding matrix progressive reading step new embedding matrix constructed refining embeddings previous step using contextual information reading step set natural language sequences texts illustration incremental refinement strategy found figure group onlookers glance watch people premise group onlookers glance person strange trick head group onlookers glance watch people hypothesis people watch another person trick group onlookers glance watch people assertions onlooker related watches people group group onlookers glance watch people task system rte group onlookers glance person strange trick head people watch another person trick entailment update weighted update figure illustration refinement strategy word representations example snli dataset comprising premise hypothesis additional external information form assertions reading architecture constructs refinements word representations incrementally conceptually represented columns series embedding matrices incrementally refined reading input text textual renderings relevant background knowledge computing representations used task model figure rte following define procedure formally denote hidden dimensionality model layer nrefined ord mbeddings first representation level consists word representations word representations depend input conceived embedding matrix whose columns indexed words word representation single word computed using gated combination fixed word vectors epw learned embeddings echar formal definition combination given epw relu epw epw char epw echar compute echar using convolutional neural network using convolutional filters width followed operation time combining character based word embeddings way common practice approach follows seo weissenborn ontextually efined ord epresentations order compute contextually refined word embeddings given prior representations assume given set texts read refinement iteration text sequence word tokens embed tokens every using embedding matrix previous layer word concatenate vector length position set indicating layer currently stacking vectors matrix obtain matrix processed bidirectional recurrent neural network bilstm hochreiter schmidhuber work resulting output projected layer followed relu relu bilstm word initially maxpool representations finally update previous embedding occurrences matching lemma every resulting finally combine representation form representation via gated addition lets model determine much revise embedding newly read information max lemma lemma note soften matching condition using lemmatization lemma pooling operation contextual information certain words usually independent current word form appear consequence minor linguistic step allows additional interaction tokens lemma important difference contextual refinement step conventional rnn architectures pooling operation performed occurrences tokens share lemma effectively connects different positions within different texts thereby mitigating problems arising dependencies importantly however allows models make use additional input relevant background knowledge xperimental etup run experiments four benchmarks two popular tasks namely recognizing textual entailment rte document question answering dqa following describe different aspects experimental setup detail models primary interest explore value refinement strategy relatively generic task architectures therefore chose basic bidirectional lstms bilstms encoders neural network top final prediction models common baselines nlu tasks considered general reading architectures opposed highly tuned nlu systems necessary adding feature lets refinement model learn update word embeddings differently different levels achieve art results however since models frequently underperform customized architectures also add refinement module reimplementation architecture rte called esim chen models trained jointly refinement module dqa baseline system add simple feature liq suggested weissenborn encoding context compare competitive baseline results provide exact model implementations bilstm baselines general training details appendix question answering apply dqa models recent dqa benchmark datasets squad rajpurkar triviaqa joshi task predict answer span within provided document given question datasets containing order examples triviaqa collected via distant supervision test set divided large noisy distant supervision part much smaller order hundreds human verified part report results see appendix implementation details recognizing textual entailment test frequently used snli dataset bowman collection sentence pairs recent multinli dataset sentence pairs williams given two sentences premise hypothesis task determine whether either entails contradicts neutral see appendix implementation details knowledge source make use speer havasi semantic network originated open mind common sense project incorporates selected knowledge various knowledge sources open multilingual opencyc presents information form relational assertion retrieval would like obtain information relations words phrases conceptnet order strengthen connection two sequences assertions conceptnet come form subject predicate object retrieve assertions appears appears vice versa still many assertions might retrieved instance rank retrievals based respective subject object end compute ranking score inverse product appearances subject object score denotes indicator function related popular idf score inverted document frequency information retrieval ranks terms higher appear less frequently across different documents training evaluation retain assertions specify individual experiments separately note although rarely might happen assertions retrieved refinement order employing strategy first read document followed question case dqa premise followed hypothesis rte additional knowledge form set assertions integrated reading input dqa rte preliminary experiments found final performance significantly sensitive order presentation decided fix order defined model squad dev exact triviaqa wiki test exact triviaqa web test exact bilstm liq reading reading knowledge reading knowledge sota results table results squad development set well test sets models triviaqa results divided distant supervision results left human verified results right model using external knowledge trained retrieved conceptnet assertions used baseline wang pan esults uestion nswering uad riviaqa table presents results two question answering benchmarks report results squad development two challenging triviaqa test sets demonstrate introduction reading architecture helps consistently additional gains using background knowledge systems even outperform current models triviaqa surprising given simplicity architecture complexity others instance system uses complex attention mechanism achieve results even baseline bilstm liq system reaches competitive results triviaqa line findings weissenborn verify additional computation gives performance boosts using reading architecture without knowledge also ran experiments bilstms baselines exhibit similar computational complexity bilstm reading found second layer even hurts performance demonstrates pooling occurrences given context layers constitutes main difference conventional stacked rnns powerful yet simple technique case important finding experiments knowledge actually helps considerably improvements measures ecognizing extual ntailment snli ulti nli table shows results rte experiments general introduction refinement strategy almost always helps without external knowledge providing additional background knowledge conceptnet bilstm based models improve substantially models improve difficult multinli dataset compared previously published systems models acquit well multinli benchmark competitively snli benchmark parallel work gong developed novel architecture rte achieves slightly better performance multinli based worth observing embedding architecture generic task model outperforms esim multinli architecturally much complex designed specifically rte task finally remark despite careful tuning esim fails http http http http exclude conceptnet assertions created one contributor verbosity reduce noise due restrictions code sharing able use public evaluation server obtain test set scores squad however remaining tasks report development accuracy test set performance refinement architecture used course new model model snli dev test mnli matched dev test mnli mismatched dev test bilstm reading reading knowledge esim reading reading knowledge sota results table results snli well models model using external knowledge trained retrieved conceptnet assertions chen gong chen snli esim reading reading knowledge esim reading reading knowledge multinli matched mismatched table development set results reducing training data embedding dimsensionality pca parenthesis report relative differences respective result directly match reported chen however multinli find implementation esim performs considerably better approximately instability results suggests well failure custom consistently perform well suggests current sota rte models may overfit snli dataset educing raining data imensionality trained ord mbeddings find little impact using external knowledge rte task using sophisticated task model esim hypothesize attention mechanisms within esim jointly powerful word representations allow recovery important lexical relations trained large dataset follows reducing number training data impoverishing word representations impact using external knowledge become larger test hypothesis gradually impoverish word embeddings reducing dimensionality pca reducing number training instances joint data dimensionality reduction results presented table show indeed slightly larger benefit employing background knowledge impoverished settings largest improvements using novel reading architecture using around examples reduced dimensionality however observe biggest overall impact baseline esim model stems contextual refinement strategy reading especially pronounced experiments highlights usefulness refinement strategy even without use additional knowledge although reducing either embedding dimensionality data individually exhibit similar less pronounced results report joint reduction results style seems amateurish look like amateur net cost operations gross cost guys file final exams men filed midterm exams look like synonym seem contradiction gross antonym net contradiction midterm antonym final contradiction look like antonym seem entailment gross synonym net entailment midterm synonym final entailment table three examples antonym synonym swapping experiment multinli assertion bilstm multinli esim multinli bilstm squad figure performance differences ignoring certain types knowledge relation predicates evaluation normalized performance differences measured subset examples assertion respective relation predicate occurs nalysis nowledge tilization additional knowledge used verify whether models make use additional knowledge conducted several experiments first evaluated models trained knowledge tasks providing knowledge test time ablation drops performance accuracy multinli squad indicates model refining representations using provided assertions useful way models sensitive semantics provided knowledge previous result show models utilize provided assertions consistent way may reflect mismatch training testing conditions therefore test models sensitivity towards semantics assertions run experiment swap synonym antonym predicate provided assertions test time heuristic retrieval mechanism counterfactuals affect truth inference still expect see significant impact performance drop multinli examples either synonym retrieved bilstm esim model large drop clearly shows models sensitive semantics provided knowledge examples prediction changes presented table demonstrate system learned trust presented assertions point make appropriate counterfactual change knowledge caused change prediction knowledge used establishing models somehow sensitive semantics wanted find type knowledge important task analysis exclude assertions including prominent predicates knowledge base individually evaluating models results presented figure demonstrate biggest performance drop total blue bars stems related assertions prominent predicate appears much frequently assertions helps connecting related parts input sequences believe related assertions offer benefits mainly modeling perspective strongly connecting input sequences thus bridging dependencies similar attention looking relative drops obtained normalizing performance differences actually affected examples green find models depend highly presence antonym synonym assertions tasks well partially derived assertions interesting finding shows sensitivity models selective wrt type knowledge task fact largest relative impact stems antonyms interesting known information hard capture distributional semantics contained word embeddings elated ork role background knowledge natural language understanding long remarked especially context classical models schank abelson minsky however recently begun play role neural network models nlu ahn long dhingra however previous efforts focused specific tasks certain kinds knowledge whereas take step towards solution integration heterogeneous knowledge nlu systems providing simple reading architecture read background knowledge encoded simple natural language statements abdication type resignation bahdanau use textual word definitions source information embeddings oov words area visual question answering utilize external knowledge form dbpedia comments short improve answering ability model marino explicitly incorporate knowledge graphs image classification model created recall mechanism standard lstm cell retrieves pieces external knowledge encoded single representation conversation model concurrently dhingra exploit linguistic knowledge using adapation grus handle graphs however external knowledge present form triples main difference approach incorporate external knowledge free text form word level prior processing task hand constitutes flexible setup ahn exploit knowledge base facts mentioned entities neural language models bahdanau long create word embeddings reading word definitions prior processing task hand pilehvar seamlessly incorporate information word senses representations solving downstream nlu task similar one step seamlessly integrating kinds assertions concepts might relevant task hand another important aspect approach notion dynamically updating wordrepresentations tracking updating concepts entities sentences dynamic memories active research direction kumar henaff kobayashi however works typically focus particular tasks whereas approach taskagnostic importantly allows integration external background knowledge related work includes storing temporary information weight matrices instead explicit neural activations word representations biologically plausible alternative onclusion presented novel reading architecture allows dynamic integration background knowledge neural nlu models solution based incremental refinement word representations reading supplementary inputs flexible used virtually existing nlu architecture rely word embeddings input results show embedding refinement using system text inputs well supplementary texts encoding background knowledge yield large improvements particular shown relatively simple task architectures based simple bilstm readers become competitive architectures augmented reading architecture acknowledgments research conducted internship first author deepmind partially supported german federal ministry education research bmbf projects sides bbdc software campus genie eferences sungjin ahn heeyoul choi tanel yoshua bengio neural knowledge language model arxiv dzmitry bahdanau tom bosc jastrzebski edward grefenstette pascal vincent yoshua bengio learning compute word embeddings fly arxiv samuel bowman gabor angeli potts christopher christopher manning large annotated corpus learning natural language inference emnlp association computational linguistics qian chen xiaodan zhu ling wei hui jiang diana inkpen recurrent neural sentence encoder gated attention natural language inference arxiv qian chen xiaodan zhu zhenhua ling wei hui jiang enhancing combining sequential tree lstm natural language inference acl bhuwan dhingra zhilin yang william cohen ruslan salakhutdinov linguistic knowledge memory recurrent neural networks arxiv yichen gong heng luo jian zhang natural language inference interaction space arxiv mikael henaff jason weston arthur szlam antoine bordes yann lecun tracking world state recurrent entity networks iclr sepp hochreiter schmidhuber long memory neural computation minghao yuxing peng xipeng qiu reinforced mnemonic reader machine comprehension arxiv yangfeng chenhao tan sebastian martschat yejin choi noah smith dynamic entity representations neural language models emnlp mandar joshi eunsol choi daniel weld luke zettlemoyer triviaqa large scale distantly supervised challenge dataset reading comprehension acl july diederik kingma jimmy adam method stochastic optimization iclr sosuke kobayashi naoaki okazaki kentaro inui neural language model dynamically representing meanings unknown words entities discourse arxiv preprint ankit kumar ozan irsoy peter ondruska mohit iyyer james bradbury ishaan gulrajani victor zhong romain paulus richard socher ask anything dynamic memory networks natural language processing icml teng long emmanuel bengio ryan lowe jackie chi kit cheung doina precup world knowledge reading comprehension rare entity prediction hierarchical lstms using external descriptions emnlp christopher manning hinrich foundations statistical natural language processing volume mit press kenneth marino ruslan salakhutdinov abhinav gupta know using knowledge graphs image classification cvpr marvin minsky interfaces communications acm bhaskar mitra nick craswell neural models information retrieval arxiv preprint karthik narasimhan adam yala regina barzilay improving information extraction acquiring external evidence reinforcement learning emnlp rodrigo nogueira kyunghyun cho query reformulation reinforcement learning emnlp boyuan pan hao zhou zhao bin cao deng cai xiaofei memen embedding memory networks machine comprehension arxiv jeffrey pennington richard socher christopher manning glove global vectors word representation emnlp mohammad taher pilehvar jose roberto navigli nigel collier towards seamless integration word senses downstream nlp applications acl pranav rajpurkar jian zhang konstantin lopyrev percy liang squad questions machine comprehension text tim edward grefenstette karl moritz hermann phil blunsom reasoning entailment neural attention iclr roger schank robert abelson scripts plans goals understanding psychology press minjoon seo aniruddha kembhavi ali farhadi hananneh hajishirzi attention flow machine comprehension iclr robert speer catherine havasi representing general relational knowledge conceptnet lrec wenhui wang nan yang furu wei baobao chang ming zhou gated networks reading comprehension question answering acl dirk weissenborn georg wiese laura seiffe making neural simple possible simpler conll adina williams nikita nangia samuel bowman challenge corpus sentence understanding inference arxiv peng wang chunhua shen anton van den hengel anthony dick ask anything visual question answering based knowledge external sources cvpr zhen bingquan liu baoxun wang chengjie sun xiaolong wang incorporating loosestructured knowledge lstm recall gate conversation modeling arxiv mplementation etails following explain detailed implementation two baseline models assume computed contextually refined word representations depending setup embedded input sequences qlq plp respectively word representation update gate initialized bias refine representations slightly beginning training following denote hidden dimensionality model layer uestion nswering encoding dqa task refers question supporting text first process sequences identical bilstms parallel followed separate linear projections bilstm bilstm initialized identity matrix prediction answer layer weissenborn first compute weighted representation processed question softmax probability distribution location answer computed mlp relu activated hidden layer follows relu fcs exp relu fce exp model trained minimize loss predicted start end positions respectively evaluation extract span best maximum token length ecognizing extual ntailment encoding analogous dqa encode input sequences bilstms however rte use conditional encoding instead therefore initially process embedded hypothesis bilstm use respective end states forward backward lstm initial states forward backward lstm processes embedded premise prediction hconcatenate outputs forward backward lstms processing premise run resulting outputs fullyconnected layer relu activation followed operation time resulting hidden state finally used predict rte label follows relu maxpool exp probability choosing category entailment contradiction neutral defined finally model trained minimize loss predicted category probability distribution raining steps lowercase inputs tokenize additionally make use lemmatization described necessary matching word representations use glove pennington employed adam kingma optimization initial halved whenever measure dqa accuracy rte dropped development set minibatches dqa rte respectively used size dqa rte additionally regularization make use dropout rate computed word representations defined dropout mask words batch models trained different random seeds top performance reported

polychronous interpretation synoptic domain specific modeling language embedded besnard gautier ouy talpin bodeveix cortier pantel strecker garcia rugina buisson dagnat besnard gautier ouy talpin inria rennes bretagne atlantique irisa campus beaulieu rennes cedex france bodeveix cortier pantel strecker paul sabatier route narbonne toulouse cedex france bodeveix cortier pantel strecker garcia thales alenia space boulevard midi cannes france rugina eads astrium rue des cosmonautes palays toulouse cedex france buisson dagnat institut bretagne brest iroise brest cedex france spacify project aims bringing advances mde satellite flight software industry advocates approach built modeling language named synoptic line previous approaches modeling statecharts simulink synoptic features hierarchical decomposition application control modules synchronous block diagrams state machines semantics described polychronous model computation synchronous language ignal introduction collaboration major european manufacturers spacify project aims bringing advances mde satellite flight software industry focuses software development maintenance phases satellite lifecycle project advocates approach built modeling language dsml named synoptic aim synoptic support aspects embedded flightsoftware design synoptic consists heterogeneous modeling programming principles defined collaboration industrial partners end users spacify project used central modeling language spacify model driven engineering process synoptic allows describe different layers abstraction highest level software architecture models bujorianu fisher eds workshop formal methods aerospace fma eptcs spacify project work licensed creative commons attribution license spacify project functional decomposition flight software mapped dynamic architecture defines thread structure software consists set threads thread characterized properties frequency priority activation pattern periodic sporadic mapping establishes correspondence software dynamic architecture specifying blocks executed threads lowest level hardware architecture permits define devices processors sensors actuators busses properties finally mappings describe correspondence dynamic hardware architecture one hand specifying threads executed processor describe correspondence software hardware architecture hand specifying data carried bus instance figure depicts layers mappings figure global view layers architecture mappings aim synthesize much mapping possible example appealing internal external schedulers however allow human intervention possible give mapping thus overriding bypassing schedules anyway consistency resulting dynamic architecture verified spacify tool suite based properties software dynamic model step development process also useful model different abstraction levels system design inside layer functional dynamic hardware architecture synoptic offers capability providing incremental design framework refinement features summarize synoptic deals diagrams mode automata blocks components dynamic hardware architecture mapping timing functional part synoptic language allows model software architecture corresponding well adapted model synchronous islands specify interaction points islands middleware platform using concept external variables synchronous islands middleware form globally asynchronous locally synchronous gals system software architecture development synoptic software architecture language tightly coordinated definition geneauto language synoptic uses essentially two types modules called blocks synoptic mutually nested diagrams polychronous interpretation synoptic mode automata nesting favors hierarchical design enables viewing description different levels detail embedding blocks states state machines one elegantly model operational modes state represents mode transitions correspond mode changes mode system may composed different connection patterns among components apart structural behavioral aspects synoptic software architecture language allows define temporal properties blocks instance block parameterized frequency worst case execution time taken account mapping onto dynamic architecture synoptic equipped assertion language allows state desired properties model development mainly interested properties permit express example coherence modes component mode component mode eventually move mode specific transformations extract properties pass verification tools main purpose paper describe formal semantics synoptic expressed terms synchronous language ignal ignal based synchronized flows synchronization process set equations elementary flows describing data control ignal formal model provides capability describe systems several clocks polychronous systems relational specifications brief overview abstract syntax synoptic provided section section describes interpretation one constructions model ignal language overview synoptic blocks main structuring elements synoptic block block defines functional unit compilation execution called many contexts different modes system design block encapsulates functionality may consist automata block implicity associated two signals signal starts execution specification may operate pace next signaled signal delivered forces reset state variables initial values blocks block dataflow automaton blocks data events trigger reset signals flow simpliy define connection event event written event combine data simple operation form flow written data feed signal back written data init feedback loop signal initially defined occurrence signal takes previous value execution controlled parent clock simultaneously executes connection composed every time triggered parent block data low data init data event actions sequences operations variables performed execution automata assignment defines new value variable current values function skip stores new values variables defined spacify project become current past conditional else executes current value true executes otherwise sequence executes action skip else automata schedule execution operations blocks performing timely guarded transitions automaton receives control trigger reset signals specified parent block automaton first triggered reset starts execution initial state specified initial state state performs action state may perform immediate transition new state written value current variable true may also perform delayed transition written waits next trigger resume execution state transition condition applies waits next trigger resumes execution state states transitions composed timed execution automaton combines behavior action execution delayed transition stutter controlled occurrence parent trigger signal execution immediate transition performed without waiting trigger reset action automaton state polychronous interpretation synoptic model computation synoptic relies polychronous language ignal section describes synoptic programs interpreted core language brief introduction ignal ignal process consists composition simultaneous equations signals delay equation init defines every time present initially defined value defined previous value sampling equation defines true finally merge equation default defines present otherwise equation use boolean arithmetic operator define nth values signal result application nth values signals synchronous composition processes consists simultaneous solution equations commutative associative process restricts signal lexical scope process ignal presence value along signal expression noted true present otherwise absent specific processes operators defined ignal manipulate clocks explicitly use simplest one synchronizes occurrences signals interpretation blocks execution block driven trigger parent block block resynchronizes trigger every time one makes explicit reference time skip action delayed transition automaton otherwise elapse time sensed outside block whose operations perceived belonging period within polychronous interpretation synoptic interpretation implements feature encoding actions automata using static single assignment result within block every sequence actions transitions defines value variables defines intermediate ones flow execution interpretation structurally similar ignal programs equally combined using synchronous composition interpretation hhpii fig parameterized reset trigger signals parent block returns process input term output term marked hhpii convenience delayed flow data init initially defines value reset value every time reset signal occurs otherwise takes previous value time dataflow data init hhx default init data hhx zii event hhx yii figure interpretation connections fig write finite product processes similarly finite merge default functional flow data defines product event flow event connects define particular cases operator convert event boolean data operator convert boolean data event write input output signals default convention synoptic synchronize input signals parent trigger however possible define alternative policies one input signals pace trigger another adapt resample trigger interpretation actions execution action starts occurrence parent trigger shall end next occurrence event execution action one may also wait synchronize event issuing skip skip behavior signal end instant newly computed values signals flushed memory execution resumed upon next parent trigger action sends signal environment execution may continue within symbolic instant unless second emission performed one shall issue skip operation takes current value define new value product conditional else executes depending current value result one new value variable defined within instant delimited start end skip therefore interpretation action consists decomposition static single assignment form end use environment associate variable definition expression guard locates time spacify project action holds internal state stores integer denoting current portion actions executed state represents start program labels skip materializes synchronized sequence actions interpretation hhpiin action fig takes parameters state variable state current section guard leads environment returns process state guard continuation updated environment write usege expression returns definition variable guard defge storing final values variables defined guard usege hhe else init gii defg usege execution started upon receipt trigger also resumed skip trigger hence signal synchronized state action signal used inform parent block automaton execution action finished back initial state end resets stores variables defined equation usege finally stops returned guard skip advances next label receives control upon guard flushes variables defined far returns new guard init resume actions past action emits guard true sequence evaluates process passes state guard environment returns state guard environment similarly conditional evaluates guard returns guard default variables defined merged environment hhpiin end pre end hhs defg skip hhs defg pre hhx giin hhx eiin usege usege gii hhp qiinb hhpiina hhqiinb else hhp qiinb default hhpiina usee hhqiinb usee figure interpretation timed sequential actions fig write merge definitions environments variables domains default note action reset parent clock synchronized sequence emissions yields one event along signal occur logical time opposed skip sends second one next trigger polychronous interpretation synoptic interpretation automata automaton describes hierarchic structure consisting actions executed upon entry state immediate delayed transitions immediate transition occurs period time allocated trigger hence synchronize conversely delayed transition occurs upon synchronization next occurrence parent trigger event result automaton partitioned regions region corresponds amount calculation performed within period trigger starting given initial state notations write immediate delayed transition relations automaton write resp predecessor successor states immediate resp delayed transitions resp state automaton write region state defined equivalence relation state written required restriction region acyclic notice still delayed transition may take place two states region interpretation automaton interpreted process automaton parameterized parent trigger reset signals interpretation defines local state synchronized parent trigger set initial state upon receipt reset signal otherwise takes previous value denotes next state interpretation states performed concurrently give states automaton unique integer label dsi designate dae number states initial state state index call action guard immediate delayed transition automaton default init interpretation states dae automaton fig implemented series mutually recursive equations define meaning state depending result obtained predecessors region since region definition acyclic system equations therefore unique solution interpretation state starts actions action defines local state synchronized parent state automaton automaton stutters evaluation action finished local state interpreting actions requires definition guard environment guard defines starts requires local state state receive control predecessor region guard environment constructed merging returned immediate predecessors parameters defined interpretation returns process together exit guard environment holding value variables defines upon evaluation delayed transition checked done definition process first checks guard delayed transition evaluates true variables defined stored defhi spacify project delayed transitions guarded one must finished evaluating moving condition defined value guard default condition stay current state mode terminated hence next state defined equation next state equation state composed form product dae merged dae dae hhpi iin defhi usefi init default use usefi default figure recursive interpretation mode automaton conclusion synoptic formal semantics defined terms synchronous language ignal one hand allows neat integration verification environments ascertaining properties system development hand formal semantics makes possible encode metamodel proof assistant sense synoptic profit formal correctness proof subsequent certification code generator way geneauto project moreover formal model ignal basis polychronous modeling environment sme sme used transform synoptic diagrams generate executable code references toom naks pantel gandriau wati geneauto automatic code generator safe subset european congress embedded real time software erts des automobile guernic talpin lann polychrony system design journal circuits systems computers special issue application specific hardware design world scientific polychrony sme available http brunette talpin gautier metamodel design polychronous systems journal logic algebraic programming elsevier

distortion abelian subgroups jan derrick wigglesworth abstract prove abelian subgroups outer automorphism group free group quasiisometrically embedded proof uses recent developments theory train track maps feighnhandel application prove rank conjecture introduction given finitely generated group finitely generated subgroup undistorted inclusion embedding respect word metrics finite generating sets standard technique showing subgroup undistorted involves finding space acts nicely constructing height function space satisfying certain properties elements large word metric change height function lot elements fixed generating set change function uniformly bounded amount paper use couple variations method let wedge circles let fundamental group free group rank outer automorphism group free group defined quotient aut inner automorphisms arise conjugation fixed element much study draws parallels study mapping class groups furthermore many theorems concerning proofs inspired analogous theorems proofs context mapping class groups groups satisfy tits alternative finite virtual cohomological dimension serre property name importantly approach study yielded classification elements analogy classification elements mapping class group along constructive ways finding good representatives elements authors proved infinite cyclic subgroups mapping class group undistorted proof also implies higher rank abelian subgroups undistorted proved infinite cyclic subgroups undistorted contrast mapping class group setting proof directly apply higher rank subgroups question whether abelian subgroups undistorted left open paper answer affirmative theorem abelian subgroups undistorted theorem implications various open problems study behrstock minsky prove geometric rank mapping class group equal maximal rank abelian subgroup mapping class group application theorem prove analogous result setting corollary geometric rank maximal rank abelian subgroup remark principle could done earlier using techniques show specific maximal rank abelian subgroup undistorted mathematics subject classification primary january author partially supported nsf grant mladen bestvina also acknowledges support national science foundation grants derrick wigglesworth course proving theorem show finite index finitely many marked graphs needed get good representatives every element abelian subgroup setting mapping class groups analogous statement surface abelian subgroup mcg thurston decomposition disjoint subsurfaces respected every element also viewed version kolchin theorem abelian subgroups prove proposition abelian subgroup exists finite index subgroup every realized one finitely many marked graphs paper outlined follows section prove translation distance arbitrary element acting outer space maximum logarithm expansion factors associated exponentially growing strata relative train track map result obtained previously independently richard wade thesis analog bers result translation distance mapping class acting space endowed metric maximum logarithms dilatation constants components thurston decomposition section use result translation distance prove main theorem special case abelian subgroup enough exponential data precisely prove result assumption collection expansion factor homomorphisms determines injective map section prove proposition use section prove main result case enough polynomial data technical part paper need obtain significantly control types occur nice circuits marked graph previously available bulk work goes towards proving proposition result provides connection comparison homomorphisms introduced defined subgroups twisting function use connection complete proof main result polynomial case finally section consolidate results previous sections prove theorem methods used sections carried minimal modification general setting would like thank advisor mladen bestvina many hours time patience would also like thank mark feighn encouragement support finally would also like express gratitude radhika gupta patiently listening completely split paths weeks end msri hospitality partial support preliminaries identify marked graph finite graph rank valence one vertices equipped homotopy equivalence called marking marking identifies homotopy equivalence determines outer automorphism say represents homotopy equivalences assumed map vertices vertices restriction edge assumed immersion let universal cover marked graph path resp either isometric immersion possibly infinite closed interval resp constant map resp constant map path called trivial finite map resp homotopic rel endpoints unique path say obtained tightening homotopy equivalence path define lift define similarly domain finite image natural decomposition edges called edge path associated circuit immersion path circuit let orientation reversed decomposition path circuit subpaths splitting denoted let graph unordered pair oriented edges turn initial endpoint paths denote edge opposite orientation path distortion abelian subgroups contains edge path say takes turn train track structure equivalence relation set edges implies initial vertex turn legal respect train track structure path legal every turn crossed associated edge path legal equivalence classes relation called gates homotopy equivalence induces train track structure follows determines map oriented edges definining first edge edge path declare filtration representative outer automorphism increasing sequence subgraphs let call stratum turn one edge called mixed turn edges called turn contain illegal turns say denote submatrix transition matrix obtained deleting rows columns except labeled edges representatives interest transition matrices come three flavors may zero matrix may identity matrix may irreducible matrix eigenvalue call zero growing neg exponentially growing stratum according possibilities stratum zero stratum called irreducible stratum definition say relative train track map representing every exponentially growing stratum following hold maps set oriented edges particular mixed turns legal nontrivial path endpoints suppose irreducible component zero stratum component eachsvertex valence least two say enveloped define hrz path circuit representative called periodic nielsen path nielsen path nielsen path indivisible written concatenation nielsen paths closed nielsen path edge wdi say linear edge call axis distinct linear edges axis call path form exceptional path scenario different signs call path path say nielsen equivalent nielsen path whose endpoints say periodic point principal neither following conditions hold endpoint periodic nielsen path exactly two periodic directions contained stratum contained component periodic points topologically circle point exactly two periodic directions relative train track map called rotationless principal periodic vertex fixed periodic direction based principal vertex fixed remark closely related notion outer automorphism rotationless need definition need following relevant facts theorem corollary exists depending rotationless every theorem corollary abelian subgroup set rotationless elements subgroup finite index stratum call path endpoints connecting path let edge irreducible stratum let maximal subpath zero stratum say taken path circuit called derrick wigglesworth completely split splitting single edge irreducible stratum indivisible nielsen path exceptional path connecting path zero stratum maximal taken say relative train track map completely split completely split every edge irreducible stratum every taken connecting path zero stratum completely split definition relative train track map filtration given said satisfies following properties rotationless rotationless completely split completely split filtration reduced core filtration element filtration element vertices endpoints indivisible periodic necessarily fixed nielsen paths necessarily principal vertices terminal endpoint neg edge principal hence fixed periodic edges periodic edge fixed endpoint fixed edge principal unique edge fixed stratum loop core graph ends contained zero strata zero stratum enveloped stratum edge vertex contained link contained linear edges linear closed nielsen path widi distinct linear edges axes neg nielsen paths highest edges indivisible nielsen path belong neg stratum linear edge linear edges exists wik nielsen paths indivisible nielsen path height composition proper extended folds defined iteratively folding composition folds involving edges homeomorphism remark several properties definition use terms defined use properties sequel main result cts following existence theorem theorem theorem every rotationless represented completely split paths circuits cancellation iteration confined individual terms splitting moreover complete splitting refines finally improved relative train track maps introduced every circuit path endpoints vertices eventually completely split culler vogtmann outer space cvn defined space homothety classes free minimal actions simplicial metric trees outer space metric defined analogy metric space distance defined logarithm infimal lipschitz constant among maps let universal cover marked graph acts covering translation hyperbolic isometry therefore axis denote projection circuit corresponding conjugacy class linear edge ewd linear edges say axis space lines denoted set denotes diagonal acts interchanging factors equipped topology space abstract lines denoted defined action resp induces action resp marking defines homeomorphism quotient action space lines denoted space abstract lines denoted distortion abelian subgroups lamination closed set lines equivalently closed subset elements lamination leaves associated finite set attracting laminations denoted coordinates given relative train track map representing attracting laminations bijection strata attracting lamination associated expansion factor homomorphism stabout studied briefly describe essential features direct reader details lines laminations expansion factor homomorphisms stab one contain neither contains let relative train track map representing stratum associated corresponding eigenvalue log conversely log eigenvalue stratum rtt representative associated image discrete subset frequently identify element paired lamination denoted paired lamination characterized fact free factor support minimal free factor carrying carries denote pair translation lengths cvn section compute translation distance arbitrary element acting outer space standard define translation distance outer space straightforward check independent cvn remainder section fixed relative train track map representing filtration lemma exponentially growing stratum exists metric every edge eigenvalue associated proof let transition matrix exponentially growing stratum let left eigenvector eigenvalue components normalize define define check condition growth edges stratum edge implies write edge path completing proof lemma define path circuit ignoring edges strata explicitly considered disjoint union note definition proof previous lemma show lemma reduced edge path metric defined lemma proof write decomposition maximal subpaths lemma applying lemma conclude thus theorem let rtt representative stratum let associated eigenvalue max log stratum derrick wigglesworth proof first show log every stratum let length function provided lemma recall logarithm factor candidate loop stretched gives lower bound distance two points cvn let circuit contained height let implies repeatedly applying lemma rearranging inequality taking logarithms using result yields log log log taking limit lower bound translation distance reverse inequality fix must find point outer space moved max log idea choose point simplex cvn corresponding relative train track map stratum much larger previous one way metric see growth every stratum let relative train track map assume neg stratum consists single edge justified example choosing let maximum edge length image edge define length function follows unique edge neg stratum edge zero stratum stratum logarithm maximum amount edge stretched difference markings map gives upper bound lipschitz distance two points check factor every edge stretched clearly stretch factor edges fixed strata single edge neg stratum max similarly edge zero stratum use notation denote length intersection path contained edge stratum normalized eigenvector since vector determined replacing max every edge thus distance moved less max log proof complete computed translation distance arbitrary acting outer space use result establish main result special case exponential case section analyze case abelian subgroup enough exponential data entire group seen called lambda map precisely given attracting lamination outer automorphism let stab expansion factor homomorphism defined corollary corollary authors prove every abelian subgroup finite index subgroup rotationless meaning every element distortion abelian subgroups subgroup rotationless distortion unaffectedsby passing finite index subgroup loss assuming rotationless let set attracting laminations elements lemma finite set laminations define taking collection expansion factor homomorphisms attracting laminations subgroup follows need interchange need following lemma lemma paired laminations constant map differ multiplicative constant determine homomorphism proof first corollary gives stab stab henceforth refer stab ratio statement always well defined determine homomorphism stab suffices show homomorphisms kernel suppose ker corollary either replacing necessary may assume paired lamination priori could different corollary says fact therefore final application corollary gives ker concludes proof theorem injective undistorted proof let rank start choosing laminations restriction function coordinates determined still injective first note contain lamination pair lemma next pass finite index subgroup choose generators reordering necessary generator satisfies let cvn arbitrary let complete proof one orthant time replacing inverses next replacing paired laminations using lemma may assume coordinates nonnegative theorem translation distance maximum eigenvalues associated strata relative train track representative necessarily attracting laminations associated strata stratum logarithm eigenvalue fact homomorphism implies thus translation distance acting outer space max log eigenvalue associated stratum max max last equality maximum taken larger set values added set let symmetric generating set let write terms generators let min rearranging combining inequalities max max thus proved image injective homomorphism undistorted conclude proof recall injective homomorphism abelian groups embedding derrick wigglesworth established result exponential setting move polynomial case first prove general result cts representing elements abelian subgroups abelian subgroups virtually finitely filtered section prove analog theorem abelian subgroups paper authors prove unipotent subgroup contained subgroup homotopy equivalences respecting fixed filtration fixed graph call subgroup filtered generic abelian subgroups unipotent prove virtually filtered namely subgroup virtually contained union finitely many first review comparison homomorphisms introduced comparison homomorphisms feighn handel defined certain homomorphisms measure growth linear edges families representative though given canonical description terms principal lifts need properties coordinates given presently define homomorphisms recall basic facts complete details comparison homomorphisms found comparison homomorphisms defined terms principal sets subgroup exact definition principal set important need know principal set abelian subgroup subset defines lift aut automorphism group let two principal sets define distinct lifts aut suppose contains endpoints axis since abelian aut defined homomorphism follows lemma ikc aut aut denotes conjugation therefore defines homomorphism hic call comparison homomorphism determined generally use letter comparison homomorphisms rotationless abelian subgroup finitely many comparison homomorphisms lemma let number distinct comparison homomorphisms let number attracting laminations map defined product comparison homomorphisms expansion factor homomorphisms injective lemma element called generic every coordinate generic representing correspondence comparison homomorphisms linear edges families described introduction briefly describe comparison homomorphism linear edge udi also comparison homomorphism family denoted udj illustrate correspondence example example let rose three petals labeled define follows bai caj determines outer automorphism denote automorphisms lie rank two abelian subgroup subgroup three comparison homomorphisms easily understood coordinates generic element element generic representing two comparison homomorphisms manifest third homomorphism denoted measures path form changes applied since sequel rely heavily correspondence comparison homomorphisms linear edges families generic element prove main result section distortion abelian subgroups proposition abelian subgroup exists finite index subgroup every realized one finitely many marked graphs proof consists restating combining results feighn handel refer reader paper relevant notation relevant results proof first replace finite index rotationless subgroup corollary proof induction rank base case follows directly lemma let let generic definitions guarantee generic admissible lemma says representing done assume claim holds abelian subgroups rank less let set generic elements complement finite lemma collection hyperplanes every element lies rank abelian subgroup kernel corresponding comparison homomorphism induction fact finitely many hyperplanes every element representative one finitely many marked graphs add single marked graph sector defined complement hyperplanes let generic let representative let disintegration defined recall finite index theorem let semigroup generic elements lie sector every every coordinate signs agree claim every element realized marked graph show explicitly reconstructing generic tuple fix let generating set generic corollary write word generators define since admissibility condition set homogeneous linear equations must preserved taking linear combinations long every coordinate must admissible see every coordinate fact positive let coordinate using fact homomorphism repeatedly applying lemma denotes coordinate vector since assumed generic lie sector conclude every coordinate positive injectivity lemma implies fact generic follows fact directly implied definitions generic tuple generic element finally apply lemma conclude thus every element representative marked filtered graph repeating argument finitely many sectors passing intersection finite index subgroups obtained way yields finite index subgroup finitely many marked graphs every generic element realized one marked graphs elements already dealt using inductive hypothesis proof complete polynomial case author introduced function measures twisting conjugacy classes axis used function prove cyclic subgroups upg undistorted order use comparison homomorphisms conjunction twisting function need establish result possible terms occuring completely split circuits establishing connection use prove theorem main result assumption enough polynomial data derrick wigglesworth last section saw correspondence comparison homomorphisms certain types paths order use twisting function goal find circuits single linear edges families subpaths moreover way control cancellation ends subpaths iteration technical section paper one heavily relies use cts main result proposition completely split circuits one main features train track maps allow one understand cancellation occurs tightening previous incarnations train track maps cancellation understood inductively based height path one main advantages completely split train track maps way cancellation occur understood directly rather inductively given representing set allowed terms completely split paths would finite following two situations linear edge gives rise infinite family inps form two linear edges axis sign give rise infinite family exceptional paths form see two subtleties one needs know one inp height stratum precisely corollary connect comparison homomorphisms twisting function would like show every linear edge exceptional family occurs term complete splitting completely split circuit fact show something stronger proposition completely split circuit containing every allowable term complete splitting complete splitting contains least one instance every edge irreducible stratum fixed neg maximal taken connecting subpath zero stratum infinite family inps infinite family exceptional paths proof proposition require careful study completely split paths aim define directed graph encodes complete splittings paths given representing define csp csp clear whose vertices oriented allowed terms completely split paths precisely two vertices edge irreducible stratum one labeled one labeled refer two vertices maximal taken connecting path zero stratum one one referred similarly two vertices family exceptional paths two vertices inp height one vertex infinite family neg nielsen paths one vertex family indivisible nielsen path whose height neg determine initial direction edge connecting two vertices csp path completely split splitting given equivalent turn legal uniqueness complete splittings lemma completely split path resp circuit endpoints vertices defines directed edge path resp directed loop csp given reading terms complete splitting conversely directed path loop csp yields quite well defined path circuit necessarily completely split ambiguity lies define path csp passes vertex labeled nielsen path neg height family example consider rose consisting two edges identity marking let defined bab representing fully irreducible outer automorphism one indivisible nielsen path abab graph csp shown figure blue edges represent fact paths completely split remark basic observation graph csp every vertex least one incoming least one outgoing edge really consequence fact every vertex least two gates bit care needed justify formally indeed let initial endpoint distortion abelian subgroups figure graph csp example legal turn edge irreducible stratum completely split edge csp possibility legal turns consist edge zero stratum case zero strata guarantees contained stratum envelops link contained particular limited number possibilities may taken connecting subpath edge inp height first two cases term complete splitting edge increasing necessary guarantee first last term splitting therefore directed edge csp terminal endpoint case inp first edge necessarily height already established directed edge csp pointed observe vertex csp directed edge ending also directed edge terminating argument shows edge csp emanating statement proposition rephrased statement graph csp namely directed loop csp passes every vertex need basic terminology study directed graphs say strongly connected every vertex connected every vertex directed edge path may define equivalence relation vertices declaring directed edge path vice versa required allow trivial edge path equivalence classes relation partition vertices strongly connected components prove csp connected one strongly connected component proposition follows directly proof proceeds induction core filtration filtration obtained given one considering filtration elements cores base case fact difficult inductive step state lemma lemma representing fully irreducible automorphism csp connected strongly connected proof assumptions two types vertices csp labeled edges labeled inps denote csp subgraph consisting vertices labeled edges recall denotes vertex csp corresponding edge leaves derrick wigglesworth attracting lamination produce path csp starting passing every vertex csp finally returning looking long segment leaf attracting lamination precisely completely split says completely split path fact train track map says complete splitting contains inps moreover irreducibility transition matrix lamination implies sufficiently large path contains every edge orientations contains edge followed every edge orientations edge path exactly shows csp connected strongly connected isolate following remark future reference remark indivisible nielsen path write edge path recall inps endpoints vertices vertex csp directed edge pointing completely split since turn must legal hence also directed edge csp argument shows edge csp vertex since csp strongly connected remark implies vertex inp directed edges coming going back csp conclude csp strongly connected case leaves attracting lamination choose orientation attracting lamination imagine ant following path determined leaf vertex see ant arrive along certain edges leave along others let edge initial vertex determines gate say departure gate occurs oriented leaf similarly say gate arrival gate edge occurs gates may arrival departure gates suppose vertex least two arrival gates vertex least two departure gates produce path csp shows subgraph one strongly connected component start edge follow leaf lamination crossed every edge forward orientation continue following leaf arrive say gate since two arrival gates edge occurs given orientation whose terminal vertex second arrival gate turn onto since distinct gates turn legal follow going backwards crossed every edge opposite direction finally continue following arrive two arrival gates going backwards use second arrival gate turn around second time follow forwards direction cross edge started construction path completely split every term complete splitting single edge associated path csp passes every vertex returns starting vertex csp strongly connected presence inp remark completes proof lemma current assumptions reduced case lamination orientable either every vertex one departure gate every vertex one arrival gate critical case latter two would like conclude situation inp example illustrates scenario edges colored red illustrate fact order turn around get vertices labeled labeled one must use inp existence inp situation provided following lemma lemma assume representing fully irreducible rotationless automorphism suppose attracting lamination orientable every vertex exactly one arrival gate inp initial edges oriented consistently orientation lamination postpone proof lemma explain conclude argument every vertex one arrival gate apply lemma conclude must inp since inps exactly one illegal turn using previous argument turn around situation one arrival gate apply lemma second time time orientation reversed obtain existence second inp allowing turn around second time distortion abelian subgroups remark since one inp stratum lemma implies lamination orientable vertex must least gates proof lemma vertex fixed since lemma guarantees every stratum contains least one principal vertex principal vertices fixed rotationless choose vertex let lift universal cover let unique arrival gate lift map fixing let infinite subtree consisting embedded rays starting leaving every vertex unique arrival gate whenever vertex unique arrival gate refer figure tree example figure tree example red path connects two vertices height first claim see notice since topological representative suffices show every vertex notice vertices characterized two things first legal second every edge edge path gate unique arrival gate initial endpoint legal train track map moreover every edge edge path occurs orientation leaf lamination since takes leaves leaves preserving orientation true gate determined every edge edge path unique arrival gate vertex thus every edge edge path unique arrival gate vertex means endow metric using left eigenvector transition matrix every edge eigenvalue transition matrix lift metric metric define height function tree measuring distance since legal paths stretched exactly let two distinct lifts height see possible take two distinct circuits based obtained following leaf lamination initial vertices lifts end distinct lifts contained height let unique embedded segment connecting lemma completely split sufficiently large moreover endpoints distinct since restriction lifts injective simply represents automorphism lifts correspond elements observe endpoints height pair distinct vertices height unique embedded segment connecting must contain illegal turn follows definition assumption every vertex unique arrival gate therefore completely split path contains illegal turn particular derrick wigglesworth must inp complete splitting initial edges oriented consistently orientation evident construction key inductive step provided moving filtration lemma explicitly describes graph change moving one element core filtration next recall core filtration filtration glk obtained restricting filtration elements cores gli stratum core sli finally let filtration defined hlci gli denote negative change euler characteristic lemma lemma hlci contain strata one following holds unique edge hlci fixed loop disjoint endpoints unique edge hlci contained two edges hlci nonfixed common initial endpoint terminal endpoints case cases hlci contains stratum hli unique stratum hlci exists following hold single nonfixed edge whose terminal vertex whose initial vertex valence one gui particular gui deformation retracts gui zero stratum words closure gli gui extended stratum hlzi component hlci disjoint gui hlci hli component gli otherwise move core filtration imagine adding new vertices csp adding new edges connecting vertices vertices already present thus define csp subgraph csp consisting vertices labeled allowable terms gli use fact restriction connected component element core filtration problem proving csp strongly connected induction core filtration csp may multiple connected components happens however gli one connected component case csp multiple connected components component gli topological circle necessarily consisting single fixed edge csp two connected components circle lemma every number strongly connected components csp equal connected components gli circles connected components gli circles following proof way difficult requires careful analysis many possible cases case real work case lemma proof lemma establishes base case exponentially growing circle csp exactly two vertices self loop lemma clearly holds proceed inductive step analysis based lemma set notation used throughout edge initial vertex terminal vertex possible denote gvli component gli containing similarly let csp vli component csp containing paths pass case gvli topological circle two components case lemma csp obtained csp adding two new vertices new vertex self loop new edges added number connected components csp increases two component strongly connected case several subcases according various possibilities edge topological types first suppose fixed edge csp obtained distortion abelian subgroups csp adding two new vertices new inps since restriction component gli inp form provided neg nielsen paths nielsen paths remark vertex incoming edge initial endpoint outgoing edge terminal endpoint moreover csp csp directed edge csp directed edge csp hence directed paths csp connecting two strongly connected subgraphs csp csp passing new vertices therefore one strongly connected component csp corresponding component gli containing component circle since contains least two edges case resp topological circle remark incoming resp outgoing edges csp vli resp csp components csp resp csp see figure csp csp figure possibility gli graph csp hlci single neg edge suppose neg edge two new vertices csp labeled argument given previous paragraph goes notice circle since would imply principal vertex gli see first bullet point definition contradicting fact vertices satisfied edge done linear new vertices csp new vertex family neg nielsen paths fact concluded inductive step vertex along remark shows new vertex strongly connected component also two vertices family exceptional paths exact reasons vertices also strongly connected component concludes proof case lemma arguments given thus far apply directly case lemma remark case neither components containing terminal endpoints new edges circles reason complicated way hence csp change hlci contains stratum case lemma component hlci disjoint gui hlci component gli restriction component fully irreducible particular csp one strongly connected component csp lemma though case lemma describes gli built three stages bottom top somehow easier prove csp correct number connected components going top bottom looking long segment leaf attracting lamination hli see lemma vertices csp labeled edges stratum hli two different strongly connected components fact show vertices strongly connected component since working assumption component hlci disjoint gui use one components gui turn around leaf lamination indeed choose component gui intersects hli let edge hli terminal vertex note deformation derrick wigglesworth retracts onto circle vertex edge hli must incident since otherwise would thus replacing necessary may assume situation circle using inductive hypothesis fact mixed turns legal connect vertex vertex csp follow leaf lamination going backwards return say along leaves lamination first place vertices labeled edges hli strongly connected component csp otherwise apply inductive hypothesis use fact mixed turns legal get path shows vertices labeled edges hli strongly connected component csp henceforth denote strongly connected component csp contains vertices csp inp height hli first last edges necessarily hli remark implies csp recall allowable terms complete splittings intersect zero strata connecting paths maximal taken particular vertex csp corresponding connecting path aforementioned strongly connected component csp let neg edge hlci terminal vertex necessarily outgoing edge csp incoming edge csp graph topological circle corresponding component csp already strongly connected directed edge graph back back csp thus subgraph contained strongly connected component csp hand topological circle directed edge back csp mixed turns legal edge hli must incident thus vertices csp labeled neg edges strongly connected component csp vertices csp argument inductive hypothesis shows component intersects hli corresponding strongly connected component csp also csp thing remaining deal neg nielsen paths families exceptional paths handled contains vertices form remark fact already established csp neg edges hlci shown every vertex strongly connected component csp coming component intersects hlci strongly connected component csp particular one strongly connected component csp component gli contains edges hlci completes proof proposition proof theorem need consider weakening complete splitting paths circuits splitting completely split path circuit coarsening complete splitting obtained considering subpath single element given define graph csp adding two vertices csp one one every vertex directed edge terminating add edge similarly every edge emanating add edge vertex beginning vertex every completely split path gives rise directed edge path csp corresponding follows immediately definition proposition corollary completely split circuit containing every allowable term ready prove main result polynomial case polynomial subgroups undistorted subsection complete proof main result polynomial case first recall height function defined given two conjugacy classes elements define twisting twu max auk cyclically reduced conjugates distortion abelian subgroups define twisting max twu proved following lemma using bounded cancellation restate convenience critical point independent lemma lemma constant conjugacy classes symmetric finite generating set since typically work train tracks similar notion twisting adapted setting let path circuit graph let circuit define twisting max path immersed define max circuit bounded cancellation lemma directly implies lemma conjugacy class ready prove polynomial abelian subgroups recall map defined taking product comparison expansion factor homomorphisms following theorem denote restriction map last coordinates corresponding comparison homomorphisms theorem let rotationless abelian subgroup assume map collection comparison factor homomorphisms injective undistorted proof first step note suffices prove generic elements uniformly undistorted set elements finite collection hyperplanes uniform bound distance point one hyperplanes generic point set constants later use emphasize depend subgroup given data handed thus far let finite set marked graphs provided proposition define maximum bcc bcc varies finitely many marked graphs lemma implies conjugacy class finitely many marked graphs let constant lemma fix minimal generating set let generic let representing chosen let comparison homomorphism largest key point given corollary provide split circuit twisting grow application map indeed let circuit provided corollary discussed section correspondence comparison homomorphisms set linear edges families assume first corresponds linear edge axis definition since splitting refines term complete splitting contains path fact splits ends subpath iteration see contains path therefore quite good enough purposes argue conclude suppose contradiction exists every using telescoping sum repeatedly applying assumption obtain combining rearranging inequalities implies contradiction establishes existence satisfying equation argument works without modification case corresponds family quasiexceptional paths address minor adjustment needed case corresponds family exceptional paths let udi udj since contains complete splitting may assume without loss problem derrick wigglesworth exponent term occuring complete splitting may negative may less case replace sufficiently high iterate exponent positive write terms generators conjugacy class repeatedly applying lemma obtain applying inequality circuit constructed letting conjugacy class second inequality justified lemma third uses property established since chosen largest coordinate injective proof complete mixed case additional difficulties mixed case since distance function cvn twisting function well suited dealing outer automorphisms whose growth neither purely exponential purely polynomial consequently element abelian subgroup image large use cvn show large image large use methods show large injectivity lemma exactly says large least one aforementioned quantities must large well theorem abelian subgroups undistorted proof assume passing finite index subgroup rotationless lemma map injective choose minimal generating set write restriction first coordinates precisely map section choose coordinates restriction coordinates injective let subset chosen coordinates corresponding expansion factor homomorphisms pass finite index subgroup choose generators proceed proofs theorems fix basepoint cvn let may assume without loss generic suffices prove generic elements uniformly undistorted replace inverses necessary ensure first coordinates replace paired lamination necessary lemma ensure look coordinates pick one largest absolute value first consider case largest coordinate corresponds expansion factor homomorphism already arranged theorem translation distance maximum eigenvalues associated strata relative train track representative since generic first coordinates associated stratum stratum logarithm eigenvalue proof theorem translation distance acting outer space max inequality may laminations theorem let min max max distortion abelian subgroups handle case largest coordinate corresponds comparison homomorphism let finite set marked graphs provided proposition let define exactly proof theorem conjugacy classes marking inverse marking finitely many marked graphs construction completely split circuit satisfying equation given polynomial case works without modification current setting comparison homomorphism equation coordinate largest absolute value using circuit defining inequalities justifications proof theorem apply verbatim present setting conclude max thus shown image undistorted since injective embedding theorem proved conclude proving rank conjecture maximal rank abelian subgroup theorem gives lower bound geometric rank rank inequality follows directly following result whose proof sketch theorem virtual cohomological dimension rank virtual cohomological dimension thus corollary geometric rank maximal rank abelian subgroup proof let finite index subgroup whose cohomological dimension since finite index subgroups rank rank well known theorem provides existence complex suffices show embedding universal cover suppose contradiction map first step replace continuous bounded distance done using argument whose proof sketched key point uniformly contractible every continuous map finite simplicial complex whose image contained contractible standard fact theorem may replaced simplicial complex dimension may assumed simplicial construct cover simplicial complex whose nerve equal barycentric subdivision cover one element cell vertex set small neighborhood define taking sufficiently small neighborhood ensure key property intersections necessarily empty dimension barycentric subdivision equal dim since arranged continuous pull back cover constructed obtain cover since elements bounded embedding elements bounded well intersection pattern elements exactly intersection pattern elements cover constructed intersection elements necessarily empty thus constructed cover bounded sets intersections contradict fact lebesgue covering dimension compact subset let compact let arbitrary cover let constant provided lebesgue covering lemma applied since elements uniformly bounded scale single constant obtain cover whose sets diameter cover necessarily refinement multiplicity contradicts fact covering dimension theorem proved derrick wigglesworth references emina translation lengths geom dedicata dedicated john stallings occasion birthday lipman bers extremal problem quasiconformal mappings theorem thurston acta mladen bestvina mark feighn michael handel tits alternative dynamics exponentiallygrowing automorphisms ann math mladen bestvina mark feighn michael handel tits alternative kolchin type theorem ann math mladen bestvina michael handel train tracks automorphisms free groups ann math jason behrstock yair minsky dimension rank mapping class groups ann math daryl cooper automorphisms free groups finitely generated fixed point sets algebra marc culler karen vogtmann moduli graphs automorphisms free groups invent samuel eilenberg tudor ganea category abstract groups ann math mark feighn michael handel abelian subgroups geom mark feighn michael handel recognition theorem groups geom mark feighn michael handel algorithmic constructions relative train track maps cts arxiv november benson farb alexander lubotzky yair minsky phenomena mapping class groups duke math stefano francaviglia armando martino metric properties outer space publ john harer virtual cohomological dimension mapping class group orientable surface invent allen hatcher algebraic topology cambridge university press cambridge michael handel lee mosher free splitting complex free group loxodromic outer automorphisms arxiv february john mccarthy subgroups surface mapping class groups trans amer math estelle souche bert wiest elementary approach tree proceedings conference geometric combinatorial group theory part haifa volume pages richard wade symmetries free artin groups phd thesis university oxford department mathematics university utah salt lake city http address dwiggles

catroid mobile visual programming system children wolfgang slany institute software technology graz university technology inffeldgasse graz austria abstract catroid free open source visual programming language programming environment image manipulation program website catroid allows casual users starting age eight develop animations games solely using android phones tablets catroid also allows wirelessly control external hardware lego mindstorms robots via bluetooth bluetooth arduino boards well parrot popular inexpensive quadcopters via wifi categories subject descriptors programming languages miscellaneous general terms design human factors languages keywords visual programming language mobile smart phone tablet programming animations educational games music kids children teenagers pedagogical introduction programming children worldwide shortage qualified software developers due rapidly increasing demand together stagnating even declining number computer science students decline even pronounced females last years seems even though younger girls interested programming degree boys age girls consistently seem lose interest late teens time society increasingly relies software thus less less understood general population moreover software development skills interest obvious professional also philosophical reasons developing software skill helps understanding fundamental mechanisms limitations underlying rational thinking attractive kids simple text success mit scratch programming environment undeniably proven practice two million times appealing note visual programming easy children motivated ready spend necessary time visual programming dumbing programming instead motivation avoiding frustration due spurious syntactic mumbo jumbo unnecessarily complicated work flows hard spot syntax errors frequently encountered mainstream programming languages drawback visual programming visual programming criticized scale well larger complex programs however practical evidence visual programming environments shows large complex programs chess engine based machine opponent multi level jump run games complex physics simulations sudoku solvers much possible hierarchical organization program elements mobile devices worldwide ten times mobile phones pcs ratio even much pronounced children think china developing countries moreover one smartphone nowadays always one pocket easily used everywhere without preparation commuting one school using public transportation backseat family car able program mobile devices also become important job qualification cheap smartphones china increasingly becoming available worldwide scale sony ericsson xperia play android smartphone playstation based portable game console particularly attractive kids visual programming use visual programming predominantly consists moving graphical elements instead typing text use visual programming based informal experiences seems aesthetically figure catroid hello world program http http figure lego mindstorm robot user interface visually programmed catroid illustrated figure figure catroid program lego mindstorm robots salient features catroid catroid runs smartphones tablets intended use children strongly inspired already mentioned scratch programming language environment thriving online developed lifelong kindergarten group mit media lab known scratch app catroid programs written visual individual commands stuck together arranging visually one fingers figure left shows catroid hello world program bricks sticking together result screen shown right speak brick bottom left figure additionally pronounces phrase via android text speech engine default language one android device android device occurs first executing program possibilities creative applications infinite especially attaching phone lego robot using many sensors built phone acceleration gyro sensors gps location based programs voice synthesis voice recognition well image recognition equally easily used build autonomous intelligent robots using similarly controlled arduino hardware arbitrary external devices controlled using catroid figure left shows main screen catroid right part list bricks appears one adds brick script object shown top three bricks used broadcasting receiving messages lower ones used loops sprite movements screen new command bricks selected user list bricks partly shown figure right visually dragged dropped using one fingers deleted dragging one fingers catroid also differs important aspects scratch app inventor compared catroid need apps written solely using smartphones tablets devices scratch intended use keyboard mouse comparatively large screen size whereas catroid focuses small devices sensitive screens thereby different user interaction usability challenges pictures often say thousand words figures show catroid action thereby illustrating features mentioned far even better session interactive system often says thousand pictures cordially invite reader try latest version figure shows parts catroid program allows controlling lego mindstorm robot left list sprite objects shown possessing scripts images right scripts shown associated object turn left one bottom screenshot left figure shows resulting user interface robot necessary bluetooth connection handshake robot http http figure main screen catroid typical command bricks figure parrot quadcopter controlled via wifi user program written using catroid waste basket figure shows parrot popular inexpensive quadcopter controlled catroid via wifi quadcopter two video cameras transmit data catroid image processing catroid uses intel opencv computer vision open source library running service android device follow simple patterns helipad photo left side figure video showing follows moving helipad available http able quickly use powerful simple use features tremendous motivator acquire necessary programming skills users age figure shows catroid running sony ericsson xperia play android smartphone playstation based portable game console plan support gamepad keys phones near future parents likely much willing buy gaming smartphones kids know children able play games moreover also empowered creatively build games animations simulations programs figure shows screen one sees executing hannah montana interactive music video animation programmed catroid created children remix original scratch project found http creating interactive music video animations tremendously motivating boys equally girls even though lot programming required kids spend days creations able upload animations videos youtube additional strong motivator kids general love show creations friends regardless mobile phone friends using youtube recorder catroid programs currently developed kids soon able upload videos youtube high quality order allow recording also android devices decrease amount data needs uploaded one android device play data transmitted phone server interpreted server exactly way user interaction additional input random seeds server records high quality optionally uploads directly youtube similar scratch catroid interpreted programming language procedural control flow objects communicate via simple broadcast messages see figure right figure sony ericsson xperia play android smartphone playstation based portable game console set scripts excecuted concurrently script running thread thus allowing real parallelism take advantage multiple cores recent smartphones children easily think terms objects actors messages style process synchronization feels totally natural main design objective language make simple understand use possible current version catroid march yet full programming language variables formulas supported time though working hard extend direction project started april small team aim quickly produce working partial solution important features implemented first contuinue started implement minimal functionality sufficient emulate highly popular creativity tool preinstalled many nintendo dsi game consoles background image change according prespecified timeline audio file playing went implement bricks used scratch programs around figure interactive music video animation http related work plethora research papers visual programming acm digital library lists papers topic google scholar reports documents related visual programming limit previous work regarding visual programming languages intended use children set languages featuring community sites supporting encouraging sharing interactive animations games created kids beside scratch catroid visual programming systems varying expressive power language include associated nintendo wario ware microsoft flipnote game youtube also seen platform share user contributed multimedia content though primarily oriented towards children contributed content made interactive acknowledgments thanks team members supporters figure catroid online community website world based statistics published scratch implemented lot currently implemented eventually make catroid general programming language catroid community website catroid system includes community website allowing children upload share projects others important integral part catroid system projects uploaded community website open source published free software license everyone download edit every project website add new functionality change current behavior project upload new version called remixing core idea behind scratch online community see figure images community website smartphone left list projects catroid community website shown right details page project shown project downloaded directly catroid reported inappropriate inappropriate content automatically detected regarding latter names projects descriptions comments user names compared extensive multilingual set cuss words well creative spelling variations recognized automatically rejected order serve needs children worldwide scale smartphone parts well website catroid available many languages crowd sourcing localization internationalization support site based pootle allows adding languages currently support several languages speakers english mandarin cantonese hindi arabian german turkish french japanese urdu russian rumanian malaysian team http references craig horton designs girls program evaluation proceedings acm technical symposium computer science education sigcse chattanooga tennessee march acm press new york kelleher motivating programming using storytelling make computer programming attractive middle school girls thesis carnegie mellon university school computer science technical report maclaurin design kodu tiny visual programming language children xbox proceedings annual acm symposium principles programming language popl acm press new york maloney resnick rusk silverman eastmond scratch programming language environment acm trans comp designing website creative learning proceedings web science society online athens greece march overmars teaching computer science game design computer ieee http http http http http

scheduling distributed clusters parallel machines approximation algorithms full version riley samir megan oct department industrial engineering operations research university california berkeley berkeley usa rjmurray department computer science university maryland college park college park usa samir department electrical engineering computer science massachusetts institute technology vassar cambridge usa megchao abstract computing framework rose prominence datasets size dozens machines single cluster needed individual jobs datasets approach exabyte scale single job may need distributed processing multiple machines multiple clusters consider scheduling problem minimize weighted average completion time jobs distributed clusters parallel machines keeping scale problems motivating work assume job divided subjobs distinct subjobs given job may processed concurrently cluster single machine concurrent open shop problem clear limitation model serial processing assumption sidesteps issue different tasks given subjob might processed parallel algorithms explicitly model clusters pools resources effectively overcome issue variety parameter settings develop two constant factor approximation algorithms problem first algorithm uses relaxation tailored problem prior work algorithm provides strong performance guarantees second algorithm exploits surprisingly simple mapping special case one machine per cluster algorithm combinatorial extremely fast first constant factor approximations problem remark shorter version paper one omitted several proofs appeared proceedings european symposium algorithms acm subject classification nonnumerical algorithms problems keywords phrases approximation algorithms distributed computing machine scheduling relaxations algorithms digital object identifier authors conducted work university maryland college park work made possible national science foundation reu grant ccf winkler foundation work also partially supported nsf grant ccf riley murray samir khuller megan chao licensed creative commons license annual european symposium algorithms esa editors piotr sankowski christos zaroliagis article leibniz international proceedings informatics schloss dagstuhl informatik dagstuhl publishing germany scheduling distributed clusters parallel machines full version introduction becoming increasingly impractical store full copies large datasets one data center result data single job may located multiple machines multiple clusters machines maintain fast avoid excessive network traffic advantageous perform computation jobs completely distributed fashion addition commercial platforms aws lambda microsoft azure service fabric demonstrating trend centralized cloud computing frameworks user manages neither data flow server allocation view converging issues following scheduling problem arises computation done locally avoid excessive network traffic individual clusters broader grid coordinate schedules maximum throughput precisely motivation hung golubchik acm symposium cloud computing paper hung modeled cluster arbitrary number identical parallel machines choose objective average job completion time problem generalizes concurrent open shop problem proposed heuristic approach heuristic called swag runs time performed well variety data sets unfortunately swag offers poor performance show section contributions problem extend model considered hung introduce first approximation algorithms general problem extensions hung model allow different machines within cluster operate different speeds incorporate release times times subjob processed support weighted average job completion time present two algorithms resulting problem combinatorial algorithm exploits surprisingly simple mapping special case one machine per cluster problem approximated time also present approach strong performance guarantees machines unit speed subjobs divided equally sized necessary unit tasks formal problem statement definition concurrent cluster scheduling set clusters set jobs job set subjobs one cluster cluster parallel machines machine cluster speed without loss generality assume decreasing ith subjob job specified set tasks performed machines cluster denote set tasks tji task tji associated processing time pjit assume pjit decreasing frequently refer subjob job cluster subjob different subjobs job may processed concurrently different clusters different tasks subjob may processed concurrently different machines within cluster write decreasing mean write increasing mean murray khuller chao subjob complete tasks complete job complete subjobs complete denote job completion time objective minimize weighted average job completion time job weight purposes computing approximation ratios equivalent minimize work equivalent objective throughout paper machine said operate unit speed complete task processing requirement units time generally machine speed processes task units time machines said identical unit speed uniform differ speed accordance graham taxonomy scheduling problems take refer concurrent cluster environment denote problem optionally may associate release time rji every subjob subjobs released time zero write example problem instances illustrate model several examples see figures tables left rows labeled identify jobs columns labeled identify clusters entry tables specifies processing requirements corresponding subjob diagrams right tables show given jobs might scheduled clusters indicated number machines figure two examples scheduling model left baseline example jobs clusters cluster identical machines cluster identical machines note job subjob cluster permitted within framework case every subjob one task right baseline example general subjob framework subjob subjob two tasks tasks shown unit length framework require subjobs divided equally sized tasks related work concurrent cluster scheduling subsumes many fundamental machine scheduling problems example restrict single cluster schedule set jobs bank identical parallel machines minimize makespan cmax total weighted completion time clever reduction even minimize total weighted lateness bank identical parallel machines see section alternatively problem reduces concurrent open shop problem problem implies particular environment objective function optional constraints esa scheduling distributed clusters parallel machines full version figure two additional examples model left baseline example variable machine speeds note benefit high machine speeds realized tasks assigned machines final schedule right problem peculiar structure clusters one single machine clusters processing requirements single job use device total weighted lateness reduction section using graham taxonomy concurrent open shop problem written three groups independently discovered using work queyranne linear program question exponential number constraints still solved polynomial time variant ellipsoid method strong algorithm concurrent cluster scheduling refines techniques contained therein well schulz see section mastrolilli developed algorithm use solvers mussq significant speed strength performance guarantee achieves approximation ratio time although mussq require solver proof correctness based fact finds feasible solution dual particular linear program fast algorithm concurrent cluster scheduling uses mussq subroutine see section hung golubchik presented framework designed improve scheduling across geographically distributed data centers scheduling framework centralized scheduler determined job ordering local dispatchers carried schedule consistent controllers job ordering hung proposed particular algorithm controller called swag performed well wide variety simulations data center assumed number identical parallel machines adopt similar framework hung show section swag performance guarantee paper outline algorithmic results although one algorithms requires solving linear program algorithms use linear program proofs correctness introduce linear program section discussing either algorithm section establishes ordering jobs processed completely specify schedule important complex work algorithms generate ordering jobs cluster section introduces strong algorithm applied instance concurrent cluster scheduling including release times rji key strong performance guarantees lay fact allows different permutations subjobs different clusters providing additional structure problem maintaining generalization concurrent open shop becomes permutation author names mastrolilli queyranne schulz svensson uhan murray khuller chao significant approximate concurrent open shop extension problem ratio combinatorial algorithm presented section algorithm fast provably accurate interesting property schedule clusters using permutation considering general case show approximation ratios obtained fully parallelizable setting zhang conclude extension maintains performance guarantees offering improved empirical performance following table summarizes results approximation ratios compactness condition refers identical machines constant condition refers rji condition refers pjit constant tji term maximum ratio fastest machine average machine speed cluster surprising results scheduling algorithms remarkably simple first algorithm solves scheduling done easily cluster second algorithm rather surprising simple reduction case one machine per cluster well understood concurrent open shop problem yields simple combinatorial algorithm proof approximation guarantee somewhat involved however addition algorithmic results demonstrate problem subsumes minimizing total weighted lateness bank identical parallel machines see section section provides additional discussion highlights novel technical contributions core linear program linear program unusual form rather introduce immediately conduct brief review prior work similar discuss paper decision variable corresponding completion objective function time job weight associated job following discussion adopt notation job processing time addition multiple machine problems discussed say machines possibly speeds earliest appearance similar linear program comes queyranne paper queyranne presents relaxation sequencing jobs single machine constraints form arbitrary subset jobs set optimal found jobs scheduled increasing order results primarily theoretical known time writing sequencing jobs single machine minimize done optimally log time call schedules see later constructive proof existence schedules instances including instances schedules strictly addressed section esa scheduling distributed clusters parallel machines full version queyranne constraint set became particularly useful problems coupling across distinct machines occurs concurrent open shop four separate groups saw used following concurrent open shop scheduling min view tremendous popularity sometimes refer linear program canonical relaxation concurrent open shop andreas schulz thesis developed queyranne constraint set greater depth part thesis schulz considered scheduling jobs identical parallel machines constraints form addition schulz showed constraints satisfied schedule jobs uniform machines schulz refined analysis several problems constructing schedule optimal schulz considered scheduling jobs increasing order statement model consider allows control job structure indicated relaxations inevitably comes expense simplicity formulations effort simplify notation define following constants give verbal interpretations pmi qji min pqji pji pjit definitions processing power cluster subjob qji maximum number machines could process subjob maximum processing power brought bear lastly pji total processing requirement subjob terms core linear program follows min pji pji pji pjit rji tji pji rji constraints carefully formulated versions polyhedral constraints introduced queyranne developed schulz use term new allows provide stronger performance guarantees framework subjobs composed sets tasks see term one primary factors allows parametrize results varying machine speeds terms maximum average machine speed rather maximum minimum machine speed constraints simple lower bounds job completion time majority section dedicated proving valid relaxation established prove solved polynomial time providing separation oracle use ellipsoid method proofs use techniques established schulz thesis murray khuller chao proof validity lemmas establish basis algorithms lemma generalizes inequality used schulz lemma relies lemma cites inequality mentioned preceding section proven queyranne lemma let set real numbers assume positive let set decreasing positive real numbers proof show case define vector ones set note terms clear given one need cite namely plug definitions see desired result lemma validity lemma every feasible schedule instance completion times define feasible solution proof constraints clear lower bounds job completion time suffices show validity constraint thus let subset fix arbitrary feasible schedule define cji completion time subjob schedule similarly define cji first time tasks subjob scheduled machine cluster finished lastly define total processing requirement job scheduled machine cluster note construction cji max cji cjf cji since pji rather innocuously write pji cji pji cji using cji cji pji cji namely pmi pmi pji cji pji cji pji cji next inequality uses bound pji cji proven queyranne subset jobs processing times scheduled single pji cji pji pji combining inequalities following pmi pji pji pji cji pmi pmi pji pji proceedings version paper stated proof cites inequality proceeds induction opted demonstrate different simpler proof discovered proceedings version finalized machine machine cluster esa scheduling distributed clusters parallel machines full version next apply lemma right hand side inequality total times pmi pmi qji pji pji arrive desired result citing cjf cji constraint theoretical complexity first two algorithms requires solving directly need address fact constraints luckily still possible solve linear programs polynomial time ellipsoid method introduce following separation oracle purpose definition oracle define violation pji let potentially feasible solution let denote ordering jobs sorted increasing order pji find violated constraint searching form maximal return violated constraint otherwise check remaining constraints directly linear time fixed finds subset jobs maximizes violation cluster finds prove correctness establishing necessary sufficient condition job lemma pji pxi proof given necessarily equal useful express terms depending whether without loss generality restrict search suppose writing similarly decomposing sum one show following pxi pxi pxi suppose strategy time writing one show pxi pxi pxi note equations hold including turning attention see implies second term equation pxi pxi murray khuller chao similarly implies second term equation pxi pxi follows iff pxi given lemma easy verify sorting jobs increasing order pxi define permutation guarantees form implies fixed finds log time procedure executed cluster leaving remaining constraints verified linear time thus runs log time equivalence separation optimization proven following theorem theorem valid relaxation solvable polynomial time explained beginning section linear programs processed appropriate sorting optimal decision variables important bounds job completion times particular ordering jobs address next section reserve first algorithm section list scheduling permutations complex work proposed algorithms generate permutation jobs procedure takes permutation uses determine start times end times machine assignments every task every subjob given single cluster machines permutation jobs introduce list ordered set tasks belonging subjob ordered longest processing time first define list list list list concatenation operator place tasks list largest task subjob smallest task subjob placing particular task assign whichever machine start time results task completed early possible without moving tasks already placed insert idle time machines necessary procedure would otherwise start job release time following lemma essential bound completion time set jobs processed proof adapted gonzalez lemma suppose jobs scheduled cluster according completion time subjob denoted satisfies max proof assume jobs released time zero let task subjob finish last denoted task least processing time construct new set tasks subjob scheduled sets potential start times machines task hence set potential completion times esa scheduling distributed clusters parallel machines full version task regardless whether subjob consisted tasks subset accordingly reassign without loss generality let denote total demand machine cluster tasks subjobs tasks set scheduled using fact sum left right sides pmi pmi dividing sum machine implies speeds using definition yields pmi estimated upward inequality completes proof case rji suppose rji take policy extreme suppose machines left idle every one jobs released note occurs precisely time clear beyond point time effectively case jobs released time zero hence bound remaining time completion right hand side inequality inequality simply adds two terms result follows lemma cited directly proof theorem lemma lemma used implicitly proofs theorems algorithm section show used construct near optimal schedules concurrent cluster scheduling rji rji although solving somewhat involved algorithm quite simple algorithm let denote instance use optimal solution define permutations sort jobs increasing order pji cluster execute theorem section characterized various assumptions help cancel additive upper bound completion time arbitrary subjob theorem general theorem perhaps surprising uniform machines theorem let completion time job using algorithm let section rji otherwise proof define max let arbitrary let pxi otherwise arbitrary define last task job complete cluster let lastly denote optimal solution feasible solution constraint implies following set see associated proofs omit customary avoid clutter notation murray khuller chao pxi pji pxi turn implies subjobs released time zero combine lemma fact pxi pxit see following transition first inequality second inequality uses pxi one subjobs released time zero lemma implies sufficient bound max constant multiple since defined increasing lji pji implies pxi combine lemma fact pxi pxit yield following inequalities complete proof identical machines theorem machines unit speed yields objective subjobs subjobs rji rji proof define theorem rji one need give careful treatment first inequality using qji pxi similarly rji first inequality implies following key refined analysis theorem lay used annihilate qxi subjobs sufficient accomplish strictly necessary theorem shows annihilate term whenever tasks given subjob length note tasks need unit lengths tasks across different subjobs differ esa scheduling distributed clusters parallel machines full version theorem suppose pjit constant tji algorithm rji otherwise proof definition pxi gives pxi pxit using assumption pjit constant tji see pxi qxi qxi qxi apply inequality proof theorem algebra yields qxi case rji uses identity pxi sachdeva saket showed approximate constant factor less theorem significant shows attain guarantee arbitrary provided pjit constant combinatorial algorithms section introduce extremely fast combinatorial algorithm performance guarantees similar unstructured inputs tji pjit call algorithm uses mussq algorithm concurrent open shop subroutine swag motivated development first address swag performance degenerate case swag prerequisite addressing performance existing algorithm procedure swag pji provide psuedocode accompanying verbal description swag swag computes queue positions every subjob every job supposing mkspn max job scheduled next job tial makespan mkspn largest nextjob mkspnj potential finish times subjobs nextjob considering current queue lengths pji subjob processing time pji end potential makespans determined return job smallest potential makespan selected scheduling point end procedure queues updated queues updated potential makespans need next iteration iterations continue last job scheduled note swag runs time theorem instance let denote objective function value swag applied let denote objective function value optimal solution exists proof let fixed arbitrary constant construct problem instance ilm follows murray khuller chao set jobs set jobs job processing time cluster zero clusters job processing time clusters chosen see figure figure left input swag example right swag resulting schedule alternative schedule easy verify swag generate schedule jobs precede jobs due savings jobs propose alternative solution jobs preceed jobs denote objective value alternative solution alt ilm noting alt ilm ilm symmetry fact clusters single machine see ilm alt ilm given following ilm alt ilm since fixed take limit respect ilm lim alt lim implies existence sufficiently large number clusters implies ilm ilm completes proof theorem demonstrates although swag performed well simulations may reliable rest section introduces algorithm superior runtime swag generating permutation jobs time rather time also performance guarantee fast approximation combinatorial algorithm concurrent cluster scheduling exploits elegant transformation concurrent open shop consider simpler problem handled mussq contributions twofold prove intuitive technique yields approximation algorithm decidedly general problem show modification made maintains theoretical bounds improving empirical performance begin defining transformation esa scheduling distributed clusters parallel machines full version definition total scaled processing time tspt transformation let set instances let set instances note total scaled processing time transformation mapping xji pjit xji total processing time required subjob scaled sum machine speeds cluster throughout section use denote arbitrary instance image tspt figure shows result tspt applied baseline example figure instance image schedules constructed using permutation take time emphasize simplicity reduction indeed tspt transformation perhaps first thing one would think given knowledge concurrent open shop problem surprising one attain performance guarantees even simple transformation algorithm execute mussq generate permutation jobs list schedule instance cluster according towards proving approximation ratio establish critical inequality lemma intuition behind lemma requires thinking every job corresponding representation job scheduled environment job scheduled environment consider results permutation used scheduling environments definitions lemma let completion time job resulting arbitrary permutation define completion time job environment optimal solution lastly define completion time job scheduling environment lemma let job corresponding job arbitrary permutation jobs proof list scheduling carried environment may determine completion time subjob bound using lemma implies nature environment implies relax bound given inequality combine inequality see last step replace final term something murray khuller chao meaningful using immediate definition desired result follows lemma true arbitrary consider ssq proof mussq correctness established first inequality chain inequalities second inequality seen substituting pji xji shows constraints weaker third inequality follows validity lemma combining inequality lemma allows bound objective way make reference inequality completes proof following theorem theorem algorithm approximation unit tasks identical machines consider concurrent cluster scheduling pjit processing times unit although size collections tji unrestricted keeping work zhang studied problem case call instances parameters fully parallelizable write graham taxonomy zhang showed scheduling jobs greedily largest ratio first decreasing results tight bound comes something surprise since largest ratio first policy optimal problem closely resembles formalize extent resembles define time resolution instance pji indeed one show time resolution increases performance guarantee lrf approaches lrf prove analogous result problem theorem proof applying techniques proof lemma hypothesis theorem next use fact definition facts together imply thus augmenting relaxation proof theorem appeals trivial lower bound namely attain performance guarantees spite natural wonder need bound might come empirical weaknesses indeed tspt make subjobs consisting many small tasks look subjobs consisting single long task additionally cluster hosting subjob single esa scheduling distributed clusters parallel machines full version extremely long task might identified bottleneck mussq even cluster machines tasks process would like mitigate issues introducing simple lower bounds seen constraints complicated fact mussq proof correctness allows constraints form without loss generality since implies pji since apply xji equivalent pji much weaker bound desire nevertheless bypass issue introducing additional clusters appropriately defined subjobs formalize augmented total scaled processing time atspt transformation conceptually atspt creates imaginary clusters imaginary cluster nonzero processing time exactly one job definition augmented tspt transformation let definition tspt augmented tspt transformation likewise mapping diagonal matrix djj valid lower bound completion time job right hand sides constraints given djj valid lower bound completion time job easy verify valid relaxation mussq returns permutation jobs use list scheduling imaginary clusters accounted beyond computations mussq reduction minimizing total weighted lateness identical parallel machines problem minimizing total weighted lateness bank identical parallel machines typically denoted lateness job deadline max reduction offer shows stated terms optimality thus approximation imply approximation reduction nevertheless provides new insights structure definition total weighted lateness reduction let denote instance set processing times set deadlines set weights number identical parallel machines given inputs transform following way create total clusters cluster machines job processing time cluster clusters consist single machine job processing time cluster zero clusters cluster cluster denote problem refer reader figure example output reduction theorem let instance let instance resulting transformation described list schedule optimal also optimal proof restrict solution space single permutations may without loss generality schedule produces value murray khuller chao additional clusters added ensure given objective written constant permutation solve optimally also solves optimally since desired result closing remarks take moment address subtle issue concurrent cluster problem price pay using permutation clusters schedules concurrent open shop shown schedules may assumed without loss optimality shown figure hold concurrent cluster scheduling general case fact precisely strong performance guarantees algorithm rely clusters possibly unique permutations figure instance exist schedule attains optimal objective value case one jobs necessarily becomes delayed one time unit compared case result see optimality gap even novel contributions came analysis first could rely processing time last task job bounded job completion time variable appealed lower bound stated need incorporate second bound critical realizing strength algorithm uncommon rounding schemes second novel introduces constraints would redundant become relevant viewing relaxation approach potential broad applications since represented effective use limited constraint set supported known algorithm take moment state open problems area one topic ongoing research developing factor purely combinatorial algorithm special case concurrent cluster scheduling considered theorem addition would broad interest determine loss optimality incurred assuming schedules simple example shows optimal schedule objective times globally optimal objective meanwhile theorem shows always exists schedule objective times globally optimal objective thus know performance ratio interval know precise value matter outside scheduling theory would valuable survey algorithms roots relaxations determine constraint sets amenable implicit modification fashion esa scheduling distributed clusters parallel machines full version acknowledgments special thanks andreas schulz sharing recent work thorough analysis linear program drives results paper thanks also hung leana golubchik sharing review ioana bercea manish purohit insights swag performance lastly sincere thanks william gasarch organizing reu led work cohort making experience unforgettable one words rick sanchez wubalubadubdub references inc amazon web services aws lambda serverless compute accessed april url https chen nicholas hall supply chain scheduling assembly systems working naveen garg amit kumar vinayaka pandit order scheduling models hardness algorithms fsttcs foundations software technology theoretical computer science teofilo gonzalez oscar ibarra sartaj sahni bounds lpt schedules uniform processors siam journal computing ronald graham eugene lawler jan karel lenstra ahg rinnooy optimization approximation deterministic sequencing scheduling survey annals discrete mathematics mohammad hajjat shankaranarayanan david maltz sanjay rao kunwadee sripanidkulchai dealer request splitting interactive cloud applications conext pages hung leana golubchik minlan scheduling jobs across geodistributed datacenters proceedings sixth acm symposium cloud computing pages acm leung haibing michael pinedo scheduling orders multiple product types minimize total weighted completion time discrete applied mathematics monaldo mastrolilli maurice queyranne andreas schulz ola svensson nelson uhan minimizing sum weighted completion times concurrent open shop operations research letters microsoft azure service fabric accessed april url https maurice queyranne structure simple scheduling polyhedron mathematical programming sushant sachdeva rishi saket optimal inapproximability scheduling problems via structural hardness hypergraph vertex cover ieee conference computational complexity pages ieee andreas schulz polytopes scheduling phd thesis andreas schulz linear programming relaxations approximation algorithms scheduling problems tour horizon working paper available upon sriskandarajah wagneur openshops jobs overlap european journal operations research qiang zhang weiwei minming resource scheduling supply constraint linear cost cocoa conference

oct sound source localization multipath environment using convolutional neural networks eric stefan williams craig jin australian centre field robotics university sydney australia computing audio research laboratory university sydney australia abstract propagation sound shallow water environment characterized boundary reflections sea surface sea floor reflections result multiple indirect sound propagation paths degrade performance passive sound source localization methods paper proposes use convolutional neural networks cnns localization sources broadband acoustic radiated noise motor vessels shallow water multipath environments shown cnns operating cepstrogram generalized inputs able reliably estimate instantaneous range bearing transiting motor vessels source localization performance conventional passive ranging methods degraded ensuing improvement source localization performance demonstrated using real data collected experiment index source localization doa estimation convolutional neural networks passive sonar reverberation introduction sound source localization plays important role array signal processing wide applications communication sonar robotics systems focal topic scientific literature acoustic array signal processing continuing challenge acoustic source localization presence interfering multipath arrivals practice conventional passive narrowband sonar array methods involve beamforming outputs hydrophone elements receiving array detect weak signals resolve sources estimate direction sound source typically sensors form linear array uniform interelement spacing half wavelength array design frequency however narrowband approach application limited band frequencies upper limit set design frequency grating lobes form due spatial aliasing leading ambiguous source directions lower limit set one octave design frequency lower frequencies directivity array much reduced beamwidths broaden alternative approach sound source localization measure time difference arrival tdoa signal array spatially distributed receivers allowing instantaneous position source estimated accuracy source position estimates found sensitive uncertainty sensor positions furthermore reverberation adverse effect time delay estimation negatively impacts work supported defence science technology group australia sound source localization approach broadband source localization reverberant environments model early reflections multipaths used subtract reverberation component signals decreases bias source localization estimates approach adopted uses minimum number sensors three localize source bearing also range using single sensor instantaneous range broadband signal source estimated using cepstrum method method exploits interaction direct path multipath arrivals observed spectrogram sensor output lloyds mirror interference pattern generalized gcc used measure tdoa broadband signal pair sensors enables estimations source bearing furthermore adding another sensor three sensor positions collinear enables source range estimated using two tdoa measurements two adjacent sensor pairs range estimate corresponds radius curvature spherical wavefront traverses receiver array latter method commonly referred passive ranging wavefront curvature however source localization performance become problematic multipath environments large number extraneous peaks gcc function attributed presence multipaths direct path multipath arrivals unresolvable resulting tdoa estimation bias also performance degrades signal source direction moves away array broadside direction completely fails endfire note case cepstrum method omnidirectional ranging performance independent source direction recently deep neural networks dnn based supervised learning methods applied acoustic tasks speech recognition terrain classification source localization tasks challenge supervised learning methods source localization ability adapt acoustic conditions different training conditions acoustic characteristics shallow water environment high levels clutter background noise multiple propagation paths making difficult environment dnn methods cnn proposed uses generalized gcc cepstral feature maps inputs estimate range bearing acoustic source passively shallow water environment cnn method inherent advantage since considers gcc cepstral values physically significant estimating source position approaches involving time delay estimation typically consider single value peak gcc cepstogram cnns trained using real acoustic recordings surface vessel underway quefrency time delay cepstrogram range output bearing output dense time seconds combined cnn fig cepstrogram surface vessel transits single recording hydrophone located sea floor corresponding pair hydrophones shallow water environment cnns operating cepstrum gcc feature map inputs also considered performances compared proposed model shown localize sources greater performance conventional passive sonar localization method uses tdoa measurements generalization performance networks tested ranging another vessel different radiated noise characteristics original contributions work development cnn passive localization acoustic broadband noise sources shallow water environment range bearing source estimated jointly range bearing estimates continuous allowing improved resolution position estimates compared passive localization networks use discretized classification approach novel loss function based localization performance bearing estimates constrained additional network regularization training unified network passive localization reverberate environments improved performance traditional methods acoustic localization cnn neural network machine learning technique maps input data label continuous value nonlinear architecture successfully applied applications image object classification hyperspectral pixelwise classification terrain classification using acoustic sensors cnns learn apply sets filters span small regions input data enabling learn local correlations architecture since presence broadband acoustic source readily observed cepstrogram fig possible create unified network estimating position vessel relative receiving hydrophone array network divided sections fig gcc cnn cepstral cnn operate parallel serve feature extraction networks gcc cepstral feature map inputs respectively next outputs gcc cnn gcc cnn dense cepstral cnn dense dense dense dense gcc input cepstral input multichannel acoustic recording fig network architecture acoustic localization cnn cepstral cnn concatenated used inputs dense layers outputs range bearing estimate gcc cnn cepstral cnn first convolutional layer filters input feature maps kernels second convolutional layer takes output first convolutional layer input filters kernels third layer also uses kernels followed two fullyconnected layers combined cnn contains two fullyconnected layers take concatenated output vectors gcc cepstral cnns input layers neurons single neuron used regression output range bearing outputs respectively layers use rectified linear units activation functions since resolution important accurate ranging acoustic source max pooling used network architecture input order localize source using hydrophone array information time delay signal propagation paths required although information contained raw signals beneficial represent way readily learned network cepstrum derived various spectra complex differential spectrum current approach power cepstrum used derived power spectrum recorded signal closely related cepstrum used frequently automatic speech recognition tasks linearly spaced frequency bands rather bands approximating human auditory system response cepstral representation signal neither time frequency domain rather quefrency domain cepstral analysis based principle logarithm power spectrum signal containing echoes additive periodic component due echoes reflections original time waveform contained echo cepstrum contain peak thus tdoa propagation paths acoustic signal measured examining peaks cepstrum useful presence strong multipath reflections found shallow water environments time delay estimation methods gcc suffer degraded performance cepstrum obtained inverse fourier transform logarithm power spectrum fourier transform discrete time signal given geometry bounded range quefrencies useful source localization separation distance decreases tdoa values position peaks cepstrum tend maximum value occurs source closest point approach sensor tdoa values greater maximum physically realizable excluded cepstral values near zero dominated source dependent quefrencies also excluded gcc used measure tdoa signal pair hydrophones useful situations spatially uncorrelated noise given array geometry bounded range useful gcc information pair recording sensors zero relative time delay corresponds broadside source whilst maximum relative time delay corresponds endfire source tdoa values greater maximum bound useful passive localization problem excluded windowing cnn inputs added benefit reducing number parameters network cepstrogram ensemble cepstrum gcc respectively vary time shown fig output example network predicts range bearing acoustic source continuous value single neuron regression output differs recent passive localization networks use classification based approach range bearing predictions discretized putting hard limit resolution estimations networks able provide joint training objective network predict range bearing acoustic source relative receiving array reverberant noisy input signals since localization acoustic source involves range bearing estimate euclidean distance network prediction ground truth minimized training range bearing output loss components jointly minimized using loss function based localization performance additional regularization expected improve localization performance compared minimizing range loss bearing loss separately total objective function minimized network training given weighted sum loss bearing loss norm polar distance given cos norm bearing loss given predicted range bearing output denoted respectively true range bearing denoted respectively inclusion term encourages bearing predictions constrained first turn providing additional regularization reducing parameter weight magnitudes two terms weighted loss term roughly equal weight training uses batch normalization stopped validation error decrease appreciably per epoch order prevent regularization dropout rate used fully connected layers training experimental results passive localization transiting vessel conducted using algorithmic method described cnns cepstral gcc inputs performances compared generalization ability networks broadband sources also demonstrated localizing additional vessel different radiated noise spectrum source level dataset acoustic data motor boat transiting shallow water environment hydrophone array recorded sampling rate khz uniform linear array ula consists three recording hydrophones interelement spacing recording commenced vessel inbound sensor array vessel transited array recording terminated vessel outbound boat equipped dgps tracker logged position relative receiving hydrophone array intervals bearing labels wrapped radians consistent bearing estimates available ulas suffer bearing ambiguity transits recorded two day period one hundred thousand training examples randomly chosen range bearing label examples uniformly distributed range labeled examples reserved cnn training validation recordings preprocessed outlined section networks implemented tensorflow trained momentum optimizer using nvidia geforce gtx gpu gradient descent calculated batches training examples networks trained learning rate weight decay momentum additional recordings vessel used measure performance methods recordings referred test dataset contain labeled examples additional acoustic data recorded different day using different boat different radiated noise characteristics acoustic recordings transit started inbound vessel array continued transit array ended outbound vessel away dataset referred generalization set contains labeled examples average bearing error deg combined cnn algorithmic method dgps fig estimates range bearing transiting vessel true position vessel shown relative recording array measured dgps combined cnn cepstral cnn gcc cnn algorithmic method average range error average bearing error deg combined cnn cepstral cnn gcc cnn algorithmic method bearing deg bearing deg combined cnn cepstral cnn gcc cnn algorithmic method fig comparison bearing estimation performance function vessels true bearing test dataset generalization dataset average range error range combined cnn cepstral cnn gcc cnn algorithmic method range fig comparison range estimation performance function vessels true range test dataset generalization dataset input network cepstral gcc feature maps used inputs cnn computed follows input example select range cepstral gcc values contain relevant tdoa information retained see section cepstral values discarded represent maximum multipath delay occur source directly sensor cepstral values less discarded since highly source dependent thus cepstrogram input liftered samples used input network cepstral feature vector calculated recording channel resulting cepstal feature map due array geometry maximum time delay pairs sensors gcc feature vector calculated two pairs sensors resulting gcc feature map gcc map size reduces number network parameters comparison localization methods algorithmic passive localization conducted using methods outlined tdoa values required algorithmic localization taken largest peaks gcc nonsensical results ranges greater discarded cnn chitectures also compared gcc cnn uses gcc cnn section combined cnn cepstral cnn uses cepstral cnn section combined cnn similar range bearing outputs fig fig shows localization results vessel one complete transit fig fig show performance localization methods function true range bearing vessel test dataset generalization set respectively cnns able localize different vessel generalization set impact performance performance algorithmic method degraded shallow water environment since large number extraneous peaks gcc attributed presence multipaths direct path multipath arrivals become unresolvable resulting tdoa estimation bias bearing estimation performance improved networks using gcc features showing time delay information pairs spatially distributed sensors beneficial networks show improved robustness interfering multipaths range estimation performance improved networks using cepstral features showing multipath information useful determining sources range combined cnn shown provide superior performance range bearing estimation conclusions paper introduce use cnn localization surface vessels shallow water environment show cnn able jointly estimate range bearing acoustic broadband source presence interfering multipaths several cnn architectures compared evaluated networks trained tested using cepstral gcc feature maps input derived real acoustic recordings networks trained using novel loss function based localization performance additional constraining bearing estimates inclusion cepstral gcc inputs facilitates robust passive acoustic localization reverberant environments methods suffer degraded performance references benesty chen huang microphone array signal processing vol springer science business media chakrabarty habets broadband doa estimation using convolutional neural networks trained noise signals arxiv preprint viberg ottersten kailath detection estimation sensor arrays using weighted subspace fitting ieee trans signal vol takeda komatani unsupervised adaptation deep neural networks sound source localization using entropy minimization proc ieee int conf speech signal process ieee zeng yang chen jin low angle direction arrival estimation time reversal proc ieee int conf speech signal process ieee krizhevsky sutskever hinton imagenet classification deep convolutional neural networks adv neural information process systems capon spectrum analysis proc ieee vol girshick donahue darrell malik rich feature hierarchies accurate object detection semantic segmentation proc ieee conf computer vision pattern carter time delay estimation passive sonar signal processing ieee trans speech signal vol carter coherence time delay estimation ieee press new york chan simple efficient estimator hyperbolic location ieee trans signal vol windrim ramakrishnan melkumyan murphy hyperspectral cnn classification limited training samples british machine vision bogert quefrency alanysis time series echoes cepstrum saphe cracking time series analysis benesty chen huang estimation via linear interpolation cross correlation ieee trans speech audio vol ferguson gao maguer aircraft flight parameter estimation using acoustic multipath delays ieee trans aerospace electronic systems vol ferguson application passive ranging wavefront curvature methods localization biosonar click signals emitted dolphins proc international conf underwater acoust measurements oppenheim schafer frequency quefrency history cepstrum ieee signal process magazine vol chen benesty huang performance gccand estimation practical reverberant environments eurasip adv signal vol jensen nielsen heusdens christensen doa estimation audio sources reverberant environments proc ieee int conf speech signal process ieee ferguson ramakrishnan williams jin convolutional neural networks passive monitoring shallow water environment using single sensor proc ieee int conf speech signal process ieee gao clark cooper time delay estimate using cepstrum analysis shallow littoral environment conf undersea defence technology vol knapp carter generalized correlation method estimation time delay ieee trans speech signal vol ferguson ramakrishnan williams jin deep learning approach passive monitoring underwater acoustic environment acoust soc vol ioffe szegedy batch normalization accelerating deep network training reducing internal covariate shift international conf machine learning ferguson modified wavefront curvature method passive ranging echolocating dolphins wild acoust soc vol srivastava hinton krizhevsky sutskever salakhutdinov dropout simple way prevent neural networks machine learning research vol xiao watanabe erdogan hershey seltzer chen zhang mandel deep beamforming networks speech recognition proc ieee int conf speech signal process ieee schau robinson passive source localization employing intersecting spherical surfaces differences ieee trans speech signal vol heymann drude christoph boeddeker patrick hanebrink beamnet training asr system proc ieee int conf speech signal process ieee valada spinello burgard deep feature learning terrain classification robotics research springer

ransac algorithms subspace recovery subspace clustering nov ery jue wang university california san diego abstract consider ransac algorithm context subspace recovery subspace clustering derive theory perform numerical experiments also draw correspondences methods hardt moitra chen lerman introduction random sample consensus acronym ransac algorithm fischler bolles many variants adaptations computer vision robustness presence gross errors outliers paper focus closely related problems subspace recovery subspace clustering presence outliers methods believed optimal yet costly terms computations fraction inliers small although limitation ransac nevertheless establish rigorously present context particular derive performance computational complexity ransac two problems perform numerical experiments corroborating theory comparing ransac methods proposed literature problem subspace recovery consider setting data consist points dimension denoted assumed points lie linear subspace points otherwise general position means following assumption place assumption data points linearly independent unless includes least points say points linearly dependent seen vectors points called inliers points called outliers setting subspace recovery without noise noise points exactly underlying subspace rather vicinity case goal recover said differently distinguish inliers outliers see figure illustration setting subspace dimension ambient dimension goal recover subspace identify inlier points present project initiated context independent study undergraduates math acknowledge support national science foundation dms problem intimately related problem robust covariance estimation dates back decades huber ronchetti maronna tyler attracted recent attention refer reader introduction zhang lerman comprehensive review literature old new subspace recovery presence outliers consider problem sometimes referred robust principal components analysis although meanings literature closely related matrix factorization component wright subspace recovery problem subspace clustering problem figure illustration two settings considered paper problem subspace clustering consider setting data consist points dimension denoted assumed points lie linear subspace subspaces total remaining points general position assumption data points linearly independent unless includes least points setting points one subspaces inliers points outliers setting subspace clustering without noise noise inliers exactly subspaces vicinity see figure illustration setting one subspace dimension two subspaces dimension ambient dimension goal cluster points corresponding problem subspace clustering applications computer vision particular movement segmentation vidal vidal contents section consider problem subspace recovery section consider problem subspace clustering cases study canonical ransac algorithm deriving theory comparing methods numerical experiments briefly discuss results section remark linear affine throughout consider case subspaces linear although applications may call affine subspaces convenience able identify point corresponding vector sometimes written subspace recovery consider setting section use notation defined particular work assumption consider noiseless setting simplicity ransac subspace recovery propose simple ransac algorithm robust subspace recovery present setting particular assumption underlying linear subspace assumed dimension determined comes subspace algorithm starts randomly selecting checking tuple forms linear subspace dimension subspace recovered algorithm stops otherwise algorithm continues repeatedly sampling random subspace discovered optionally algorithm made stop maximum number tuples sampled formulation detailed algorithm known input data points dimension output linear subspace dimension containing least points repeat randomly select data points tuple linearly dependent return subspace spanned tuple algorithm ransac subspace recovery design procedure exact noiseless setting noisy setting method shown essentially optimal however researchers shied away ransac approach time complexity formalize folklore following proposition algorithm exact number iterations geometric distribum thus expected number iterations success probability order held fixed note iteration requires order operations requires computing rank matrix proof algorithm sample independently uniformly random tuple linearly dependent assumption linearly dependent total fit points tuple draws bill probability drawing suitable tuple independent total number draws algorithm stops geometric distribution success probability assumed fixed know mean distribution large consider variant geometric distribution supported positive integers reader invited verify still holds true long applications number outliers fraction sample meaning close ransac number iterations depends exponentially dimension subspace confirms folklore least setting remark simplicity analyzed variant algorithm tuples drawn replacement number iterations infinite however practice one draw tuples without replacement equally easy present setting recommended schattschneider green variant time comn moreover proposition still applies understood upper plexity number iterations negative hypergeometric distribution case remark dimension unknown possible strategy start run algorithm maximum number iterations pair points found aligned origin move continue fashion increasing dimension satisfactory tuple found algorithm would start algorithm succeed eventually algorithm hardt moitra subspace recovery said researchers avoid ransac procedures running time saw prohibitive recently however hardt moitra proposed ransactype algorithm strikes interesting compromise running time precision algorithm designed case sample size larger ambient dimension namely described follows repeatedly draws random tuple found linearly dependent tuple found algorithm returns set linearly dependent points tuple see description algorithm virtue procedure require knowledge dimension underlying subspace input data points output linear subspace repeat randomly select data points tuple linearly dependent return subspace spanned subset linearly dependent points tuple algorithm subspace recovery proposition algorithm exact number iterations geomet thus expected number ric distribution success probability iterations note iteration requires order operations requires computing rank matrix proof assumption place linearly dependent contains least points subspace thus repeat statement stops exactly found contains least points moreover also assumption points within tuple linear dependent must belong therefore algorithm returns therefore exact turn number iterations number iterations obviously geometric success probability probability drawn uniformly random contains least points probability indeed probability drawing balls without replacement urn red balls total sample contains least red balls present context balls course points red balls points linear subspace hardt moitra analyze algorithm slightly different setting goal finding maximum fraction outliers tolerated algorithm breaks sense run polynomial time particular show algorithm number iterations geometric distribution success probability least expected number iterations bounded obviously polynomial fact better following consequence proposition corollary addition holds fixed bounded positive quantity depends consequently algorithm expected number iterations order proof let denote random variable hypergeometric distribution parameters described depends show bounded irrespective parameters long conditions met noting increasing decreasing suffices consider varies along sequence varying way makes expected number iterations largest define var condition implies along sequence parameters consideration moreover along sequence standard normal limit using slutsky theorem last line numerical experiments performed numerical experiments comparing ransac form algorithm procedure algorithm geometric median subspace gms zhang lerman appears one best methods market used code available teng zhang website inlier uniformly distributed intersection unit sphere underlying subspace outlier simply uniformly distributed unit sphere result algorithm averaged repeats performance measured first principal angle returned subspace true subspace fair gms two algorithms exact results reported table parameters average system time difference angle ransac gms ransac gms table numerical experiments comparing ransac gms problem subspace recovery text dimension subspace ambient dimension number inliers number outliers sample size performed another set experiments corroborate theory established proposition complexity ransac results shown figure setting repeated times expected dimensionality problem increases ransac complexity becomes quickly impractical simulated simulated simulated theory theory theory number iterations intrinsic dimension figure average number iterations ransac form algorithm function subspace dimension ratio sample size number inliers dashed lines averages simulation lines derived theory proposition subspace clustering consider setting section use notation defined particular work assumption consider noiseless setting simplicity also assume subspaces dimension denoted ransac subspace clustering propose simple ransac algorithm subspace clustering linear subspaces determined comes subspace algorithm starts randomly selecting checking tuple forms linear subspace dimension one subspaces recovered points subspace extracted data otherwise algorithm continues repeatedly sampling random condition met algorithm continues fashion subspaces recovered formulation detailed algorithm assumed known input data points dimension number subspace output linear subspaces dimension containing least points repeat randomly select data points tuple linearly dependent return subspace spanned tuple remove points subspace data end algorithm ransac subspace clustering procedure exact design since noiseless setting researchers embraced ransac approaches running time confirm folklore following assume simplicity subspaces number points proposition algorithm exact number iterations distribution independent geometric distribution success probability stochastically bounded negative binomial parameters thus expected number iterations bounded order held fixed proof similar proposition omitted remark dimensions subspaces unknown strategy analogous described remark course possible number subspaces unknown stopping rule help decide whether remains subspace discovered details omitted approach although natural could prove complicated adapting algorithm hardt moitra subspace clustering algorithm consists applying algorithm subspace recovered removing points subspace continuing subspaces recovered algorithm subspace clustering based algorithm hardt moitra algorithm instead resulting algorithm suited case based fact algorithm expected number iterations bounded resulting algorithm subspace clustering expected number iterations bounded see algorithm assume number subspaces known assume dimensions subspaces known need input data points number subspace output linear subspaces number points exceeding dimension repeat randomly select data points tuple linearly dependent repeat find smallest number linearly dependent points tuple return subspace spanned points remove points subspace data linearly dependent points tuple end algorithm subspace clustering based algorithm reason extract smallest number linearly dependent points step avoid situation contains points points case assuming points linearly dependent span one subspaces particular step however computationally challenging amounts finding sparsest solution linear system problem known challenging tropp wright one possibility replace finding solution minimum norm tropp wright use constraint central method proposed elhamifar vidal algorithm chen lerman spectral curvature clustering scc algorithm chen lerman fact ransac type method designed noisy setting therefore based function quantifies close spanning subspace dimension less equal case strictly less case algorithm draws number random tuple denoted computes matrix wij wij applies form spectral graph partitioning algorithm closely related method method assumes subspaces dimension assumed known noiseless setting one could take return tuple linearly dependent otherwise case wij simply number among drawn linearly dependent chen lerman analyzes method setting reduces situation show method exact case chen lerman consider case subspaces affine adapt method case area linear parameters average system time rand index ransac ssc scc tsc ransac ssc scc tsc table numerical experiments comparing ransac ssc scc tsc problem subspace clustering text dimension subspaces assumed ambient dimension number subspaces number inliers per subspace assumed number outliers sample size numerical experiments performed numerical experiments compare various methods subspace clustering specifically ransac sparse subspace clustering ssc elhamifar vidal spectral curvature clustering scc chen lerman subspace clustering tsc reinhard heckel inlier uniformly distributed intersection unit sphere corresponding subspace outlier simply uniformly distributed unit sphere result algorithm averaged repeats performance measured rand index results reported table discussion conclusion small scale experiments ransac seen competitive methods least intrinsic dimensionality large many outliers many underlying subspaces present data observed context subspace recovery context subspace clustering references wright robust principal component analysis journal acm jacm chen lerman foundations spectral clustering framework hybrid linear modeling foundations computational mathematics chen lerman spectral curvature clustering scc international journal computer vision elhamifar vidal sparse subspace clustering computer vision pattern recognition cvpr ieee conference ieee fischler bolles random sample consensus paradigm model fitting applications image analysis automated cartography communications acm hardt moitra algorithms hardness robust subspace recovery conference learning theory colt volume huber ronchetti robust statistics edition wiley maronna robust multivariate location scatter annals statistics jordan weiss spectral clustering analysis algorithm advances neural information processing systems reinhard heckel helmut bolcskei robust subspace clustering via thresholding ieee transactions information theory schattschneider green enhanced ransac sampling based combinations proceedings conference image vision computing new zealand acm tropp wright computational methods sparse solution linear inverse problems proceedings ieee tyler multivariate scatter annals statistics vidal subspace clustering ieee signal processing magazine vidal sastry generalized principal component analysis gpca ieee transactions pattern analysis machine intelligence wright ganesh rao peng robust principal component analysis exact recovery corrupted matrices via convex optimization advances neural information processing systems zhang lerman novel robust pca journal machine learning research

idle programs idle sensors parallel composition sequential composition termination composition install method invocation module update sensor broadcast sensor sense else field sensing conditional execution arxiv dec network sensors field net modules method collection values variable targets field measure position broadcast local battery capacity module figure syntax csn section addresses syntax semantics calculus sensor networks syntax calculus given grammar figure calculus encompasses structure networks programs networks flat unstructured collections sensors values sensor represents abstraction physical sensing device located position running program module collection methods sensor makes available internal external usage typically collection methods may interpreted library functions tiny operating system installed sensor sensors may broadcast values neighborhood sensors radius defines transmitting power sensor specifies border communication circle centered position position sensor radius likewise radius defines sensing capability sensor meaning sensor may read values inside circle centered position radius values define field measures may sensed value consists tuple denoting strength measure given position plane values managed environment csn primitives manipulating values besides reading sensing values assume environment inserts values network update contents networks combined using parallel composition operator processes built inactive process idle idle denotes terminated thread sensing values environment sensed programs may combined sequence parallel sequential composition designates program first executes proceeds execution contrast represents simultaneous execution however consider sensors support limited form parallelism interact execution mutually recursive method definitions makes possible represent infinite behaviours values data exchanged sensors basic values method labels positions modules notice calculus sense communication modules example consider programming examples section present examples programmed csn typical operations performed networks sensors goal show expressiveness csn calculus presented also identify aspects networks may interesting model following examples denote msensor msink modules installed anonymous sensors network modules installed sink respectively note also sensors assumed builtin method deploy responsible installing new modules intuition method part tiny operating system allows sensors react first placed field finally assume small examples network layer supports scoped flooding shall see next section supported via software inclusion state sensors ping start simple ping program sensor ping method invoked calls method forward network position battery charge arguments method forward invoked sensor network triggers another call forward network sink distinct implementation method incomming invocation logs position battery values given arguments overall result call sink reachable sensors network principle receive call flood network positions battery charge values values eventually reach sink get logged msensor ping net net forward net cin dom net cout net net install cin cin sense figure reduction semantics processes msink forward log position power net msink msensor msensor querying example shows program network sink periodically queries network readings sensors sensor sample method samples field using sense construct calls method forward neighbourhood position value sampled arguments call queries neighbourhood recursively replica original call original call course made sink method start sample calls method sample network within cycle note sink method named sample instead start sample might get call sample elsewhere network could interfere sampling control cycle msensor sample net net forward net msink start sample net forward log position value msink msensor msensor pbnn polling example cycle sampling done sensor instead sink previous example sink invokes method start sample method propagates call network invokes sample sensor method samples field within cycle forwards result network implementation requires less broadcasts previous one sink call start sample network hand increases amount processing per sensor msensor start sample net sample net forward net msink forward log position value net msink msensor msensor pbnn code deployment examples assume means deploying code sensors example address problem show programmed csn code wish deploy execute one previous example achieve goal sink first calls deploy method network install new module methods start sample sample forward call recursively deploys code sensors network sink calls start sample start sampling waits forwarded results method forward msensor deploy net msink forward log position value net start sample net sample net forward net net msink msensor msensor pbnn refined version code one avoids start sample method completely programmed deploy code sensors sending methods sample forward sensors network invoking deploy deployed code activated call sample sink instead using start sample method msensor deploy net msink forward log position value net sample net net sample sample forward net net msink msensor msensor pbnn notice implementation method sample changed method executed first time sensor starts propagating call neighborhood changes install call newly installed code sample one first implementation example method continues execute calls new version sample starts sampling field forwarding values sealing sensors example shows install sensor network module contains method seal prevents dynamic sensors preventing anyone tampering installed code module also contains method unseal restores original deploy method thus allowing dynamic sink installs module containning methods network broadcasting method call deploy sensor receives call installs module floods neighborhood replica call another message sink replaces deploy method idle prevents instalation software sensors thus effectively seals network external interaction one allowed remainder methods modules sensors msensor deploy msink net seal unseal net msink msensor net deploy idle net msensor pbnn september calculus sensor networks miguel francisco departamento computadores liacc faculdade universidade porto portugal arxiv dec departamento faculdade universidade lisboa portugal abstract consider problem providing rigorous model programming wireless sensor networks assuming collisions packet losses errors dealt lower layers protocol stack propose calculus sensor networks csn captures main abstractions programming applications class devices besides providing syntax semantics calculus show expressiveness providing implementations several examples typical operations sensor networks also included detailed discussion possible extensions csn enable modeling important features networks sensor state sampling strategies network security keywords sensor networks networks ubiquitous computing programming languages ntroduction sensor network challenge sensor networks made tiny devices capable sensing physical world communicating radio links significantly different wireless networks design sensor network strongly driven particular application sensor nodes highly constrained terms power consumption computational resources cpu memory sensor applications require distributed software updates without human intervention previous work fundamental aspects wireless sensor networks mostly focused models sensor nodes assumed store process data coordinate transmissions organize routing messages within network relay data remote receiver see draft fig wireless sensor network collection small devices deployed target area organize network collect measurements physical process transmit data wireless medium data fusion center processing references therein although models provide useful insights connectivity characteristics overall power efficiency sensor networks strong need formal methods capture inherent processing memory constraints illuminate massively parallel nature sensor nodes processing well adapted specific characteristics sensor networks formalism kind specifically process calculus likely strong impact design operating systems communication protocols programming languages class distributed systems terms hardware development well represented class sensor nodes called originally developed berkeley deployed tested several research groups companies currently available trademark crossbow technology mentations sensor nodes controlled operating systems tinyos programming languages like nesc view programming models underlying tools one following drawbacks provide rigorous model calculus sensor network programming level would allow formal verification correctness programs among useful analysis provide global vision sensor network application specific distributed application making less intuitive error prone programmers require programs installed sensor individually something unrealistic large sensor networks allow dynamic network recent middleware developments deluge agilla address drawbacks providing higher level programming abstractions top tinyos including massive code deployment nevertheless still far comprehensive programming solution strong formal support analytical capabilities previous observation motivates design sensor network programming model scratch beyond meeting challenges programming code deployment model capable producing quantitative information amount resources required sensor network programs protocols also providing necessary tools prove correctness related work given distributed concurrent nature sensor network operations build sensor network calculus thirty years experience gathered concurrency theorists programming language designers pursuit adequate formalism theory concurrent systems first steps towards goal given milner development ccs calculus communicating systems ccs describes computations concurrent processes may interact simple synchronization without otherwise exchanging information allowing processes exchange resources links memory references sockets code besides synchronizing considerably increases expressive power formal systems systems known able model mobility patterns resources thus constitute valuable tools reason concurrent distributed systems first system built milner work later developments initial proposal allowed simplification provided asynchronous form calculus since several calculi proposed model concurrent distributed systems many prototype implementations programming languages systems join tyco nomadic pict previous work prasad established first process calculus approach modeling broadcast based systems later work prasad taha established basis calculus broadcasting systems focus line work lies protocol layer networks trying establish operational semantics associated theory allows assertions made networks recently mezzetti sangiorgi discuss use process calculi model wireless systems focusing details lower layers protocol stack collision avoidance establishing operational semantics networks contributions main contribution sensor network programming model based process calculus name calculus sensor networks csn calculus offers following features specifically tailored sensor networks approach csn focuses programming managing sensor networks assumes collisions losses errors dealt lower layers protocol stack system architecture distinguishes csn generic wireless network calculus presented scalability csn offers means provide sensor nodes abilities thus meeting challenges programming managing sensor network broadcast communication instead unicast communication typical process calculi csn captures properties broadcast communication favored sensor networks strong impact energy consumption topology network topology required programmed processes would unrealistic case sensor networks communication constraints due power limitations wireless interface sensor nodes communicate direct neighbors network thus notion neighborhood sensor node set sensor nodes within communication range introduced directly calculus memory processing constraints typical limitations sensor networks terms memory processing capabilities captured explicitly modeling internal processing intelligence individual sensors local sensing naturally sensors able pick local measurements environment thus geographically limited sensitivity provide features devise csn calculus offering abstractions data acquisition communication processing top layer formed network sensor nodes immersed scalar vector field representing physical process captured sensor nodes sensor nodes assumed running parallel sensor node composed collection labeled methods call module represents code executed device process executed sensor node result remote procedure call module sensor seen point view callee result reception message sensor nodes multithreaded may share state example finally adding notions position range able capture nature broadcast communication geographical limits sensor network applications remainder paper structured follows next section describes syntax semantics csn calculus section iii presents several examples functionalities implemented using csn commonly required sensor networks section discuss design options made extend csn model aspects sensor networks finally section presents conclusions directions future work alculus section addresses syntax semantics calculus sensor networks simplicity remainder paper refer sensor node sensor device network sensor syntax provided grammar figure operational semantics given reduction relation depicted figures sensors field idle programs idle parallel composition sensors sequential composition termination method invocation composition install module update sensor sense field sensing broadcast sensor else conditional execution net fig network modules values method collection variable field measure targets position broadcast battery capacity local module syntax csn syntax let denote possible empty sequence elements syntactic category assume countable set labels ranged letter used name methods within modules countable set variables disjoint set labels ranged letter variables stand communicated values battery capacity position field measures modules given program context syntax cns found figure explain syntactic constructs along informal intuitive semantics refer next section precise semantics calculus networks denote composition sensor networks scalar vector field field set pairs position measure describing distribution physical quantity temperature pressure humidity space position given coordinate system sensors measure intensity field respective positions sensor networks flat unstructured collections sensors combined using parallel composition operator sensor represents abstraction physical sensing device parametric position describing location sensor coordinate system transmission range specified radius circle centered position battery capacity position sensors may vary time sensor mobile way transmission range hand usually remains constant time sensor battery exhausted designated inside sensor exists running program module module collection methods defined sensor makes available internal external usage method identified label defined abstraction program parameters method names pairwise distinct within module mutually recursive method definitions make possible represent infinite behavior intuitively collection methods sensor may interpreted function calls tiny operating system installed sensor communication sensor network happens via broadcasting values one sensor neighborhood sensors inside circle centered position position sensor radius broadcast sensor stands sensor broadcast phase already communicated sensors broadcasting fundamental keep track sensors engaged communication far thus preventing delivery message sensor one broadcasting operation target sensors collected bag sensor emitting message upon finishing broadcast bag emptied target sensors released network construct construct available programmer programs ranged idle program denotes terminated thread method invocation selects method arguments either local module broadcasts request neighborhood sensors depending whether keyword keyword net respectively program sense reads measure surrounding field binds within installing replacing methods sensor module performed using construct install calculus also offers standard form branching else construct programs may combined sequence parallel sequential composition designates program first executes proceeds execution contrast represents simultaneous execution values data exchanged sensors comprise field measures positions battery capacities modules notice calculus communicating module means ability transfer code retransmit install remote sensor examples first example illustrates network sensors sample field broadcast measured values special node known sink sink node may different sensors network except usually possesses distinct software module allows collect process values broadcasted network behavior want program following sink issues request network sample field upon reception request sensor samples field position broadcasts measured value back sink sink receives processes values extended version example may found section code modules sensors msensor sink msink given modules parametric position broadcasting range sensor module equipping sensors method sample invoked propagates call neighborhood samples field sense forwards value network notice sensor propagates original request sink required since general sensors network broadcasting range sink therefore sensor echos request hopefully covering network message forwarding recurrent pattern found examples another method sensors module forward simply forwards values sensors network module sink contains different implementation forward method since sink gather values sent sensors log leave unspecified processing done log position value program network sensors idle except sink requests sampling msensor sample net sample sense net net msink forward net sample msink msensor msensor next example illustrates broadcast deployment installation code example runs follows sink node deploys module network seals sensors henceforth preventing dynamic network extended version current example may found section code modules sensors sink given module one wish deploy network carries method seal forwards call network installs new version deploy nothing executed msensor deploy net deploy msink seal net deploy net msink msensor deploy net msensor semantics calculus two name bindings field sensing method definitions displayed occurrence name binding scope sense occurrence name free scope binding otherwise occurrence name bound set free names sensor referred following milner present reduction relation help structural congruence relation structural congruence relation depicted figure allows manipulation term structure adjusting reduce relation defined smallest congruence relation sensors programs closed rules given figure parallel composition operators programs sensors taken commutative associative idle neutral elements respectively vide rules monoid rogram monoid ensor rule idle seq asserts idle also neutral respect sequential composition programs rule program stru incorporates structural congruence programs sensors sensor broadcasting message uses bag collect sensors become idle monoid rogram monoid ensor idle fig max cin cout idle seq program stru broadcast exhausted structural congruence processes sensors engaged communication rule broadcast allows sensor start broadcasting operation terminated sensor sensor insufficient battery capacity performing internal external reduction step vide rule exhausted reduction relation networks notation describes sensors evolve reduce sensors sensing field reduction defined top reduction relation sensors notation inductively defined rules figure reduction sensors parametric field two constants cin cout represent amount energy consumed performing internal computation steps cin broadcasting messages cout computation inside sensors proceeds invoking method either method rno method broadcast release sensing values rule sense updating method collection sensor rule install invocation local method arguments evolves differently depending whether definition part method collection sensor rule method describes invocation method module defined result program values bound variables definition present decided actively wait definition see rule method usually invoking undefined method causes program get stuck typed programming languages use type system ensure invocations undefined methods ruling programs compile time runtime another possible choice would simply discard invocations undefined methods choice provides resilient applications coupled procedure deploying code sensor cin method dom method cout broadcast net net net release cin install install sense cin fig sense parallel structural network reduction semantics processes networks network envision invoke method network code deployed see example may sensors method invocation arrives deployed code semantics propose call actively waits code installed sensors communicate network broadcasting messages message consists remote method invocation unspecified sensors neighborhood emitting sensor words messages targeted particular sensor communication neighborhood sensor defined communication radius guarantee message broadcasted given sensor arrives surrounding sensors might instance landscape obstacles prevent two sensors otherwise within range communicating also broadcast operation message must reach neighborhood sensor notice saying message reach sensor multiple times fact might result echoing message subsequent broadcast operations model broadcasting messages two stages rule broadcast invokes method remote sensor provided distance emitting receiving sensors less transmission radius sensor receiving message put bag emitting sensor thus preventing multiple deliveries message broadcasting observe rule enforce interaction sensors neighborhood rule release finishes broadcast consuming operation net emptying contents emitting sensor bag broadcast operation starts application rule broadcast proceeds multiple eventually none applications rule broadcast one target sensor terminates application rule release installing module sensor module rule install amounts add methods absent replace methods common rigorously operation installing module top denoted may defined operator reminiscent abadi cardelli operator updating methods imperative object calculus sensor senses field immersed rule sense sampling value field position continues computation replacing value bound variables program rule parallel allows reduction happen networks sensors rule structural brings structural congruence reduction relation operational semantics illustrated illustrate operational semantics cns present reduction steps examples discussed end section reduction suppress side annotations writing sensors due space constraints consider rather simple network sink another sensor net sample msink msensor assume sensor within range sink network may reduce follows msink idle msensor broadcast msink idle msensor broadcast monoid ensor msink idle msensor release monoid rogram idle msink msensor method idle msink sense msensor broadcast idle msink sense msensor release idle msink sense msensor idle msink msensor sense monoid ensor msensor idle msink broadcast msensor idle msink broadcast monoid ensor msensor idle msink release monoid rogram idle msensor msink idle msensor log position value msink method monoid ensor log position value idle msensor msink reduction steps sink gets field values sensor position logs sensor idle waiting interaction following present reduction step second last example section illustrate broadcast deployment installation code due space restrictions use simple network sink another sensor within reach net deploy net msink msensor network may reduce follows msink idle msensor broadcast msink idle msensor broadcast monoid ensor msink idle msensor release monoid rogram msink msensor msink install msensor method broadcast msink install msensor release monoid ensor msink install msensor install msink idle msensor broadcast msink idle msensor broadcast monoid ensor msink idle msensor release monoid rogram idle msink msensor method idle msink install deploy idle msensor broadcast idle msink install deploy idle msensor release monoid ensor idle msink install deploy idle msensor install idle msink idle msensor deploy idle reductions sink idle deploying code sensor sensor also idle waiting interaction code module installed deploy method disabled iii rogramming xamples section present examples programmed csn typical operations performed networks sensors goal show expressiveness csn calculus presented also identify aspects networks may interesting model following examples denote msensor msink modules installed anonymous sensors network modules installed sink respectively note also sensors assumed builtin method deploy responsible installing new modules intuition method part tiny operating system allows sensors react first placed field finally assume small examples network layer supports scoped flooding shall see next section supported via software inclusion state sensors ping start simple ping program sensor ping method invoked calls method forward network position battery charge arguments method forward invoked sensor network triggers another call forward network sink distinct implementation method incomming invocation logs position battery values given arguments overall result call sink reachable sensors network principle receive call flood network positions battery charge values values eventually reach sink get logged msensor ping net net ping forward net msink forward log position power net ping msink msensor msensor querying example shows program network sink periodically queries network readings sensors sensor sample method samples field using sense construct calls method forward neighbourhood position value sampled arguments call queries neighbourhood recursively replica original call original call course made sink method start sample calls method sample network within cycle note sink method named sample instead start sample might get call sample elsewhere network could interfere sampling control cycle msensor sample sense net net sample forward net msink start sample net sample forward log position value msink msensor msensor polling example cycle sampling done sensor instead sink previous example sink invokes method start sample method propagates call network invokes sample sensor method samples field within cycle forwards result network implementation requires less broadcasts previous one sink call start sample network hand increases amount processing per sensor msensor start sample net sample sample sense net sample forward net msink log position value forward net msink msensor msensor code deployment examples assume means deploying code sensors example address problem show programmed csn code wish deploy execute one previous example achieve goal sink first calls deploy method network install new module methods start sample sample forward call recursively deploys code sensors network sink calls start sample start sampling waits forwarded results method forward msensor deploy net deploy log position value start sample net sample sample sense net sample forward net msink forward net deploy net msink msensor msensor refined version code one avoids start sample method completely programmed deploy code sensors sending methods sample forward sensors network invoking deploy deployed code activated call sample sink instead using start sample method msensor deploy net deploy msink forward log position value net deploy sample net sample sample sense net sample sample forward net net sample msink msensor msensor notice implementation method sample changed method executed first time sensor starts propagating call neighborhood changes install call newly installed code sample one first implementation example method continues execute calls new version sample starts sampling field forwarding values sealing sensors example shows install sensor network module contains method seal prevents dynamic sensors preventing anyone tampering installed code module also contains method unseal restores original deploy method thus allowing dynamic sink installs module containning methods network broadcasting method call deploy sensor receives call installs module floods neighborhood replica call another message sink replaces deploy method idle prevents instalation software sensors thus effectively seals network external interaction one allowed remainder methods modules sensors msensor deploy net deploy msink net deploy seal deploy idle uns eal deploy net deploy net msink msensor msensor iscussion previous sections focused attention programming issues sensor network presented core calculus expressive enough model fundamental operations local broadcast messages local sensing environment software module updates csn allows global modeling sensor networks sense allows design implement sensor network applications distributed applications rather giving programmer view programming task also provides tools manage running sensor networks namely use software deployment capabilities important features sensor networks consciously left csn sequel discuss features sketch ideas would include support state programming point view adding state sensors essential sensors limited computational capabilities may perform data processing sending sink processing assumes sensor capable buffering data thus maintain state way csn sensors state indeed atributes may viewed sensor state since characteristic sensor usually controlled hardware level chose represent state parameters sensors programmer may read values time builtin method calls change data performed transparently programmer hardware operating system mentioned clear value changes time position may also change time envision sensors endowed form mobility sensors dropped atmosphere flowing ocean allow systematic extension sensors state variables assume sensor heap values variables stored model chosen heap orthogonal sensor calculus discussion assume enrich values language set keys ranged heap may thus defined map keys values intuitively think associative memory usual operations put get lookup hash programs running sensors may share state exchanging keys assume also operations atomic thus race conditions arise basic model heap ping example section scoped flooding thus eliminating echos software associating unique key remote procedure call broadcast network key created hash function takes arguments position battery sensor sensor receiving call ping propagates call neighborhood generates new key send position battery charge forward call stores key heap avoid forwarding forward call hand time sensor receives call forward checks whether key associated call heap nothing forwards call stores key heap avoid future msensor ping net ping hash net put forward lookup net put msink msensor net ping msink idle msensor events another characteristic sensors modus operandi sensors sample field result instructions implemented software controls case csn sensors programmer responsible controlling sensing activity sensor network course possible sensor nodes activated different ways example may sensing routines implemented hardware operating system level thus directly controllable programmer classes sensor nodes tipically sample field periodically activated given condition arises temperature given threshold detection given threshold detection strong source infrared light way certain environmental conditions events activate sensor triggering execution handler procedure processes event support kind sensors csn could achieved assuming sensor builtin handler procedure say handle events handler procedure activated receives value field triggered event note point view sensor occurrence event equivalent deployment method invocation processing core field value associated event sensor control deployment may programmed react different ways calls providing adequate implementations handle routine events could included semantics given section following rule event case builtin method code deployment handler could programmed change behavior network presence events one could envision default handler handle idle ignores events could change default behavior event triggers alarm gets sent sink possible implementation dynamic network default handlers seen code msensor handle idle msink handle idle net deploy handle alarm net alarm net alarm alarm msink sing bell msensor msensor default implementation handle procedure superseded one eventualy triggers alarm sink complex behavior could modeled sensors take multiple readings handler associated event security finally another issue outmost importance management sensor networks security important note many potential applications sensor networks high risk situations examples may monitorization ecological disaster areas volcanic sismic activity radiation levels contaminated areas secure access data fundamental establish credibility correctly assessing risks management episodes csn taken security issues consideration goal time however one feature calculus may provide interesting solutions future fact csn computation within sensor results invocation methods modules sensor either originating network within sensor sense modules sensor work firewall used control incomming messages implement security protocols thus remote method invocations software updates might first validated locally methods sensor modules actions would performed idea equipping sensors general domains kind membrane filters interactions surrounding network explored instance kell calculus brane calculi miko one possible development incorporate features membrane model csn current formulation calculus also assumes methods module sensor visible network possible implement access policy methods way methods private sensor invoked within sensor allows example complete encapsulation state sensor onclusions uture ork aiming providing sensor networks rigorous adequate programming model upon operating systems programming languages built presented csn calculus sensor networks developed specifically class distributed systems identifying necessary sensing processing wireless broadcasting features calculus opted base work abstraction physical link layer communication issues contrast previous work wireless network calculi thus focusing system requirements programming applications approach resulted csn syntax semantics whose expressiveness illustrated series implementations typical operations sensor networks also included detailed discussion possible extensions csn account important properties sensors state sampling strategies security part ongoing efforts currently using csn establish mathematical framework reasoning sensor networks one major objective work consists providing formal proofs correctness data gathering protocols commonly used current sensor networks whose performance reliability far evaluated computer simulations experiments practical point view focus set development prototype implementation csn prototype used emulate behavior sensor networks software ultimately port programming model natural development architecture sensor network applications acknowledgements authors gratefully acknowledge insightful discussions gerhard maierbacher departamento computadores faculdade universidade porto eferences tinyos documentation project available http abadi cardelli imperative object calculus tapsoft theory practice software development number lncs pages akyildiz sankarasubramaniam cayirci survey sensor networks ieee communications 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mobile processes parts information computation prasad taha towards primitive higher order calculus broadcasting systems ppdp international conference principles practice declarative programming prasad calculus broadcasting systems tapsoft volume pages scaglione servetto interdependence routing data compression sensor networks proc acm mobicom atlanta schmitt stefani distributed process calculus proceedings popl pages acm press stefani calculus kells proceedings fgc volume elsevier science vasconcelos lopes silva distribution mobility lexical scoping process calculi workshop high level programming languages hlcl volume entcs pages elsevier science wojciechowski sewell nomadic pict language infrastructure design mobile agents concurrency ieee calculus sensor networks miguel francisco departamento computadores liacc faculdade universidade porto portugal arxiv dec departamento faculdade universidade lisboa portugal abstract consider problem providing rigorous model programming wireless sensor networks assuming collisions packet losses errors dealt lower layers protocol stack propose calculus sensor networks csn captures main abstractions programming applications class devices besides providing syntax semantics calculus show expressiveness providing implementations several examples typical operations sensor networks also included detailed discussion possible extensions csn enable modeling important features networks sensor state sampling strategies network security keywords sensor networks networks ubiquitous computing programming languages ntroduction sensor network challenge sensor networks made tiny devices capable sensing physical world communicating radio links significantly different wireless networks design sensor network strongly driven particular application sensor nodes highly constrained terms power consumption computational resources cpu memory sensor applications require distributed software updates without human intervention previous work fundamental aspects wireless sensor networks mostly focused models sensor nodes assumed store process data coordinate transmissions organize routing messages within network relay data remote receiver see references therein although models provide useful insights draft fig wireless sensor network collection small devices deployed target area organize network collect measurements physical process transmit data wireless medium data fusion center processing connectivity characteristics overall power efficiency sensor networks strong need formal methods capture inherent processing memory constraints illuminate massively parallel nature sensor nodes processing well adapted specific characteristics sensor networks formalism kind specifically process calculus likely strong impact design operating systems communication protocols programming languages class distributed systems terms hardware development well represented class sensor nodes called originally developed berkeley deployed tested several research groups companies currently available implementations sensor nodes controlled operating systems tinyos programming languages like nesc view programming models trademark crossbow technology inc draft underlying tools one following drawbacks provide rigorous model calculus sensor network programming level would allow formal verification correctness programs among useful analysis provide global vision sensor network application specific distributed application making less intuitive error prone programmers require programs installed sensor individually something unrealistic large sensor networks allow dynamic network recent middleware developments deluge agilla address drawbacks providing higher level programming abstractions top tinyos including massive code deployment nevertheless still far comprehensive programming solution strong formal support analytical capabilities previous observation motivates design sensor network programming model scratch beyond meeting challenges programming code deployment model capable producing quantitative information amount resources required sensor network programs protocols also providing necessary tools prove correctness related work given distributed concurrent nature sensor network operations build sensor network calculus thirty years experience gathered concurrency theorists programming language designers pursuit adequate formalism theory concurrent systems first steps towards goal given milner development ccs calculus communicating systems ccs describes computations concurrent processes may interact simple synchronization without otherwise exchanging information allowing processes exchange resources links memory references sockets code besides synchronizing considerably increases expressive power formal systems systems known able model mobility patterns resources thus constitute valuable tools reason concurrent distributed systems first system built milner work later developments initial proposal allowed simplification provided asynchronous form calculus since several calculi proposed model concurrent distributed systems many draft prototype implementations programming languages systems join tyco nomadic pict previous work prasad established first process calculus approach modeling broadcast based systems later work prasad taha established basis calculus broadcasting systems focus line work lies protocol layer networks trying establish operational semantics associated theory allows assertions made networks recently mezzetti sangiorgi discuss use process calculi model wireless systems focusing details lower layers protocol stack collision avoidance establishing operational semantics networks contributions main contribution sensor network programming model based process calculus name calculus sensor networks csn calculus offers following features specifically tailored sensor networks approach csn focuses programming managing sensor networks assumes collisions losses errors dealt lower layers protocol stack system architecture distinguishes csn generic wireless network calculus presented scalability csn offers means provide sensor nodes abilities thus meeting challenges programming managing sensor network broadcast communication instead unicast communication typical process calculi csn captures properties broadcast communication favored sensor networks strong impact energy consumption topology network topology required programmed processes would unrealistic case sensor networks communication constraints due power limitations wireless interface sensor nodes communicate direct neighbors network thus notion neighborhood sensor node set sensor nodes within communication range introduced directly calculus memory processing constraints typical limitations sensor networks terms memory processing capabilities captured explicitly modeling internal processing intelligence individual sensors draft local sensing naturally sensors able pick local measurements environment thus geographically limited sensitivity provide features devise csn calculus offering abstractions data acquisition communication processing top layer formed network sensor nodes immersed scalar vector field representing physical process captured sensor nodes sensor nodes assumed running parallel sensor node composed collection labeled methods call module represents code executed device process executed sensor node result remote procedure call module sensor seen point view callee result reception message sensor nodes multithreaded may share state example finally adding notions position range able capture nature broadcast communication geographical limits sensor network applications remainder paper structured follows next section describes syntax semantics csn calculus section iii presents several examples functionalities implemented using csn commonly required sensor networks section discuss design options made extend csn model aspects sensor networks finally section presents conclusions directions future work alculus section addresses syntax semantics calculus sensor networks simplicity remainder paper refer sensor node sensor device network sensor syntax provided grammar figure operational semantics given reduction relation depicted figures syntax let denote possible empty sequence elements syntactic category assume countable set labels ranged letter used name methods within modules countable set variables disjoint set labels ranged letter variables stand communicated values battery capacity position field measures modules given program context syntax cns found figure explain syntactic constructs along informal intuitive semantics refer next section precise semantics calculus draft sensors field idle programs idle parallel composition sensors sequential composition termination method invocation composition install module update sensor sense field sensing broadcast sensor else conditional execution net fig network modules method collection values variable field measure targets position broadcast battery capacity local module syntax csn networks denote composition sensor networks scalar vector field field set pairs position measure describing distribution physical quantity temperature pressure humidity space position given coordinate system sensors measure intensity field respective positions sensor networks flat unstructured collections sensors combined using parallel composition operator sensor represents abstraction physical sensing device parametric position describing location sensor coordinate system transmission range specified radius circle centered position battery capacity position sensors may draft vary time sensor mobile way transmission range hand usually remains constant time sensor battery exhausted designated inside sensor exists running program module module collection methods defined sensor makes available internal external usage method identified label defined abstraction program parameters method names pairwise distinct within module mutually recursive method definitions make possible represent infinite behavior intuitively collection methods sensor may interpreted function calls tiny operating system installed sensor communication sensor network happens via broadcasting values one sensor neighborhood sensors inside circle centered position position sensor radius broadcast sensor stands sensor broadcast phase already communicated sensors broadcasting fundamental keep track sensors engaged communication far thus preventing delivery message sensor one broadcasting operation target sensors collected bag sensor emitting message upon finishing broadcast bag emptied target sensors released network construct construct available programmer programs ranged idle program denotes terminated thread method invocation selects method arguments either local module broadcasts request neighborhood sensors depending whether keyword keyword net respectively program sense reads measure surrounding field binds within installing replacing methods sensor module performed using construct install calculus also offers standard form branching else construct programs may combined sequence parallel sequential composition designates program first executes proceeds execution contrast represents simultaneous execution values data exchanged sensors comprise field measures positions battery capacities modules notice calculus communicating module means ability transfer code retransmit install remote sensor examples first example illustrates network sensors sample field broadcast measured values special node known sink sink node may different sensors draft network except usually possesses distinct software module allows collect process values broadcasted network behavior want program following sink issues request network sample field upon reception request sensor samples field position broadcasts measured value back sink sink receives processes values extended version example may found section code modules sensors msensor sink msink given modules parametric position broadcasting range sensor module equipping sensors method sample invoked propagates call neighborhood samples field sense forwards value network notice sensor propagates original request sink required since general sensors network broadcasting range sink therefore sensor echos request hopefully covering network message forwarding recurrent pattern found examples another method sensors module forward simply forwards values sensors network module sink contains different implementation forward method since sink gather values sent sensors log leave unspecified processing done log position value program network sensors idle except sink requests sampling msensor sample net sample sense net net msink forward net sample msink msensor msensor next example illustrates broadcast deployment installation code example runs follows sink node deploys module network seals sensors henceforth preventing dynamic network extended version current example may found section code modules sensors sink given module one wish deploy network carries method seal forwards call network installs new version deploy nothing executed draft idle monoid rogram monoid ensor idle fig max cin cout idle seq program stru broadcast exhausted structural congruence processes sensors msensor deploy net deploy msink seal net deploy net msink msensor deploy net msensor semantics calculus two name bindings field sensing method definitions displayed occurrence name binding scope sense occurrence name free scope binding otherwise occurrence name bound set free names sensor referred following milner present reduction relation help structural congruence relation structural congruence relation depicted figure allows manipulation term structure adjusting reduce relation defined smallest congruence relation sensors programs closed rules given figure parallel composition operators programs sensors taken commutative associative idle neutral elements respectively vide rules monoid rogram monoid ensor rule idle seq asserts idle also neutral respect sequential composition programs rule program stru incorporates structural congruence programs sensors sensor broadcasting message uses bag collect sensors become engaged communication rule broadcast allows sensor start broadcasting operation draft cin method dom method cout broadcast net net net release cin install install sense cin parallel structural fig sense network reduction semantics processes networks terminated sensor sensor insufficient battery capacity performing internal external reduction step vide rule exhausted reduction relation networks notation describes sensors evolve reduce sensors sensing field reduction defined top reduction relation sensors notation inductively defined rules figure reduction sensors parametric field two constants cin cout represent amount energy consumed performing internal computation steps cin broadcasting messages cout computation inside sensors proceeds invoking method either method rno method broadcast release sensing values rule sense updating method collection sensor rule install draft invocation local method arguments evolves differently depending whether definition part method collection sensor rule method describes invocation method module defined result program values bound variables definition present decided actively wait definition see rule method usually invoking undefined method causes program get stuck typed programming languages use type system ensure invocations undefined methods ruling programs compile time runtime another possible choice would simply discard invocations undefined methods choice provides resilient applications coupled procedure deploying code sensor network envision invoke method network code deployed see example may sensors method invocation arrives deployed code semantics propose call actively waits code installed sensors communicate network broadcasting messages message consists remote method invocation unspecified sensors neighborhood emitting sensor words messages targeted particular sensor communication neighborhood sensor defined communication radius guarantee message broadcasted given sensor arrives surrounding sensors might instance landscape obstacles prevent two sensors otherwise within range communicating also broadcast operation message must reach neighborhood sensor notice saying message reach sensor multiple times fact might result echoing message subsequent broadcast operations model broadcasting messages two stages rule broadcast invokes method remote sensor provided distance emitting receiving sensors less transmission radius sensor receiving message put bag emitting sensor thus preventing multiple deliveries message broadcasting observe rule enforce interaction sensors neighborhood rule release finishes broadcast consuming operation net emptying contents emitting sensor bag broadcast operation starts application rule broadcast proceeds multiple eventually none applications rule broadcast one target sensor terminates application rule release installing module sensor module rule install amounts add methods absent replace methods common rigorously operation installing module top denoted may defined draft operator reminiscent abadi cardelli operator updating methods imperative object calculus sensor senses field immersed rule sense sampling value field position continues computation replacing value bound variables program rule parallel allows reduction happen networks sensors rule structural brings structural congruence reduction relation operational semantics illustrated illustrate operational semantics cns present reduction steps examples discussed end section reduction suppress side annotations writing sensors due space constraints consider rather simple network sink another sensor net sample msink msensor assume sensor within range sink network may reduce follows draft msink idle msensor broadcast msink idle msensor broadcast monoid ensor msink idle msensor release monoid rogram idle msink msensor method idle msink sense msensor broadcast idle msink sense msensor release idle msink sense msensor idle msink msensor sense monoid ensor msensor idle msink broadcast msensor idle msink broadcast monoid ensor msensor idle msink release monoid rogram idle msensor msink idle msensor log position value msink method monoid ensor log position value idle msensor msink reduction steps sink gets field values sensor position logs sensor idle waiting interaction following present reduction step second last example section illustrate broadcast deployment installation code due space restrictions use simple network sink another sensor within reach net deploy net msink msensor draft network may reduce follows msink idle msensor broadcast msink idle msensor broadcast monoid ensor msink idle msensor release monoid rogram msink msensor msink install msensor method broadcast msink install msensor release monoid ensor msink install msensor install msink idle msensor broadcast msink idle msensor broadcast monoid ensor msink idle msensor release monoid rogram idle msink msensor method idle msink install deploy idle msensor broadcast idle msink install deploy idle msensor release monoid ensor idle msink install deploy idle msensor install idle msink idle msensor deploy idle reductions sink idle deploying code sensor sensor also idle waiting interaction code module installed deploy method disabled draft iii rogramming xamples section present examples programmed csn typical operations performed networks sensors goal show expressiveness csn calculus presented also identify aspects networks may interesting model following examples denote msensor msink modules installed anonymous sensors network modules installed sink respectively note also sensors assumed builtin method deploy responsible installing new modules intuition method part tiny operating system allows sensors react first placed field finally assume small examples network layer supports scoped flooding shall see next section supported via software inclusion state sensors ping start simple ping program sensor ping method invoked calls method forward network position battery charge arguments method forward invoked sensor network triggers another call forward network sink distinct implementation method incomming invocation logs position battery values given arguments overall result call sink reachable sensors network principle receive call flood network positions battery charge values values eventually reach sink get logged msensor ping net net ping forward net msink forward log position power net ping msink msensor msensor querying example shows program network sink periodically queries network readings sensors sensor sample method samples field using sense draft construct calls method forward neighbourhood position value sampled arguments call queries neighbourhood recursively replica original call original call course made sink method start sample calls method sample network within cycle note sink method named sample instead start sample might get call sample elsewhere network could interfere sampling control cycle msensor sample sense net net sample forward net msink start sample forward log position value net sample msink msensor msensor polling example cycle sampling done sensor instead sink previous example sink invokes method start sample method propagates call network invokes sample sensor method samples field within cycle forwards result network implementation requires less broadcasts previous one sink call start sample network hand increases amount processing per sensor msensor start sample net sample sample sense net sample forward net msink forward log position value net msink draft msensor msensor code deployment examples assume means deploying code sensors example address problem show programmed csn code wish deploy execute one previous example achieve goal sink first calls deploy method network install new module methods start sample sample forward call recursively deploys code sensors network sink calls start sample start sampling waits forwarded results method forward msensor deploy net deploy log position value start sample net sample sample sense net sample forward net msink forward net deploy net msink msensor msensor refined version code one avoids start sample method completely programmed deploy code sensors sending methods sample forward sensors network invoking deploy deployed code activated call sample sink instead using start sample method msensor deploy net deploy msink forward log position value draft net deploy sample net sample sample sense net sample sample forward net net sample msink msensor msensor notice implementation method sample changed method executed first time sensor starts propagating call neighborhood changes install call newly installed code sample one first implementation example method continues execute calls new version sample starts sampling field forwarding values sealing sensors example shows install sensor network module contains method seal prevents dynamic sensors preventing anyone tampering installed code module also contains method unseal restores original deploy method thus allowing dynamic sink installs module containning methods network broadcasting method call deploy sensor receives call installs module floods neighborhood replica call another message sink replaces deploy method idle prevents instalation software sensors thus effectively seals network external interaction one allowed remainder methods modules sensors msensor deploy net deploy msink net deploy seal deploy idle uns eal deploy net deploy draft net msink msensor msensor iscussion previous sections focused attention programming issues sensor network presented core calculus expressive enough model fundamental operations local broadcast messages local sensing environment software module updates csn allows global modeling sensor networks sense allows design implement sensor network applications distributed applications rather giving programmer view programming task also provides tools manage running sensor networks namely use software deployment capabilities important features sensor networks consciously left csn sequel discuss features sketch ideas would include support state programming point view adding state sensors essential sensors limited computational capabilities may perform data processing sending sink processing assumes sensor capable buffering data thus maintain state way csn sensors state indeed atributes may viewed sensor state since characteristic sensor usually controlled hardware level chose represent state parameters sensors programmer may read values time builtin method calls change data performed transparently programmer hardware operating system mentioned clear value changes time position may also change time envision sensors endowed form mobility sensors dropped atmosphere flowing ocean allow systematic extension sensors state variables assume sensor heap values variables stored model chosen heap orthogonal sensor calculus discussion assume enrich values language set keys ranged heap may thus defined map keys values intuitively think associative memory usual operations put get lookup hash programs running sensors may share state exchanging keys assume also operations atomic thus race conditions arise draft basic model heap ping example section scoped flooding thus eliminating echos software associating unique key remote procedure call broadcast network key created hash function takes arguments position battery sensor sensor receiving call ping propagates call neighborhood generates new key send position battery charge forward call stores key heap avoid forwarding forward call hand time sensor receives call forward checks whether key associated call heap nothing forwards call stores key heap avoid future msensor ping net ping hash net put forward lookup net put msink msensor net ping msink idle msensor events another characteristic sensors modus operandi sensors sample field result instructions implemented software controls case csn sensors programmer responsible controlling sensing activity sensor network course possible sensor nodes activated different ways example may sensing routines implemented hardware operating system level thus directly controllable programmer classes sensor nodes tipically sample field periodically activated given condition arises temperature given threshold detection given threshold detection strong source infrared light way certain environmental conditions events activate sensor triggering execution handler procedure processes event support kind sensors csn could achieved assuming sensor builtin handler procedure say handle events handler procedure activated receives value field triggered event note draft point view sensor occurrence event equivalent deployment method invocation processing core field value associated event sensor control deployment may programmed react different ways calls providing adequate implementations handle routine events could included semantics given section following rule event case builtin method code deployment handler could programmed change behavior network presence events one could envision default handler handle idle ignores events could change default behavior event triggers alarm gets sent sink possible implementation dynamic network default handlers seen code msensor handle idle msink handle idle net deploy handle alarm net alarm net alarm alarm msink sing bell msensor msensor default implementation handle procedure superseded one eventualy triggers alarm sink complex behavior could modeled sensors take multiple readings handler associated event security finally another issue outmost importance management sensor networks security important note many potential applications sensor networks high risk situations examples may monitorization ecological disaster areas volcanic sismic activity radiation levels contaminated areas secure access data fundamental establish credibility correctly assessing risks management episodes csn taken security issues consideration goal time however one feature calculus may provide interesting solutions future fact csn computation within sensor draft results invocation methods modules sensor either originating network within sensor sense modules sensor work firewall used control incomming messages implement security protocols thus remote method invocations software updates might first validated locally methods sensor modules actions would performed idea equipping sensors general domains kind membrane filters interactions surrounding network explored instance kell calculus brane calculi miko one possible development incorporate features membrane model csn current formulation calculus also assumes methods module sensor visible network possible implement access policy methods way methods private sensor invoked within sensor allows example complete encapsulation state sensor onclusions uture ork aiming providing sensor networks rigorous adequate programming model upon operating systems programming languages built presented csn calculus sensor networks developed specifically class distributed systems identifying necessary sensing processing wireless broadcasting features calculus opted base work abstraction physical link layer communication issues contrast previous work wireless network calculi thus focusing system requirements programming applications approach resulted csn syntax semantics whose expressiveness illustrated series implementations typical operations sensor networks also included detailed discussion possible extensions csn account important properties sensors state sampling strategies security part ongoing efforts currently using csn establish mathematical framework reasoning sensor networks one major objective work consists providing formal proofs correctness data gathering protocols commonly used current sensor networks whose performance reliability far evaluated computer simulations experiments practical point view focus set development prototype implementation csn prototype used emulate behavior sensor networks software ultimately port programming model natural development architecture sensor draft network applications acknowledgements authors gratefully acknowledge insightful discussions gerhard maierbacher departamento computadores faculdade universidade porto draft figure available jpg format http figure available jpg format http

filter smoother improved covariance matrix approximation aug henri nurminen tohid ardeshiri robert fredrik gustafsson count smoothing algorithms linear models measurement noise presented presented algorithms use variational bayes based posterior approximation coupled location skewness variables reduce error caused variational approximation although variational update done suboptimally simulations show proposed method gives accurate approximation posterior covariance matrix earlier proposed variational algorithm consequently novel filter smoother outperform earlier proposed robust filter smoother existing alternatives accuracy speed present simulations tests based navigation data particular gps data urban area demonstrate performance novel methods moreover extension proposed algorithms cover case distribution measurement noise multivariate outlined finally paper presents study theoretical performance bounds proposed algorithms error fig error histogram uwb ranging experiment described shows positive skewness edge bars show errors outside figure limits anchor range true position likelihood index skew skewness robust filtering kalman filter variational bayes rts smoother truncated normal distribution lower bound ntroduction asymmetric noise processes present many inference problems radio signal based distance estimation example obstacles cause large positive errors dominate symmetrically distributed errors sources example error histogram distance measurements collected indoor environment given fig asymmetric outlier distributions predicted normal distribution equivalent second order moments normal distributions symmetric distributions skew generalization modeling flexibility capture skewness noise processes illustrate fig shows contours likelihood function three range measurements measurements positive outliers example normal measurement noise models compared due additional modeling flexibility based likelihood provides apposite spread probability mass normal based likelihoods nurminen department automation science engineering tampere university technology tut box tampere finland nurminen receives funding tut graduate school foundation nokia corporation tekniikan ardeshiri division automatic control department electrical engineering university sweden received funding swedish research council project scalable kalman filters work ardeshiri currently department engineering university cambridge trumpington street cambridge gustafsson division automatic control department electrical engineering university sweden email fredrik fig contours likelihood function three range measurements normal left middle right measurement noise models based likelihoods handle one outlier upper row model handles two positive outlier measurements bottom row due asymmetry measurement model parameters selected values first two moments coincide applications skew distributions limited radio signal based localization biostatistics skewed distributions used modeling tool handling heterogeneous data involving asymmetric behaviors across subpopulations psychiatric research skew normal distribution used model asymmetric data economics skew normal skew used models describing claims insurance examples describing approaches analysis modeling using multivariate skew normal skew econometrics environmetrics presented various algorithms dedicated statistical inference time series data exhibit asymmetric distribution particle filters easily adapted skew noise distributions computational complexity filters increases rapidly state dimension increases skew kalman filter proposed filter extended robust filter using monte carlo integration solutions based models measurement noise dependent process skewed marginals article proposes filtering independent skew measurement process noises cost increasing filter state dimension time skew filters sequential processing requires numerical evaluation multidimensional integrals inference problem skew likelihood distributions also cast optimization problem proposes approach model measurement noise uwb based positioning problem using tailored distribution skewness also modeled mixture normal distributions gaussian mixtures many filtering algorithms distributions gaussian sum filter interactive multiple model imm filter however gms exponentially decaying tails thus sensitive outlier measurements furthermore order keep computational cost gaussian sum filter practicable mixture reduction algorithm mra required mras computationally expensive involve approximations posterior density filtering smoothing algorithms linear discretetime models measurement noise using variational bayes method presented tests real uwb indoor localization data filter shown accurate computationally inexpensive paper proposes improvements robust filter smoother proposed analogous measurement noise modeled skew proposed filter smoother use approximation filtering smoothing posteriors however main contributions paper new factorization approximate posterior distribution derivation rao lower bound crlb proposed filter smoother application existing method approximating statistics truncated multivariate normal distribution tmnd proof optimality truncation ordering approximation moments tmnd tmnd multivariate normal distribution whose support restricted truncated linear constraints renormalized integrate unity aforementioned contributions improve estimation performance filter smoother reducing covariance underestimation common inference algorithms chapter knowledge approximations applied skew earlier works wand rest paper structured follows section filtering smoothing problem involving univariate skew posed section iii solution based formulated problem proposed proposed solution evaluated using simulated data well realworld data sections respectively essential expressions extend proposed filtering smoothing algorithms problems involving multivariate mvst distribution given section performance bound time series data measurement noise derived evaluated simulation section vii concluding remarks given section viii nference roblem formulation consider linear gaussian state evolution model axk iid denotes probability density function pdf multivariate normal distribution mean covariance matrix rnx state transition matrix rnx indexed state estimated initial prior distribution subscript read time using measurements time rnx process noise measurements rny assumed governed measurement equation cxk measurement matrix measurement noise independent process noise product independent univariate skew pdf iid rii model also nonstationary sake lighter notation subscripts rii omitted univariate skew parametrized location parameter spread parameter shape parameter degrees freedom pdf pdf student gamma tion also denotes cumulative distribution function cdf student degrees freedom expressions first two moments univariate skew found model independent univariate measurement noise components justified applications data different sensors assumed statistically independent noise extension comparison multivariate noise discussed section independent univariate noise model induces hierarchical representation measurement likelihood cxk rny diagonal matrix whose diagonal elements square roots rii spread parameters skew rny diagonal matrix whose diagonal elements shape parameters vector whose elements degrees freedom operator gives entry argument diagonal matrix priori independent random diagonal elements also tmnd closed positive orthant support location parameter matrix furthermore gamma distribution shape parameter rate parameter bayesian smoothing means finding smoothing posterior smoothing posterior approximated factorized distribution form subsequently approximate posterior distributions computed using approach approach minimizes divergence kld dkl log true posterior factorized approximation dkl minimized numerical simulations manifest covariance matrix underestimation known weakness approach chapter one contributions paper reduce covariance underestimation filter smoother proposed removing independence approximations posterior approximation proposed filter smoother presented section iii iii roposed ilter moother using bayes theorem state evolution model likelihood joint smoothing posterior pdf derived posterior analytically tractable propose seek approximation form qxu factors specified argmin dkl qxu qxu hence approximated independent highly correlated posteriori analytical solutions obtained cyclic iteration log qxu log cxu log log qxu expected values right hand sides taken respect current qxu chapter also cxu constants respect variables respectively furthermore joint pdf written axl cxk computation expectation relegated appendix requires first two moments tmnd support orthant moments computed using formulas presented require evaluating cdf general multivariate normal distributions atlab function mvncdf implements numerical quadrature dimensional cases carlo method dimensionalities however methods prohibitively slow therefore approximate tmnd moments using fast sequential algorithm suggested method initialized original normal density whose parameters updated applying one linear constraint time constraint mean covariance matrix normal distribution computed analytically distribution approximated normal updated moments method illustrated fig bivariate normal distribution truncated positive quadrant approximated normal distribution result sequential truncation depends order constraints applied finding optimal order applying truncations problem combinatorial complexity hence adopt greedy approach whereby constraint applied chosen among remaining constraints resulting normal closest true tmnd lemma optimal constraint select one truncates probability optimality respect kld measure example fig vertical constraint truncates probability applied first lemma let tmnd support argmin dkl argmin ith element ith diagonal element iverson bracket proof dkl log log log log means equality additive constant since increasing function proof follows obtained algorithm optimal processing sequence computing mean covariance matrix given multivariate normal distribution truncated positive orthant given table many programming languages numerically robust method implement line algorithm table using scaled complementary error function erfcx erfcx recursion convergent local optimum chapter however proof convergence available moments tmnd approximated spite lack convergence proof iterations diverge numerical simulations presented section smoother update includes forward filtering step smoother rtss compute approximate filtering posterior posterior tmnd variables restricted positive orthant tmnd approximated multivariate normal distribution whose parameters contour normal truncation contour normal approximation linear constraint truncated area fig sequential truncation method approximating truncated normal distribution normal distribution original normal distribution contour ellipse contains probability truncated area gray first applied truncation gray contour resulting normal approximation second applied truncation gray contour normal approximation final normal approximation table moothing measurement noise table ptimal sequential truncation positive orthant inputs set truncated components indices argmin underflow pdf cdf else end end outputs obtained using sequential truncation approximation enables recursive forward filtering use rtss backward smoothing step gives normal approximations marginal smoothing posteriors qxu uxkk iterations converge variables integrated get approximate smoothing posteriors parameters output smoother sts algorithm table sts restricted online recursive algorithm synthesize filter summarized table iii filter output filtering step also tmnd analogy sts approximated multivariate normal distribution recursive algorithm using sequential truncation tmnd qxu approximated normal distribution parameters marginals outputs filter stf algorithm table iii imulations numerical simulations use satellite navigation pseudorange measurements model iid ksi ith satellite position bias prior parameter model linearized using first order taylor polynomial approximation linearization error negligible satellites far satellite constellation global positioning system first second year provided international gnss service used visible satellites error rmse computed computations made atlab inputs initialization iny repeat update qxu given blockdiag czt end end update given qxu czt end end converged outputs computation tmnd statistics subsection study computation moments untruncated components tmnd one state one measurement vector per monte carlo replication generated model degrees freedom corresponding likelihood prior diag replications compared methods sequential truncations optimal truncation order topt random order trand variational bayes analytical formulas using atlab function mvncdf mvncdf trand constraints chosen randomly truncation update skew variational bayes filter stvbf iteration terminated position estimate changes less iteration reference solution expectation value bootstrap particle filter update samples table iii iltering measurement noise mvncdf trand topt inputs initialization iny repeat update qxu given blockdiag czt update given qxu czt end converged end outputs fig topt outperforms trand one negative outlier added measurement noise vector one truncation truncates much probability rest fig shows distributions estimates differences estimate errors given per cent estimation error box levels quantiles asterisks show minimum maximum values topt outperforms trand cases high skewness reflects result lemma mvncdf accurate topt cases high skewness mvncdf computational load roughly times topt justifies use sequential truncation approximation order truncations sequential truncation algorithm affects performance clear differences amounts probability mass truncation truncates fig presents example measurement noise realization generated skew normal distribution modified min min random index parameter large generates one negative outlier measurement vector results one truncation significantly larger truncated probability mass rest truncations fig shows difference trand error topt error positive difference means topt accurate errors refer distance estimate figure shows large topt accurate trand thus effect truncation ordering accuracy sequential truncation approximation pronounced one truncation truncates much rest justifies greedy approach result lemma ordering truncations approximation posterior covariance matrix tested studying normalized estimation error squared nees values nees values shown fig covariance matrix correct expected value nees dimensionality gets large nees values large indicates underestimates covariance matrix apart mvncdf topt trand give nees values closest sequential truncation provides accurate covariance matrix approximation inference section proposed filter stf compared filters using numerical simulations trajectory state model random walk process noise covariance diag parameter compared methods bootstraptype stvbf variational bayes filter tvbf kalman filter measurement validation gating discards measurement components whose normalized innovation squared larger distribution quantile used parameters mean variance used skew tvbf parameters obtained matching degrees freedom skew computing maximum likelihood location scale parameters set numbers generated skew results based monte carlo replications mvncdf trand topt nees fig large values topt outperforms trand mvncdf accurate computationally heavy average nees trand error topt error error error filter output mean covariance matrix true state distributions filters fig topt nees closer optimal value sequential truncation gives accurate posterior covariance matrix corresponds symmetric hence comparison methods become identical tvbf stvbf stf rmse difference tvbf stvbf median rmse median rmse nvb fig stf converges five iterations required number particles left right fig stf outperforms comparison methods noise rmse differences stf rmse per cent stf rmse differences increase increased upper lower used state evolution model velocity model user position receiver clock error used section thus filter state lkt state evolution model time difference measurements used parameter values initial prior normal distribution mean given method first measurement large covariance matrix measurement model pseudorange model used simulations section ksi median rmse ests real data pseudorange positioning two gnss global navigation satellite system positioning data sets collected central london test filters performance challenging satellite positioning scenario numerous measurements data include toa based pseudorange measurements gps global positioning system satellites set contains trajectory collected car using gnss receiver lengths tracks durations hour measurements received onesecond intervals first track used fitting filter parameters second track used studying filters positioning errors ground truth measured using applanix system improves gnss solution tactical grade inertial measurement units gps satellites locations obtained broadcast ephemerides provided international gnss service algorithms computed atlab nvb fig illustrates filter iterations convergence measurement noise components generated independently univariate skew figure shows proposed stf converges within five iterations outperforms filters already two iterations except solution furthermore fig shows stf converged state close converged state rmse require many particles outperform stf stf also converges faster stvbf process noise variance parameter large fig shows distributions rmse differences stf rmse percentages stf rmse stf tvbf use five iterations stvbf uses iterations stf clearly smallest rmse unlike stvbf new stf improves accuracy even small explained improved covariance matrix approximation proposed smoother also tested measurements generated compared smoothers proposed smoother sts variational bayes smoother stvbs variational bayes smoother tvbs rtss measurement validation gating fig shows sts lower rmse smoothers based symmetric distributions furthermore sts iteration converges five iterations less faster stvbs rtss tvbs stvbs sts median rmse rmse difference fig five sts iterations give converged state rmse left right normal rmse rmse ekf tvbf stf nvb fig measurement error distributions fitted real gnss data normal error models modes fixed zero normal empirical cdf ilter parameters real gnss data ekf tvbf stf error position ith satellite time transmission measurement model linearized respect prior mean using first order taylor series approximation compared filters based three different models measurement noise fig rmse horizontal left vertical right position real gnss data function number iterations table nvb empirical cdf model basis stf stvbf model basis tvbf normal model basis extended ekf measurement validation gating pseudoranges unbiased case location parameters fixed furthermore degrees freedom fixed compromise outlier robustness performance based inlier measurements deviation parameter normal model fitted data using algorithm parameter model well parameters model fitted metropolis algorithm location parameter obtained numerically finding point sets mode noise distribution zero three error distributions parameters given table pdfs plotted fig fig shows filter rmses function number iterations stf tvbf converge within five iterations empirical cdf graphs user position errors five iterations shown fig rmses well relative running times given table results show modelling skewness improves positioning accuracy important especially accuracy vertical direction explained sensitivity vertical direction large measurement errors due bad measurement geometry accuracy vertical direction low even measurements correct downweighting erroneous altitude information requires careful modelling noise distribution tails computational burden stf implementation five iterations three times tvbf fig shows two stf iterations would already enough match tvbf average rmse error fig empirical error cdfs real gnss data horizontal error left vertical error right band indoor positioning another application stf indoor localization using toa measurements band uwb radios collected five test tracks laboratory environment optical reference positioning three test tracks real university campus indoor environment measurement equipment spoonphone smartphone android operating system uwb channel pulse radio mhz mhz bandwidth six bespoon uwb tags system uses toa ranging thus clock synchronization required uwb measurement update localization error computation done frequency algorithms computed atlab novelty article compared previous article stf algorithm toa measurements used state evolution model random walk blockdiag state position coordinates process noise parameters initial position assumed known scaled speed light toa gives direct measurement distance uwb beacon user thus measurement model kbi distance vector position ith uwb beacon measurement noise measurement function linearized prior mean test three alternative models measurement noise estimation algorithm skewt model stf table iii filter model tvbf normally distributed high accuracy reference measurements provided use vicon tracking system courtesy uas technologies lab artificial intelligence integrated computer systems division aiics department computer information science ida http rmsehorizontal rmsevertical running time ekf tvbf rmse table rmse relative running times real gnss data stf ekf tvbf stf histogram distr normal nvb fig filter rmses uwb indoor positioning function number iterations table vii average rmse meters uwb indoor positioning route laboratory campus campus campus ekf tvbf stf fig real uwb ranging measurement error histogram distribution distributions fitted data normal error models noise ekf measurement validation gating degrees freedom parameters fixed parameters optimized maximize likelihood laboratory measurements maximum likelihood parameter values given table pdfs fitted distributions compared error histogram fig laboratory data used parameter calibration positioning tests obtain fair comparison optimal filtering algorithm eliminates effect possible differences calibration positioning data evaluate performances independent data set also measured three test tracks campus tampere university technology rough reference position based interpolation timestamped locations table distribution families include multivariate canonical fundamental skew cfust multivariate unified skew comprehensive review different variants mvst given mvst variant used article based cfust discussed general variant mvst variant parameter matrix rnz square matrix rnz arbitrary matrix pdf mvst inz det maximum likelihood parameter values uwb positioning based laboratory data set normal filters also number iterations nvb parameter fig shows filters rmses averaged data sets different values nvb five iterations sufficient stf uwb indoor positioning stf matches tvbf accuracy already two iterations rmses filters given table vii tvbf stf use five iterations show significantly lower rmse ekf measurement validation gating furthermore proposed stf lower rmse tvbf campus track measured avoiding condition difference stf tvbf small track xtension mvst skew several multivariate versions pdf multivariate skew mvst involves cdf univariate definition skew given involves cdf multivariate versions mvst special cases general multivariate pdf cdf inference algorithms proposed paper extended cover case elements measurement noise vector statistically independent jointly multivariate measurement noise follows mvst mvst filtering smoothing algorithms presented tables iii apply slight modifications core convenient extension fact mvst represented similar hierarchical model however shape matrices required diagonal matrix form iny scalar prior notice admits small value measurement components potentially outliers simultaneously unlike independent univariate components model difference illustrated pdf contour plots fig coincides covariance matrix update smoother backward recursion fisher information matrix multivariate measurement noise specific modification required mvst measurement noise sts algorithm table replacing line iny vii erformance ound lower bound bayesian lower bound crlb lower bound mse matrix state estimator random variable using observations sense matrix difference positive semidefinite state estimator regularity conditions necessary hold integrability first two partial derivatives joint distribution asymptotically unbiased estimator conditions satisfied likelihood normal prior distribution even though hold hierarchical model used proposed variational estimator due restriction positive orthant filtering crlb model follows recursion ert mvst iny matrix iny iny similarly specific modification required mvst measurement noise stf algorithm table iii replacing line fig pdf bivariate measurement noise independent univariate components model mvst model gradient respect derivation given appendix evaluation expectation challenging measurements due requirement evaluate cdf multivariate tdistribution partial derivatives woodbury matrix identity recursion equivalent covariance matrix update kalman filter measurement noise covariance model measurement noise components independently univariate case fisher information obtained applying conditionally independent measurement component summing resulting formula matches matrix diagonal matrix diagonal entries eii univariate distributed random variable rii substituted formula matches fisher information formula obtained univariate skew case integrals respect one scalar variable evaluated numerically simulation fisher information matrix measurement noise distribution furthermore smoothing crlb model follows recursion study crlb linear model measurement noise generating realizations model state process noise covariance matrix measurement parameters determine parameters formulas thus measurement noise distribution variance generate realizations process compute crlb mse bootstrap particles stf crlb mses computed first component state last time instant fig shows crlb model figure shows increase skewness well decrease crlb significantly suggests nonlinear filter significantly better gives mse fig shows mses stf expected mse approaches crlb stf slightly worse figures also show although crlb becomes looser distribution becomes skewed correctly indicates modeling skewness still improves filtering performance crlb fig crlb time instant model fixed measurement noise variance skewness decreases crlb significantly fig mse mses left stf right close viii onclusions proposed novel approximate filter smoother linear models skewed measurement noise distribution derived rao lower bounds filtering smoothing estimators algorithms based variational bayes approximation posterior independence approximations removed earlier versions algorithms avoid significant underestimation posterior covariance matrix removal independence approximations enabled sequential truncation algorithm approximating mean covariance matrix truncated multivariate normal distribution optimal processing sequence given sequential truncation simulations tests gnss outdoor positioning uwb indoor positioning data show proposed algorithms outperform methods eferences gustafsson gunnarsson mobile positioning using wireless networks possibilities fundamental limitations based available wireless network measurements ieee signal processing magazine vol july chen yang liao liao mobile location estimator rough wireless environment using extended imm data fusion ieee transactions vehicular technology vol march kok hol indoor positioning using ultrawideband inertial 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economics vol marchenko multivariate distributions econometrics environmetrics dissertation texas university december doucet godsill andrieu sequential monte carlo sampling methods bayesian filtering statistics computing vol july naveau genton shen skewed kalman filter journal multivariate analysis vol kim ryu mallick genton mixtures skewed kalman filters journal multivariate analysis vol rezaie eidsvik kalman filter variants closed skew normal setting computational statistics data analysis vol alspach sorenson nonlinear bayesian estimation using gaussian sum approximations ieee transactions automatic control vol fortmann tracking data association ser mathematics science engineering series academic press williams maybeck hypothesis control techniques multiple hypothesis tracking mathematical computer modelling vol may nurminen ardeshiri piche gustafsson robust inference models skewed measurement noise ieee signal processing letters vol november bishop pattern recognition machine learning springer wand ormerod padoan mean field variational bayes elaborate distributions bayesian analysis vol lee mclachlan finite mixtures canonical fundamental skew unification restricted unrestricted skew models statistics computing cover thomas elements information theory john wiley sons tzikas likas galatsanos variational approximation bayesian inference ieee signal processing magazine vol beal variational algorithms approximate bayesian inference dissertation gatsby computational neuroscience unit university college london tallis moment generating function truncated multinormal distribution journal royal statistical society series methodological vol genz numerical computation rectangular bivariate trivariate normal probabilities statistics computing vol genz bretz comparison methods computation multivariate probabilities journal computational graphical statistics vol positioning filters floor plan information international conference advances mobile computing multimedia momm new york usa acm simon simon constrained kalman filtering via density function truncation turbofan engine health estimation international journal systems science vol rauch striebel tung maximum likelihood estimates linear dynamic systems journal american institute aeronautics astronautics vol august dow neilan rizos international gnss service changing landscape global navigation satellite systems journal geodesy vol february kirubarajan estimation applications tracking navigation theory algorithms software john wiley sons hartikainen recursive filtering smoothing nonlinear systems using multivariate distribution ieee international workshop machine learning signal processing mlsp september axelrad brown gps navigation algorithms global positioning system theory applications parkinson spilker eds washington aiaa hartikainen gaussian optimal smoothing nonlinear state space models ieee transactions automatic control vol august spoonphone online available http sahu dey branco new class multivariate skew distributions applications bayesian regression models canadian journal statistics vol genton fundamental skew distributions journal multivariate analysis multivariate extended distributions related families metron international journal statistics vol van trees detection estimation modulation theory part detection estimation linear modulation theory new york john wiley sons filtering predictive smoothing bounds nonlinear dynamic systems automatica vol ciccio monti inferential aspects skew tdistribution quaderni statistica vol bayesian filtering smoothing cambridge cambridge university press lower bound linear filtering measurements international conference information fusion fusion july ppendix erivations smoother derivations qxu gives log qxu log log axl log cxk log log log axl log log axl log log axl log axk log log log log derived section means components required nonnegative truncation components qxu thus form joint smoothing posterior linear model process noise covariance state transitionh matrix measurement model matrix matrix measurement noise covariance matrix denote pdfs related model would possible compute truncated multivariate normal posterior joint smoothing distribution account truncation positive orthant using sequential truncation however would impractical large due large dimensionality feasible solution approximate filtering distribution striebel smoother rtss forward filtering step multivariate normal distribution iverson bracket notation normalization factor epe ukk varpe ukk approximated using sequential truncation given multivariate normal approximations filtering posteriors lemma backward recursion rtss gives multivariate normal approximations smoothing posteriors quantities required derivations section expectations smoother posteriors eqxu eqxu covariance matrices varqxu uxkk varqxu lemma let process measurement process known distribution standard markovianity assumptions filtering posterior multivariate normal distribution implies statement holds induction argument derivations gives mean covariance matrix filtering posterior proof proof mostly similar proof theorem first assume joint conditional distribution markovianity azk log qxu log use formula markovianity log log therefore log log qxu log thus model independent univariate measurement noise components diagonal entries separate random variables given therefore conditioning rule multivariate normal distribution derivations section required diagonal matrix diagonal elements model multivariate measurement noise form scalar random variable one parameter given therefore log qxu log qxu log given thus required expectation iny ppendix erivation isher information mvst consider multivariate measurement model mvst rny rny rny logarithm pdf log log det log iny log function iny denote pdf cdf scaled multivariate degrees freedom root matrix hessian matrix term log iny derived log iny rrt rrt term log differentiated twice using chain rule log gives log function antisymmetric second derivative function antisymmetric additive constant rregularity conditions given integral exists terms products positive powers rational expressions denominator higher degree nominator derivat tives evaluated bounded continuous function integral det also exists bounded continuous positive power rational expression denominator higher degree nominator similar arguments show integrability first second derivative likelihood guarantees regularity conditions crlb satisfied thus expectation log iny det det drt prt iny iny rrt iny function antisymmetric symmetric function integral exists outline proof integrability certain functions show crlb exists fulfils mvst iny mvst implies mvst arat gives log iny iny iny thus fisher information measurement model mvst log ert iny rrt mvst iny defined

based consensus analysis networks link failures may xue lin yuanshi zheng long wang abstract paper system presented formulated terms delta operator proposed system unify systems network practice communication among agents acted upon various factors communication network among faulty agents may cause link failures modeled randomly switching graphs first show delta representation system reaches consensus mean probability almost surely expected graph strongly connected results induce system random networks also reach consensus sense second influence faulty agents consensus value quantified original network using matrix perturbation theory error bound also presented paper finally simulation example provided demonstrate effectiveness theoretical results index terms consensus systems delta operator link failures error bound ntroduction distributed cooperative control problem systems captured great attention interest motivated diverse applications various fields biology sociology work supported nsfc grant nos fundamental research funds central universities grant young talent fund university association science technology shaanxi china grant corresponding author long wang lin zheng center complex systems school engineering xidian university china xuelinxd wang center systems control college engineering peking university beijing china email longwang may draft control engineering computer science order finish different cooperative tasks variety protocols established systems lots criteria concerning coordination provided etc fundamental problem coordination consensus characterizes phenomenon multiple agents achieve common decision agreement consensus problem studied long time management science rise consensus problem control filed influenced vicsek model model multiple agents agent updates state using average state well neighbors theoretical analysis consensus vicsek model finished authors proposed classical consensus protocols systems provided several sufficient conditions solve consensus problem inspired results many researchers devoted studying consensus problems system analyzed two perspectives one dynamic model interaction network viewpoint dynamic model related researches include dynamics dynamics hybrid dynamics switched dynamics heterogeneous dynamics etc viewpoint interaction network related researches fixed networks switching networks antagonistic networks random networks development digital controller many cases system obtains input signal discrete sampling instants according actual factor researchers investigated sampled control control systems well known systems obtained directly systems based sampled control described shift operator however applications may possess higher sampling rate lead problems shift operator applied represent system shift operator show intuitive connection system system overcome limitations goodwin used delta operator represent dynamics sampled data system compared shift operator approach delta operator several advantages superior finite world length coefficient representation convergence counterpart sampling period tends zero worth pointing delta operator makes may draft smooth transition representation underlying system sampled period tends zero therefore used unify systems due advantages delta operator existed many related research results inspired researches apply delta operator describe system sampled data representation proposed systems well known communication may destroyed realistic network due link failures node failures etc thus consensus systems random networks also studied based delta operator consider consensus systems random networks paper assume exist faulty agents receive information send information lead link failures network original network without faulty agents undirected connected graph phenomenon often occurs practice instance receiver emitter agent failure leads link failure communication network different consider consensus system directed random networks due variation networks however consensus value changed therefore analyze influence faulty agents original network main contribution paper twofold first show delta representation system reaches consensus different sense mean probability almost surely expected graph strongly connected based delta operator get consensus conditions also appropriate system random networks second analyze influence faulty agents consensus value original network using matrix perturbation theory error bound consensus values network link failures original network presented structure paper given follows section based delta operator system established section consensus different sense studied section provide error bound caused faulty agents section simulation example presented finally give short conclusion section notation let denote column vector ones column vector zeros set real numbers real matrices respectively ith eigenvalue matrix denoted denotes standard euclidean norm may draft write vector induced matrix norm write kak bij max kaxk bij say nonnegative matrix moreover row sums said row stochastic matrix given vector matrix denotes transpose let max dii max min max max denotes group inverse matrix reliminaries graph theory communication relationship agents modeled graph vertex set edge set eij nonnegative matrix aij agents adjacent aij set neighbors agent denoted degree matrix dij diagonal matrix dii aij laplacian matrix graph defined lij aij lij eigenvalues denoted lii graph said strongly connected exists path two distinct vertices path connects directed graph sequence distinct vertices vim vim vir undirected connected graph positive simple zero eigenvalue moreover exists min throughout paper always assume undirected connected graph exist faulty agent agent able receive send information problem statement paper consider system consists agents continuoustime dynamics ith agent described state control input ith agent respectively may draft system representation obtained using traditional shift operator given hui sampling period worth noting sampling period lose information underlying system moreover difficult describe next value difficulty avoided using delta operator introduced delta operator defined follows using delta operator representation system described seen know hence smooth transition ensures discretetime system converges system system apply classic consensus protocol aij using hold protocol given aij denote system protocol represented based discussion analysis know system converges system paper original network undirected connected know agent influenced information neighbours however may exist agent may draft unable receive send information network call type agent faulty agent paper without loss generality assume agents faulty agent agents normal always receive send information network two scenarios considered scenario four cases considered agent receive information agent receive information agents receive information simultaneously agents normal networks correspond four cases respectively assume randomly switches among distinct networks networks correspond occurrence probabilities respectively moreover scenario four cases considered agent send information agent send information agents send information simultaneously agents normal networks correspond four cases respectively assume randomly switches among distinct networks networks correspond occurrence probabilities respectively moreover system scenario written ltk laplacian matrix time point note graph gtk invariant time interval corresponding adjacent matrix time point atk throughout paper sampling period satisfies two main objectives considered paper first consensus system considered second error bound consensus values system system presented remark simplicity focus two faulty agents however analytical methods concerning error bound paper extended scenario two faulty agents left interested readers exercise definition system reaches consensus mean holds lim may draft probability holds lim almost surely holds lim definition let denote transition matrix markov chain markov chain called regular chain exist positive elements lemma transition matrix regular chain lemma ergodic chains transition matrices limiting probability vectors respectively lemma property delta operator time function represented lemma assume sampling period dmax system reach average consensus graph undirected connected proof let due one consider lyapunov function lemma holds hlt since graph undirected connected implies laplacian matrix positive hence eigenvalues repsented gersgorin disk theorem get min dmax owing may draft min due proves therefore converge system achieve average consensus asymptotically remark lemma exists implies lim lim seen reduced note system converges system consequently system reaches average consensus undirected connected graph iii onsensus analysis section shown system reaches consensus despite existence faulty agents supposed scenario scenario expression pattern network hence following results viewed unified conclusions system scenarios theorem assume sampling period dmax system reaches consensus mean expected graph strongly connected furthermore lim lim hltk lim vectors left eigenvectors matrices respectively proof pointed solution system due invariance graph gtk time interval get may draft hltk lim lim hltk lim lim according dmax hli hdi hai positive diagonal elements since immediate hltk also nonnegative matrix positive diagonal elements follows hltk positive diagonal matrix since expected graph strongly connected matrix hltk nonnegative irreducible positive diagonal elements moreover easy verify hltk disc theorem one hltk hence theorem deduced hltk algebraically simple eigenvalue consequently matrix primitive virtue theorem obtain lim hence system reaches consensus mean next give consensus value system proposition follows hltk hence lim lim hltk lim lim let ltk dmax atk hence matrix nonnegative irreducible positive diagonal elements similar previous discussion lim theorem assume sampling period dmax system reaches consensus probability expected graph strongly connected may draft proof since expected graph strongly connected theorem follows lim wij since matrix let hltk wij hltk row stochastic matrix get verified nonincreasing let hence yields lim hence result chebyshevs inequality follows therefore lim shown theorem lim matrix also row stochastic matrix similar proof proved also holds theorem assume sampling period dmax system reaches consensus almost surely expected graph strongly connected proof follows theorem probability theorem exists subsequence converges almost surely hence exists almost surely since nonincreasing holds almost surely therefore holds almost surely implies system reaches consensus almost surely remark pointed theorem one since network invariant time interval partial state system represented shown theorem sequence achieves consensus mean using conclude achieves consensus mean therefore system random networks reaches consensus mean expected graph strongly connected may draft indicates consensus result system random networks reduces consensus result system random networks moreover theorems also appropriate system rror analysis section consider error bound consensus value lim consensus value original network lim solve problem matrix perturbation theory property finite markov chains applied analysis consensus problem apply suppose expected graph strongly connected dmax theorem shows row stochastic matrix property row stochastic matrix element wij matrix satisfies wij hence definition regarded transition matrix regular chain moreover transition matrix ergodic chain noteworthy following analysis results appropriate scenario scenario theorem assume sampling period dmax expected graph strongly connected hli correspond respectively proof pointed written theorem derive lim row stochastic matrix hence proves row sums equal moreover follows theorem lim let analyze error bound kek may draft using lemmas holds holds algebraic manipulations following equation implies kek due get kek follows kek hence analyze solve problem introduce vector obvious therefore owing kki kky theorem exists eigenvector corresponding implies moreover similar analysis due deduced know matrix consequently max dmax follows kky account max obtain kek may draft theorem know lim analyze kek clear lim similar regarded transition matrix regular chain moreover using lim lim hence kek similar analysis get therefore kek remark agent receive information considered scenario agent send information considered scenario assume exist agents receive send information scenario iii scenario similar theorem error bound consensus value calculated dmax theorem assume sampling period expected graph strongly connected scenario max max proof due matrix expressed similar analysis theorem kek calculate may introduce vector draft utilizing following equation obtained due let substituting equation leads max using get hence max due holds therefore kek max max corollary assume sampling period dmax expected graph strongly connected scenario may max draft max proof due matrix expressed utilizing max similar analysis theorem max kek imulation section simulation presented illustrate effectiveness theoretical results example consider communication network chosen figure interaction topology among agents randomly switches among networks correspond occurrence probabilities respectively calculation get sampling period choose initial value figure depicts state trajectories system random networks seen agents reach consensus may draft fig network topologies state time fig state trajectories system random networks original network denoted graph state trajectories system network shown figure shown agents reach consensus calculation obtain obtain therefore based theorem follows figures easy verify error bound less may draft state time fig state trajectories system network onclusion paper based delta operator system proposed pointed proposed system converge continuoustime system sampling period tends zero assume exist faulty agents send receive information network communication network described randomly switching networks random networks proved consensus mean probability almost surely achieved expected graph strong connected furthermore influence faulty agents consensus value analyzed error bound consensus values network link failures original network presented future based delta operator consider formation control containment control systems etc eferences murray consensus problems networks agents switching topology ieee transactions automatic control vol zheng wang novel group consensus protocol heterogeneous systems international journal control vol gao wang based consensus systems topology ieee transactions automatic control vol lin zheng consensus switched multiagent systems ieee transactions systems man cybernetics systems published doi may draft wang xiao consensus problems networks dynamic agents ieee transactions automatic control vol zheng wang containment control heterogeneous systems international journal control vol xiao wang chen gao formation control systems automatica vol degroot reaching consensus journal american statistical association vol vicsek czirok jacob cohen schochet novel type phase transition system particles physical review letters vol jadbabaie lin morse coordination groups mobile autonomous agents using nearest neighbor rules ieee transactions automatic control vol ren beard consensus seeking multiagent systems dynamically changing interaction topologies ieee transactions automatic control vol xiao wang asynchronous consensus systems switching topology delays ieee transactions automatic control vol xie wang consensus control class networks dynamic agents international journal robust nonlinear control vol zheng wang consensus hybrid systems ieee transactions neural networks learning systems published zheng wang consensus switched systems ieee transactions circuits systems express briefs vol zheng zhu wang consensus heterogeneous systems iet control theory applications vol altafini consensus problems networks antagonistic interactions ieee transactions automatic control vol hatano mesbahi agreement random networks ieee transactions automatic control vol lin zheng wang consensus switched systems random networks international journal control vol xie liu wang jia consensus networked systems via sampled control fixed topology case proceedings american control conference cao xiao wang consensus control systems via synchronous periodic event detection ieee transactions automatic control vol goodwin middleton poor digital signal processing control proceedings ieee vol feuer goodwin sampling digital signal processing control boston middleton goodwin improved finite word length characteristics digital control using delta operators ieee transactions automatic control vol may draft bayat johansen explicit model predictive control formulation approximation ieee transactions automatic control vol ginoya shendge phadke extended disturbance observer applications ieee transactions industrial electronics vol kar moura sensor networks random links topology design distributed consensus ieee transactions signal processing vol fagnani zampieri randomized consensus algorithms large scale networks ieee journal selected areas communications vol meyer role group generalized inverse theory finite markov chains siam review vol goodwin yuz cea sampling models american control conference buckley eslami fuzzy markov chains uncertain probabilities mathware soft computing vol meyer condition finite markov chain perturbation bounds limiting probabilities siam journal algebraic discrete methods vol horn johnson matrix analysis cambridge university press bernstein matrix mathematics theory facts formulas princeton university press ash probability measure theory academic press may draft

identifiability skew normal mixtures one known component shantanu jaina michael levineb predrag radivojaca michael trossetc dec department computer science indiana university bloomington indiana department statistics purdue university west lafayette indiana department statistics indiana university bloomington indiana abstract give sufficient identifiability conditions estimating mixing proportions mixtures skew normal distributions one known component consider univariate case well two multivariate extensions multivariate skew normal distribution msn azzalini dalla valle canonical fundamental skew normal distribution cfusn genton characteristic function cfusn distribution additionally derived introduction study identifiability mixing proportion mixture two skew normal distributions one components known problem direct implications estimation mixing proportions given sample mixture sample one components sample mixture typically collected set objects study whereas component sample collected set objects verified satisfy property interest setting common domains absence property easily verified due practical systemic constraints social networks molecular biology etc social networks example users may allowed email addresses shajain shantanu jain mlevins michael levine predrag predrag radivojac mtrosset michael trosset preprint submitted elsevier january click like particular product thus data collected one component samples sample users clicked like mixture sample users accurate estimation mixing proportions setting fundamental implications false discovery rate estimation storey storey tibshirani context classification estimating posterior distributions ward jain recovering true classifier performance menon jain identifiability estimation mixing proportions extensively studied yakowitz sprag dempster tallis chesson mclachlan peel recently case one known component considered nonparametric setting bordes ward blanchard jain patra sen though nonparametric formulation highly flexible also problematic due issues irreducibility assumption violated blanchard jain patra sen addition often reasonable practice require unimodality density components difficult ensure nonparametric formulation guarantee unimodality components allow skewness model components skew normal family generalization gaussian family good theoretical properties tractability inference genton although family introduced recently see azzalini azzalini dalla valle gained practical importance econometrics financial domains genton recently literature identifiability parametric mixture models emphasized identifiability respect subset parameters cases single location parameter location scale parameters change furthermore previous results address case univariate mixture distributions studies considered identifiability mixtures general multivariate densities respect parameters holzmann browne mcnicholas work concerns identifiability respect mixing proportions mixtures two skew normal distributions one known component show section setting identifiability respect mixing proportions equivalent identifiability respect parameters consider univariate multivariate families skewnormal distributions establishing identifiability respect parameters begin univariate skew normal family introduced azzalini extend results two forms multivariate skew normal families msn cfusn introduced azzalini dalla valle genton respectively families discussed section main contribution theorems state sufficient conditions identifiability mixing proportion mixture msn cfusn components respectively also derive concise formula characteristic function cfusn appendix problem statement let families probability density functions pdfs let family pdfs form densities referred component pdfs referred mixture pdf referred mixing proportion therefore family mixtures setting study identifiability density mixture respect parameter first univariate two different multivariate families distribution families defined genton start first studying general identifiability conditions section identifiabilty mixtures known component section discuss identifiability mixtures context problem show general notion identifiability equivalent identifiability mixing proportion lemma however main contribution section lemma gives useful technique prove identifiability tailored setting applied skew normal mixtures later paper lemma lemma restatements results jain terms densities instead measures consider mixture distribution equation let known component distribution equivalent restricting singleton set minor abuse notation denote family mixtures note equation treated pdf parametrized reflect parameterization rewrite function given family distributions said identifiable mapping therefore identifiable lack identifiability means even different target density contains information tell apart interested estimating need identifiable identifiability might seem weaker requirement compared identifiability however lemma shows two notions identifiability equivalent lemma identifiable identifiable proof definition identifiability necessary condition identifiable prove sufficient condition well let assume identifiable also suppose definition identifiability follows therefore implies thus identifiable technically require bijection ignore obvious onto requirement simplicity consider largest possible contains pdfs except pdf equal almost everywhere contains non trivial two component mixtures one components lemma section shows family identifiable next establish necessary sufficient condition identifiability lemma identifiable proof first prove identif iable give proof contradiction suppose identifiable lemma identifiable thus without loss generality assume therefore equality obtain using simple algebra means turn follows since selected conclude contains empty completes proof statement prove identif iable give proof contradiction suppose identifiable let common member follows let follows belong show indeed immediately implies therefore greater thus follows identifiable lemma follows statements next lemma gives sufficient condition identifiability mathematically convenient relies notion span set functions denoted span contains finite linear combinations functions span lemma consider family pdfs assume pair pdfs exists linear transformation possibly depending choice maps function span function domain denote value transformation function span thus function denote value function exists sequence lim lim identifiable proof give proof contradiction suppose conditions theorem satisfied identifiable lemma follows exists common element say exists since exists linear transform sequence satisfying condition follows consequently tnn contradiction satisfies condition invoke lemma later paper two linear transforms namely moment generating function mgf transform characteristic function transform observe transforms main ideas lemma linear transforms limits come theorem teicher identifiability finite mixtures skew normal mixtures contains pdfs without family identifiable lemma section therefore desirable choose smaller family makes mixture model identifiable rich enough model real life data paper take parametric approach normal family presents limited option since normal mixtures typically require large number components capture asymmetry real life data skew normal asymmetric convenient alternative univatiate multivariate settings thus restrict attention mixture families unknown known components skew normal contribution section theorem theorem theorem give rather large identifiable family skew normal mixtures similar approach reported ghosal roy mixtures normal skew normal distributions result however results much extensive family giving results first introduce univariate skew normal family well two common multivariate generalizations univariate skew normal family azzalini introduced skew normal family distributions generalization normal family allows skewness location scale shape parameter controls skewness distribution right skewed left skewed reduces normal distribution pdf given probability density function pdf cumulative distribution function cdf standard normal distribution respectively pewsey genton kim genton derived mgf family table multivariate skew normal families azzalini dalla valle proposed extension skew normal family multivariate case particular generalization useful property marginals skew normal well recently several families multivariate skew normal distributions proposed discussed lee mclachlan paper consider alternate parametrization azzalini multivariate skew normal family denoted msn msn pdf covariance matrix location parameter parameter pdf normal distribution mean zero covariance cdf standard univariate normal azzalini dalla valle kim genton derived mgf distribution table table alternate parametrization identifiability results algorithms better formulated terms alternate parameters table gives relationship alternate canonical parameters well related quantities identity matrix family msn cfusn alternate parametrization canonical alternate alternate canonical sign related quantities lin studied maximum likelihood estimation finite multivariate skew normal mixtures another family canonical fundamental skew normal distribution cfusn introduced genton cfusn pdf covariance matrix defined table location parameter cdf multivariate normal distribution zero mean covariance mgf given table mgf obtained lin best knowledge expression available literature derived theorem appendix purposes study identifiability subset family univariate skew normal pdfs also univariate skew normal pdf written contains univariate skew normal mixtures theorem gives sufficient condition family identifiable genton define general form cfusn family allows nonsquare matrices table skew normal families expression characteristic function moment generating function parameters defined table denotes imaginary number rxp exp denote cdfs standard univariate multivariate normal distributions respectively expression cfusn ith column family msn cfusn mgf exp exp exp exp exp exp notation notation theorem lemma lemma let denote partition defined multiset column vectors column vectors direction set relationship formally defined following equivalence relationship let denote canonical vector direction vectors defined let space orthogonal vector let null null space matrix let complement set theorem family pdfs defined table identifiable proof consider partition sets defined follows show given pair pdfs conditions lemma parameters corresponding defined satisfied let table use lemma statements prove statement first select applying lemma statements obtain therefore sequence satisfies conditions lemma use choose lemma statement basis proof first select applying lemma statement moreover owing fact mgf always obtain sequence positive know satisfies conditions lemma thus conditions lemma satisfied consequently identifiable theorem family pdfs msn msn defined table identifiable proof consider partition sets defined follows msn standard partial order relationship space matrices specifically implies positive definite note also contains pdfs whose matrix unrelated partial ordering show given pair pdfs conditions lemma parameters corresponding defined table satisfied let choose characteristic function transform linear transform pick existence guaranteed lemma applying lemma statements notice sequence obtain satisfies conditions lemma choose moment generating function transform linear transform pick existence guaranteed scalar value choose otherwise choose easy see applying lemma statement obtain moreover owing fact mgf always positive know sequence satisfies conditions lemma thus conditions lemma satisfied consequently identifiable theorem let give concise representation cfusn parameters family pdfs cfusn identifiable cfusn kvv defined table addition representing skewness matrix also represents multiset containing column vectors exp exp set indexes containing entries note arbitrary property used multiple times proof also compute limit note limit primarily determined sign quadratic form either however limit determined sign still well oscillates undefined limit unless case limit use notation throughout proof give proof contradiction supposing family identifiable lemma implies exists characteristic function linear transform proof first define show equation leads contradiction possible values consider partition sets defined follows cfusn standard partial order relationship space matrices precisely implies positive consider following cases cover contingencies proceed follows equation implies term goes since applying lemma statement limit term exists limit entire rhs consequently lhs follows since limit term exists also exist limit entire lhs exist summarizing lim lim pick existence guaranteed shown lemma either contradicts equation proceed follows use equation get term goes since limit term exists lemma statement since applying lemma statement limit rhs exists limit entire lhs consequently term summarizing lim lim lim lemma statement lim lim lim lim lim equation last step follows consequently exp equation summarizing since lemma statement follows kvv thus hence contradiction equation implies notice term goes since applying lemma statement limit term exists limit entire rhs consequently lhs follows since limit term exists also exist limit entire lhs exists summarizing lim lim pick existence guaranteed lemma ensures either contradicts equation comment extension theorem speculate theorem strengthened removing condition kvv definition removal condition breaks current proof case notice case implies satisfies exp integer definition shown equation implications reduce equation exp exp exp using lemma statement positive integer exp looking definition seems term negative integer except special cases would imply rhs exp still goes yet lhs leads contradiction auxiliary results lemma contains pdfs except identifiable proof contains pdfs except note either since let mixture follows also consequently mixture therefore last expression equivalent thus however hence identifiable lemma symmetric matrices either exists vector proof suppose exist vector thus immediately contradicts hence implies exists hand implies combination implies however impossible since summarizing exists either true give recipe find let vector existence already proved choose else let existence guaranteed choose picked see exists notice first second small enough thus picking small enough ensures lemma let matrices symmetric positive semidefinite matrix let also denote multiset containing column vectors respectively using notation let let imaginary number assume following statements true even number elements equal contribution constant kvv constant null proof first partition elements three sets notice either singleton empty vectors collected single component set vector equivalent column vectors implicitly means column vectors direction equivalent consequently matrix column vectors row vectors symmetric equivalent words expressed constant ensures positive summarizing moreover vector appear inside equivalent consequently also singleton set empty properties implicitly used rest proof next show following result used multiple times proof given vector finite multiset vectors prove notice choosing guarantees choosing ensures follows set obtained removing satisfies see indeed notice removing finite number dimensional linear spaces either dimensional dimensional reduces lebesgue measure set provided coincide guaranteed using result show existence two vectors let vector whose existence shown using result follows given let exists existence definition let vector whose shown using result follows prove statement break argument three exhaustive cases picking follows since source column vectors consequently follows since two possibilities source column vectors thus satisfy must true column vectors element singleton already know case true consequently satisfy must true well since element either covered cases consequence remaining set equal number sets belong must true column vectors proves statement prove statement rewrite formula follows statement proves statement let given defined earlier since equation thus kvv equation existence justified statement fact contains proves statement prove statement notice null set definition follows null either notation landau notation use landau asymptotic notation next lemmas defined follows functions defined subset lim lim lemma consider two univariate skew normal distributions let related given table let let characteristic functions corresponding two distributions refer table lim lim provided limit exists extended real number line let mgf mgf moment generating functions corresponding two distributions refer table mgf mgf lim proof use landau notation defined notation statement instead working directly complex circum vent complication working ratio absolute value squared always real multiplying ratio conjugate obtain expression absolute value squared follows property complex conjugate fraction exp exp previous expression using asymptotic upperconsider ratio bound numerator lower bound denominator obtained lemma statement get thus exp exp consequently exp exp exp exp lim lim follows statement similar derivation asymptotic ratio equation derive asymptotic using lemma statement exp consequently lim lim follows provided limit exists combining result statement proves statement statement definition mgf table get mgf exp mgf previous expression apply asymptotic upperconsider ratio bound numerator lower bound denominator obtained lemma statement asymptotic applicable exp exp thus mgf exp exp mgf exp exp term dominates term exponential asymptotic goes irrespective relation consequently lim mgf mgf lemma consider two skew normal distributions msn msn let related given table let let characteristic functions corresponding two distributions refer table lim lim provided limit exists extended real number line let mgf mgf moment generating functions corresponding two distributions refer table mgf mgf lim proof use landau notation defined notation statement use approach lemma expression squared absolute value characteristic function ratio obtained multiplying ratio conjugate given exp exp previous expression using asymptotic upperconsider ratio bound numerator lower bound denominator obtained lemma statement get thus exp exp consequently exp exp exp exp lim lim follows statement similar derivation asymptotic ratio equation derive asymptotic using lemma statement exp consequently lim lim follows provided limit exists combining result statement proves statement statement definition msn mgf table get mgf exp mgf consider ratio previous expression apply asymptotic bound numerator lower bound denominator obtained lemma statement asymptotic applicable exp exp thus mgf exp exp mgf exp exp term dominates term exponential asymptotic goes irrespective relation consequently lim mgf mgf lemma consider two skew normal distributions cfusn cfusn let related given table let let characteristic functions corresponding two distribuexp exp tions refer table let let nary number ith column lim using landau notation defined notation exp exp positive integer exp proof exp exp exp exp exp exp exp exp exp exp exp exp using lemma statement get lim proves statement using equation exp exp exp exp exp using lemma statement exp exp exp exp exp proves statement rxp lemma let standard normal cdf exp let finite using landau notation defined notation exp exp exp lim exp exp defined recursively integer follows exp exp proof statement consider function exp prove statements first derive limits evaluate limit apply rule numerator denominator end take limit ratio derivative numerator denominator applying leibniz integral rule get lim lim lim exp exp exp lim exp consequently exp thus also holds true thus exp exp ment moreover follows equation since true well approaches exp proves statement statement performing integration parts gives exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp exp thus exp exp exp exp exp exp notice term order since lim lim lim exp exp exp applying rule exp exp exp exp lim lim applying leibniz integral rule exp exp lim term exp term since lim lim exp lim lim exp exp exp exp exp exp lim applying rule applying leibniz integral rule consequently exp exp proves statement consequently statement exp statement implies thus exp consequently exp notice trivially exp well completes proof statement exp thus statement also implies exp exp consequently notice exp well completes proof statement trivially conclusions give meaningful sufficient conditions ensure identifiability mixtures msn cfusn components proved identifiability terms parameter contains scale skewness information consistent interpretation across three skew normal families results strong sense set parameter values covered sufficient condition lebesgue measure set parameter space ghosal roy study identifiability mixture standard normal one components second component given uncountable mixture skew normals treating work point distribution make valid comparison identifiability result concluding superiority results owing larger coverage parameter space conditions references references genton fundamental skew distributions multivar anal azzalini class distributions includes normal ones scand stat azzalini results class distributions includes normal ones statistica azzalini dalla valle multivariate distribution biometrika blanchard lee scott novelty detection mach learn res bordes delmas vandekerkhove semiparametric estimation mixture model one component known scand stat browne mcnicholas mixture generalized hyperbolic distributions stat dempster laird rubin maximum likelihood incomplete data via algorithm stat soc pages genton distributions applications journey beyond normality crc press ghosal roy identifiability proportion null hypotheses models distribution electron statist holzmann munk gneiting identifiability finite mixtures elliptical distributions scand stat jain white radivojac estimating class prior posterior noisy positives unlabeled data advances neural information processing systems nips pages jain white trosset radivojac nonparametric learning class proportions arxiv preprint url http jain white radivojac recovering true classifier performance learning proceedings aaai conference artificial intelligence aaai pages kim genton characteristic functions scale mixtures multivariate distributions multivar anal lee mclachlan mixtures skew normal skew adv data anal classif lin maximum likelihood estimation multivariate skew normal mixture models multivar anal mclachlan peel finite mixture models john wiley sons menon van rooyen ong williamson learning corrupted binary labels via estimation proceedings international conference machine learning icml pages patra estimation mixture model applications multiple testing statist soc pewsey characteristic functions wrapped distributions congreso nacional estadistica operativa pages storey direct approach false discovery rates stat soc storey positive false discovery rate bayesian interpretation ann stat storey tibshirani statistical significance genomewide studies proc natl acad sci tallis moment generating function truncated distribution stat soc pages tallis chesson identifiability mixtures austral math soc ser teicher identifiability finite mixtures ann math stat ward hastie barry elith leathwick data algorithm biometrics yakowitz spragins identifiability finite mixtures ann math statist appendix cfusn characteristic function theorem characteristic function cfusn given exp dimensionality cfusn distribution imaginary number rwhere exp column proof use stochastic representation usn obtained lin given standard multivariate normal distribution truncated dimensions symmetric positive matrix follows expressed terms normal distribution truncated normal distribution precisely cfx cfg using expression multivariate normal cfx exp basic properties connection corresponding mgf gives cfh mgfh using expression mgfh derived tallis replacing covariance matrix tallis identity matrix get exp pdf exp exp exp exp exp dwi exp applying substitution integral numerator changes domain integration real line complex plane define integral correctly one needs specify path complex plane across integration performed using path parallel real line get exp exp dui dui exp using lemma kim genton simplify integral term get exp exp dui exp exp dvi substituting exp substituting expression equation completes proof

complexity frictional interfaces complex network perspective department civil engineering lassonde institute university toronto canada sharifzadeh faculty engineering kyushu university hakozaki japan evgin department civil engineering university ottawa ontario canada abstract shear strength behavior rough rock joint analyzed using complex network approach develop network approach correlation patterns void spaces evolvable rough fracture crack type correlation among networks properties hydro attributes obtained experimental tests fracture slip direct result revealed networks joint distribution locally globally filtered correlation gives close relation contact zones detachment sequences evolution shear strength rock joint especially spread node degree rate spread clustering coefficient rate yielded possible stick slip sequences displacements method developed investigate complexity dept civil engineering college behavior faults well energy localization crumpled ridge networks controlling energy distribution key words frictional interface rock joint shear strength complex networks correlation contact areas introduction last decade complex networks used increasingly different fields science technology initial applications complex networks geosciences mostly related earthquakes complexity recursive events main objective related research understanding topological complexity events based field measurements disclose facets woven events characterization spatial temporal structural studies pertaining topological complexity application geoscience fields reveals acquisition gathering direct information especially temporal scale difficult many cases impossible least current technologies addition complex earthquake networks recently analysis climate networks volcanic networks river networks highway networks large scale measurements taken account geoscience fields gradation soil particles fracture networks aperture fractures granular materials initial step refers organizational step tries find possible dominant structures within system next step mentioned works provide suitable simple method yield similar structure algorithm may support evolution structure spatial temporal cases small scales topological complexity evaluated relation may important structural complexity geological fields related fracture networks fracture networks dilatancy joint networks excavation damaged zones cracking pavements structures fault networks large scale recognized analysis networks characterization fractures proper space space essential step furthermore taking direct relationship void spaces contact areas account one may interest considering induced topological complexity opening elements frictional contacts fracture behavior using linear elastic fracture mechanics know aperture aspect ratio generally index available energy growth rupture crack like behavior rupture frictional interfaces also support role contact areas equivalently apertures addition variations fluid flow features permeability tortuosity directly controlled aperture spaces order characterize main attributes fractured systems mechanical hydraulic properties several methods suggested literature recently authors proposed implementation complex network analysis evolution apertures rough rock fracture based euclidean measure results confirmed dependency properties attributes characterized aperture networks present study also related complex aperture networks however current study presents analysis frictional forces shearing based correlation apertures rock joint analysis associated set network attribute aperture distribution area aforementioned method also employed analysis coupled partial differential equations related flow respect behavior collective motion ensemble discrete contacts vicinity phase transition step try characterize collective behavior aperture strings using networks paper answer following three questions hidden complex structure experimentally observed apertures effect specific structural complexity apertures mechanical response fracture apertures regulate show curve words relate topological complexity apertures evolution path fracture organization paper follows section includes brief description networks characterization addition construction procedure aperture networks explained section covers summary experimental procedure last section presents evaluation behavior rock joint followed analysis constructed network network evolving apertures section describe general method setting network fracture surface surface property superposition narrow profiles ribbons one attribute system words one attribute system granulated strings profiles ribbons relationship discrete strings long range correlation elastic results interwoven network topological complexity interactions frictional behavior including response joint related sum real contact areas fluctuates changes apertures also occurs based collective motion spatially coupled contact zones shown structural complexity dynamic aperture changes controlling regulating joint behavior unstable response order explain details work need characterize topological complexity network consists nodes edges connecting set nondirected network considered string measured aperture node aperture string pixels pixel shows void size cell depending direction strings length profiles varies maximum numbers strings cases perpendicular direction shear minimum one parallel direction make edge two nodes correlation measurement cij aperture profiles used main point selection space explore explicit implicit hidden relations among different distributed elements system pair signals profiles containing elements pixels correlation coefficient written cij obviously noted cij restricted cij related perfect correlations correlations perfect respectively selection threshold make edge seen different views choosing constant value may associated current accuracy accumulated data maximum threshold system loses dominant order fact unique way selection constant value however preservation general pattern evolution must considered hidden patterns related several characters network characters express different facets relations connectivity assortivity hubness centrality grouping properties nodes edges generally seems obtaining stable patterns evolution absolute variation give suitable reasonably formed network also different approaches used density links dominant correlation among nodes space distribution edges clusters study set cij considering definition filtering uncorrelated profiles metric space previous study sensitivity observed patterns associated euclidean distance profiles max distinguished clustering coefficient describes degree neighbors particular node connected mean neighbors connected nodes particular node clustering coefficient shows collaboration connected nodes assume node neighboring nodes exist edges neighbors define actual number ratio edges neighbors clustering coefficient given average network node nodes define closer one larger interconnectedness network connectivity distribution degree distribution probability finding nodes edges network large networks always fluctuations degree distribution large fluctuations average value refers highly heterogeneous networks homogeneous networks display low fluctuations another perspective clustering networks closely related degree correlations vertex degree correlations measures statistical dependence degrees neighbouring nodes network correlation criterion complex networks related network assortativity concept correlation included within conditional probability distribution node degree connected node degree words degrees neighbouring nodes independent meaning degree correlation also defined average degree nearest neighbours knn high degree nodes hubs tend make link high degree nodes otherwise knn disassortative point view fractal complex networks degree correlation may used tool distinguish network structures fact fractal networks large degree nodes hubs tend connect small degree nodes fractality disassortativity also clustering nature network drawn average nodes degree giving clustering distribution spectrum many knn increases decreases high degree nodes hubs tend make link low degree nodes networks internet clustering spectrum decreasing function degree may interpreted hierarchical structures network contrast networks social networks scientific collaborations also see complex aperture networks showing assortative behaviour shown spreading crack like behaviour due shearing fracture followed patterns proper spectrum similarly using degree correlation one may define virtual weight edge average number edges connected nodes average characteristic path length mean length shortest paths connecting two nodes graph shortest path pair nodes network assumed geodesic distance gij mean geodesic distance given gij gij geodesic distance shortest distance node number nodes use well known algorithm finding shortest paths presented dijkstra based mentioned characteristics networks two lower upper bounds networks recognized regular networks random networks networks regular networks high clustering coefficient long average path length random networks construction based random connection nodes low clustering coefficient shortest possible average path length however watts strogatz introduced new type networks high clustering coefficient small much smaller regular ones average path length called small world property summary laboratory tests study small world properties rock joints results several laboratory tests used joint geometery consisting two joint surfaces aperture two surfaces measured shear flow tests performed later rock granite unit weight uniaxial compressive strength mpa artificial rock joint made mid height specimen splitting using special joint creating apparatus two horizontal jacks vertical jack sides joint cut creating joint final size sample length width height using special mechanical units various mechanical parameters sample measured virtual mesh square element size spread surface height position measured laser scanner details procedure found different cases normal stress mpa used variation surfaces recorded shows shear strength evolution different normal loads study focus patterns obtained test mpa normal stress implementation analysis complex aperture networks section set designated complex network aperture profiles perpendicular shear direction using correlation measure distribution correlation values along profiles successive shear displacements obtained plotting correlation distribution shows transition near poisson distribution gaussian distribution change type distribution followed phenomena tailing inducing homogeneity correlation values towards high correlation values words tailing procedure tied residual part states joint thus described reducing entropy system clusters information correlation space formed another point view considering correlation patterns inferred throughout shear procedure relatively high correlation profile profiles certain neighborhood radius radius correlation increasing anisotropic development shear displacement using method described previous section complex aperture network developed correlation patterns seen figure formation highly correlated nodes clusters distinguishable near peak point estimated controlling factor evolution path system related formation cliques communities show locality properties clusters intera structures much discriminated last displacements rather initial time steps global variations structures sensitive reduction shear stress fact forming hubs constructed networks may give key element synchronization aperture profiles collective motion discrete contact zones along shear process words reaching one multiple attractors rate reaching peak point organized spreading stabilizing clusters unfortunately due low rate data sampling exact evolution patterns possible however discussion joint degree correlations general concept proposed three characteristics constructed networks namely total degree nodes clustering coefficient mean shortest path length depicted function shear displacement figure followed nearly monotonic parameters considerable sharp change transition shear displacement observed three illustrated parameters transition assumed state transition post peak state taking account rate variation parameters transformation step discriminated also despite clustering coefficient trend show shape number edges mean short length shear displacement roughly exhibiting trend results provide necessary information classification aperture networks rock joint high clustering coefficient low average characteristic path length clearly show aperture networks properties development shear stress networks much faster peak point states feature explained understanding concept contact areas interlocking asperities maximum static point correlation shows relatively uniform shape rather former later cases also current configuration implies homogeneity revealed network nodes high degree tending absorb nodes low edges indicates property similarity within network structures shear displacements immediately near peak figure point destroy homogeneity network spreading slow fronts dropping frictional coefficient accompanied trial make stable cliques inducing heterogeneity network structures using microscopic analysis proven homogenous topologies many small clusters spread network merge together form giant synchronized cluster event predicted reaching peak threshold heterogeneous graphs however one central cores hubs driving evolutionary path figuring synchronization patterns absorbing small clusters seen figure figure two giant groups recognizable displacement shows attractors states dynamic system however two discriminated clusters showing structures within proper networks hubs high degree nodes separated hubs low degree nodes general one may overestimate internal structures networks means entire steps least small branch fractility followed attributed weight distribution associated correlation concept shows virtual heaviness edges increasing simultaneously joint degree distribution also growing indicates networks assortative distribution weights unveiled hubs also clearly followed figure two general discriminated patterns recognizable contrary patterns correlation clustering coefficients drawn eruption local synchronization generally closed least near peak point dropping shear strength variation local clusters continue especially point near critical step local clusters present much uniform percolation rather states final steps stable state regime regional structures clear worth stressing rate variation local joint clustering patterns apparently much higher global patterns joint degree distribution also must noticed peak point structures joint triangles density approximately unchangeable conclusion burst much dense local hubs scaled disclosing slow fronts spreading following spectrum networks collective view shows nearly uniform growing trend third degree polynomial may fitted however respect individual analysis local analysis negative trend pursued spectrum networks related correlation concept expresses probability selecting node certain degree connected two nodes definite degrees evolution spectrum aperture networks euclidean space using clustering analysis accumulated objects come details fracture evolution either mechanical analysis case detecting explicit scaling difficult let transfer calculated network properties variation rate space depicting clustering coefficient mean degree rates shows similar trend evolution shear strength however displacement variation edges clustering coefficient unravels different fluctuations negative scaling large anisotropy dci dki space expressed dki followed figure congestion objects makes general elliptic approximately covers points details correlation among two components presents expansion contraction patterns fall final attractors thus emerged patterns related correlation variation rate edges rate clustering coefficient proposing certain core time step absorbing objects within black hole residual part much obvious rather states definition anisotropy deviation rate changes profiles new space reference pre post peak behaviours obtained transferring interlocking step coulomb threshold level accompanying maximum anisotropy immediate dropping standard starting fluctuate reaching uniform decline fluctuation anisotropy may associated behaviour rock joint main reason shallow earthquakes noticed results later new space completely matching analysis joint degree joint clustering distribution figure illustrated new variable regard durability entropy system initiating post behavior scaled minus zero variation parameter analyzed structures frequencies parallel perpendicular aperture networks also directed network based contact strings preferentiality possible energy flow rupture tips introduced also inspected synchronization strings using kuromoto model fact definition parameter fluctuation anisotropy filtered conclusions study presented special type complex aperture network based correlation measures main purpose study make connection apparent mechanical behavior rock joint characterized network incorporation correlation apertures evaluation continuously changing contact areas growth aperture within networks showed effects structural complexity evolution path rock joint results showed main characteristics aperture networks related shear strength behavior rock joint residual shear strength corresponded formation giant groups nodes networks addition based joint correlation upon edges triangles post peak behaviour rock joint shear analyzed results may used approach insert complex aperture networks surface growth methods general understanding conditions sudden movement shock fault references newman structure function complex networks siam review baiesi 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sharifzadeh http evgin complex aperture networks adler thovert fractures fracture networks kluwer academic alava nukala zapperi statistical models fracture advances physics volume issue pages knopoff organization seismicity fault networks pnas april vol zimmerman chen cook effect contact area permeability fractures hydrol lanaro random field model surface roughness aperture rock fractures int rock mech min sci wilson introduction graph theory fourth edition prentice hall harlow gao jin identification nonlinear dynamics twophase flow complex networks review colizza flammini serrano vespignani detecting ordering complex networks nature physics issue pages song havlin makse origins fractality growth complex networks nature physics pages kim goh salvi kahng kim fractality complex networks critical supercritical skeletons phys rev brown kranz bonner correlation surfaces natural rock joints geophys res lett hakami einstein genitier iwano characterization fracture aperturesmethods parameters proc int congr rock mech tokyo lanaro stephansson unified model characterization mechanical behavior rock fractures pure appl ghaffari complexity analysis unsaturated flow heterogeneous media using complex network approach http newman assortative mixing networks phys rev lett sokolov changing correlations networks assortativity dissortativity acta phys pol korniss synchronization weighted uncorrelated complex networks noisy environment optimization connections transport efficiency phys rev dijkstra note two problems connexion graphs numerische mathematik newman barabasi watts structure dynamics princeton university press watts strogatz collective dynamics networks nature mitani esaki zhou nakashima experiments simulation shear flow coupling properties rock joint rock mechanics conference essen mitani esaki sharifzadeh vallier shear flow coupling properties rock joint modeling geographical information system gis isrm conference south african institute mining metallurgy sharifzadeh experimental theoretical research coupling properties rock joint thesis kyushu university japan valentini perugini poli nature networks next term possible implications disequilibrium transport magmas beneath ridges journal volcanology geothermal research gnecco enrico meyer ernst fundamentals friction wear springer rubinstein cohen fineberg contact area measurements reveal dependence static friction phys rev lett sharifzadeh mitani esaki joint surfaces measurement analysis aperture distribution different normal shear loading using gis rock mechanics rock engineering volume number april pages xia rosakis kanamori laboratory earthquakes rupture transition science rubinstein cohen fineberg detachment fronts onset dynamic friction nature rubinstein cohen fineberg visualizing experimental observations processes governing nucleation frictional sliding phys appl phys sornette critical phenomena natural sciences berlin heidelberg arenas kurths moreno zhou synchronization complex networks reports strogatz exploring complex networks nature vol brace byerlee mechanism earthquakes science volume issue scholz earthquakes friction laws nature http sharifzadeh http motifs networks shear fractures submitted journal barahona pecora synchronization systems phys rev lett figures mpa figure variation shear strength different cases normal stresses mpa mpa mpa without control upper shear box figure evolution correlation values aperture profiles shear displacement figure correlation patterns throughout shear displacements figure visualization adjacency matrix achieved networks mean geodesic distance figure clustering displacement number average path number bins number bins figure joint degree distribution top row top right row shear slip figure attributed weight distribution links related joint degree distribution figure joint clustering coefficient distribution plus attributed weight histograms based averages triangles connected link sequence figures well figure data fit figure spectrum complex aperture networks evolution mean degree node clustering coefficient fitness polynomial function dki dci dki dci space respect shear displacements figure data accumulation data related shear displacements std shear displacements anisotropy figure variation evolution rate spectrum networks space

oct mapping class groups linear positive characteristic button abstract orientable surface finite topological type genus least possibly closed possibly number punctures boundary components show mapping class group faithful linear representation dimension field positive characteristic introduction common question ask given infinite finitely generated group whether linear instance consider braid groups automorphism group aut free group mapping class group mod closed orientable surface genus linearity first case open known hold showed aut linear whereas third case open however whereas definition linearity group embeds field practice one tends concentrate case fact finitely generated group embeds embeds field characteristic zero enough restrict case characteristic zero representations considered however still ask faithful linear representations positive characteristic instance three examples unknown braid group admits faithful linear representation dimension field positive characteristic aut proof applies field characteristic zero case also faithful representations positive characteristic proof mapping class groups show faithful linear representations mod dimension field positive characteristic orientable surface finite topological type genus least might closed might number punctures boundary components idea comes considering analogy finitely generated group linear positive characteristic nice geometric action showing gersten free cyclic group faithful linear representation positive characteristic looking closely see definition nice aligns closely linearity positive characteristic struck similarities notion finitely generated group acting properly semisimply properly cocompactly complete cat space bridson shows surfaces mentioned mapping class group mod admit action result first credited proof consists finding obstruction existence action one groups obstruction involves taking element infinite order centraliser said group applying condition abelianisation centraliser show condition holds verbatim groups linear positive characteristic thus obtaining obstruction leave open question whether mapping class group closed orientable surface genus linear positive characteristic note shown using braid group results group linear characteristic zero anyway proof following crucial point distinguishes treatment linear groups positive characteristic classical case proposition algebraically closed field positive characteristic exists elements matrix diagonalisable proof characteristic take power least proof put jordan normal form indeed form matrix splits blocks corresponding generalised eigenspaces upper triangular block taking eigenvalue block corresponding form upper triangular zeros diagonal modulo power thus block equal block making diagonal matrix mapping class group mod surface proposition proposition orientable surface finite type genus least number boundary components punctures dehn twist simple closed curve abelianisation centraliser mod finite contrast theorem suppose linear group field positive characteristic centraliser infinite order element image abelianisation also infinite order proof abelianisation universal abelian quotient group enough find homomorphism abelian group maps element infinite order use determinant first replace field algebraic closure proposition tells diagonalisable element whereupon showing infinite order abelianisation could course smaller centraliser establish take basis actually diagonal group together repeated eigenvalues idk references means element thus also form block structure consequently homomorphisms multiplicative abelian group determinant also subdeterminant functions detk define deti determinant ith block expressed respect basis diagonalises indeed homomorphism could deti finite order implies thus also finite multiplicative order however true finite order means contradiction thus know map elements infinite order abelian group homomorphism deti corollary orientable surface finite type genus least number boundary components punctures mod linear field positive characteristic proof combine proposition theorem get contradiction dehn twists infinite order references bigelow braid groups linear amer math soc bigelow budney mapping class group genus two surface linear algebr geom topol references bridson semisimple actions mapping class groups cat spaces geometry riemann surfaces london math soc lecture note cambridge univ press cambridge button minimal dimension faithful linear representations common finitely presented groups http formanek procesi automorphism group free group linear algebra kapovich leeb actions discrete groups nonpositively curved spaces math ann korkmaz linearity certain mapping class groups turkish math krammer braid groups linear ann math selwyn college university cambridge cambridge address

packing circles within circular containers new heuristic algorithm balance constraints case washington alves luiz leduino salles antonio carlos ednei felix corresponding author aplicadas universidade estadual campinas limeira brazil instituto tecnologia universidade federal paulo dos campos brazil departamento universidade federal ponta grossa brazil oliveira salles neto moretti moretti edneif reis faculdade abstract work propose heuristic algorithm layout optimization disks installed rotating circular container unequal circle packing problem additional balance constraints proved problem heuristics methods resolution larger instances main feature heuristic based selection next circle placed inside container according position system center mass approach tested series instances circles compared literature computational results show good performance terms solution quality computational time proposed algorithm keywords packing problem layout optimization problem nonidentical circles heuristic algorithm introduction study install unequal disks rotating circular container adaptation model unequal circle packing problem balance behavioral constraints problem arises engineering applications development satellites rockets multiple spindle box rotating structure low cost high performance equipment require best internal among different geometric devices problem known layout optimization problem lop consists placing set circles circular container minimum envelopment radius without overlap minimum imbalance circle characterized radius mass original threedimensional case equipment must rotate around axis different circles see figure represent cylindrical objects placed inside circular container figure illustrates physical problem figure shows rotating cylindrical container symbol arrow illustrate rotation around axis equipment angular velocity another viewpoint figure shows interior equipment distinct circular devices need placed example six cylinders placed radii masses heights necessarily equal research packing circles circular container documented used obtain good solutions heuristic metaheuristic hybrid methods used publications discussing disk problems balance constraints lop combinatorial problem proved lenstra rinnooy kan problem proposed teng mathematical model series intuitive algorithms combining method constructing initial objects topomodels iteration method described validity proposed algorithms numerical examples tang teng presented genetic algorithm called decimal coded adaptive genetic algorithm solve lop developed version genetic algorithm called positioning technique best ordering placing circles container compare two existing natureinspired methods qian extended work tang teng introducing genetic algorithm based intervention human expert examines best solution obtained loops many generations designs new solutions methods based particle swarm optimization pso applied lop developed pso method operator approach escape local minima maintaining characteristic fast speed convergence zhou proposed hybrid approach based constraint handling strategy suit pso improvement made using direct search increase local search ability algorithm xiao presented two approaches based gradient search hybrid simulated annealing method second hybrid pso method lei presented adaptive pso better search performance employs strategies plan space global search local search obtain global optimum huang chen proposed improved version algorithm proposed wang solving disk packing problem equilibrium constraints strategy accelerating search process introduced steepest descends method shorten execution time liu lop converted unconstrained optimization problem solved basin algorithm presented together improved energy landscape paving method gradient method based local search heuristic update mechanism liu presented simulated annealing heuristic solving lop incorporating neighborhood search mechanism adaptive gradient method neighborhood search mechanism avoids disadvantage blind search simulated annealing algorithm adaptive gradient method used speed search best solution liu developed tabu search algorithm solving lop algorithm begins random initial applies gradient method adaptive step length search minimum energy proposed hybrid approach based quasiphysical optimization method improvement made adapting descent tabu search procedures algorithm approach takes account diversity search space facilitate global search also search corresponding best solution promising local area liu presented heuristic based energy landscape paving lop converted unconstrained optimization problem using strategy penalty function method subsequently heuristic approach combines new updating mechanism histogram function improved energy landscape paving local search solving lop paper propose new heuristic solve lop basic idea approach called placing technique cmpt place circle according current position center mass system results selected set instances found huang chen xiao lei liu liu validate approach compare results heuristic instances computational results show good performance terms solution quality computational time paper organized follows section presents formal unequal circle packing problem balance constraints established section describes heuristic section present analyze experimental results section concludes paper problem formulation consider following layout optimization disks installed rotating circular container given set circles necessarily equal minimal radius circular container circles packed without overlap shift dynamic equilibrium system minimized decision problem stated follows consider circular container radius set circles radii mass let coordinates container center center coordinates circle let objective function second objective function measures shift dynamic equilibrium system caused rotation container without loss figure circular devices inside rotating circular container feasible solution generality consider problem determine exists dimensional vector following mathematical formulation lop minimize subject max pair preset weights constraint states circle placed inside container extend outside container constraints require two circles placed inside container overlap figure illustrates typical feasible solution lop circles numbered radius circle overlap circles seven circles completely placed larger circle radius radius container develop constructive heuristic guided simple strategy suboptimal solution reached gradually placing circle time inside container circle placed euclidean coordinates system following evaluation criteria select new position circle according current center mass system without overlapping circles placed earlier attempt wasted spaces placing circle end select new coordinates container center completely eliminate dynamic imbalance system perform criteria need notations denote center coordinates circle euclidean distance center coordinates circles set points line segment whose endpoints figure illustrates set contact pair say contact pair circles layout partial layout denoted partial pattern layout formed subset circle centers already placed inside container without overlap assume addition container complete layout solution figure illustrates partial layout formed circles placed inside container among others contact pairs placed cyclic order let cyclic order circles already placed inside container without overlap addition intersection two sets one endpoint common say placed cyclic order contact cyclic order let placed cyclic order circles two two contact pairs say contact cyclic order given say circles counterclockwise order relation circle circles clockwise order relation circle main area let placed cyclic order say area bounded union line segments main area denoted figure illustrates contact cyclic order formed circles placed inside container note two two contact pairs figure addition illustrated contact cyclic order formed circles note circles dashed lines completely placed feature approach since several contact cyclic order obtained circling approach important requirement since yield compact layout xcm figure two contact cyclic orders partial layout let subset circles placed inside container cardinality denote centroid coordinates border let partial layout contact cyclic order center circle belongs say addition border partial layout figure illustrates border circles partial layout circles note circle centers belong hand border since consider two cases inclusion placing circles case require circle included must touch least two previously placed circles second case require another circle included occupies wasted spaces placing circle reasonable requirement since generally yield compact layout one separate circles two cases inclusion explained partial layout lop example seven existing circles illustrated figure figure shown case inclusion two positions place circle dashed lines touching contact pair two positions place circle dashed lines touching contact pair position obtained solutions following particular case problems apolonio coxeter denote coordinates solution system belong note system two real solutions whenever riq figure choosing coordinates circle obtain feasible layout however enough choose coordinates circle circle overlaps circles approach always select coordinates order place new circle touching contact pair border figure due potentially large differences radii possible occur overlap circles border illustrated figure get around situation repositioning circle coordinates solution new system circle touches circles figure case inclusion possible reposition following placement approach external placement let partial layout border external placement placement circle inside container overlap center belong becomes contact pair least two circles denoted external placement external placement always selected outside however overlap repositioning new circle explained done following routine procedure external placement routine input circle contact pair partial layout border output external placement step calculate system circle overlap circles stop otherwise step step overlap circle circles repeat circle overlaps circle furthest respect counterclockwise order border circle overlaps circle furthest respect clockwise order border choose solution system furthest centroid circles respect euclidean distance first new circle overlap circles border external placement routine selects approach case convenient way place next circle however overlap step routine circles order reposition circle centroid partial layout eventually avoiding kind overlap obtain compact layout including circle checked possibility including another circle occupy wasted spaces placing circle check among remaining circles outside container preferably largest one circle placed container centralized position without overlap centralized position centroid coordinates certain set circles includes circle two circles touching circle figure illustrates second case inclusion investigate possibility positioning circle wasted space placing circle touching contact pair centroid circles wasted space placing circle touching circles centroid circles internal placement let partial layout border contact pairs previous circle included internal placement placement circle inside container center belongs overlap circle center placed centroid coordinates subset denote internal placement meaning placed centroid coordinates let partial layout border algorithm positioning case inclusion always done looking contact pairs border suppose remaining circle selected placed touching contact pair placement circle causes addition one element one index perhaps removal indices represented following operation figure two cases inclusion external placement internal placement choice fewer indices operation applied means circle placed inside container touching circles without overlap index added subset indices removed coordinates added partial layout note removal indices index inserted indices possible placement circle placement circle causes possible addition coordinates partial layout approach require imbalance system zero seems intuitive requirement may result good solution center mass system xcm xcm ycm one shift center rotating circular container center mass system zero imbalance shift made outer iteration end algorithm may increase envelopment radius thus layout represents complete solution lop denote radius container max rik xcm moreover index reached denoted kmax arg placing technique cmpt present new placing technique yields compact layouts quality solutions manner let permutation place circles partial layout one one according order permutation given order inclusion circles must positioned follows procedure initial layout routine input circles output initial layout initial border place circle coordinates choose coordinates touches circle solve system place coordinates without overlap figure illustrates initial initial border among many optional positions choose example coordinates suppose already placed circles describe approach placing circle verify possibility placing another circle place circle see procedure require circle touches least two previously placed circles see figure procedure generally yield compact layout however increase compactness layout wasted spaces placing circle occupied another circle see procedure xcm xcm figure example cmpt routine observe additional circle envelopment radius layout generally enlarged order minimize rate growth radius inclusions must properly choose new position circle yields smaller envelopment radius strategy cmpt attempts reduce rate growth envelopment radius including every circle around coordinates center mass system updated outer iteration strategy consists shifting origin euclidean plane current center mass system require circle touches circles contact pair arbitrarily chosen among elements border taking consideration quadrants euclidean plane approach performed according following routine procedure cmpt routine input partial layout border output sets step calculate coordinates center mass xcm circles translate origin euclidean plane xcm step include contact pair set center belongs quadrant euclidean plane given border procedure separates contact pairs according quadrants euclidean plane origin shifted current center mass system figure illustrates procedure figure observe coordinates center mass xcm system coincide coordinates origin wish place next circle around coordinates xcm order mitigate growth envelopment radius dividing plane quadrants obtain border circular shape see figure place new circle different quadrant euclidean plane origin shifted xcm layout evenly distributed choice different quadrants contact pair position next circle operation border lead updated border similar circular shape generally yield compact layout wasted space main area envelopment radius minimized see example figure next describe two cases inclusion following routine procedure inclusion routine input circle permutation sets contact pair set partial layout border output partial layout border sets step obtain external placement new values procedure fewer indices border step note step possible obtain internal placement set circle preferably largest permutation exclude inclusion routine attempts place new circles compact layout first computes external placement next circle procedure updates values order obtain fewer indices step border updated operation indices removed step indices correspond internal circles border compared indices added circle placed inside container contact pairs including removed sets choice guarantees operation exclude internal circles border finally search place another circle internal placement performed performed algorithm builds complete solution represented layout contemplates improvements via circle repositioning border process causes changes index removed repositioned operation removal index represented following operation operation applied means circle deleted position delete current layout test new position improves radius container main routine choose position circle inside container according following main procedure main routine input permutation output layout complete solution step initialization obtain initial layout initial border procedure step cmpt obtain sets procedure step layout construction contact pairs circles outside container repeat choose arbitrary include circle possible circle procedure step circles outside container return step otherwise step step compute kmax kmax step delete current repeat step contact pair excluding step obtain external placement new values procedure radius container improved return step step routine complete solution whose container center center mass system given permutation main routine builds initial layout step placing four circles procedure next step main aspect approach performed procedure cmpt routine euclidean plane divided four parts subsets contact pairs obtained next step repeated looking subset circles remaining placed step arbitrary contact pair chosen two cases inclusions performed procedure placing circles inside container obtain complete solution border step performed via circle repositioning border attempts improvements envelopment radius end center container shifted center mass system achieves zero imbalance order placement circles previously described permutation used input algorithm generate layout specifying order circles placed since exist possible permutations circles need appropriate technique order search large space preliminary tests show wasted spaces placing circles minimized greater order addition circles favors larger radii let sequence obtained considering radii descending order circles choose integer subdivide terms sequence blocks thus possible obtain subsequence used input algorithm permuting positions elements elements permute positions last elements procedure several subsequences place different circles may generated actually possibilities thus obtain sequence xcm ikmax figure suboptimal solution circles inside container generate distinct subsequences numerical experiments instance dimension greater equal chose generate sequences complexity analysis real computational time main routine depend number circles also diversity circle radii number circles current border well implementation analyze upper bound complexity main routine complete solution border including process recall circles border two two contact pair given partial layout circles already placed inside container circles outside let number circles border partial layout strategy cmpt procedure checks position circles euclidean plane done position circle see procedure touching two circles border existing circles positions since two existing circles two possible positions third determine external placement must check overlap circles check overlap circle circles good implementation reduce number checks assess positions place circle time checking overlaps times complexity obtain external placement placing circle must check circle outside container placed internal placement must check overlaps among circles subset done process select one circle assess contact pairs try improve envelopment radius checking external placements thus complexity bounded therefore complexity placing circles main routine bounded placing new circle operation border operation controls size iterations since theoretical upper bound experimental results section measure quality performance algorithm series instances circles literature tested three sets instances literature compare approach series hybrid approaches based simulated annealing particle swarm optimization xiao lei hybrid approach based simulated annealing neighborhood search mechanism adaptive gradient method liu hybrid tabu search algorithm gradient method liu series heuristics based energy landscape paving gradient method local search liu liu series algorithms based approaches gradient method local search huang chen liu size size size table data instance first set instances liu radii mass size radii mass second set instances huang chen radii mass size radii mass third instances xiao radii mass size radii mass methods search optimal layout directly evolving positions every circle well considering imbalance use benchmark suite instances problem table numerical results set instances instance instance instance huang chen lei liu time time time liu liu time liu times algorithm time time time described table test algorithm instance present range detailed description instances found huang chen liu xiao routines implemented matlab language executed intel core ram linux operating system except instances circles decide generate distinct permutations input algorithm instance number executions main routine instance best solution found selected amount tests proved adequate comparisons results second third sets instances presented tables respectively compare approach described indicated reference results shown size instances best radius container obtained objective function imbalance obtained second objective function running time seconds table shows approach proved competitive obtained best value instance tied instance obtained results worse best result instances respectively similar results obtained second set test seen table algorithm tied instance obtained results worse best result instances table compare approach three algorithms set data version simulated annealing second set data reference one version particle swarm optimization pso third set data table numerical results second set instances instance instance huang chen liu liu time time time liu algorithm time time time heuristic based energy landscape paving approach proved competitive relation envelopment radius obtained better results instances obtained worse results cases average approximately worse results literature instances overall running time obtained algorithm considered good since center rotating circular container shifted center mass system always making solutions interesting others set instances figure illustrates typical solution obtained algorithm instance circles note large border circles size carefully read cmpt routine see initial border iteratively transformed border latter iteratively transformed border example two inclusions internal placement computational results show proposed algorithm effective method solving circular packing problem additional balance constraints conclusions presented new heuristic called placing technique packing unequal circles circular container additional balance constraints main feature algorithm use euclidean plane origin center mass system select new circle placed inside container evaluate approach series instances literature compare existing algorithms computational results show approach competitive outperforms published methods solving problem conclude approach simple high performance future work focus problem packing spheres table numerical results third set instances instance instance xiao xiao pso time time liu algorithm time time acknowledgements authors indebted anonymous reviewers helpful comments author wishes thank capes grant second author grateful fapesp third author thank cnpq references coxeter problem apollonius american mathematical monthly huang optimization method solving circle packing problem equilibrium constraints computers industrial engineering huang chen note improved algorithm packing unequal circles within larger containing circle computers industrial engineering lei constrained layout optimization based adaptive particle swarm optimizer berlin heidelberg lenstra rinnooy kan complexity packing covering partitioning problems schrijver packing covering combinatorics econometric institute mathematisch centrum amsterdam liu sun study particle swarm optimization mutation operator constrained layout optimization chinese journal computers chinese apud xiao amos liu jiang xue liu zhang energy landscape paving circular packing problem performance constraints equilibrium physica statistical mechanics applications liu basin algorithm circular packing problem equilibrium behavioral constraints science china information sciences liu chen liu wang equilibrium constraint layout using simulated annealing computers industrial engineering liu geng new heuristic algorithm circular packing problem equilibrium constraints science china information sciences qian teng sun interactive genetic algorithm application constrained layout optimization chinese journal computers apud xiao amos tang teng genetic algorithm application layout optimization journal software chinese apud xiao amos teng sun zhong layout optimization dishes installed rotating table packing problem equilibrium behavioural constraints science china mathematics series wang huang zhang improved algorithm packing unequal circles within larger containing circle european journal operational research xiao amos two hybrid compaction algorithms layout optimization problem biosystems xiao amos novel genetic algorithm layout optimization problem evolutionary computation cec ieee congress ieee zhou gao gao particle swarm optimization based algorithm constrained layout optimization control decision

counting racks order may matthew ashford oliver may abstract rack thought set maps permutation blackburn showed number isomorphism classes racks least paper improve upper bound matching lower bound proof involves considering racks loopless directed multigraphs edge colour applying various combinatorial tools introduction rack pair set binary operation exists whenever call order rack note conditions equivalent statement map bijection mentioned blackburn racks originally developed correspondence conway wraith structures known quandles racks introduced independently joyce matveev invariants knots fenn rourke provide history concepts nelson gives overview structures relate areas mathematics mathematical institute university oxford andrew wiles building radcliffe observatory quarter woodstock road oxford riordan example note set obtain rack known trivial rack group resulting quandle known conjugation quandle example let abelian group aut automorphism binary operation alexander quandle affine let racks map rack homomorphism bijective homomorphism isomorphism concerned racks isomorphism rack order clearly isomorphic rack take underlying ground set denote set racks set isomorphism classes racks order several published results concerning enumeration quandles small order nelson henderson macedo nelson enumerated isomorphism classes quandles order work clauwens vendramin gives enumeration isomorphism classes quandles order whose operator group transitive operator group section recently pilitowska gave enumeration medial quandles type quandle order far aware previous asymptotic enumeration result general racks due blackburn giving lower upper bounds respectively theorem improves upper bound case dial quandles authors conjecture upper bound restriction main result paper proves upper bound general racks hence medial quandles theorem let sufficiently large integers lower bound follows construction theorem focus upper bound bound given theorem obtained applying group theoretic results operator group associated rack arguments apply combinatorial results graph associated rack graph next section graphical representations racks rack set bijections setting following result see example gives correct conditions collection maps rack throughout paper write maps right proposition let set let collection functions domain define binary operation rack bijection proof noted earlier conditions rack hold bijection remains show condition equivalent essentially reworking omit simple details light proposition well rack set set maps providing bijections satisfy move freely two unless otherwise stated operator group rack subgroup sym generated following standard lemma see example lemma shows proposition extended elements operator group lemma let rack let operator group rack represented directed multigraph give vertex colour put directed edge colour vertex vertex remove loops graph edge colour incident helpful recast representation racks directed multigraphs slightly setting let set let sym simple loopless directed graph setting considering decomposition disjoint cycles see consists collection disjoint directed cycles isolated double edges corresponding cycles length two isolated vertices extend case multiple permutations natural way definition suppose sym set permutations set directed loopless multigraph putting directed edge colour also reduced graph directed graph obtained letting least one directed edge note reduced graph contains two edges also observe subgraph namely continuing let clarify terminology path directed multigraph need respect orientation edges underlying undirected graph component directed graph component underlying undirected graph directed sequence vertices let return racks definition let rack let associated maps mean directed multigraph sense thus associated although may necessarily consider coloured also write indicating graph whole rack describing racks graphical context may refer elements vertices following two observations straightforward crucial lemma let sym family permutations let distinct directed proof need prove statement let path edge part directed cycle thus replaced directed replacing edge gives directed shortest walk path lemma let sym family permutations orbit natural action spans component proof let orbit natural action exists sequence elements sequence exactly equivalent edges successively coloured value indicating direction edge thus partition orbits coincides partition components applying last result rack shows orbits operator group natural action coincide components illustrate notions simple example let rack subrack rack thus subset forms subrack bijection also thus separated graph notation context always abbreviated restriction subset left implicit outline proof short section give brief outline count number racks shall reveal information unknown rack several steps counting number possibilities revealed information step end rack determined completely obtain upper bound number racks principle behind argument follows choose set reveal maps consider components graph key lemma shows set vertices component contains precisely one element revealing maps determines entire rack set big relatively components actually need consider two sets choose threshold consider set vertices degree strictly greater underlying graph choose probabilistically relatively small set vertex high degree also high degree degree vertex high number components contained small allows determine maps vertices degree construct greedily set given size adding vertices one time revealing corresponding maps time choosing vertex whose map joins components follow every join limited number components reveal restriction components complex nature argument store revealed maps count racks consistent main term argument comes considering maps acting within components control action maps revealing extra information corresponding neighbours controlled consists low degree vertices section formally information requires straightforward graph theory section show number possibilities section complete proof theorem important information rack degrees graphical representations racks let rack set vertices definition notation respect simple graph course similarly show subrack components lemma let rack subrack let span component hence also proof first suppose directed edge colour take arbitrary exists subrack observe suppose contradiction thus contradicting fact hence arbitrary bijection let arbitrary lemma directed path instead considering directed result follows multigraph theory construction information requires straightforward graph theory multigraph vertex set edge multiset unordered pairs multisets multiset obtained including element multiplicity sum multiplicity multiplicity multiset unordered pairs multigraph vertex set edge multiset subsection consider undirected multigraphs clarity paths components directed multigraph underlying undirected multigraph results remain true directed multigraphs write number components multigraph let distinct following result standard following results formulated using simple graphs use multigraphs consistent definition graph rack proposition let multigraph multisets unordered pairs elements proof case follows observation since general case follows induction definition let multigraph multiset unordered pairs elements let span say merged exist edge denote set vertex sets components merged note multisets edges single edge merge two components unordered pair lemma let multigraph multisets unordered pairs elements proof write component merging edge suppose components write vertex set component containing containing contained single component follows easily follows cases thus corollary let multigraph multiset unordered pairs elements proof order write precisely edges eia eij eij write always consider adding edges order given lemma edges follows words vertex set component component multigraph set vertices information introduce following terminology rack let denote set vertices write set vertices strictly greater show partition actually partition subracks lemma let rack subracks proof lemma two vertices component hence separated thus subracks large given rack construct set following procedure subgraph described construction otherwise order vertices follows choose given partial ordering choose next vertex take otherwise introduce notation write set edges colour set vertex sets components merged edges colour note key property set given next lemma lemma let rack proof note statement trivial thus assume write empty graph note also suppose proposition contradicting ordering vertices hence arbitrary conclude decreasing sequence fact follows take ordering construction noting may rewrite thus combine corollary see showing result formally information associated rack need notation firstly write set vertices components merged write figure representation components edges colour blue precisely five components shaded light blue merged edges colour set vertices shaded components restriction included information see figure following lemma gives key property set slightly general setting say write instead lemma let rack let span component proof take set described let arbitrary edge colour merged must arbitrary bijection follows apply lemma spanning component consists vertices components merged thus formally information associated rack definition let rack let notation let order vertices arbitrary way let second entry equivalent set maps alternatively graph fourth entry equivalent set maps lemma image maps contained within knowing fourth entry determines necessary entry note also entry equivalent set maps thus graph seventh entry equivalent think map set form set considered detail section next section show image small considering map determining information random subsets part argument relating vertices high degree requires probabilistic tools particular require result known chernoff bounds use following workable version see example theorems corollary theorem let pindependent random variables taking values range let let rack ease notation write lemma let rack let construct random subset retaining element probability independently elements proof let sop ber variables independent apply theorem showing statement second statement take vertex let choose element jvi fjvi put jvi elements clearly distinct xjvi apply theorem see since second result follows high degree part need following crucial lemma lemma let rack let span component let let knowledge maps sufficient determine maps maps conjugate sym proof let lemma directed let colours edges path fil thus fil fij lemma thus fil maps determined thus determined also note conjugate map fil proving result show main result section proposition let rack sufficiently large exists set determined sets maps proof construct set mixture probabilistic deterministic arguments let consider random subset described lemma let event item lemma since follows large enough call log let event let denote number vertices hence item lemma log let event every vertex markov inequality hence large enough large enough thus hence set vertex whenever means graphical log terms vertex adjacent least vertices lemma edges disjoint union vertex sets components component contained within size least hence components write vertex sets components take set vertices shown lemma knowledge maps fvk determine maps applying component turn shows knowledge maps determine second entry put result corollary let exists positive integer proof take large previous result hold determined equivalently maps suitable set depending follows number distinct triples arising racks clearly possibilities choices map possibilities maps hence hence exists positive integer components graph order prove choices need following lemma recall construction lemma let rack let span component let knowledge maps vertex sufficient determine map maps determine set thus set proof let lemma directed note graph determined maps knowledge assumed let denote length shortest directed show determined induction graph distance base case true assumption take suppose result true smaller take shortest directed path length let edge penultimate vertex path exists know determined map also determined inductive hypothesis vertex determined hence determined know vertex map result follows induction show second main result section proposition let exists positive integer proof corollary equal number distinct values ranges racks order produce bounds entries clearly choices set recall construction follows thus possibilities vertex possibilities maps possibilities maps hence possible values three entries ranges racks order consider rack suppose three entries determined fix must consider possibilities set components merged restricted map crudely naj possibilities components lemma number possibilities large take choose arbitrary maps determined already lemma restriction determined entirely possibilities vertex considering components making possibilities restriction note determined elements regardless set possibilities possibilities combining bounds log log possibilities ranges racks order large thus log large thus large hence exists positive integer proving result maps acting within components preparatory results need following easy claim claim real numbers proof simply observe claim used prove following key technical lemma notation chosen match quantities next subsection lemma let npbe positive integer sequence integers set proof expanding product set since positive integers similarly positive integers thus hence bound sum using claim elaborate version argument gives corresponding stability result saying informally speaking close close close full details see proof theorem end section introduced information rack explained thought map set let call set section showed image size section image consider racks show information corresponding known many possibilities consider set detail subsequent discussion form described set elements sym indexed arbitrary order subset injective maps sequence elements sym indexed set arbitrary order last two entries graphical nature relate abstract formally graph associated definition let write sense multigraph write set vertex sets components describe form namely sequence mji subsets indexed arbitrary order sequence indexed yji sym yji avoid later inconvenience extend mji yji follows set ordered pairs think edges colour setting mji yji suppose rack recall entry also determined terms maps corresponding comparing see also thus finally mji yji means information known need determine maps determine entire rack noting set yji determined follows upper bound number possibilities maps also upper bound number racks reduce number maps left determine considering components graph lemma let fixed let rack let cci let vci set vertices determined maps proof set vertex sets components take knowledge determines maps thus lemma knowledge map fvj determine maps applying components shows determine entire rack determining set maps restrictions determined equal otherwise need determine restrictions proving upper bound need notation let denote number vertices components size exactly components size exactly thus proposition let define racks proof write cci let vci set vertices let rack lemma determined maps follows upper bound number possibilities maps also upper bound number racks let span component let lemma particular possibilities determined lemma determined thus possibilities components size possibilities map considering components together possibilities maps follows racks proving result proposition allows prove main result proof theorem recall denotes set isomorphism classes racks order upper bound let corollary proposition large take large possibilities proposition lemma racks thus extremal result positive integer let partition let rpn denote set racks components exactly parts let denote number parts size exactly two extension methods used paper used prove unless exists constant many words informally speaking almost exponentially strong sense racks almost components size idea proof function similar proposition taking account size components rather components size two special case symmetric group two elements small abelian full proof see references matthew ashford graphs algebraic objects phd thesis university oxford simon blackburn enumerating racks quandles kei electronic journal combinatorics clauwens small connected quandles arxiv preprint roger fenn colin rourke racks links codimension two journal knot theory ramifications richard henderson todd macedo sam nelson symbolic computation quandles journal symbolic computation benita sam nelson matrices quandles homology homotopy applications wassily probability inequalities sums bounded random variables journal american statistical association svante janson tomasz luczak andrzej random graphs john wiley sons agata pilitowska david anna zamojskadzienio structure medial quandles journal algebra david joyce classifying invariant knots knot quandle journal pure applied algebra sergei vladimirovich matveev distributive groupoids knot theory sbornik mathematics sam nelson combinatorial revolution knot theory notices ams leandro vendramin quandles low order journal knot theory ramifications pages

generalized minimum distance estimators linear regression dependent errors jan jiwoong kim university notre dame abstract paper discusses minimum distance estimation method linear regression model dependent errors strongly mixing regression parameters estimated minimum distance estimation method asymptotic distributional properties estimators discussed simulation study compares performance minimum distance estimator well celebrated estimator simulation study shows superiority minimum distance estimator another estimator koulmde package used simulation study available online see section detail keywords dependent errors linear regression minimum distance estimation strongly mixing introduction consider linear regression model xip xij non random design variables parameter vector interest methodology estimators obtained minimizing dispersions pseudo distances data underlying model referred minimum distance estimation method paper estimate regression parameter vector estimation method collection model dependent process let independent identically distributed random variables distribution function unknown classical estimator obtained minimizing following mises cvm type empirical integrating measure multiple reasons cvm type distance preferred including asymptotic normality corresponding estimator see parr schucany parr millar many researchers tried various obtain estimators anderson darling proposed estimator obtained using another important example includes giving rise hodges lehmann type estimators integrand replaced kernel density estimator assumed density function hellinger distance estimators obtained see beran departing one sample setup koul dewet extended domain application estimation regression setup assumption known proposed class estimators minimizing weighted empirical error koul extended methodology case error distribution unknown symmetric around zero furthermore shown therein regression model independent nongaussian errors estimators regression parameters obtained minimizing various integrating measures least asymptotic variance among estimators including wilcoxon rank least absolute deviation lad ordinary least squares ols normal scores estimators estimators obtained degenerate integrating measure display least asymptotic variance errors independent laplace however efficiency estimators depends assumption errors independent errors dependent estimation method less efficient estimators examples efficient methods include generalized least squares gls gls nothing regression transformed transformed prominent advantage using gls method decorrelation errors result transformation motivated efficiency estimators demonstrated case independent errors desirable property gls decorrelation dependent errors author proposes generalized estimation method mixture gls methods estimation applied transformed variables generalized means domain application method covers case dependent errors extent main result paper generalizes work koul efficiency method demonstrated case independent errors main goal paper show generalized estimation method still competitive linear regression model dependent errors indeed simulation study empirically shows main goal achieved rest article organized follows next section characteristics dependent errors used paper studied also cvm type distance various processes need order obtain estimators introduced section describes asymptotic distributions optimal properties estimators findings finite sample simulations described section proofs deferred appendix remainder paper italic boldfaced variable denotes vector boldfaced variable denotes matrix identity matrix carry suffix showing dimension denotes identity matrix function let denote real vector kuk denotes euclidean norm denotes real matrix means entries functions strongly mixing process cvm type distance generated sequence said let satisfy strongly mixing condition sup referred mixing number chanda gorodetskii koul withers investigated decay rate mixing number roots works section defines decay rate assumed paper see assumption hereinafter errors assumed strongly mixing mixing number addition assumed stationary symmetric around zero next introduce basic processes distance required obtain desired result recall model let denote design matrix whose ith row vector model expressed let real matrix inverse positive definite symmetric matrix note diagonalization positive definite symmetric matrix guarantees existence also symmetric matrix let qin denote ith row vector define transformed variables yei gls method obtained covariance matrix transforms dependent errors uncorrelated ones decorrelates errors however gls obtains slightly different manner instead using gls equates inverse covariance matrix gls uses cholesky decomposition empirical result section describes diagonalization yields better estimators propose class generalized estimators regression parameter upon varying impose noether condition let denote jth column let dik matrix real numbers denote jth column stated koul qxa dik xak noether condition max next define cvm type distance generalized estimator obtained let denote density function respectively analogue replaced empirical reasonable candidate however rarely known since original regression error assumed symmetric transformed error also symmetric therefore introduce koul definition dik dik indicator function measure symmetric around subsequently define inf next define ith row vector observe matrices respectively define matrix entry entry entries zeros finally define following matrices qxa let needed asymptotic properties note fijh asymptotic distribution current setup note section investigate asymptotic distribution minimizing closed form solutions numerical solutions tried hence would impracticable derive asymptotic distribution redress issue define qxa matrix next define inf unlike minimizing closed form solution therefore unb one reasonable approximate asymptotic distribution approximated idea plausible certain conditions called uniformly locally asymptotically quadratic see koul detail converges zero probthese conditions shown difference ability see theorem basic method deriving asymptotic properties similar sections koul method amounts showing uniformly locally asymptotically quadratic belonging bounded achieve goals need following assumptions set turn roots section koul matrix nonsingular satisfies lim sup max kdj integrating measure symmetric around real sequences lim sup define max let xak kuk lim sup xau xau depend dik xau xaufi continuous density respect lebesgue measure fir model strongly mixing mixing number satisfying lim sup remark note implies noether condition implies corollary koul note case errors asymptotic established weaker conditions noether condition normality dependence errors forces assume two stronger conditions remark discuss examples satisfy clearly satisfied finite measure next consider measure given dfi continuous symmetric around zero dfi another useful example measure given measure satisfied many symmetric error including normal logistic laplace example normal closed form integral using well celebrated tail bound normal distribution see theorem durrett obtain exp corresponding extensions one recall koul sample estimator location parameter regression model remark consider condition bounded implies two conditions measure normal logistic laplace densities particular dfi logistic condition also satisfied ready state needed results first theorem establishes needed uniformly locally asymptotically quadraticity corollary shows boundedness theorem corollary counterparts conditions suitably standardized theorem koul respectively note condition theorem met appendix condition theorem trivial theorem let model assume hold sup proof see appendix corollary suppose assumptions theorem hold exists inf proof see appendix theorem assumptions theorem therefore proof note first term side nothing proof follows theorem corollary case illustrated theorem koul next define symmetry around yields let denote covariance matrix define matrix write observe qxa qxa ready state asymptotic distribution lemma assume positive definite addition assume sup identity matrix proof prove claim suffices show asymptotically normally distributed note sum theorem mehra rao cni note also observe kax max max max assumption finally obtain lim inf lim sup assumption terms denominator hence desired result follows theorem mehra rao corollary addition assumptions theorem let assumption lemma hold proof claim follows lemma upon noting denote asymptotic variance remark let asym asym qxa qxa qxa observe transformed errors distribution qxa simplified therefore asym qxa moreover transformed errors uncorrelated result transformation simplified asym simulation studies section performance generalized estimator compared one denote covariance matrix errors gls estimators let estimate respectively consequently obtain gls estimator gls order obtain generalized estimator try two different refer generalized estimators corresponding estimators respectively order generate strongly mixing process dependent errors several restrictive conditions required mixing number decays fast enough assumption met withers proposed upperbound decay rate mixing number shake completeness reproduce theorem corollary lemma let independent characteristic functions max max let sequence complex numbers omax min assume max sequence strongly mixing mixing number max generate strongly mixing process lemma consider four independent normal laplace logistic mixture two normals mtn note finite second moments hence set easily seen hence assumption satisfied satisfies strongly mixing condition let equivalently laplace distribution density function fla exp density function logistic innovation given flo exp exp generate set mean normal laplace logistic innovations since assumed sum symmetric set standard deviation normal set laplace logistic respectively mtn consider subsequently generate using next set true obtain xik random sample uniform distribution subsequently generated using models estimate generalized gls methods report empirical bias standard error mean squared error mse estimators use lebesgue integrating measure obtain generalized estimators author used package koulmde package available comprehensive archive network cran https table report biases mse estimators sample sizes repeated times author used high performance computing center hpcc accelerate simulations simulations done bias gls mse bias mse bias mse denote normal laplace logistic mtn respectively table bias mse estimators expected biases estimators decrease increases first consider normal normal gls estimators display best performance gls show similar biases hence mse estimators show slightly worse performance aforementioned ones display similar smaller bias estimators corresponding always larger turn cause larger mse therefore conclude gls show similar performance better one normal come different conclusion estimators outperform estimators gls estimators display similar bias gls mse bias mse bias mse table bias mse estimators performance note weighing merits gls estimators terms bias hard example laplace gls estimators show almost biases estimator show smaller larger bias gls estimators consider estimators display least regardless gls estimators show somewhat similar laplace logistic however estimators smaller gls ones mtn result estimators display least mse gls corresponding laplace logistic show similar mse estimators show smaller mse gls ones mtn appendix proof theorem section koul illustrates holds independent errors proof theorem therefore similar one theorem section define dik xau dik xau rewrite dik xaufi dik xaufi dik xaufi note last term integrand kth coordinate vector show suprema norms first four terms integrand applying inequality cross product terms complete proof therefore prove theorem suffices show sup sup sup dik xaufi dik xaufi sup taken kuk consider proof case similar facts hold case observe implies max kdi max therefore immediately follows proof involve dependence errors hence proof koul thus shall prove thereby completing proof theorem begin let jku yku wku denote dik replaced dik define rewrite wku xau bni xau xau note bni kuk recall lemma deo lemma suppose strongly mixing random variables mixing number suppose two random variables respectively measurable respect assume kxkp kxkp consequently addition consider following lemma lemma proof given let note therefore inequality last inequality follows assumption thereby completing proof lemma consider cross product terms wku dik djk xau xau dik djk bni bnj dik djk max max kbnj second inequality follows lemma convergence zero follows lemma consequently fubini theorem together obtain every fixed kuk lim sup lim sup xau lim sup max maxi complete proof suffices show exists lim sup sup kkv kku follows koul thereby completing proof theorem proof corollary proof independent errors found section koul difference proof section one arises part involves dependence error thus present proof analogue koul let dik dik note symmetry fubini theorem obtain addition lemma yields max together fact lemma obtain eklk dik max max max using chebyshev inequality exists klk rest proof proof lemma koul references beran minimum hellinger distance estimates parametric models ann deo note empirical processes strong mixing sequences ann koul behavior robust estimators regression model dependent errors ann koul minimum distance estimation linear regression unknown error distributions statist probab koul minimum distance estimation tests autoregression ann koul weighted empirical process nonlinear dynamic models springer berlin vol koul wet minimum distance estimation linear regression model ann mehra rao weak convergence generalized empirical processes relative strong mixing ann millar robust estimation via minimum distance methods zeit fur noether theorem wald wolfowitz ann math parr schucany minimum distance robust estimation amer statist prescitt schucany theory ahead business cycle measurement conference public policy

static analysis programs using file format specifications raveendra raghavan apr indian institute science bangalore tata consultancy services indian institute science bangalore raghavan narendran abstract programs process data reside files widely used varied domains banking healthcare analysis precise static analysis programs context software verification transformation tasks challenging problem key insight static analysis programs made useful knowledge input file formats programs made available analysis propose generic framework able perform given underlying abstract interpretation program restricting attention analysis program paths potentially feasible program input conforms given file format specification describe implementation approach present empirical results using real realistic programs show approach enables novel verification transformation tasks also improves precision standard analysis problems introduction processing data resides files documents central aspect computing many organizations enterprises standard file formats document formats developed evolved various domains facilitate storage interchange data banking enterpriseresource planning erp billing analysis wide adoption standard formats led extensive development software reads processes writes data formats however lack tool support developers working domains specifically targets idioms commonly present programs address issue proposing generic approach static analysis programs takes program well specification input file format program input analyzes program context behaviors program compatible specification motivating example work motivated particular batch programs context enterprise legacy systems programs typically executed periodically run process input file contains transaction records accumulated since last run order motivate challenges analyzing programs introduce running example small batch program well sample file meant process figure data division input file buffer output file buffer char digit procedure division open read end move itm record processing move itm record processing batch header else itm record processing diff batch header write else record processing move move else move rest header record processing else trl record processing trl record processing else terminate program error read end move close goback hdr itm itm itm trl hdr itm itm trl diff typ pyr tot src hdr typ rcv amt itm typ pyr tot src rcv amt main type payer account number total batch amount source bank receiver account num item amount fig example program sample input file input file record layouts input file format although example toy one sample file shown figure adheres simplified version real banking format record shown row fields demarcated vertical lines file format records grouped logically batches batch representing group payments one customer customers batch consists header record value hdr first field contains information paying customer followed one item payment records itm first field identify recipients followed trailer record trl figure gives names fields header well item records first field typ discussed another field particular relevance discussions src field header records identifies whether paying customer customer bank running program different bank diff meanings fields explained part figure code figure shows example program syntax data division contains declarations variables used program including input file buffer output file buffer basically overlay union following terminology language two record layouts shown figure record read buffer program interprets contents using appropriate layout based value typ field output buffer assumed fields pyr rcv amt well fields relevant discussions field declarations elided figure brevity statements program appear within procedure division program main loop lines record read input file first outside main loop line end iteration loop line iteration recent record read processed according whether header record lines item record lines trailer record lines sole write statement program processing block line writes processed payment record using information current item record well previously seen header record lines represent code details elided populates certain fields distinct ways depending whether paying customer bank different bank analysis issues challenges programs typically employ certain idioms distinguish programs domains programs read unbounded number input records rather fixed input size furthermore typically program designed process arbitrary inputs input files adhere known domain related state variables used program keep track types records read current point execution state variables used decide process new record read file instance program figure variable set lines different based src field header record read variable used line decide process item records batch subsequently read certain cases state variables could also used identify unexpected sequences reject analyzing understanding transforming programs precise ways requires unique form path sensitive analysis program point distinct information program state needs tracked corresponding distinct pattern record types could read far control reaches point illustrate using example tions program figure answers would enable various verification transformation activities program silently accept inputs natural important verification problem context programs running example input file happens contain item record first record without preceding header record variable would uninitialized item record read control reaches line variable initialized header record seen lines therefore condition line could evaluate nondeterministically furthermore output buffer written line could contain garbage pyr field field also initialized header record seen line words programs could silently write garbage values output files databases given inputs undesirable ideally running example programmer ought employed additional state variable keep track whether header record seen every item record ought emitted warning aborted program upon identifying violation requirement words state tracking programs complex prone done erroneously therefore need automated analysis check whether program accepts bad files files adhere specification files program behaviors possible inputs situations interested information program states arise prefixes files read instance developer might interested knowing possible uses unitialized variables runs files without clutter caused warning reports pertaining runs files intuitively first category warnings mentioned signifies genuine errors program many cases developers try ensure meaningful outputs corrupted input files example program fact instances uninitialized variables used runs files related note one might want know program falsely issue input warning even run file sort acceptance problem could happen either due programming error due misunderstanding developer part inputs expected could checked asking whether statements program issue warnings line example program reachable runs inputs example program turns happen program behaviors possible restricted scenarios interest situations need identify paths program taken runs certain narrower files instance running example might interested parts program required input files contain batches whose header records always src field parts constitute lines program except lines variable longer need set used input batches guaranteed batches essentially classical program specialization problem specialization criterion rather standard criterion parameters program program specialization various applications example program comprehension decomposition monolithic programs collections smaller programs internally cohesive functionality reducing overhead approach contributions approach static analysis based file formats primary contribution paper generic approach perform given underlying abstract interpretation interest based abstract lattice manner maintaining point distinct abstract fact element distinct patterns record types could read far typically lift analysis domain finite set predicates would required analysis domain would essentially example program set six predicates one formed conjuncting one three predicates hdr itm trl one two predicates would natural candidates however coming set predicates manually would tedious requires detailed knowledge state variables program usage automated predicate refinement might able generate predicates complex iterative process might potentially generate many additional predicates would increase running time analysis qsh shdr itm shdr itm itm dhdr qdh trl eof record type shdr dhdr itm trl constraint typ typ typ typ hdr src hdr src diff itm trl dhdr fig input automaton input record types specifications usually readily available even standards used previous programming languages researchers context tasks parser validator generation testing key insight specification represented input automaton whose transitions labeled record types set states automaton call file states directly used lift analysis using domain intuition abstract fact mapped file state program point possible concrete states arise point runs consume sequence records concatenation types records accepted file state automaton figure shows input automaton used running example figure showing associated input record type descriptions dependent types sample input file figure file per automaton sequence consists types records file namely shdr itm itm itm trl dhdr itm itm trl accepted automaton intuitively statements read statements affect program execution therefore lifted transfer functions straightforward use underlying transfer functions analysis transfer function read plays key role enforcing ordering among record types files instance consider qsh figure represents situation wherein header record read therefore output read transfer function qsh mapped join abstract facts predecessors qsh namely mapped input transfer function applications addition basic approach propose two applications address two natural problems analysis programs knowledge explored previously literature first application sound approach check program potentially accepts files accepts files second sound technique specialize program wrt given specialization criterion represents restriction full represented input automaton program file state graph pfsg propose novel program representation pfsg graph derived graph cfg program given input automaton program pfsg basically exploded version cfg original program controlflow paths pfsg subset paths cfg certain paths infeasible given input automaton omitted cfg existing static analysis applied pfsg without modifications benefit infeasible paths end ignored analysis describe modify basic approach emit pfsg also discuss formally results analysis differ performed pfsg compared performed original cfg implementation empirical results implemented approach applied several realistic well real cobol batch programs approach found related conformance issues certain real programs also able verify absence errors programs program specialization context observed approach surprisingly precise able identify statements conditionals need retained specialized program found analysis used identify references possibly uninitialized variables reaching definitions gave improved precision standard analysis many cases rest paper structured follows section introduce key assumptions definitions section present approach well two applications mentioned section introduces pfsg section presents implementation result section discusses related work section concludes paper assumptions definitions definitions records record types files record contiguous sequence bytes file field labeled record record zero fields zero fields record taken leaflevel record record type intuitively specification length record names fields lengths constraint contents record example consider record types shown figure row shows name record type associated constraint say record type iff satisfies length well value constraints type instance first record file figure type shdr see figure note general record could multiple types definitions files read operations file sequence records possibly different lengths successive records file assumed demarcated explicitly either markers captures length record run time file pointer associated open file read statement upon execution retrieves record pointed pointer copies file buffer program associated file advances file pointer definition input automaton input automaton tuple finite set states refer file states eof set record types set transitions file states transition labeled element designated start state set designated final states transition labeled eof iff transition final state outgoing transitions final states note input automaton may two different senses multiple transitions file state could label also possible record two distinct types two types labels two outgoing transitions file state let state define type language set sequences types elements take automaton start state final state defined union type languages states transitions define record language follows consists sequences records exists sequence types sequences equal length record type recall file nothing sequence records say file conforms input automaton accepts final state let sequence records possibly empty say file state input automaton exists execution trace given program starts program entry consumes records via read statements passes reaches program point say trace due prefix trace reaches point input automaton define sequence nodes graph cfg given program visited trace sequence always begins entry node cfg contains least two nodes trace ends point last node sequence input automaton often abbreviate automaton input automaton accepts files expected given input program specialization automaton input automaton accepts subset files automaton full automaton input automaton accepts every possible file program accesses multiple sequential input files situation could handled using two alternative approaches concurrently using multiple input automatons analysis one per input file modeling one input files primary input file associated automaton modeling reads remaining files always returning undefined record adopt latter approaches experimental evaluation approach section describe primary contribution generic approach lifting given abstract interpretation wrt specification discuss soundness precision following present details two applications generic approach mentioned section finally present extension approach enables specification data integrity constraints contents input files relation contents persistent tables abstract interpretation lifted using input automatons inputs approach program input automaton arbitrary underlying abstract interpretation join set transfer functions signature associated statements conditionals objective described introduction use provided input automaton compute least solution considering paths program potentially feasible wrt given input automaton lattice use lifted analysis partial ordering lattice point wise ordering based underlying lattice initial value supply entry program input approach initial value used context underlying analysis discuss transfer functions lattice consider following three categories cfg nodes statements read statements conditionals read statements let node neither read statement conditional let fln underlying transfer function node since file state trace point node change trace executes node transfer function node fln let conditional node true successor false successor let underlying transfer functions since conditional node modify trace either transfer function conditionals stands finally consider case node read node interesting case executing read statement change file state trace firstly note terminology dataflow value said read qsh qdh lsh ldh itm itm shdr itm dhdr eof trl shdr qdh qsh shdr shdr lsh itm ldh itm itm trl eof fig illustration transfer function read statements represent concrete state element concretization written secondly make following assumption underlying transfer function flr read statements rather simply signature function flr ought signature record type element intuitively flr return dataflow fact represents set concrete states result execution read assuming concrete state execution read state represented read statement retrieves record type input file places input buffer correspondingly flr eof return dataflow fact represents set concrete states result execution read assuming concrete state execution read state represented input buffer program gets populated undefined value statement within end clause executes read operation illustration say underlying analysis constant propagation analysis flr would return fact obtained performing following transformations remove existing facts associated input buffer obtain suitable new facts input buffer using constraints associated type hand flr eof would perform step ready present transfer read node transfer function flr label label returns label type eof transition intuition behind transfer function follows qsh qdh qsh qdh itm close qsh qdh hdr qsh qdh read qsh qdh fig fix point solution program figure using analysis abbreviations used hdr diff itm trl file state trace executing read trace one predecessor states automaton executing read therefore fact lattice associated point read statement obtained follows file state transition labeled automaton transfer fact flr fact mapped point read statement take join transferred facts figure sketches transfer function schematically edge column read statement column read denotes transfer happens due step label edge denotes label corresponding transition automaton abbreviated instance flr figure also omitted columns compactness presentation limited setting however analysis extended setting using standard techniques details discuss section illustration figure shows solution certain program points example program figure computed analysis using wellformed automaton shown figure assume underlying lattice product possibly uninitialized variables uninit lattices table figure denotes solution function program point precedes statement indicated table column table shows underlying dataflow value associated file state columns underlying dataflow value represents unreachability basically omitted tables brevity first component dataflow value within angle brackets indicates constant values variables second component within curly braces indicates set variables possibly uninitialized empty sets omitted figure brevity abbreviate variable names well constant values sake compactness described caption figure interest space focus attention one program points point condition line execution trace reaching point one following four file states qsh qdh file states associated facts indicate value hdr hdr itm trl respectively additionally state variable possibly uninitialized columns qsh qdh lines initialize variable may visited yet whereas initialized two columns fact associated flows true branch conditional line conditional tests contains itm therefore inferred definitely initialized time referenced line desired precise result soundness precision complexity approach soundness result intuitively underlying analysis sound lifted analysis modulo assumption input file given program conforms given input automaton said sound program point program solution computed represents concrete states result due possible execution traces begin concrete state represented given initial value following theorem states soundness characterization analysis formally theorem assuming sound abstract interpretation solution produced approach program point given program starting initial value represents concrete states result point due possible executions begin concrete state represented input file conforms given automaton proof theorem straightforward following observations follow theorem given input automaton automaton solution program point represents concrete states result point executions files given input automaton full automaton solution program point represents concrete states result point possible executions including files given input automaton approach produce solution least precise one would produced directly underlying analysis however choice input automaton impact precision approach intuitively input automaton subautomaton input automaton obtained deleting certain states transitions constraining types label transitions result precise solution also accept language structurally refines result precise solution formalize notions appendix time complexity analysis used automaton set states worst case times time complexity underlying analysis two applications analysis section describe use analysis described section address two new problems mentioned introduction file format conformance checking program specialization file format conformance checking mentioned introduction verification question developers would like answer whether program silently accept input file possibly write corrupted output file acceptance conversely could program reject file via abort warning message acceptance different programs use different kinds idioms reject input file generating warning message continuing processing usual ignoring erroneous part input file processing remaining records aborting program via exception order target modes generic manner approach relies developer identify related rejection points program statements program format violations flagged using warnings aborts etc detecting detect warnings applying analysis using automaton using given programstate abstraction domain interval analysis issuing warning fact associated rejection point intuition simply rejection points unreachable program run files since analysis conservative never produces dataflow facts approach miss issues long developer fail mark actual rejection point rejection point illustration say line program figure marked rejection point using automaton figure using underlying analysis analysis find line unreachable therefore warnings issued detecting intuitively program acceptance errors respect given file format program reach end main procedure without going rejection point run input file conform automaton check property follows first extend given automaton full automaton accepts input files systematically adding new final state new states new transitions lead new states original states intent new states accept record sequences accepted file state original automaton provide full details construction appendix secondly modify transfer functions lifted analysis rejection points map file states output intuitively idea behind block paths rejection points apply analysis using full automaton using programstate abstraction domain flag warning dataflow value associated file state final state original automaton final point main procedure clearly since analysis dataflow facts program points miss scenarios long developer wrongly mark point rejection point program specialization based file formats mentioned introduction would natural developers want specify specialization criteria programs patterns sequences record types input file propose use input automatons purpose example automaton figure modified removing file state qdh well transitions incident would obtained would specialization automaton accepts files batches begin headers approach program specialization using specialization automaton follows apply analysis section using given specialization automaton input automaton using abstraction domain identify program points every file state mapped per solution computed step basically program points unreachable executions input files conform specialization criterion immediately follow points projected program yield specialized program details projection operation focus paper easy see approach sound marks point unreachable definitely unreachable runs input files adhere given criterion illustration using specialization automaton mentioned using underlying analysis lines code figure marked unreachable worthwhile noting one use specialization automaton criterion instead simply specified header records value src field line would identified unreachable intuuitively path consisting lines along uninitialized would found infeasible discussed section subsequently step part core approach following simplifications could done program make lines unconditional remove respective controlling conditions would safe else branches two conditions become empty remove line entirely would safe conditional line removed variable used anywhere program imposing data integrity constraints input files core approach discussed section used input automatons constrain sequences types records appear input file however many situations file also needs satisfy certain data integrity constraints wrt contents certain persistent tables constraints also specified conjunction input automaton certain paths program execute upon violation constraints identified pruned analysis time potential improve precision usefulness approach running example say requirement receiver payment input file represented field item record necessarily account holder bank requirement could enforced code figure adding logic right line check value appears primary key accounts database table bank execute lines check fails however user approach wishes assert input files never contain items refer invalid account numbers logic mentioned could identified redundant enable users give specifications allow predicates form isintable tab field isnotintable tab field associated record type definitions tab name persistent table field name field record type example itm type figure could augmented typ itm isintable accounts rcv accounts master accounts table semantics value rcv field guaranteed primary key row table accounts similarly isnotintable tab field asserts value field guaranteed primary key row tab assume programming language following construct keybased lookup table tab read tab buffer key variable invalid key semantics statement follows table row key matching value variable found table tab copied buffer control given matching key found buffer content undefined control given enhancement record type specifications extend analysis framework follows new lattice use given original underlying lattice set possible predicates two kinds mentioned describe changes required transfer functions transfer function normal read statements read input files described section augmented follows whenever record certain type read predicates incoming fact refer fields input buffer removed predicates associated included outgoing fact transfer function statement move copies outgoing fact predicates incoming fact refer variables predicate incoming fact refers creates copy predicate makes refer instead adds outgoing fact transfer functions conditionals need change finally need handle lookups interesting case consider statement read buffer key invalid key transfer function first checks predicate form isintable present incoming dataflow fact essentially treats statementsn unreachable else predicate form isnotintable present incoming fact essentially treats unreachable otherwise treats reachable formal presentation transfer functions omitted paper interest space program file state graph pfsg section introduce program representation programs program file state graph pfsg formalize properties pfsg finally discuss pfsg serves basis performing program analyses without modifications enabling ignore certain cfg paths infeasible per given input automaton structure construction pfsg pfsg representation based cfg program well given input automaton pfsg basically exploded cfg set states file states node cfg nodes pfsg words pfsg nodes number nodes set structural property pfsg edge present nodes pfsg edge cfg words path pfsg corresponds path cfg let entry node cfg node regarded entry node pfsg pfsg constructed straightforward manner using basic approach described section precision pfsg linked precision underlying analysis selected example standard analysis could used precision required powerful lattice instance relational domain wherein lattice value represents set possible valuations variables octagon domain could used solution obtained approach edges added pfsg per following procedure edge cfg rule applicable read node transition automaton add edge pfsg rule applicable read node add edge pfsg rules add edge solutions respectively follow restriction resp execution trace reach resp due sequence records resp intuition behind first rule read statement executes modifies program could intuitively input automaton could start simulating automaton program starts executing transition appropriate target state upon execution read statement based type record read executing read statement program could transition transition present input automaton second rule switch program statements read statements affect program internal state valuation variables program qsh qdh dhdr itm shdr eof eof trl fig pfsg program fig input automaton fig illustration pfsg illustration consider pfsg shown figure corresponding program figure input automaton figure recall automaton describes files headers item records trailers appear correct positions visually figure laid six columns corresponding six file states input automaton nodes pfsg labeled corresponding line numbers program therefore node labeled column actually node represents open statement line program related note start state input automaton line entry node program node mentioned fact entry node pfsg certain parts pfsg elided brevity represented using cloud patterns pfsg generated using solution approach section using constant propagation underlying analysis fragment solution shown figure sets possibly uninitialized variables figure ignored current context discuss detail portion pfsg figure emphasis elides certain infeasible paths present original cfg line program read statement per given input automaton outgoing transitions qsh qdh therefore per rule pfsg approach see section outgoing edges copies node qsh qdh columns clarity labeled edges types corresponding transitions qsh column second column essentially consists copy loop body specialized situation wherein last record read type shdr shdr type transitions coming qsh particular note true edge qsh column elided solution see figure constant propagation fact associated qsh file state point indicates value hdr fact abbreviated figure therefore solution underlying fact associated qsh file state edge ends results rule adding false edge line program read statement edge node qsh bottom qsh column entry column sole successor qsh input automaton column consists copy loop body specialized situation wherein previous record read type itm type transitions coming end column control goes column end column back beginning qsh qdh columns notable structure pfsg inherited cfg input automaton mentioned discussion control transfers one column another mirror transitions input automaton paths within column inherited cfg specialized wrt type record last read program analysis using pfsg program analysis performed using cfg naturally performed unmodified using pfsg simply letting analysis treat node underlying node analysis less precise original cfg program construction every path pfsg corresponds path original cfg words extra paths pfsg contrary certain cfg paths infeasible per given input automaton could omitted pfsg words precision analysis improved ignoring executions due certain infeasible inputs instance example discussed due omitted edge qsh column path pfsg visits copies nodes order even though path exists original cfg words pfsg encodes fact given input file format item record occur first record input file however general due possible imprecision given underlying analysis paths infeasible per given input automaton would necessarily excluded pfsg illustrate benefits program analysis using pfsg discuss two example analyses say wish perform possibly uninitialized variables analysis due path original cfg use line would declared possibly uninitialized however given input automaton since every path reaches line pfsg reaches via lines defined use mentioned would declared definitely initialized analysis performed pfsg analysis done pfsg figure would indicate point line would constant value however analysis done pfsg analysis would indicate hdr value hdr diff value constant otherwise correlations identified pfsg exploded hence segregates cfg paths end program point due record sequences accepted different input automaton correlations one mentioned identified general using cfg unless expensive domains relational domains used two instances precision improvement mentioned also obtained using approach section use uninit underlying domain uninit analysis first instance simply use underlying domain second instance however general several scenarios pfsg serves better foundation performing program analysis approach section approach section applies forward dataflow analysis whereas pfsg used forward well backward dataflow analysis problems pfsg basis applying static analysis techniques dataflow analysis symbolic execution assertional reasoning etc implementations techniques designed cfgs applied unmodified pfsg analyses likely benefit pruning paths pfsg infeasible per given input automaton formal properties pfsg soundness characterize paths original cfg necessarily present pfsg result forms basis soundness static analysis applied pfsg theorem let given underlying sound abstract interpretation consider execution trace program begins concrete state represented given initial dataflow fact let sequence records due executes number nodes encounter upon read final last read encountered path pfsg first node ith node node ith node form last node form intuitively theorem states execution traces due record sequences accepted given input automaton paths taken traces present pfsg specific scenario pfsg used perform dataflow analysis theorem instantiated follows corollary let sound dataflow analysis framework based lattice let given dataflow fact program entry element lattice node original cfg let denote final solution computed using cfg using initial value consider pfsg obtained using given input automaton node pfsg let denote final solution computed byfd applied pfsg using initial value let precision set concrete states arise node program run input files conform soundness precision ordering among pfsgs clear discussion section pfsg produced approach given cfg input automaton fixed depends selected underlying abstract interpretation theorem given states matter abstract interpretation used pfsg sound elide paths executed due record sequences accepted long sound however precision pfsg depends precision given cfg input automaton define precision ordering set pfsgs obtained using different sound underlying domains etc pfsg said precise another pfsg every edge also present note implies every path also present theorem underlying domain consistent abstraction another underlying domain pfsg obtained using least precise pfsg obtained using implementation evaluation prog name acctran dtap clieopp loc cfg nodes automaton full automaton fig benchmark program details targeted implementation cobol batch programs prevalent large enterprises based variety standard well proprietary file formats another motivating factor choice one authors paper extensive professional experience developing maintaining cobol batch applications implemented analysis using proprietary program analysis framework prism implementation java use call strings approach precise analysis cobol programs use recursion therefore place apriori bound lengths analysis code primarily consists implementation generic analysis framework described section currently implemented extension data integrity constraints described section implemented pfsg construction approach section also lightweight scripts process solution emitted analysis compute results specialization problem well file conformance problem see section ran tool laptop intel ghz cpu ram benchmark programs used set eight programs benchmarks evaluation figure lists key statistics programs second third columns give sizes programs terms lines code including variable declarations terms number executable nodes cfg constructed prism program acctran toy program used running example previous paper example inventory management program used textbook showcase typical sequential file processing program program dtap developed authors paper payments validation program uses validation rules implements taken widely used standard specification program clieopp payment validation transformation program developed professional developer large consulting services company training purposes format validation rules uses another standard specification programs used bank validating reporting return payments sent branches bank programs major multinational financial services companies program format translator translates various kinds input records corresponding output records reads data sequential master file collects data required computing monthly interest fee account writes data various output files file formats used four programs proprietary columns figure give statistics automaton program programs acctran respective original sources programs also give expected input file formats real programs derived record types well automatons going programs guessing intended formats input files programs program maintainers provided file format specification case programs dtap clieopp constructed record types well automatons respective standard specifications cases employed creating automatons namely incoming transitions file state labeled type programs also constructed full automaton use context acceptance analysis created full automaton using corresponding automaton basis following basic procedure described section statistics full automatons presented last two columns figure create full automaton clieopp full automatons program turn large unwieldy specify instead used automatons place full automatons analysis cause potential unsoundness evaluate approach three different contexts effectiveness detecting conformance violations programs usefulness specializing programs ability improve precision standard dataflow analysis file format conformance checking first step experiment manually identified rejection points program actually task program idioms rejecting files programs wrote warnings messages log files others used system routines terminating program others used cobol keywords goback stop run furthermore since every instance warning output termination necessarily due file format related issues exercise care selecting instances prog file format conformance warnings name acceptance acceptance acctran dtap clieopp fig conformance checking results due issues also manually added summary functions analysis calls certain system routines terminate program summary functions treat calls returning dataflow value file states thus simulating termination experiment use constant propagation underlying analysis lifted approach figure summarizes results analysis program second column captures number instances file state automaton value rejection point basically warnings third column depicts number file states full automaton excluding final states original automaton reach final point main procedure value basically warnings running time analysis seconds less programs except large program analysis took seconds discussion results noteworthy aspect results four eight programs namely acctran dtap verified errors case clieop warnings turned true positives manual examination code contained programming errors cause rejection files also manually examined one program warnings although program textbook program follows complex idiom certain fields certain record types input file format program supposed contain values appear primary keys sorted persistent table accessed program however automaton created capture constraint hence overapproximated caused false warnings reported discussion results clear table implementation reports warnings programs numbers marked potentially lower really mentioned section actually use full automaton program uses sequential lookup persistent table idiom extension section support two programs manually examined four programs report findings warnings reported two programs dtap turned genuine input dtap similar one shown figure difference uses single state place qsh qdh program happens accept files contain batches header record trailer record occur without intervening item records violation specification case discussed warnings maintainers program agreed genuine however present another program runs standard workflow ensures files supplied case automatons overapproximated one challenging idiom also contributes imprecision routines emit warnings emit fileconformance warnings called certain kinds warnings called since currently scheme mark rejection points left routines unmarked conservative gesture program specialization program name acctran acctran dtap dtap dtap dtap clieopp clieopp criterion name deposit withdraw add change delete ddbank ddcust ctbank ctcust payments directdebit edit update form telex modified trancopy daccts maccts criterionspecific common nodes nodes fig specialization criteria results objective experiment evaluate effectiveness approach identifying program statements relevant given criteria specified specialization automatons experiment used underlying analysis ran tool multiple times program time different specialization criterion identified represents meaningful functionality perspective instance consider program input file program consists sequence request records request either add item inventory stored persistent table change details item inventory delete item inventory meaningful criterion program would one concerned add requests similarly change delete meaningful criteria figure summarizes results experiment row figure corresponds programcriterion pair third column figure indicates mnemonic name given criteria note criterion trancopy specializes program process one twelve kinds input record types done specialization criteria brevity report one figure trancopy results criterion sum numbers fourth fifth columns figure number cfg nodes determined analysis relevant criterion reached value file state specialization automaton instance acctrandeposit number relevant cfg nodes total nodes program see figure fifth column indicates number common nodes relevant criteria supplied fourth column indicates number nodes relevant corresponding individual criterion common criteria note case dtap show commonality across four criteria within two subgroups contains two related criteria also case common nodes depicted across twelve criteria notable programs commonality among statements relevant different criteria high statements specific individual criteria fewer number belief program comprehension setting ability developer separately view common code code would let appreciate better way processing logic underlies criteria manual examination manually examined output tool determine precision programs except difficult well logic made manual evaluation difficult surprise tool precise every criterion four remaining programs acctran clieop dtap fail mark unreachable cfg node actually unreachable per human judgment executions input files conformed given specialization automaton basically evidence specialization automatons conjunction underlying analysis sufficiently precise mechanism specialize programs remaining one program discussed section input automaton although specialized program contain extra statements ideally removed result actually turns precise relative given automaton precision improvement existing analyses discussed section scenarios one interested performing standard analyses program restricted paths taken runs files evaluate scenario implemented two analyses one possibly uninitialized variables analysis whose abstract domain call uninit wherein one wishes locate references variables either initialized initialized using computations turn refer possibly uninitialized variables second reaching definitions analysis whose abstract domain call ran two analyses two modes direct mode analysis run lifted mode analysis done lifting automaton lifted mode uninit used uninit underlying analysis used underlying analysis component required enable illustrated figure interest space summarize results uninit variable references labeled uninitialized direct mode whereas labeled lifted mode dtap analogous numbers programs lifted mode performed marginally better direct mode case total number edges computed lifted mode computed direct mode dtap clieopp programs reduction marginal numbers experiments large program domains mentioned yet scale programs sizes programs direct analyses took anywhere hundreds second seconds lifted analyses took anywhere tenths second seconds limited study programs lifted mode give significant benefit causes imprecision observed array references calls external programs handle conservatively confounding factors programs could offset precision improvement afforded input automatons discussion summary encouraged experimental results except two smaller programs acctran benchmark programs either real work real formats implement real specifications conformance checking program specialization two novel problems whose context evaluated tool tool verified four programs rejecting files found genuine related errors several programs tool precise program specialization context finally enabled improvement precision context uninitialized variables reaching definitions analysis four eight programs related work discuss related work broadly several categories analysis programs exists body literature work godefroid saxena representatives testing programs whose inputs described grammars regular expressions via concolic execution approaches suited bug detection high precision approach aimed conservative verification well program understanding transformation tasks various approaches proposed literature recover record types file types programs program analysis approaches complement potentially able infer input automatons programs situations file formats available report auguston shows decidability verifying certain kinds assertions programs program specialization blazy describe approach specialize fortran programs using constant propagation significant body literature technique partial evaluation sophisticated form program specialization involving loop unrolling arbitrary depths simplification expressions etc approaches typically support criteria fixed sized program inputs launchbury extend partial evaluation allow criteria data structures consel provide interesting variant partial evaluation wherein propose based framework specialize functional programs abstract values signs types ranges approach could potentially framed instantiation approach lifted lattice corresponding lifted transfer functions program slicing program slicing widely applicable software engineering tasks usually locate portion program relevant criterion constrained variants program slicing provide good precision general cost potentially expensive existing approaches constrained slicing specifically support constraints record sequences may appear input files programs lifted analysis described section enables sort slicing typestates rich body literature specifying using type states seminal work strom context analyzing programs automatons used capture state file open closed error knowledge first work space use automatons encode properties prefix records read file shape analysis shape analysis precise technique verifying shapes properties data structures high level data file similar list operations used traverse files data structures different knowledge shape analysis used literature model contents states files read programs would interesting topic future work explore feasibility approach conclusions future work presented paper novel approach apply given abstract interpretation program associated input basically enables approach elide certain paths program infeasible per file format hence enhance precision usefulness underlying analysis demonstrated value approach using experiments especially context two novel applications file format conformance checking program specialization key item future work allow richer constraints data input file persistent tables instance general logical constraints constraints expressing sortedness would useful many settings obtain enhanced precision usefulness also would like investigate techniques domains batch programs programs xmlprocessing programs applications references auguston decidability program verification achieved replacing equality predicate constructive one technical report new mexico state university blazy facon sfac tool program comprehension specialization proc ieee workshop program pages nov caballero yin liang song polyglot automatic extraction protocol message format using dynamic binary analysis proc acm conf computer comm security ccs pages canfora cimitile lucia conditioned program slicing information software technology apache common log format http html brain drain cobol systems computerworld may http consel hornof marlet muller thibault volanschi lawall partial evaluation software engineering acm comput consel khoo parameterized partial evaluation acm transactions programming languages systems toplas cousot cousot abstract interpretation unified lattice model static analysis programs construction approximation fixpoints proc acm symp principles programming languages popl pages cui peinado chen wang tupni automatic reverse engineering input formats proc acm conf comp comm security ccs pages das lerner seigle esp program verification polynomial time proc conf prog langs design impl pldi pages david krop clieop client orders file description http devaki kanade static analysis checking data format compatibility programs proc foundations softw tech theor comput science fsttcs pages driscoll burton reps checking conformance producer consumer proc foundations softw engg fse pages retail payment system rps deutsche bundesbank http field ramalingam tip parametric program slicing proc sym principles prog langs popl pages fischer jhala majumdar joining dataflow predicates acm sigsoft int symp foundations softw engg fse pages fisher walker pads project overview proc int conf database theory icdt pages godefroid kiezun levin whitebox fuzzing proc acm conf prog lang design impl pldi pages harman hierons fox danicic howroyd conditioned slicing proc int conf software maintenance icsm pages introduction standards http jones gomard sestoft partial evaluation automatic program generation prentice hall international khare saraswat kumar static program analysis large embedded code base experience proc india software engg conf isec komondoor ramalingam recovering data models via guarded dependences working conf reverse engg wcre pages launchbury project factorisations partial evaluation volume cambridge university press mine octagon abstract domain higher order symbol march murach prince menendez work sequential files murach structured cobol chapter murach sagiv reps wilhelm parametric shape analysis via logic proc symp principles prog langs popl pages saxena poosankam mccamant song symbolic execution binary programs proc int symposium softw testing analysis issta pages sharir pnueli two approaches interprocedural data flow analysis muchnick jones editors program flow analysis theory application prentice hall professional technical reference sinha ramalingam komondoor parametric process model inference working conf reverse engg wcre pages strom yemini typestate programming language concept enhancing software reliability software engineering ieee transactions united nations rules electronic data interchange administration commerce transport draft directory http dependent types practical programming phd thesis university qian zhang chen brief survey program slicing sigsoft softw eng notes mar appendix precision approach discuss precision approach alluded section detail function program points given program dataflow values lattice function program points functions domain set file states input automaton given solution program say precise iff program point note actually using precise shorthand equally precise precise let obtained program directly using underlying analysis first key result follows theorem input automaton set files states solution computed approach using using underlying analysis precise solution computed directly intuitively theorem captures fact results tracking different dataflow values lattice different file states causes increase precision note theorem touch upon soundness order ensure soundness would additionally need accept files files depending notion soundness sought different question naturally arises multiple candidate automatons accept set files give equally precise results used part analysis answer general turns also shown input automaton accepts smaller set files another automaton first automaton need necessarily give precise results second one programs fact precision linked set files accepted well structure automatons order formalize intuition first define formally notion precision ordering different solutions program using different input automatons solution said precise another solution iff program point file state exists file state define notion refinement among input automatons program say automaton refinement automaton iff exists mapping function transition labeled symbol exists one transitions furthermore label transitions either eof types constraint implies constraint input automaton refinement input automaton following two properties shown hold accepting state mapped accepting state set files accepted subset files accepted main result precision ordering input automatons follows theorem program given underlying analysis input automaton refinement input automaton solution computed approach using precise solution computed approach using important take away theorem choice two input automatons accept set files two different automatons accepting set files one refinement refined automaton give precision one refinement checking errors discuss procedure extend given automaton eof full automaton first create new type named none associate constraint lets cover records covered types original set types also add new file state automaton denote discussion let every state every type transition labeled add transition labeled finally add one new file state automaton make final state add eof transitions states state intuition behind construction accepts files accepts record sequences prefixes files

achievability performance bounds source coding elad domanovitz uri erez dec abstract source coding proposed method compression distributed correlated gaussian sources scheme encoder quantizes observation using fine lattice reduces result modulo coarse lattice rather directly recovering individual quantized signals decoder first recovers set judiciously chosen integer linear combinations quantized signals inverts observed method works well source covariance matrices present work quantifies measure bad covariance matrices studying probability source coding fails function allocated rate probability respect random orthonormal transformation applied sources prior quantization important case signals compressed correspond antenna inputs relays rayleigh fading environment orthonormal transformation viewed performed nature hence results provide performance guarantees distributed source coding via integer forcing scenario ntroduction source coding proposed scheme distributed lossy compression correlated gaussian sources minimum mean squared error distortion measure similar channel coding counterpart scheme encoders use nested lattice codebook encoder quantizes observation using fine lattice quantizer reduces result modulo coarse lattice plays role binning rather directly recovering individual quantized signals decoder first recovers set judiciously chosen integer linear combinations quantized signals inverts appealing feature source coding shared previously proposed practical methods coding distributed source coding problem inherent symmetry supporting equal distortion quantization rates potential application source coding distributed compression signals received several relays suggested explored similar channel coding source coding works well gaussian vector sources following approach present work quantify measure bad source covariance matrices considering randomized version source coding random unitary transformation applied sources prior quantization general transformation implies joint processing encoders note natural scenarios including distributed compression signals received relays rayleigh fading environment random transformation actually performed fact already empirically observed source coding performs well latter scenario rest paper organized follows section formulates problem distributed compression gaussian sources compound vector source setting provides relevant background source coding section iii describes randomly precoded source coding empirical performance section derives upper bounds probability failure function excess rate section deterministic linear precoding considered bound excess rate needed derived number sources correlation matrix spacetime precoding derived determinant codes used show bound significantly tightened case uncorrelated sources concluding remarks appear section roblem ormulation background section provide problem formulation briefly recall achievable rates source coding developed work domanovitz erez supported part israel science foundation grant heron consortium via israel ministry economy industry domanovitz erez department electrical engineering systems tel aviv university tel aviv israel email domanovi uri follows since left right singular vector matrices gaussian matrix equal eigenvector matrices wishart ensembles kkt respectively latter known uniformly haar distributed see chapter distributed compression gaussian sources start recalling classical problem distributed lossy compression jointly gaussian real random variables quadratic distortion measure specifically consider distributed source coding setting encoding terminals one decoder encoders access vector realizations random variable random vector corresponding different sources assumed gaussian zero mean covariance matrix kxx xxt encoder maps observation index using encoding function sends index decoder decoder equipped decoding functions upon receiving indices one terminal generates estimates vector achievable exist encoding functions decoding functions kxk focus symmetric case denote sum rate best known achievable scheme symmetric setting berger tung following general suboptimal sum rate achievable log det kxx rbt shown rbt lower bound achievable rate source coding refer rbt benchmark simplify notation note absorbed kxx hence without loss generality assume throughout compound source model scheme outage formulation consider distributed lossy compression vector gaussian sources kxx define following compound class gaussian sources value rbt via covariance matrix rbt kxx log det kxx rbt quantify measure set source covariance matrices considering outage events events sources integer forcing fails achieve desired level distortion even though rate exceeds rbt broadly given quantization scheme denote necessary rate achieve given covariance matrix kxx rscheme kxx given target rate rbt covariance matrix kxx rbt scheme outage occurs rscheme kxx quantify measure bad covariance matrices follow apply random orthonormal precoding matrix vector source samples prior encoding mentioned amounts joint processing samples hence problem longer distributed general nonetheless scenario described section certain statistical settings precoding operation redundant viewed performed nature applying precoding matrix source vector obtain transformed source vector pkxx covariance matrix follows achievable rate quantization scheme precoded source rscheme drawn random latter rate also random scheme outage probability defined turn pout scheme rbt time axis suppressed sequel vector notation reserved describe samples taken different sources rscheme rbt sup kxx rbt probability ensemble unitary precoding matrices considered gap benchmark sequel quantify tradeoff quantization rate equivalently excess rate rbt outage probability pout rbt defined source coding manner similar equalization channel coding applied problem distributed lossy compression approach based standard quantization followed binning however framework decoder first uses bin indices recovering linear combinations integer coefficients quantized signals recovers quantized signals purposes suffices state achievable rates source coding refer reader derivation proofs recall theorem stating covariance matrix kxx source coding achieve sum rate satisfying rif kxx log min max atk kxx rif kxx atk kxx det atk kth row integer matrix denote effective rate achieved equation matrix kxx symmetric positive definite therefore admits cholesky decomposition kxx fft rif kxx log min max kft notation det denote lattice spanned matrix problem finding optimal matrix equivalent finding shortest set linearly independent vectors denoting kth successive minimum lattice log rif kxx successive interference cancellation significantly improves achievable rate equalization channel coding analogous scheme implemented case source coding specifically given integer matrix let defined cholesky decomposition kxx llt denote mth element diagonal shown achievable rate successive source coding denote choice given kxx max kxx kxx finally optimizing choice obtain kxx min det log kxx gap rif even rbt quite small covariance matrices nevertheless arbitrarily large next quantify measure bad covariance matrices considering source coding iii recoded ource oding mpirical erformance recalling slight abuse notation rate source coding given precoding matrix denoted rif kxx rif pkxx log min max atk pkxx kxx udut pkxx pudut det since kxx symmetric allows orthonormal diagonalization unitary precoding applied quantify measure bad sources consider precoding matrices uniformly haar distributed group orthonormal matrices matrix ensemble referred circular real ensemble cre defined unique distribution orthonormal matrices invariant left right orthonornal transformations given random matrix drawn cre orthonormal matrix equal distribution since put equal distribution cre precoding sake computing outage probabilities may simply assume also drawn cre specific choice integer vector define slight abuse notation rif log atk udut log correspondingly rif log min max det let lattice spanned may rewritten rif log let denote set diagonal matrices value rbt rbt det may thus rewrite outage probability source coding defined pout rbt sup rif rbt rbt probability respect random selection drawn cre illustrate performance present empirical performance case twodimensional compound gaussian source vector outage probability computed via simulation figure depicts results different values rbt rather plotting outage probability complement depicted plot probability rate falls rbt seen figure outage probability function converges limiting curve rbt increases figure depicts results single high rate rbt required compression rate required support given outage probability constraint marked several outage probabilities observe outage probability gap bits bits per source required empirical outage empirical outage empirical outage empirical outage empirical outage empirical outage empirical outage empirical outage fig empirical results complement outage probability source coding applied compound gaussian source vector function various values rbt outage probability gap bits bits per source required outage probability gap bits bits per source required pper ounds utage robability recoded nteger orcing ource oding section develop achievability bounds source coding derivation much along lines results analogous problem channel coding developed refer results latter many points next lemma provides upper bound outage probability precoded source coding function rbt rate gap well number sources dmax defined denote kak lemma gaussian sources rbt drawn cre rif rbt rbt dmax dmax max set defined defined rbt empirical outage fig zoom empirical outage probability compound gaussian source vector rbt proof let denote dual lattice note spanned matrix successive minima related theorem max denoting hermite constant tightest known bound hermite constant derived since increasing function follows smaller combining latter exact values hermite constant dimensions known define otherwise therefore may bound achievable rates via dual lattice follows log rif hence rif rbt rbt log rbt denote rbt wish bound equivalently wish bound given matrix rbt note event equivalent event applying union bound yields note whenever dmax therefore substituting set relevant vectors follows rest proof follows footsteps lemma given appendix lemma provides explicit bound outage probability order calculate one needs diagonal matrices rbt diagonal matrix sum relevant integer vectors hence bound evaluated moderate compression rates small number sources following theorem may viewed counterpart theorem provides looser yet simple bound another advantage bound depend achievable rate theorem sources rbt drawn cre rif cmax cmax note constant depends number sources proof see appendix similarly case channel coding section analyzing theorem reveals two main sources looseness may tightened union bound inherent loss union bound fact terms summation may completely specifically using corollary set appearing summation may replaced smaller set kak dual lattice bounding via dual lattice induces loss reflected may circumvented case source vector using accomplished lemma theorem present next lemma gaussian source vector rbt drawn cre rbt dmin min dmin min dmin defined theorem gaussian source vector rbt drawn cre proof see appendix figure depicts bounds derived well results monte carlo evaluation case gaussian cre calculating empirical curves lemmas assumed high quantization rates rbt lemmas calculated going grid values satisfying application distributed compression cloud radio access networks since described source coding well precoding reals outline application source coding cloud radio access network scenario assuming real channel model comment adaptation scheme realistic scenario complex channel consider scenario depicted figure transmitters send data modeled gaussian source vector mimo broadcast channel data received receivers relays wish compress forward processing decoding central node via noiseless bit pipes wish minimize distortion central node subject rate constraints distributed lossy source coding problem see depiction figure covariance matrix received signals relays given kxx snrhht note absorb snr channel hence set snr kxx hht assume entries channel matrix gaussian mentioned introduction svd matrix similar derivation section simple factor deduced regardless rate number sources noting result outcome hence need account cases bounds computed applying factor lemmas accordance footnote rather plotting outage probability plot complement empirical outage empirical outage thm lemma thm lemma thm thm fig upper bounds outage probability source coding various values source dimension optical distribution network managing control unit fig cloud radio access network communication scenario two relays compress forward correlated signals receive several users satisfies belong cre may therefore express random covariance matrix kxx drawn cre follows precoding matrix redundant assumed also drawn cre thus analysis holds also considered scenario specifically assuming encoders subject equal rate constraint given distortion level relation compression rate source coding guaranteed outage probability meeting prescribed distortion bounded using theorem note precoded channel coding applied complex channels described precoded source coding extended complex gaussian sources describing outage event case assume precoding matrix drawn circular unitary ensemble cue bounds derived replacing derivations relation compression rate source coding outage probability hold scenario complex gaussian channels cue precoding viewed performed nature erformance uarantees nteger orcing ource oding eterministic recoding section consider performance source coding used conjunction judiciously chosen deterministic precoding performance measured stricter sense previous sections namely outage allowed achieving goal outage case general gaussian sources requires performing precoding whereas case parallel independent sources precoding still suffices note away outage events desirable come price namely precoding assumed section requires joint processing different sources prior quantization thus precluded distributed setting hand precoded scheme considered section several advantages respect traditional source coding correlated sources namely traditional approach requires utilizing statistical characterization source encoder side via prediction filter via applying transformation dct dft along bit loading contrast considered section applied applying universal transformation transformation independent source statistics addition samples quantized bit rate knowledge statistics source needs course utilized decoder side propertry may advatageous certain applications similar case compression whereas latter requires general use bit allocation unless source stationary source coding additive bound general sources similar case channel coding derive additive bound gap benchmark achieving guaranteed performance requires joint algebraic number theoretic based precoding encoders following theorem due ordentlich theorem ordentlich sources covariance matrix kxx benchmark rbt excess rate respect benchmark normalized per number used source coding nvd precoding matrix minimum determinant bounded rif rbt log log proof see appendix note similarly case channel coding gap benchmark large thus limited applicability uncorrelated sources special case uncorrelated gaussian sources much tighter bounds comparison theorem quantization rate given distortion level may obtained first precoding may replaced precoding space allows obtain tighter counterpart theorem derived section next derive yet tighter performance guarantees also following ideas developed numerically evaluating performance source coding densely quantized set source diagonal covariance matrices belonging compound class bounding excess rate evaluated ones possible source vector compound class case uncorrelated sources covariance matrix kxx diagonal hence becomes kxx denote diag compound set channels may parameterized rbt log rbt note may associate diagonal element rate corresponding individual source log thus compound class sources may equivalently represented set rates rbt rbt define quantized set follows interval rbt divided length rbt thus resolution determined parameter quantized belong grid rbt rbt rbt may similarly define quantized set rbt diagonal matrices diagonal entries satisfy rbt theorem gaussian vector independent sources covariance matrix rbt rate source coding given precoding matrix upper bound rif max rif log rbt rbt rbt proof assume covariance matrix compound class hence associated satisfies rbt assume without loss generality rate source coding associated specific integer linear combination vector rif log denote need following two lemmas whose proofs appear appendices respectively lemma diagonal covariance matrix associated rbt exists rbt diagonal covariance matrix associated defined lemma consider gaussian vector diagonal covariance matrix let integer vector rif log rif using lemma denoting covariance matrix associated whose existence guaranteed lemma follows rif log log rif using lemma rif log rif combining obtain rif rif log rif recalling rif log min max rif det denoting optimal integer matrix quantized channel necessarily optimal matrix follows rif rif assuming atk rows rif max rif max log rif log rif rif rif log thus conclude therefore rbt rif max rif log concludes proof theorem example achievable performance show fig gives empirical performance two three real sources achieved using source coding fixed precoding matrix grid rbt precoding matrix taken explicit precoding matrix used two sources three sources rif rather plotting gap benchmark plot efficiency ratio rate source coding rbt also plot upper bound rif given theorem cyclo precoding bound performance cyclo precoding bound performance rbt fig empirical guaranteed upper bound efficiency source coding two three uncorrelated gaussian sources using precoding matrices given respectively taking calculation theorem ppendix roof heorem following footsteps lemma adopting geometric interpretation described may interpret ratio surface area ellipsoid inside ball radius surface kak area entire ellipsoid axes ellipsoid defined denote vector okak vector drawn cre norm kak using lemma since assume drawn cre right hand side equal okak capell capell capell kakk kak kak dmax substituting obtain capell capell max recalling see rbt obtain rbt dmax ppendix roof heorem establish theorem follow footsteps proof theorem obtain dmax defined respectively noting kak summation bounded dmax apply lemma bound number integer vectors contained ball given radius using bound noting kak exactly integer vectors right hand side may bounded rbt vol dmax max note trivially holds left hand side evaluates zero since empty set case hence right hand side rewritten rbt vol dmax max iii search independent since follow iii note since assuming max max max thus follow max iii max max max explicit derivation appears appendix replaced obtain also shown appendix holds observe since implies hence indeed recalling denoting cmax follows iii cmax using substituting volume unit ball vol follows right hand side upper bounded rbt cmax dmax cmax finally substitute defined obtain rbt cmax rbt cmax cmax defined ppendix roof heorem first recall theorem minkowski theorem upper bounds product successive minima theorem minkowski lattice spanned full rank matrix det prove theorem need following two lemmas lemma gaussian source vector covariance matrix kxx rbt integer matrix satisfies kxx rbt log det kxx defined proof kxx log log log det llt log det kxx rbt log det theorem shows successive used channel coding loss terms achievable rate restricting class unimodular matrices note claim holds also framework successive source coding replacing matrix spanning lattice defined theorem snrhht ggt defined noting optimal expressed cases aopt min max det diagonal elements corresponding matrix derived cholesky decomposition case established optimal unimodular follows kxx aopt suc rbt ready prove following lemma analogous theorem lemma gaussian source vector covariance matrix kxx rbt optimal integer matrix upper bounded rif kxx rbt log kxx rate mth equation corresponding mth row defined proof log log log det rbt log inequality due theorem minkowski theorem rif kxx case two sources lemma kxx kxx rbt kxx rbt kxx kxx kxx rbt equivalently lemma note optimal integer matrix used general different optimal matrix used nonetheless applying one decodes first equation lowest rate since equation suc effect follows first row cases hence kxx kxx since source decoded first follows hence kxx rbt therefore kxx max kxx kxx rbt max rbt kxx max rbt kxx henceforth analyze outage rbt target rates smaller rbt inequality kxx rbt satisfied thus consider excess rate values satisfying goal bound kxx rbt kxx rbt rbt kxx log let wish bound equivalently given matrix corresponding kxx via relation note event equivalent event applying union bound yields using derivation lemma get rbt dmin min since analyzing case two sources rbt dmin min applying similar argument appears appendix part proof theorem noting implies cmax get rbt rbt finally substituting defined obtain rbt rbt ppendix dditive ase ound nvd pace ime recoded ources combining precoding integer forcing context channel coding suggested next briefly recall present necessary modifications case source coding derive additive bound using unitary precoding matrix satisfying determinant nvd property theory nvd codes developed complex field prove convenient employ complex precoding matrices end may assume stack samples two time slots samples stacked first time slot represent real part complex number samples stacked second time slot represent imaginary part complex number hence kxx kronecker product note benchmark stacked source vector rbt kxx next order allow precoding stack times complex outputs sources let denote effective source vector corrlation matrix refer effective covariance matrix takes form assume precoding matrix principle either deterministic random applied effective source vector analyze performance case precoding matrix pst deterministic specifically precoding matrix induced perefect block code operating stacked source covariance matrix given explanation extract precoding matrix code found section denote corresponding precoding matrix reals pst denote pst kph assume precoding matrix unitary benchmark normalized total number time extensions used remains unchanged log det pst kph log det rbt noted assume generating matrix perfect code employed precoding matrix code called perfect full rate satisfies determinant nvd condition code generating matrix unitary let denote minimal determinant code codes use minimal number time extensions possible thus total stacked complex samples subsisting dimension number real samples jointly processsed rate source coding samples given rif pst log bound note every lattice using minkowski theorem theorem appendix follows det hence rate source coding normalized number time extensions bounded rif pst log log det log det log log log rbt log next use results derived channel coding using nvd precoding note since covariance matrix positive may written kxx hht covariance matrix stacked source vector may written may take many choices choice corresponds channel matrix viewed real representation complex channel matrix present case real imaginary part context effective covariance matrix similarly rewritten hht using channel coding terminology define minimum distance receiver dmin dmin min khak qam setting snr real representation lemma states min hpst min snrdmin hpst using corollary get hpst min cwi min min cwi min min cwi min cwi log det mutual information since cwi rate real matrix resulting complex matrix equals rbt defined hence obtain min hph min turn yields min pst hht ptst min finally plugging bound arrive rif pst rbt log log log rbt log log ppendix roof emma gaussian source component specific rate transformed different gaussian source rate appropriate scaling specifically scaling source component results parallel uncorrelated sources variances therefore log log log log log associate rate tuple rbt rate tuple rbt according following transformation follows scaling factor needed achieve bounded rbt rbt follows since readily verified function denoting follows rbt monotonically increasing rbt hence trivially holds since thus lemma follows observing follows well ppendix roof emma recalling note written rif log therefore scaling gaussian input vector factor rif log log log rif log rif eferences ordentlich erez source coding ieee transactions information theory vol aguerri guillaud integer forcing conversion massive mimo systems signals systems computers asilomar conference ieee domanovitz erez outage behavior integer forcing random unitary ieee transactions information theory vol edelman rao random matrix theory cambridge university press vol tung multiterminal source coding thesis information theory ieee transactions vol november fischler universal precoding parallel gaussian channels master thesis tel aviv university tel aviv online available https nazer source coding successive cancellation duality information theory isit ieee international symposium ieee mehta random matrices statistical theory energy level academic press lagarias lenstra schnorr bases successive minima lattice reciprocal lattice combinatorica vol blichfeldt minimum value quadratic forms closest packing spheres mathematische annalen vol ordentlich private communication oggier viterbo algebraic number theory code design rayleigh fading channels foundations trends communications information theory vol ordentlich erez simple proof existence good pairs nested lattices ieee transactions information theory vol micciancio goldwasser complexity lattice problems cryptographic perspective springer science business media vol ordentlich erez nazer successive optimality corr vol online available http approximate sum capacity symmetric gaussian interference channel ieee transactions information theory vol domanovitz erez combining block modulation integer forcing receivers electrical electronics engineers israel ieeei ieee convention nov ordentlich erez precoded universally achieves mimo capacity within constant gap information theory ieee transactions vol jan oggier rekaya belfiore viterbo perfect block codes ieee transactions information theory vol elia kumar pawar kumar feng explicit space time codes achieving diversity multiplexing gain tradeoff information theory ieee transactions vol sept

coding convex splitting private communication quantum channels aug mark march abstract wiretap channel communication model involving classical sender legitimate quantum receiver quantum eavesdropper goal private communication protocol uses channel sender transmit message way legitimate receiver decode reliably eavesdropper learns essentially nothing message transmitted private capacity wiretap channel equal maximum number bits transmitted channel privacy error larger present paper provides lower bound private classical capacity exploiting recently developed techniques anshu devabathini jain warsi called coding convex splitting lower bound equal difference hypothesis testing mutual information alternate smooth lower bound leads lower bound coding rate private classical communication memoryless wiretap channel introduction among many results information theory ability use noise wiretap channel purpose private communication stands one great conceptual insights classical wiretap channel modeled conditional probability distribution sender alice access input channel legitimate receiver bob access output eavesdropper eve access output goal private communication alice bob use wiretap channel way alice communicates message reliably bob time eve able determine message transmitted author proved mutual information difference max achievable rate private communication wiretap channel alice bob allowed use many independent times since interest wiretap channel waned many increasingly refined statements achievable rates private communication wiretap channels hearne institute theoretical physics department physics astronomy center computation technology louisiana state university baton rouge louisiana usa many years contribution protocol quantum key distribution developed proposal private communication quantum channel quantum information theory started becoming field right many researchers revisited several known results shannon information theory quantum lens merely academic exercise revealed remarkable improvements communication rates could attained physical channels practical interest strategies exploited one important setting revisited wiretap channel quantum case simplest extension classical model given wiretap channel abbreviated wiretap channel described following map classical symbol alice input channel joint output quantum state bob eve system represented density operator acting tensorproduct hilbert space bob eve quantum systems goal private communication wiretap channel similar classical wiretap channel however case bob allowed perform collective quantum measurement output quantum systems order determine alice message time would like difficult eve figure anything transmitted message even access quantum computer memory store quantum systems receives channel output authors independently proved quantum generalization formula achievable rate private communication quantum wiretap channel alice bob allowed use many independent times namely proved following holevo information difference achievable rate max information quantities formula holevo information bob eve respectively formally defined later present paper since developments increasing interest quantum information community determine refined characterizations communication tasks strongly motivated fact experimentally difficult control large number quantum systems practice one access finite number quantum systems anyway one scenario interest discussed quantum wiretap channel hitherto work offering achievable oneshot rates private communication wiretap channels however work consider bounding coding rate private communication wiretap channel main contribution present paper lower bound private capacity wiretap channel namely prove mpriv represents maximum number bits sent mpriv alice bob using wiretap channel privacy error defined formally later exceed quantities side inequality particular generalizations holevo information bob eve defined later worthwhile note information quantities computed using programming computational runtime polynomial dimension channel thus channels reasonable dimension quantities efficiently estimated numerically constants chosen substituting independent identically distributed wiretap channel side inequality using expansions holevo informations picking find following lower bound coding rate private classical communication mpriv log represents maximum number bits sent mpriv alice bob using wiretap channel times privacy error exceed holevo informations make appearance term proportional number channel uses side second order term proportional consists quantum channel dispersion quantities defined later additionally feature inverse cumulative gaussian distribution function thus bound leads lower bound coding rate comparable bounds appeared classical information theory literature prove bound use two recent remarkable techniques coding convex splitting main idea coding conceptually simple communicate classical message alice bob allow share quantum state communication begins number messages bob possesses systems alice systems alice wishes communicate message sends mth system channel reduced state bob systems nam nam quantum channel reduced state systems product state however reduced state systems generally correlated state bob binary measurement distinguish joint state product state sufficiently well base decoding strategy scheme reliable long number bits communicated chosen roughly equal mutual information known hypothesis testing mutual information exactly used coding authors thus forged transparent intuitive link quantum hypothesis testing communication case communication convex splitting rather intuitive well thought dual coding scenario mentioned suppose instead alice bob means generating state perhaps strategy mentioned suppose alice chooses variable uniformly random state perspective someone ignorant choice following mixture lemma guarantees long roughly equal mutual information known alternate smooth information state nearly indistinguishable product state coding convex splitting used recently effectively establish variety results quantum information theory present paper use approaches conjunction construct codes wiretap channel main underlying idea follows original approach allowing message variable local key variable local randomness latter selected uniformly random used confuse eavesdropper eve communication begins alice bob eve allowed share copies common randomness state think copies partitioned blocks contain copies state alice wishes send message picks uniformly random sends system wiretap channel long roughly equal bob use decoder hypothesis testing mutual information figure long roughly equal alternate smooth lemma guarantees overall state eve information iemax systems regardless message chosen nearly indistinguishable prodk uct state thus scheme bob figure eve figure anything intuition behind coding scheme gives sense achievable number bits max sent privately alice bob main purpose present paper develop details argument furthermore show scheme derandomized copies common randomness state fact necessary rest paper proceeds follows section review preliminary material includes several metrics quantum states pertinent information measures section develops coding approach communication channels coding developed highlight different approach communication show section approach used shared communication also show therein derandomize codes case shared randomness actually necessary classical communication channels section represents main contribution present paper lower bound private classical capacity wiretap channel last development section show lower bound leads lower bound coding rate private classical communication memoryless wiretap channel therein also show lower bounds simplify wiretap channels using binary keying coding strategy private communication bosonic channel section concludes summary open questions future work preliminaries use notation concepts standard quantum information theory point reader background rest section review concepts less standard set notation used later paper trace distance fidelity purified distance let denote set density operators acting hilbert space let denote set subnormalized density operators trace exceeding one acting let denote set positive operators acting trace distance two quantum states equal operator direct operational interpretation terms distinguishability states prepared equal probability task distinguish via quantum measurement optimal success probability equal fidelity defined generally use formula define uhlmann theorem states max ira ira ira fixed purifications respectively optimization respect unitaries statement holds generally fidelity invariant respect isometries monotone respect channels sine distance two quantum states defined proven metric later name purified distance shown metric subnormalized states via embedding following inequality relates trace distance purified distance relative entropies variances quantum relative entropy two states defined whenever supp supp equal otherwise quantum relative entropy variance defined whenever supp supp hypothesis testing relative entropy states defined inf entropy states defined dmax inf smooth entropy states parameter defined dmax inf following expansions hold max evaluated states log dmax log expansion features cumulative distribution function standard normal random variable exp inverse defined sup mutual informations variances quantum mutual information information variance bipartite state defined paper exclusively interested case system classical written probability distribution orthonormal basis set quantum states hypothesis testing mutual information defined follows bipartite state parameter smooth entropy one define mutual quantity state follows dmax note following expansions direct consequence definitions log dmax log another quantity related follows iemax inf dmax recall relation lemma quantities giving slight modification useful purposes lemma state following inequality holds imax dmax proof see recall claim states exists state dmax dmax let denote optimizer dmax taking find exists state dmax dmax triangle inequality purified distance conclude since quantity side includes optimization states satisfying conclude inequality operator inequality key tool analyzing error probabilities communication protocols operator inequality given operators following inequality holds lemma lemma key tool used recent developments quantum information theory state variant lemma helpful obtaining bounds privacy ensuing lower bound coding rate proof closely follows proofs available slight differences completeness appendix contains proof lemma lemma convex split let state let following state let iemax state public classical communication definition classical capacity begin defining classical capacity channel write channel fully quantum form following quantum channel orthonormal basis let classical communication code consists collection probability distributions one message decoding positive measure povm refer side inequality decoding error orthonormal basis define state measurement channel equality follows direct calculation ihm ihm given channel classical capacity equal mpub largest satisfied fixed mpub one allow shared randomness alice bob communication begins case one obtains shared randomness assisted capacity channel could allow decoding povm consisting extra operator needed lower bound classical capacity first consider protocol randomness assisted public classical communication goal alice use channel send one messages error probability larger next section shows derandomize shared randomness needed main result section lower bound classical capacity channel although result already known development section important building block wiretap channel result section full detail sake completeness also approach given uses decoding channel fix probability distribution channel input alphabet consider following state think representing shared randomness let denote following state results sending system channel coding scheme works follows let alice bob share copies state shared state alice systems labeled bob systems labeled alice would like communicate message bob simply sends system channel case reduced state bob follows related first one permutation observe state systems unitary representation permutation bob way distinguishing joint state product state high probability able figure message communicated let txb denote test measurement operator satisfying txb ixb think identifying high probability complementary operator ixb identifies highest probability subject constraint txb test form following measurement operator txm ixm think test figure whether reduced state systems observe message operator related first one permutation systems message transmitted measurement operator acts probability accepting txb however measurement operator acts probability accepting txb measurement operators form measurement follows message operator related first one permutation systems called decoder analyzed case entanglementassisted communication error probability coding scheme follows message irm error probability fact message due observations irm irm irm irm let analyze error probability first message applying operator inequality cii find error probability bounded irm cii ixb txb cii trb ixb txb cii txb consider hypothesis testing mutual information inf take test txb bob decoder optimal measurement operator error probability bounded ixb cii cii pick get last line indeed consider would like cii rewriting find satisfy cii picking implies algebra cii quantity represents lower bound randomnessassisted classical capacity channel bound holds average error probability maximal error probability coincidence due protocol assistance shared randomness lower bound classical capacity show derandomize code main result section following lower bound shot classical capacity channel holding mpub note although result already known development section important building block wiretap channel result section stated previously approach given uses decoding channel reasoning previous section following bound average error probability code let analyze expression definition follows implies also recall optimal consider qxb define qxb similarly qxb demonstrates suffices take optimal measurement operator qxb qxb defined achieve optimal value taking consider ixm ihxm qxbm ixm qxbm implies qbm qbm qbm qbm qxbm qbm completed observe povm support povm full space adding employing find average error probability follows last line usual shannon trick exchanging average messages expectation random choice code employing bound find exists particular set values sequence constitutes codewords corresponding povm used decoder number bits code transmit equal shared randomness required code derandomized remark achieve maximal error probability one remove worst half codewords lower bound achievable number bits private classical communication definition private classical capacity suppose alice bob eve connected cqq channel following form bob system eve system fully quantum version channel follows orthonormal basis define private classical capacity following way let private communication code consists collection probability distributions one message decoding povm refer side inequality privacy error orthonormal basis state state define state measurement channel given channel private classical capacity equal mpriv largest satisfied fixed mpriv condition combines reliable decoding security conditions single average error criterion see represents generalization error criterion public classical communication channel one could different definition private capacity two separate criteria approach beneficial purposes case code satisfying satisfies two separate criteria well easily seen invoking monotonicity trace single error criterion private capacity approach taken latter paper shown notions asymptotic private capacity equivalent using either single error criterion two separate error criteria lower bound private classical capacity main result section following lower bound shot private capacity wiretap channel holding mpriv begin allow alice bob eve shared randomness following form bob system alice system eve system natural let eve share randomness well amounts giving knowledge code used let denote state resulting sending system channel coding scheme alice bob use follows message local key local key represents local uniform randomness alice accessible bob eve assume alice bob eve share copies state communication begins denote state indeed starting applying monotonicity trace distance partial trace system get recalling interpret asserting decoding error probability exceed considering partial trace system implies security criterion get conventional two separate criteria satisfied code satisfies single privacy error criterion send message alice picks uniformly random set sends system channel thus chosen reduced state bob eve systems state bob systems xmk bob decode uses decoder decode message local key let denote decoding povm reasoning section long following bound holding xmk xmk defined sections reasoning section also xmk write defined section define following measurement channels clear consider ihk ihk averaging quantity applying condition get applying convexity trace distance bring average inside monotonicity respect partial trace system side find let define state trb consider write using observations finally write ihm ihm combining development implies consider state eve systems analysis privacy fixed state xmk simplicity notation follows labeling systems however chosen uniformly random conditioned message fixed state eve systems follows xmk would like show state invariance trace distance respect states find lemma relation trace distance purified distance find pick guaranteed state consider rewrite applying find putting together find triangle inequality gives achievable shared randomness difference average maximal error shared randomness allowed show derandomize code take average messages find conclude exist particular values thus final conclusion number achievable bits sent privacy error larger equal asymptotics private classical communication section show lower bound private capacity leads lower bound coding rate private communication wiretap channel also show bounds simplify wiretap channels using binary keying coding strategy private communication bosonic channel applying lemma take still achieving performance substituting wiretap channel bounds evaluating case using expansions max taking sufficiently large get mpriv log example wiretap channel let consider applying inequality wiretap channel following form classical input leads pure quantum state bob pure quantum state eve channel may seem bit particular discuss next section one induce channel practically relevant channel known bosonic channel order apply inequality channel fix distribution input symbols leading following state well known straightforward calculate following simplifications occur denotes quantum entropy state proposition demonstrates similar simplification occurs information variance quantities special case wiretap channel employing find following lower bound coding rate wiretap channel mpriv log defined proposition let state corresponding ensemble holevo information variance equal entropy variance expected state takes special form following equality holds proof state consider furthermore holds eigenvectors direct calculation observe projection onto space orthogonal find furthermore direct calculation ihx equality used expansion fact orthogonal finally putting together conclude example bosonic channel induce wiretap channel bosonic channel follows consider coding scheme called binary keying bpsk let recall basic facts needed gaussian quantum information support argument follows curious reader consult details channel transmissivity sender inputs coherent state outputs bob eve coherent states respectively note overlap two coherent states equal fact main quantity need evaluate information quantities average photon number coherent state equal scheme induces following wiretap channel channel sender would like transmit symbol prepares coherent state input physical channel prepares coherent state bob eve similar explanation holds sender inputs symbol scheme distribution unbiased equal probability pick selecting codewords thus expected density operators output bob eve respectively follows straightforward computation reveals eigenvalues function overlap equal similarly eigenvalues given immediately plug find lower bound coding rate private communication bosonic channel mpriv log respectively denote binary entropy binary entropy variance benchmark compare performance bpsk code private capacity bosonic channel given figure plots normal approximation lower bound coding rate bpsk coding various parameter choices comparing asymptotic performance bpsk actual private capacity normal approximation consists terms besides log term typically serves good approximation capacity even small values necessarily valid previously observed conclusion paper establishes lower bound private classical capacity wiretap channel turn leads lower bound coding rate private communication wiretap channel main techniques used decoding order guarantee bob decode reliably convex splitting guarantee eve determine message alice transmitted opinion two methods represent powerful approach quantum information theory already used effectively variety contexts future work would good improve upon lower bounds given extensions methods might helpful endeavor note completion results present paper naqueeb warsi informed author unpublished result establishes lower bound private capacity wiretap channel terms difference hypothesis testing mutual information smooth information acknowledgements grateful anurag anshu saikat guha rahul jain haoyu qingle wang naqueeb warsi discussions related topic paper acknowledge support office naval research national science foundation private communication rate private communication rate normal approximation bpsk asymptotic private capacity normal approximation bpsk asymptotic private capacity number channel uses normal approximation bpsk asymptotic private capacity normal approximation bpsk asymptotic private capacity private communication rate private communication rate number channel uses number channel uses number channel uses figure figures plot normal approximation bpsk private communication using asymptotic limit bpsk asymptotic private capacity various values channel transmissivity mean photon number proof lemma sake completeness appendix features proof lemma let optimizer iemax inf dmax take marginal define following state think approximation fact good approximation small consider joint concavity root fidelity turn implies inequality definition purified distance fact imply let probability distribution set states following property holds quantum relative entropy state supp supp applying follows first term side equality simplifies lower bound last term consider partial trace systems gives equality follows thus averaging inequality implies putting together find definition means important property quantum relative entropy applying side get well known inequality imply turn implies pick inf dmax guaranteed triangle inequality purified distance get concludes proof references anurag anshu vamsi krishna devabathini rahul jain quantum message compression applications february anurag anshu rahul jain naqueeb ahmad warsi one shot entanglement assisted classical quantum communication noisy quantum channels hypothesis testing convex split approach february charles bennett gilles brassard quantum cryptography public key distribution coin tossing proceedings ieee international conference computers systems signal processing pages bangalore india 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implementation travelling salesperson problem using genetic algorithm comparative study python php ruby aryo novanto fajar program teknologi informasi dan ilmu universitas brawijaya malang indonesia aryo yudistira world connected internet abundance internet users connected web popularity cloud computing research need artificial intelligence demanding research genetic algorithm optimization method natural selection genetic evolution utilized many applications web mining load balancing routing scheduling web service selection hence challenging task discover whether code mainly server side web based language technology affects performance travelling salesperson problem tsp non problem provided problem domain solved many scientists prefer python implementation another popular interpreter programming language php php hypertext preprocessor ruby benchmarked line codes file sizes performances based implementation runtime found varies among programming languages based result use ruby implementation recommended genetic algorithm language introduction refers intelligent behaviour unexceptionally webbased application combination web application becoming future trends application moreover trend cloud computing already risen many applications use various purposes well known algorithm use heuristic approach gather fully optimized solutions used widely various web applications search engine web mining application become effective algorithm terms pattern recognition recently found new trend like social graph technology optimization promising scientific point view data processing analysis scripts often time consuming require many hours computed computer device iteration process along debugging process longer moreover scientists different focus work compared professional programmers keen methodology rather tools utilizing faster completion programming task surely dreamed many scientists even beginner programmers research supported program teknologi informasi dan ilmu komputer ptiik universitas brawijaya consideration effective naturally choose kind programming language also based psychological review narrated catch rapidly progressive research speed simplicity become necessary programming therefore needed give scientists pls could quickly iterate preserve tidiness simplicity easily used even though compiled faster run time interpreter simple recent emerging python php ruby advantages using python basically lie ease use interpreted object oriented programming language bridge many scientists need without loosing sense object oriented style however effectiveness measured many lines code written much syntaxes initiated implement another drawback using compiled looking denial service type attacks computational many researches use benchmarking purposes paper benchmarks interpreter supporting problem domain solved case travelling salesperson problem tsp become benchmark several heuristics performance test use tsp varies based domain problems tsp problem solution optimized based natural selections genetic evolutions solve complete domain problems tsp idea finding route given number cities visiting city exactly return starting city length route minimized path visits every city returns starting city creating closed circuit called route simplest direct method solve tsp number cities enumerating every possible route calculating every route length choosing one route shortest length possible every city may become starting point route one route may length regardless direction route taken logically problem set using integer distance every pair cities represented matrix possible route represented permutation number cities thus number possible routes factorial turns computationally burdened difficult enumerate find length every route may become larger polynomial complexity hence tsp needs algorithm able find route produces minimum length without enumerate possible routes cities given genetic algorithm basically search algorithm used find solutions evolution search space solve problems tsp end solution called individual set solutions normally called population individuals evolving many generations populations important thing must initiated genetic representation solution domain fitness function order evaluate quality individuals moreover selection stochastic function selects individuals based fitness value generated population yields called individual even though utilizes crossover mutation steps define end iteration every generation probability obtain better individuals ancestors terms fitness cost solving tsp problem using optimal depends crossover mutation method used research crossover crossover step produce various chromosomes individuals chosen chromosomes previous generation research selecting two best chromosomes best fitness cost crossover crossover steps randomly picking gene point chromosomes finally pick rest remaining genes randomly unique genes picked new population later generated previously selected chromosomes therefore recombination process two parent individuals generate new chromosomes example producing individual two individuals parents represented using crossover explained abcdefgh efghabcd gene new individual taken parent genes crossover process done generates new individual like following example aebcgfdh mutation mutation extension crossover process executed probability rate used avoid local optimum mating process depends crossover probably yields local optimum chance approach fitness value relatively high mutation process given cover problem way reach global optimum solution mutation method used research done choosing gene index switch pointed gene first index along rest genes follows sequence example offspring produced crossover process abcdefgh pointed gene mutation resulting mutation offspring efghabcd iii interpreted languages pls related programmable dynamic environment components bound together high level emergence compiled languages java led world wide web www transformation however leave drawbacks parsed codes produced compilers stored scattered files hence compiled environment files must included instruction codes parsed codes header files executable linked codes altogether nonetheless significant distinction compilers interpreters compilers parse execute different actions sequentially though speed standalone executable existence compiled increasing complexity process nevertheless many recent compilers able compile execute code directly memory giving interpreted language fashion use interpreted beneficial since interpreted environment every instruction executed right away parsed eased programmers since result presented quickly moreover interpreted code run without compilers linkers produce executable codes however disadvantages related poor performance executable program interpreter dependence compared compiled ils php python ruby merely made hobbyists without research goal lisp widely applicable interpreter support many implementations built python pyga exact result effectiveness python php ruby implementation still domain interest explored despite pros cons quite easy implemented yields slow process usage solve problems methodology tsp based research implemented three commonly used object oriented programming language without frameworks python php ruby codes represent written based given pseudo code implemented using variable names methods initialization logic implementation tested using one data source parameter values codes implemented close possible parts pseudo code implemented one using multiple methods functions variables implemented manner keeping code close possible supposedly yields objective measurement results one many important functions used random number generator random number generator functions utilized generate random number order compensate probability crossover copy mutate parents random number generator implemented using number generator prng function prng random number generator function given seed number always return random value technique used ensure programs method loop inside program run circumstances thus giving result prng function used research implemented using separate script run system call implemented performance measurement implemented small modification added scripts adding current time micro time function several points code taking account current generation best fitness cost value carrying measurement values testing units made using specified value data ensure implemented scripts using seed number random values therefore implemented mate equal parents generating candidate population every generation return best individual generation end script execution scripts run measured macbook pro computer running mac ghz intel core processor mhz memory intel graphics version python ruby php interpreter used measurement processes respectively main objective paper automatically infer precise bounds execution times best fitness depends input data sizes three different thus recommendation widely used implementation given genetic algorithm pseudo code tsp table shows solve tsp written python php ruby initial population individuals generated comma separated value csv file contains names cities along coordinates table genetic algorithm pseudo code pseudo code class city function initialize name initialize name city along coordinate class individual function initialize route initialization individual consists cities nodes function makechromosome file generate individual chromosomes file function evaluate calculate length given population set cities using euclid distance formula function crossover recombination return new offspring given spouse class environment function initialize initialization population data population size maximum number generations crossover rate mutation rate optimum number generation related best fitness cost convergence value function makepopulation create initial population consists two parents file parent parent generated shuffled parent function evaluate sorts individual based calculated length value starting least length best largest individual current population getbestindividual function getbestindividual best length obtained far less best individual population obtained crossover best fitness cost remains unchanged else best fitness cost obtained updated first individual order population current crossover offspring population function run goal achieved generation step increase generation number function goal current generation reached maximum generation current generation reached optimum current generation minus best individual generation larger optimum number given goal achieved otherwise continue iteration function step crossover evaluate fitness costs individual current population function crossover initialize candidate population select two best parent candidates parent parent current population candidate population still less given size randomise crossover rate crossover rate offspring crossover parent parent else offspring copy parent randomise mutation rate mutation rate offspring mutate current offspring evaluate offspring fitness cost offspring exists candidate population add offspring individual candidate population candidate population become new population function mutate individual mutating individual manner switching half given individual orderly run environment parameters csv file containing population data number population number maximum generation crossover rate mutation rate optimum best generation implementation pseudo code pseudo code used solve tsp research implemented python php ruby every method variables ensuring results values across implementations given codes utilizing population data impossible implement methods variables exact ways due different natures programming style however implemented codes methods variables across ensured behave values environment variables population data random number generation previously mentioned random number generator prng used implemented using separate code called system call function implemented therefore implemented scripts random number performance value circumstances prng code implemented ruby known good performance shown table native random number generator functions across different behave differently separation prng code main script required ensure using random number running time therefore execution time measurement would objective real world implementation use native random function recommended research proved use system call causes bottleneck program execution table prng implementation ruby prng code implementation ruby seed argv max argv srand seed argv puts rand else puts rand argv end else puts end seed number generation seed number used generate random values incremented one previous seed value calling prng script seed implemented global variable therefore value always available fed prng value maintained runtime data experiment genome data individual used unit tests measurements shown table iii stored csv file plain text values stored line csv file consist name cities along coordinates file read initialization environment values used population initial genome data table iii initial populations gnome data city name balikpapan malang jayapura manado bandung banjarmasin pontianak jakarta medan makassar execution time measurement workarounds used order portable time measurement timing difficult possible achieve synthetic benchmark approach followed purpose repeatedly execute instructions estimation large enough time later averaging total execution time number times run generally possible run single instruction repeatedly within abstract machine since resulting sequence would legal may break abstract machine run memory etc therefore complex sequences instructions must constructed repeated instead previous measurement research conducted case knapsack problem measured using native timing function execution time measurements tsp problem also getting program execution time difference start time end time script execution calculated measurements done several times environment circumstances measurement done different number population data csv file program execution times measured using five six seven eight nine ten cities consecutive ways environmental parameters set maximum generations populations within generation crossover rate mutation rate generations limit best fitness cost convergence termination limit environment measured times average values standard deviations estimated result analysis implementation pseudo code provided resulting python php ruby codes python php ruby codes implemented lines code respectively python php ruby file size bytes respectively shortest line codes python python scripting nature require closing tags method function loop implementation php ruby comes code file size python consumes bytes implement ruby programming style characteristic shorter simpler compared thus resulting smallest file size implementation results program execution measurement research shown table python used basis performance measurement widely used research purposes therefore compare execution time python comes web environment research conducted jafar php outperform python based last seed best fitness cost best generation measurement results table infer tests returning best individual generation number cities therefore proving executions flow run exactly tests ruby proved outperform python php execution time ruby performance improvements vary php performance slower python table measurement result php coefficient variant best fitness cost length python ruby php python ruby php python ruby php python ruby php python ruby php python ruby number cities maximum minimum average standard deviation fig program execution time milliseconds number cities tsp expected solution fitness cost minimum smaller value result better execution time grows longer number cities individual increase shown fig relation fitness cost total distance selected routes number cities individual shown fig implemented way resulting fitness cost value number cities regardless used regression analysis using analysis variance anova conducted measurement result shown table regression analysis result infer script execution times highly correlated data sizes implemented environment cities included individual mean higher number inputs longer program execution time would best generation last seed performance python fig fitness cost route total distance number cities proved fitness cost route length highly correlated data size implemented environment higher number inputs cities included individual larger fitness cost would individual larger fitness cost worse conclusion based measurement analysis process program execution time best fitness cost highly relied data size number cities larger data size longer execution time worse outcome tsp solution research pseudo code implementation shows ruby code smallest file size compared python php python least line codes testing overall program execution time ruby faster python php therefore usage ruby recommended gain performance implementation tsp web environment python php acknowledgment authors would like acknowledge assistance program teknologi informasi dan ilmu komputer ptiik universitas brawijaya research facilities financial support authors also acknowledge assistance colleagues ptiik great assistance improving manuscript references andaur rios roman velasquez best web site structure users based genetic algorithm approach department industrial engineering university chile santiago chile ren cloud computing language engineering effective computing advanced intelligence international journal advanced intelligence vol number punch iii using genetic algorithms data mining optimization educational systems michigan state university gursel improving search social networks agent based mining guo engler toward practical incremental recomputation scientists implementation python language stanford university python http php http ruby http branigan risk web programming technologies lucent technologies dunlop varrette pascal use genetic algorithm high performance computer benchmark tuning university luxembourg najera tsp three evolutionary approaches local search university birmingham sarmady investigation genetic algorithm parameters universiti sains malaysia strachey fundamental concepts programming languages higher order symbolic computation kluwe academic publishers riley interpreted compiled languages internet http november august purer php python ruby web scripting language shootout vienna university technology pospichal parallel genetic algorithm solving knapsack problem running gpu brno university technology koepke reasons python rocks research reasons jafar anderson abdullat comparison dynamic web content processing language performance lamp architecture west texas university canyon view publication stats

qualitative assessment recurrent human motion andre ebert michael till beck andy mattausch lenz belzner claudia nov mobile distributed systems group institute computer science munich germany email belzner linnhoff applications designed track human motion combination wearable sensors physical exercising raised huge attention recently commonly provide quantitative services personalized training instructions counting distances qualitative monitoring assessment still missing detect malpositions prevent injuries optimize training success address issue presenting concept qualitative well generic assessment recurrent human motion processing continuous time series tracked motion sensors therefore segmentation procedure extracts individual events specific length propose expressive features accomplish qualitative motion assessment supervised classification verified approach within comprehensive study encompassing athletes undertaking different body weight exercises able recognize six different exercise types success rate assess qualitatively average success rate assessment activity recognition physical exercises segmentation ntroduction regular physical exercising improves athlete health sufferings chronic diseases even alzheimer disease lowered context mobile phone applications training support running crossfit etc became popular provide customized workout plans detailed exercise instructions well quantitative statistical functions providing challenging exercises without supervision athletes arises new problems wrong execution exercises malpositions absence sufficient warming phases may lead less training success even serious injuries especially athletes likely harm unsupevised workout believe automated monitoring reduces injuries drastically training success could improved significantly moreover generic concept capable recognizing assessing various recurrent human motions also applicable areas medical observations gait analysis optimization workflows address unsolved issue previously introduced sensx distributed sensor system capturing processing human motion established paradigm qualitative analysis human motion consisting four fundamental steps see figure detection motion event recognition qualitative assessment characterization personal use preprint copy permitted republication redistribution uses require permission ieee paper published within proceedings european signal processing conference eusipco kos greece doi ieee fig four fundamental steps human motion analysis within logical layer hardware layer functions sensor feedback provider proposed reasons specific assessment step treated within paper focuses step using existing sensx architecture basis thereby contributions within paper follows propose novel concept qualitative assessment complex recurrent human motion covers extraction motion events segments individual length expressive feature set well system supervised classification selected implemented validate concept conducted comprehensive exemplary study athletes executing repetitions six different types body weight exercises present results concerning assessment human motion well human activity recognition basis motion sensor data elated ork following provide brief overview related work concerning segmentation recognition assessment human motion thereby focusing complex motion sequences described multiple coordinated movements conducted several extremities time body weight exercises instead simple activities often subject activity recognition within previous research walking sitting analysis assessment reoccurring events within time series become feasible need extracted segments first bulling name sliding windows segmentation restposition based segmentation use additional sensors context resources applicable procedures sliding window approaches move window static size sequentially across incoming stream data extract window current content analysis authors recofit climbax used sliding window moved discrete steps across motion data stream approaches offer valuable ideas segmentation concept still due absence length adjustment events actual duration cover needs actual start endpoints short events pushup captured accurately leads noise within event segment fragments preceding following events noise may disturb qualitative assessment process specific event significantly energybased solutions perform well segmentation long term activities describable different energy potentials sitting running examine individual repetitions short movements energy potential diverse enough suitable segmentation restposition based segmentation also feasible rest positions within continuous event set use external information sources gps suitable movements targeted approaches also facilitated manual segmentation suitable great numbers events realtime analysis review procedures led necessity developing individualized segmentation process considers requirements concerning dynamic accurate extraction complex motion events quantitative counting repetitive activities well recognition classification fields research present much work bound topic within paper jiang recognize simple activities like lying walking sitting morris dealing complex exercises facilitated techniques neuronal networks well typical classifiers supervised learning combinations contrast qualitative assessment human motion data examined sparsely yet ladha well pansiot assessed performance climbers extracting analyzing features power control stability speed without examining individual climbing moves work provides valuable information concerning handling preprocessing motion data still allow assessment individual movements specific extremities within chain multiple climbing features velloso assessed repetitions recurrent motion recording five wrongly executed exercises classified afterwards template comparison though able classify exercise mistakes success rate approach able identify fixed number predefined error cases thus suitable generic assessment human motion gymskill system qualitative evaluation exercises conducted balance board exercises examined assessed individualized principal component analysis gymskill bound analysis combination balance board therefore also capable generic motion analysis concluding aware concept enables qualitative assessment human motion generic way without bound predefined motions equipment solution present novel approach tracking recognizing assessing human motion iii approach qualitative assessment human motion following explain advance extracting recurring events time series afterwards describe preprocessing selection expressive features prepare qualitative assessment via supervised learning data input analysis procedure individual streams motion data see technical details therefore five sensor devices tracking acceleration rotation information two fastened tracked person ankles two attached wrist fifth worn chest combination processing unit preprocessing segmentation extract individual events incoming multidimensional signal set developed dynamic segmentation algorithm capable segmenting heterogeneous sets motion events individually signal segment may specific length contains information exactly one rotation acceleration axis exactly one specific event extracted event segments also individual temporal length comparison identify individual motion events first examine meaningful signal vector within signal set typically signal contains highest dynamics variance within values allows distinct identification segment start point end point calculating standard deviation signals whereby defines current signal measured value expectancy value signal highest deviation taken analysis assume every type motion event described individual set local extrema use sets identify distinct events figure depicts whole segmentation process set bicycle crunches within setting acceleration ankle sensors along proved provide meaningful signal depicted figure signals individual repetitions contain high amount noise well unique peaks fig step step procedure segmenting recurrent motion events individual length set bicycle crunches fig signal set mountain climbers determination ideal cutoff factor representative specific class movements peaks may contain information critical assessing movement terms quality irrelevant segmentation designed applied aggressive butterworth low pass filter signal see figure thus information unnecessary segmentation extinguished essential periodicities left filter determined multiplying sampling frequency cutoff factor essential filter effect onto signal figure shows results empirical determination indicates sensor setup must within range order recognize individual event occurrences within set repetitions thereby different setup used individual exercise due low pass filtering combination usage extrema patterns identification individual duration segment well starting point tsx ending point tex see figure becomes feasible achieved follows bicycle twist described set one local minimum one local maximum movements may characterized differing combinations multiple local extrema depicted set mountain climbers figure least two different signal parts identifiable example filtered signal scanned sequentially pattern new occurrence detected window size estimated event length applied signal first phase estimated event length derived event sets depicted figure since every repetition individual length content need adjust segments start end points individually within second phase precondition following assumption origin terminus repeated motion event located signal zero crossing rest periods check segment window encompasses demanded number types extrema sequentially add sub segments predefined length relevant extrema pattern matched matching phase fine tuning undertaken capture exact segment ending last element within segment positive value wind forward add single samples reach next zero crossing otherwise last element within segment negative value wind back last zero crossing remove values way determining individual length current segment keep timestamps starting endpoint tsx tex subsequently used cut specific segment slightly smoothed original signals see figure still contain important movement information output procedure quantity event segments differing length whereby segment exact borders contains information exactly one motion event feature selection labeling commonly feature vector within machine learning scenarios consists fixed number features describing one instance due fact activity segments individual length consist individual signals issue challenging use segmented time series directly feature set creation length would need trimmed interpolated match fixed length feature vector interpolation would result unwanted artificial noise trimming could lead loss important information finally preceding efforts extract event segment individual length would worthless furthermore one event dataset recorded evaluation see section consists sample values average roughly per signal building feature vectors length greater leads massive computational load classification overcome challenges exploited observations made examination dataset figure visualizes standard deviations acceleration rotation signals randomly selected lunges labeled quality class good lunges labeled quality class poor general lunges labeled class show much higher deviation rotation acceleration values users feet class values significantly lower proper dataset fig standard deviations rotation acceleration signals set lunges labeled class class fig components feature vector describing one individual motion event confusion matrix visualizing results qualitative assessment bicycle crunches lunge described big step forward well bringing one knee nearly ground results greater movement energy decently conducted lunges result less energy within signals happens forwards downwards wrist acceleration placed onto users hip workout contrast rotation low proper lunges higher improper ones related smooth movement conducted skilled athletes unsteady movements conducted unskilled athletes due relatively smooth steady movement athletes torso chest sensor provide significant information concerning activity observations show even individual signal standard deviation contains enough information assess activity qualitative way based cognition designed feature vector describe individual activity instance contains standard deviations signals plus time interval specific instance milliseconds see figure one event evaluation data set see section consists average sampling values utilizing procedure described able compress information ratio additionally added label concerning individual motion events quality rating good poor see section evaluation recorded six body weight exercises crunches lunges jumping jack mountain climber bicycle crunches squats conducted athletes male female sex aged till athlete complete sets repetitions exercise individual sets scheduled mandatory break instruction video shown athletes exercise prior execution tracked motion data individual exercise repetitions additionally conducted exercises taped video later labeling experts labeling range good bad data labeled follows exercises labeled initially class mistake specific deviation video instructions steps small mountain climber initial class gets added small deviation severe deviation error points final class rounded result overall error score hence completely different errors performance motion event may lead error score therefore quality class qualitative motion assessment used two different classification approaches supervised learning one manual one automated hyper parameter optimization within manually configured four popular classification algorithms human activity recognition see section decision tree driven random forest support vector machine svm classifier naive bayes algorithm table classifier svm average duration table correct classification rates qualitative assessment manual classifier selection configuration presents performance manual classifier selection within cross validation provides best results average correct classification rate taking building evaluation model performed way faster worse results success rate training time classifier ibk average table correct classification rates facilitating hyper parameter optimization valuation section first describe setup study subsequently present results evaluation give insights performance segmentation algorithm approach facilitates hyper parameter optimization layer automated selection appropriate classifiers hyper parameter tuning table shows significantly improved results facilitating neighbor ibk also cross validation four six exercise types assessed correctly success rate average rate despite varying time spans different classifiers test models except one trained within less second thousands event instances demonstrates efficiency scalability light weight feature vector design offers promising chances mobile realtime usage volatile classification rates different exercise types may explained discrete value domain labels well subjective labeling procedure assumption also indicated figure shows confusion matrix qualitative assessment bicycle crunches incorrectly classified events became assigned neighboring quality classes event label originates rounded error score may occur label score border value event gets label although quality rated contrast classifier may decide feature vector looks like member class finally leads wrong classification segmentation results able extract recorded motion events individual segments specific length using segmentation approach described section makes total extracted events activity recognition preliminary study evaluated automated recognition eight different body weight exercises basis acceleration data paper feature vectors built basis individual acceleration rotation signals standard deviations rather time series fixed significantly bigger length applied new design dataset see section achieved correct recognition rate manual classifier configuration reached applying automated hyper parameter optimization within cross validation training evaluation model took instances compared preliminary studies related approaches performance regarded within field complex human motion recognition onclusion uture ork paper presented generic approach dynamic individual segmentation recurrent human motion events well qualitative assessment complex human motion evaluation recorded exemplary dataset containing repetitions six different different body weight exercises extracted segments individual length additionally segments tagged quality label able estimate generic quality class individual event occurrence average correct classification rate individual exercise types adapting expressive heavily compressed feature vector sheer recognition activities actually reach correct classification rate automatic hyper parameter optimization performed significantly better manual approaches concept features generic analysis approach conjecture applicable various recurrent human motions transferable multiple operational areas sports medical observation even workflow optimization results offer promising options future work assessment process adapting compressed feature vectors information valuable identifying tangible errors positioning malpositions gets lost thereby conducted movement rated good bad qualitative manner neither exact reason identified carve concrete characteristics specific quality assessment new features principal components may crucial explore issues dynamic generic analysis approaches neural networks may bring new insights subject ongoing research eferences radak hart sarga koltai atalay ohno boldogh exercise plays preventive role alzheimer disease journal alzheimer disease vol koplan siscovick goldbaum risks exercise public health view injuries public health reports vol ebert kiermeier marouane sensx sensing assessment complex human motion networking sensing control icnsc ieee international conference ieee bulling blanke schiele tutorial human activity recognition using inertial sensors acm computing surveys csur vol morris saponas guillory kelner recofit using wearable sensor find recognize count repetitive exercises proceedings annual acm conference human factors computing systems acm ladha hammerla olivier climbax skill assessment climbing enthusiasts proceedings acm international joint conference pervasive ubiquitous computing acm ding shangguan yang han zhou yang zhao femo platform exercise monitoring rfids proceedings acm conference embedded networked sensor systems acm mortazavi pourhomayoun alsheikh alshurafa lee sarrafzadeh determining single best axis exercise repetition recognition counting smartwatches wearable implantable body sensor networks bsn international conference ieee jiang yin human activity recognition using wearable sensors deep convolutional neural networks proceedings acm international conference multimedia acm pansiot king mcilwraith yang climbsn climber performance monitoring bsn medical devices biosensors international summer school symposium ieee velloso bulling gellersen ugulino fuks qualitative activity recognition weight lifting exercises proceedings augmented human international conference acm roalter diewald scherr kranz hammerla olivier gymskill personal trainer physical exercises pervasive computing communications percom ieee international conference ieee thornton hutter hoos combined selection hyperparameter optimization classification algorithms proceedings acm sigkdd international conference knowledge discovery data mining acm

earth system grid supporting next generation climate modeling research david bernholdt shishir bharathi david brown kasidit chanchio meili chen ann chervenak luca cinquini bob drach ian foster peter fox jose garcia carl kesselman rob markel middleton veronika nefedova line pouchard arie shoshani alex sim gary strand dean williams invited paper understanding earth climate system might changing preeminent scientific challenge global climate models used simulate past present future climates experiments executed continuously array distributed supercomputers resulting data archive spread several sites currently contains upwards simulation data growing rapidly looking toward beyond must anticipate prepare distributed climate research data holdings many petabytes earth system grid esg collaborative interdisciplinary project aimed addressing challenge enabling management discovery access analysis critically important datasets distributed heterogeneous computational environment problem fundamentally grid problem building upon globus toolkit variety technologies esg developing environment addresses authentication authorization manuscript received march revised june work supported part department energy scientific discovery advanced computation scidac program grant bernholdt chanchio chen pouchard oak ridge national laboratory oak ridge usa bernholdtde chanchiok chenml pouchardlc bharathi chervenak kesselman usc information sciences institute marina del rey usa shishir annc carl brown cinquini fox garcia markel middleton strand national center atmospheric research boulder usa dbrown luca pfox jgarcia strandwg drach williams lawrence livermore national laboratory livermore usa drach foster nefedova argonne national laboratory argonne usa foster nefedova shoshani sim lawrence berkeley national laboratory berkeley usa shoshani asim digital object identifier data access data transport management services abstractions remote data access mechanisms scalable data replication cataloging rich semantic syntactic information data discovery distributed monitoring portals using system modeling data management earth system grid esg grid computing introduction global climate research today faces critical challenge deal increasingly complex datasets fast becoming massive current storage manipulation archiving navigation retrieval capabilities simulations performed advanced components atmosphere oceans land sea ice biosphere produce petabytes data useful output must made easily accessible researchers nationwide national laboratories universities research laboratories institutions thus need create deploy new tools allow data producers publish data secure manner allow data consumers access data flexibly reliably way increase scientific productivity climate researchers turning climate datasets community resources goal earth system grid esg project create virtual collaborative environment linking distributed centers users models data esg research development program designed develop deploy technologies required provide scientists virtual proximity distributed data resources need perform research participants esg include ieee proceedings ieee vol march national center atmospheric research ncar boulder lawrence livermore national laboratory llnl livermore oak ridge national laboratory ornl oak ridge argonne national laboratory anl argonne lawrence berkeley national laboratory lbnl berkeley usc information sciences institute marina del rey recently los alamos national laboratory lanl los alamos past three years esg made considerable progress towards goal providing collaborative environment earth system scientists first developed suite metadata technologies including standard schema automated metadata extraction metadata catalog service second developed deployed security technologies include user registration authentication community authorization data transport technologies developed esg include data transport access robust multiple file transport integration transport mass storage systems support dataset aggregation subsetting finally developed web portal integrates many capabilities provides interactive user access climate data holdings cataloged close climate data rich scientific metadata provides beginnings digital scientific notebook describing experiments conducted demonstrated series increasingly functional versions software events ncar community climate system model ccsm workshop supercomputing conference deployed system spring provide community access assessment datasets produced intergovernmental panel climate change ipcc esg aims improve greatly utility shared community climate model datasets enhancing scientist ability discover access datasets interest well enabling increased access tools required analyze interpret data current global climate models run supercomputers within beyond earth simulator center japan produce terabytes data stored local archival storage analyses data contribute understanding planet influencing climate change policy makers react respond scientific information global community produce future trends climate modeling increase computational storage requirements scientific advances require massive increase computing capability sublinear still extremely substantial increase volume distribution climate modeling data see increases physical resolution models elevation number ensemble runs enhanced quality terms clouds aerosols biogeochemical cycles parameters broadening overall scope extends regions project future clear global earth system modeling activities going produce petabytes data increasingly plex distributed due nature computational resources climate modeling community creating shared data archives time however archives help enable work specific groups limit access community large prohibit analysis comparison across archives current infrastructure analysis large distributed datasets confined separate storage systems geographically distributed centers daunting task esg goal dramatically improve situation leveraging emerging developments grid technology break artificial barriers creating environment encompasses multiple computational realms spanning organizational boundaries goal delivering seamless access diverse climate modeling data broad user base task challenging technical cultural standpoints paper describe esg project accomplished last three years discuss ongoing future efforts increase utility esg climate community coming decade functional objectives next present functional objectives use cases related climate model dataset management access various climate model simulations performed output simulations stored archival storage simulation output comprising several thousand files several hundred gigabytes data resulting data great interest many researchers policy makers educators others however current technology must data manager spend considerable time managing process data creation user wishes access data must engage difficult tedious process requires considerable knowledge services resources necessary metadata search discovery specifics archival system software system accounts analysis software extracting specific subsets complexity data access tends restricted privileged specialists goal esg system simplify data management task making easier data managers make data available others data access task making climate data easy access web pages via web browser making data available esg community first requirement tools allow climate model data managers make model data available esg community tools include means create searchable databases metadata provide catalogs data locate given piece data archival system online storage make catalogs data accessible via web prior advent esg capabilities exist potential users model data contact data proceedings ieee vol march managers personally begin laborious process retrieving data wanted important tools data publishers curators robust easy use data publishers familiar implementation details system thus tools must intuitive reflect user perspective data accessible user local workstation although climate simulation datasets relatively static change time errors corrected existing runs extended new postrun analyses added thus sufficient able publish dataset must easy publishers update metadata time publishing tools must allow sufficiently privileged users search update metadata database ensuring metadata physical datasets stay synchronized defining virtual datasets addition physical datasets generated directly climate simulations maintained archival storage data producers want define virtual datasets datasets defined set transformation operations physical virtual datasets virtual datasets instantiated yield physical datasets esg system either proactively fashion following user request physical datasets discarded following use alternatively cached avoid cost subsequent regeneration virtual datasets important allow data producers define abstractions physical files may convenient efficient users access transformations used define virtual datasets may include concatenation files one datasets subsetting along one dimensions computation reduction operations work esg consider concatenation subsetting architecture extended future support arbitrary transformations chimera system following example virtual dataset defined via concatenation subsetting dataset contains two datasets field time net effect provide data consumer virtual view dataset hides underlying organization complexity thousands related files providing simple convenient access member esg community able browse search discover access distributed climate model data easily using esg web portal browse esg data holdings hierarchically simple text search capability allows perform searches data catalogs subsequent browsing search select data wants possibly narrowing choices via additional possibilities presented search results also select individual fields regions interest specifying selection ultimately come multiple physical files web portal creates efficient means finding accessing published model data user states wants access esg performs tasks required deliver desired result retrieving large numerous datasets esg user wishes access large amounts climate model data particularly data located archival system esg tool called datamover efficiently robustly accomplishes task example esg user may wish access many simulated years model data store data local online storage datamover allows user move large volume data without requiring personally monitor steps involved manages processes takes corrective actions needed transmit data archival storage system location user choosing using climate analysis application access esg holdings esg user may choose utilize common data analysis package like climate data analysis tools cdat ncar graphics command language ncl access data published via esg example web portal search give information needs pull data directly software package implies requirement interfaces compatible applications distributed data access protocols provide remote access capability allows esg user community far greater access climate model data previously available well ability extensive complicated analyses large sets data efficiently since need retrieve store data locally beginning analysis iii esg architecture next present description overall esg architecture fig shows major components involved esg system bottom top following database storage application servers basic resources esg constructed include computational resources used creation virtual data disk caches mass storage systems used hold data assets database servers used store metadata application servers hosting esg portal services infrastructure provides remote authenticated access shared esg resources enabling access credentials data metadata replica location services rlss submission management computational jobs services services span esg resources provide capabilities site site data movement distributed reliable metadata access data aggregation filtering esg applications important esg portal provides convenient interface esg services applications include user level tools data publication bernholdt earth system grid supporting next generation climate modeling research fig esg architecture schematic showing major components well assorted clients data analysis visualization accomplishments bringing user community climate datasets first three years esg development reached milestone achievements following areas speed access transport data user request via integration advanced gridftp data transfer technologies widely used project network data access protocol opendap software robust multifile movement various storage systems via development datamover technology standardization data formats metadata schemas tools discovery use analysis managing sharing scientific data security access controls supporting user registration authentication purposes auditing highly secure access control sensitive operations development portal enabling interactive searching esg data holdings browsing associated metadata retrieval datasets may large first official release esg greater climate community began spring portal used deliver community data produced intergovernmental panel climate change ipcc assessment program first production release esg vision providing foundation data publication analysis applications portals collaborative environments climate research demonstrated esg web portal developed esg web portal provides entry point esg data holdings purpose allow climate scientists around world authenticate esg services browse search subset retrieve data esg portal uses esg services associate metadata datasets determine data location deliver data location desired user portal includes aggregated data selection option allows user select variable subregion time level ranges metadata role data descriptions metadata critical discovery analysis management sharing scientific data developed standardized metadata descriptions metadata services support important functionality esg developed metadata schema oriented toward types metadata generated atmospheric general circulation coupled models schema defined terms network common data format netcdf markup language ncml provide representation metadata appears netcdf encoded simulation data esg metadata schema supported range services used provide access specific metadata values esg metadata catalog based open grid services architecture data access integration service uses relational technology browsing searching esg metadata holdings also deployed separate metadata service called replica location service rls provide information location datasets allowing data replicated migrated without modify entries main catalog ncml data proceedings ieee vol march esg generates certificate stores myproxy service gives user myproxy registration system also administrator interface allows administrator accept reject user request important benefits system users never deal certificates portal get user certificate myproxy service needed shared data access requires ability specify enforce access control individual datasets implemented prototype authorization service integrating community authorization service cas gridftp server used implement data queries system supports access control based community specified policy stored cas server policy database data transport access services fig registration system architecture provide ability bridge esg metadata dataset catalogs conforming thematic realtime environmental data distributed services thredds specification thredds catalogs automatically generated data description information contained database location information retrieved distributed rls databases security esg large diverse user community multiple laboratories participate esg make security challenging problem devoted considerable effort analyzing documenting security requirements esg adopted security approach includes user authentication user registration esg portal access control grid security infrastructure gsi used common underlying basis secure single mutual authentication machinery provides secure access control esg resources using public key infrastructure pki credentials doe grids certificate authority esg recognized doe accredited virtual organization participates policy management authority board overhead associated obtaining public key credentials online registration process used access pki credentials without overhead normally associated certificate generation mechanism allows broad base users access public data providing auditing access remote users esg existing trust relationship required research sponsors esg developed registration system allows users register easily esg portal registration system architecture shown fig focus system ease use developed portal extensions cgi scripts automate user registration requests system solicits basic data user generates certificate request opendap protocol suite technologies provide flexible easy access remote data based widely accepted community opendap software esg extended opendap create opendap server clients support gsi security use gridftp data transport mechanism addition seamless joining data stored different opendap servers aggregation part esg production release esg also built prototype data access transport system demonstrating use cdat ncl access remote datasets using gridftp protocol via client continued work necessary goes production including support aggregation goal client software foundation distributed climate applications gain access esg resources multiple file transport esg developed tools efficient reliable transfer multiple files among storage sites transport thousands files tedious extremely important task climate modeling analysis applications scientists run models powerful supercomputers often need massive volume data generated moved reliably another site often source destination storage systems specialized mass storage systems high performance storage system hpss national energy research scientific computing center nersc berkeley mass storage systems mss ncar analysis phase subsets data need accessed various storage systems moved reliably destination site analysis takes place automation file replication task requires automatic space acquisition reuse monitoring progress thousands files staged source mass storage system transferring files network archiving target mass storage system leveraged software developed storage resource manager srm project bernholdt earth system grid supporting next generation climate modeling research achieve robust multifile transport esg srms software components placed front storage systems accept multifile requests manage incremental access files storage system use gridftp transport files destination two important results related multiple file transport achieved first phase esg project first motivated practical need mss ncar able store files get files mss directly remote sites achieve goal adapted version srm ncar mss allowed multifile movement various sites ncar system used repeatedly move reliably thousands file esg sites second reliability achieved srm monitoring staging transfer archiving files recovering transient failures tool deployed dynamically monitor progress multifile replication response esg user requirements ability move entire directories single command software module called datamover developed datamover interacts source destination storage systems srms setup target directory instructs srms perform robust multifile movement datamover srm used esg portal copy files portal disk space files source locations may remote storage systems user able access files directly portal capability makes possible portal act behalf user get files remote sites without requiring user direct access sites monitoring built monitoring infrastructure allows keep track state resources distributed across seven esg institutions monitoring required provide users robust reliable environment depend upon part daily research activities monitor hardware software components make esg provide status information esg operators users monitoring infrastructure builds upon globus toolkit grid information services ontologies esg metadata developed prototype esg ontology available esg portal ontology contains following classes scientific investigation subclasses simulation observation experiment analysis datasets subclasses campaign ensembles pedigree subclasses identity provenance project service access relationships supported ontology include ispartof isgeneratedby isderivedfrom hasparameter usesservice climate simulation data current focus esg scientific investigation metadata defined ontology accommodates types scientific investigation example observational data collected oceanographers dataset metadata includes time space coverage parameters pedigree provenance metadata trace origins dataset successive operations performed recording conditions dataset interest produced instance provenance describes models versions model software hardware configurations used run interest service metadata associates earth science data formats servers providing functionality subsetting coordinate space visualization expression evaluation esg ontology clearly separates metadata potentially reusable scientific applications project information metadata specific application iterative work detailed concept definition rigor needed specifying relationships entities required ontology authoring substantially improved esg schema current esg data holdings one key goals esg serve critical mass data interest climate community steadily increasing amount data provided climate community esg first phase project esg data holdings consisted small set ccsm parallel climate model pcm data number early users accessing data via prototype esg system deliberately kept small ranged one five concurrent users end pcm data totaled approximately several sites including ncar program climate model diagnosis intercomparison pcmdi nersc ornl lanl users requested accounts pcmdi access pcm data holdings representing around science projects including university researchers doe collaborators various groups interested climate simulation regional scale ncar focused publishing ccsm data llnl published climate simulation data generated major climate modeling groups worldwide intergovernmental panel climate change ipcc fourth assessment report far ipcc jointly established world meteorological organization wmo united nations environment program carries periodic assessments science climate change serve governments others interested potential impact future climate change fundamental effort production collection analysis climate model simulations carried major international research centers although centers individually often produce useful analyses individual results collective model output several modeling centers often yields better proceedings ieee vol march fig schematic esg data portals services archives understanding uncertainties associated models toward end wmo working group coupled modeling wgcm consultation ipcc leadership coordinates set standard simulations carried major modeling centers order tremendous commitment computational scientific effort maximum benefit essential wide community climate scientists critically analyze model output ipcc wgcm therefore asked pcmdi collect model output data ipcc simulations distribute community via esg late ccsm ipcc model runs totaled approximately data ccsm one major climate simulation efforts worldwide interest data scientific community others significant order hundreds scientists researchers policy makers others esg providing access considerable store data based code configured differently address esg ipcc requirements deployment portals services archives section discuss future work esg project next two years goal esg collaborative increase amount data made available scientists esg portal enhance functionality performance robustness esg components critical role ipcc data commitments made llnl pcmdi provide community access data led adopt overall deployment strategy depicted fig lowest level data maintained archives pcmdi ipcc data ncar climate simulation data ccsm pcm services enable remote access data archives maintaining metadata used data discovery implementing authentication authorization copy services located primarily ncar available use another copy pcmdi provide access ipcc data currently software installed pcmdi site yet supporting users ipcc data similarly two web data portals data access one based pcmdi ipcc data portal provide access ipcc ccsm pcm lanl eventually data ncar pcmdi instantiations data portal services external collaborations appropriate developed esg tools services collaboration national international groups currently esg close collaboration british atmospheric data centre national oceanic atmospheric administration noaa operational model archive distribution system nomads earth science portal esp consortium thredds noaa geophysical fluid dynamics laboratory opendap project also held discussions linked environments atmospheric discovery lead project geosciences cyberinfrastructure network geon project earth system modeling framework esmf future work increasing utility esg climate modeling community additional deployment data archives services portals esg data services web data portal installed pcmdi support ipcc data distribution soon registering users want access data eventually plan include sites data archives providing climate datasets esg users including lbnl ornl web portal overall system integration esg data portal integrates different esg services authentication authorization metadata data processing data transport revise existing web portal required support new user communities ocean bernholdt earth system grid supporting next generation climate modeling research modelers needed either providing areas portal developing customized virtual portals community also engaged formal usability testing select members research community security services current esg security architecture deals primarily authentication users various grid resources grid services storage systems several key points remain addressed including authorization access control resources support specific site requirements passwords otps authorization authorization infrastructure build upon community authorization service cas developed part globus toolkit building globus toolkit grid security infrastructure gsi cas allows resource providers specify access control policies delegate access control policy management community specifies policies via cas resource providers maintain ultimate authority resources spared policy administration tasks adding deleting users modifying user privileges esg distinguishes two types authorization authorization makes access control decisions based file access permissions groups users data server could accessed one group users inaccessible another group authorization makes access control decisions based group permission access particular services example privileged users might allowed access data efficiently hierarchical storage systems using datamover service less powerful users would able access data server types authorization users given access rights initial esg portal via esg registration system changes user access rights would done resource providers using special tools developed password otp authentication various options supporting use passwords including online generates gsi credentials automatically user authenticated otp system thus allowing access esg sites without authentication another possibility gsi credentials generated online one site following otp authentication accepted another site requirement third option would include changing gsi credentials specify whether authentication generate user certificate used otp investigate options implement one best fits esg metadata schema services esg focused strongly metadata remaining tasks include following metadata schema start aim esg develop system potential longevity interoperation emerging data systems worldwide new standards like emerging facilitate interoperation multiple data systems plan spend effort evolving metadata direction work undertaken primarily llnl ncar concert number projects british atmospheric data center thredds etc face future needs metadata changes required esg researchers incorporated existing ogsa metadata service metadata catalog support virtual data work required extend metadata catalogs support virtual data definitions data producers define virtual datasets data consumers discover request virtual datasets intend work ncml virtual data definition language multiple metadata catalog services implement replicated metadata catalog avoid single point failure initial goal work improve reliability also note replication improve performance load balancing metadata query workload initially plan use metadata catalog ncar master catalog periodic updates second catalog located lawrence livermore national laboratory deploy general distributed metadata services developed also done exploratory work using open archive initiative oai protocols accomplish function evaluate relative effectiveness level effort required browse search query support richer metadata catalog query capabilities esg scientists federation provide interoperation two heterogeneous metadata services esg metadata services thredds catalogs provide simple distributed queries across two catalog types datamover services work thus far datamover application resulted unique capability copies directories effectively providing equivalent unix copy command across heterogeneous storage environment including ncar locally developed mss strong critical prerequisite required thus use datamover largely restricted power users users formal accounts sites involved data transfer leaves average user may want use web portal locate identify large number files must moved back local system web portal provides workable interface selection small number files limited gigabytes however user wants large number files need different sort capability transfers require automation deal queuing proceedings ieee vol march fig steps involved defining discovering requesting instantiating virtual dataset management cache disk space network problems recovery transient failures continue refine enhance increase robustness scaling datamover tool beyond evolve current datamover application work seamlessly web portal allowing user make selection large number files trigger launch application plan provide data movement capabilities two types users refer casual users frequent casual users grid credentials frequent users willing process acquiring grid credentials achieve efficient transport files casual users requested files moved disk cache managed web portal first pulled users systems frequent users plan develop pull files directly source location user location avoiding transport web portal disk cache efficient method file transport necessary moving large volume data also conform security policies work behind firewall adapt local security policy aggregation services additional development enhance virtual data services web portal native client access performance robustness clients esg clients include web portal user desktop applications cdat ncl plan complete netcdf client interface applications providing full functionality api transparently handling aggregation subsetting virtual data services return useful data users necessary client applications operate various implementations esg security utilize url access data cases client may need generate urls thus requiring access esg catalogs servers migrate services latest release globus toolkit web components new striped gridftp server integration esg security models connections catalog services client interface expected require modifications core libraries addition plan complete integration data access transport rlss srms hierarchical storage systems monitoring provide enhanced monitoring capabilities improve robustness esg infrastructure include monitoring larger number esg services components providing better interfaces querying historical information resource availability virtual data support provide support virtual data next phase esg fig depicts tasks involved publishing discovering accessing virtual dataset include following data provider publishes esg metadata catalog definition new dataset indicating name associated metadata either constituent component files physical dataset definition virtual dataset client user program issues query metadata catalog datasets interest names datasets matching user query returned figure query happens match properties associated virtual dataset name returned client requests dataset requesting virtual data service shown data service figure dataset subset virtual data service retrieves recipe metadata catalog virtual data service instantiates specified subset fetching appropriate data assembling pieces create new dataset finally virtual data service returns requested data user step shown virtual data service may also publish location new physical dataset metadata catalog bernholdt earth system grid supporting next generation climate modeling research accelerate processing subsequent requests dataset step also shown related work esg worked closely several existing efforts include doe science grid project whose work authentication infrastructure related security issues particularly useful esg esg uses doe science grid certification authority basis authentication esg also worked closely unidata several initiatives including joint development ncml specification provides standard xml encoding content netcdf files ncml extended support coordinate system information logic gis interoperability ncml still developed established community standard esg also makes use thredds specification developed within unidata esg collaborative includes members globus alliance esg made extensive use globus toolkit components example esg acted early adopter stringent beta tester gridftp code identifying number subtle errors implementation esg collaborated globus compelling technology demonstrations involving data movement close wide area networks esg also made use gsi rls remote job submission capabilities monitoring information systems cas provided globus toolkit esg also includes members srm project lbnl srms provide dynamic storage management support multifile requests lbnl team adapted srm developed hpss mss ncar part esg srm team also developed datamover providing ability move entire directories robustly recovery failures diverse mass storage systems ncar lbnl ornl well disk systems ncar llnl datamover used repeatedly members esg team move thousands files robustly mass storage systems sites esmf project engaged staff ncar nasa doe laboratories develop standard framework construction modular climate models next phase scope esmf esg expand embrace environments earth system researchers see strong opportunities mutually beneficial interaction esg esmf models extended access remote data publish data number scientific projects face challenges similar explored esg grid physics network griphyn project uses grid technologies support physicists emphasis support virtual data data management workflow management particle physics data grid also employs grid technologies support physics research lead project developing scientific grid infrastructure support mesoscale meteorological research geon project supports geoscientists emphasis data modeling indexing semantic mediation visualization esg technology based globus toolkit srm middleware scientific grid projects use storage resource broker srb middleware unlike layered approach taken globus toolkit srb provides tightly integrated functionality includes extensive data management capabilities including support metadata organizing data collections containers maintaining consistency among replicas vii conclusion increasingly complex datasets produced global climate simulations fast becoming massive current storage manipulation archiving navigation retrieval capabilities goal esg provide data management manipulation infrastructure virtual collaborative environment overcomes challenges linking distributed centers users models data important role esg project provide critical mass data interest climate community including ccsm pcm ipcc simulation model output past three years esg project made considerable progress toward goal community esg developed suite technologies including standard metadata schema tools metadata extraction metadata services data discovery security technologies provide authentication registration authorization capabilities version data access robust multiple file transport using srm datamover monitoring infrastructure development web portal interactive user access climate data holdings date catalogued close climate data rich scientific metadata next phase esg increase utility esg climate modeling community expanding data holdings provide capabilities system next two years esg provide enhanced performance reliability well richer authorization metadata data transport aggregation capabilities support virtual data references earth system grid esg online available http climate system model online available http chervenak remote access climate simulation data challenge problem data grid technologies parallel vol intergovernmental panel climate change ipcc online available http foster chimera virtual data system representing querying automating data derivation presented int conf scientific statistical database management edinburgh proceedings ieee vol march sim shoshani natarajan datamover robust replication networks presented int conf scientific statistical database management ssdbm santorini island greece climate data analysis tools cdat online available http ncar command language ncl online available http foster earth system grid turning climate datasets community resources presented annu meeting amer meteorological orlando caron netcdf markup language ncml unidata boulder atkinson data access integration management grid blueprint new computing infrastructure san mateo morgan kaufmann chervenak giggle framework constructing scalable replica location services presented conf high performance networking computing baltimore chervenak palavalli bharathi kesselman schwartzkopf performance scalability replica location service presented high performance distributed computing honolulu thematic realtime environmental data distributed services thredds online available http butler authentication infrastructure ieee computer vol foster security architecture computational grids proc acm conf computer communications security pearlman community authorization service group collaboration proc ieee int workshop policies distributed systems networks allcock gridftp protocol extension ftp grid global grid forum online available http opendap online available http allcock data management transfer computational grid environments parallel vol storage resource management project online available http http czajkowski grid information services distributed resource sharing proc ieee int symp high performance distributed computing pouchard ontology scientific information grid environment earth system grid presented symp cluster computing grid ccgrid tokyo japan pouchard data discovery semantic web technologies earth sciences int dig libraries published blackmon community climate system model bull amer meteorol vol parallel climate model pcm online available http earth system modeling framework online available http linked environments atmospheric discovery lead online available http geon geosciences network online available welch security grid services proc ieee int symp high performance distributed computing british atmospheric data center badc online available http sompel lagoze open archives initiative protocol metadata harvesting open archives initiative online available http doe science grid project online available http fulker bates jacobs unidata virtual community sharing resources via technological infrastructure bull amer meteorol vol globus toolkit online available http shoshani sim storage resource managers essential components grid grid resource management state art future trends nabrzyski schopf weglarz eds new york kluwer griphyn project toward petascale virtual data grids avery foster online available http particle physics data grid ppdg project online available http storage resource broker online available http baru sdsc storage resource broker presented annu ibm centers advanced studies toronto canada authors photographs biographies available time publication bernholdt earth system grid supporting next generation climate modeling 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high performance codes flash memories ahmed hareedy homa esfahanizadeh lara dolecek mar electrical eng department university california los angeles los angeles usa ahareedy hesfahanizadeh dolecek dense flash memory devices operate low error rates require powerful error correcting coding ecc techniques emerging class ecc techniques broad applications class spatiallycoupled codes block code partitioned components rewired multiple times construct code focus codes underlying structure paper present approach design high performance nbsc codes optimized practical flash channels aim minimizing number detrimental general absorbing sets type two gasts graph designed code first stage deploy novel partitioning mechanism called optimal overlap partitioning acts protograph code produce optimal partitioning corresponding smallest number detrimental objects second stage apply new circulant power optimizer reduce number detrimental gasts third stage use weight consistency matrix framework manipulate edge weights eliminate many possible gasts remain code first two stages operate unlabeled graph code simulation results reveal nbsc codes designed using approach outperform codes used flash channels ntroduction excellent performance codes among attractive error correction techniques deployed modern storage devices codes offer superior performance binary codes thus well suited modern flash memories nature detrimental objects dominate error floor region nonbinary codes depends underlying channel device unlike case canonical channels recent research revealed general absorbing sets type two gasts objects dominate error floor codes practical inherently asymmetric flash channels analyzed gasts proposed combinatorial framework called weight consistency matrix wcm framework removes gasts tanner graph codes results least order magnitude performance gain asymmetric flash channels particular class codes received recent attention class codes codes constructed via partitioning underlying ldpc code components coupling together multiple times recent results codes include asymptotic analysis finite length designs codes designed using cutting vector partitioning optimized magnetic recording applications introduced idea partitioning underlying block code minimizing overlap rows circulants called minimum overlap recently introduced applied awgn channels paper present first study codes designed practical flash channels underlying block codes focus codes combinatorial approach design codes comprises three stages first two stages aim optimizing unlabeled graph code graph code edge weights set third stage aims optimizing edge weights three consecutive stages operate binary protograph code express number subgraphs want minimize terms overlap parameters characterize partitioning block code solve discrete optimization problem determine optimal overlap parameters call new partitioning technique optimal overlap partitioning given optimal partitioning apply new heuristic program optimize circulant powers underlying block code reduce number problematic subgraphs unlabeled graph code call heuristic program circulant power optimizer cpo optimized underlying topology using first two stages last stage focus edge weight processing order remove many possible remaining detrimental gasts code achieve goal use wcm framework also enumerate minimum cardinality sets edge weight changes candidates gast removal three stages necessary code design procedure demonstrate advantages code design approach approaches use partitioning partitioning context column weight codes rest paper organized follows section present preliminaries section iii detail theory partitioning context column weight codes cpo described section next section propose discussion wcm framework code design steps simulation results presented section finally paper concluded section vii reliminaries section review construction codes well partitioning techniques furthermore recall definition gasts key idea wcm framework throughout paper column row paritycheck matrix corresponds variable node check node equivalent graph matrix moreover entry matrix corresponds edge equivalent graph matrix let matrix underlying regular code column weight degree row weight degree binary image consists circulants circulant form row group index column group index identity matrix cyclically shifted one unit left circulant permutation matrix circulant powers codes codes prime paper underlying block codes use design codes codes zero circulants code constructed follows first partitioned disjoint components size hbm defined memory code component hby contains circulants zero circulants elsewhere work focus second coupled together times see construct binary image matrix code hbsc size replica submatrix hbsc contains hbt zero circulants elsewhere see replicas denoted overlap parameters partitioning well circulant powers selected enhance properties hbsc third matrix generated replacing value focus fourth matrix code hsc constructed applying partitioning coupling scheme described binary protograph matrix bpm general binary matrix matrix resulting replacing circulant zero circulant bpms hbp hbp respectively size bpm hsc hbp size hsc also replicas circulants technique partitioning construct hbsc partitioning technique vector ascending integers used partition matrix circulants indices zero circulants elsewhere matrix another recently introduced partitioning technique partitioning partitioned overlap pair rows circulants minimized moreover partitioning assumes balanced partitioning also balanced distribution circulants among rows partitioning significantly outperforms partitioning paper demonstrate new technique outperforms technique gasts objects dominate error floor codes asymmetric channels practical flash channels recall definitions gasts unlabeled gasts definition consider subgraph induced subset vns tanner graph code set vns values set vns set said general absorbing set type two gast size number unsatisfied cns connected number cns connected unsatisfied cns connected either degree degree connected strictly satisfied unsatisfied neighboring cns set given values definition let subset vns unlabeled tanner graph code let set cns connected graphical configuration unlabeled gast ugast satisfies following two conditions connected strictly neighbors examples gasts ugasts shown fig wcm framework removes gast careful processing edge weights key idea framework represent gast terms set submatrices gast adjacency matrix submatrices wcms property edge weights gast processed force null spaces wcms particular property gast completely removed tanner graph code see iii partitioning heoretical nalysis order simultaneously reduce number multiple ugasts determine common substructure minimize number instances substructure unlabeled tanner graph code graph hbsc propose new partitioning scheme context codes scheme extended higher column weights overwhelming majority dominant gasts encountered codes simulated flash channels ugast occurs common substructure frequently see fig thus focus removal ugasts fig two dominant gasts codes flash gasts appropriate edge weights assumed ugast cycle length graph hbp binary protograph code defined entries hbp results cycles length graph hbsc mod power circulant indexed hbsc otherwise cycle results cycle length graph hbsc integer divides clear fig ugast cycle length thus motivated fact partitioning aims deriving overlap parameters hbp result minimum number cycles length graph hbp binary protograph code run cpo reduce number ugasts graph hbsc unlabeled graph code breaking condition cycles length many cycles optimized graph hbp possible goal minimize number cycles length binary protograph code via partitioning hbp also partitioning achieve goal establish discrete optimization problem expressing number cycles length graph hbp function overlap parameters standard code parameters solve optimal overlap parameters start following lemma lemma tanner graph code parameters binary protograph number cycles length given lfs number cycles length vns spanning one particular replica say number cycles length vns spanning two particular consecutive replicas say proof lemma maximum number consecutive replicas spanned vns cycle length code thus vns cycle length span either one replica two consecutive replicas since exist replicas distinct pairs consecutive replicas repetitive nature code follows let overlapping set rows binary matrix set positions rows simultaneously overlap define overlap parameters follows number row hbp definitions size overlapping set rows hbp size overlapping set rows hbp definitions size overlapping set rows hbp moreover let max define following functions used theorem theorem uses combinatorics give exact expressions terms overlap parameters theorem tanner graph code parameters binary protograph computed follows proof term represents number cycles length vns spanning one ireplica nont zero submatrix replica hbpt hbpt four possible cases arrangement cns cycle length vns spanning one replica cases listed three cns within hbp number cycles length cns inside hbp denoted three cns within hbp number cycles length cns inside hbp denoted two cns within hbp one within number cycles length case denoted two cns within hbp one within number cycles length case denoted four different cases arrangement illustrated upper panel fig next find number cycles length four cases terms overlap parameters standard code parameters particularly case cycle length comprised overlap rows overlap rows overlap rows hbp note overlap must distinct associated column index position result valid cycle length overlap rows selected among possible choices among overlaps exist overlaps associated column indices overlaps pairs rows thus overlaps need considered separately avoid incorrect counting argument applies choose overlap two pairs rows result number different ways choose overlaps form cycle length defined case number cycles length computed exactly case using overlap parameters matrix hbp thus case one overlap solely belongs hbp two overlaps cross hbp see fig overlap three options choose two rows three hbp example suppose overlap chosen rows hbp cross overlaps row hbp row also row row note since hbp result partitioning overlaps row hbp row based option three chosen number cycles length computed using overlap parameters hbp total number cycles length case defined case number cycles length computed case difference case one overlap solely belongs hbp two overlaps cross hbp see fig consequently hand term represents number cycles length vns spanning two consecutive replicas submatrix two consecutive replicas hbp hbp hbp four possible cases arrangement cns vns cycle length vns spanning two consecutive replicas cases listed three cns within hbp two vns belong first replica one belongs second replica number cycles length case denoted three cns within hbp one belongs first replica two vns belong second replica number cycles length case denoted one within hbp two cns within hbp besides two vns belong first replica one belongs second replica number cycles length case denoted two cns within hbp one within besides one belongs first replica two vns belong second replica number cycles length case denoted four different cases arrangement illustrated lower panel fig next find number cycles length four cases terms overlap parameters standard code parameters particularly case two overlaps belong hbp first replica one overlap belongs hbp second replica see fig overlap hbp three options choose two rows three option two overlaps inside hbp must distinct associated column indices positions result valid cycle length overlap inside hbp column index two overlaps thus number different ways choose overlaps form cycle length given defined case number cycles length computed case difference case one overlap belongs hbp first replica two overlaps belong hbp second replica see fig thus case one overlap solely belongs hbp second replica two overlaps cross hbp first replica see fig overlap second replica three options choose two rows three two overlaps belong first replica must distinct corresponding column indices positions consequently total number cycles length case given defined case number cycles length computed case difference case one overlap solely belongs hbp first replica two overlaps cross hbp second replica see fig thus note operator used avoid counting options valid number distinct solutions optimal vectors fig different cases cycle length red binary protograph upper panel lower panel case vns spanning main idea theorem computed decomposing four tractable terms term represents distinct case existence cycle length binary protograph union cases covers existence possibilities case characterized locations cns vns comprising cycle respect hbp replica replicas fig illustrates eight cases along terms corresponds case remark consider special situation rows hbp overlap reduces simply number ways select one position overlapping set pair define minimum number cycles length graph hbp binary protograph thus discrete optimization problem formulated follows min lemma total number partitioning choices code parameters binary protograph given optimal vector given constraints optimization problem conditions overlap parameters valid thus constraints seven parameters last constraint guarantees balanced partitioning hbp needed prevent case group elements group either hbp hbp involved significantly cycles remaining elements remaining solution optimization problem unique however since solutions result number partitioning choices work one solutions call optimal vector lemma gives total number partitioning choices proof goal find number partitioning choices achieve general set overlap parameters necessarily optimal particular need find number different partitioning choices code number row size overlapping set rows hbp size overlapping set rows overlap hbp factorize number partitioning choices three tractable factors choose positions row hbp positions number choices choose positions row hbp positions among positions exist positions row simultaneously number choices choose positions row hbp positions among positions exist positions rows simultaneously positions rows simultaneously positions rows simultaneously number choices conclusion number partitioning choices achieve general set overlap parameters solution optimization problem unique distinct solutions optimal vectors achieve symmetry optimal vectors corresponds partitioning choices factors obtianed replacing optimal vector equations respectively thus total number partitioning choices given optimal vector proves lemma remark first seven constraints optimization problem stated easily verified lemma replacing irculant ower ptimization picking optimal vector partition design hbp run heuristic cpo reduce number ugasts graph hbsc steps cpo initially assign circulant powers codes hbp results cycles length hbsc design hbp hbp using contains two replicas circulant powers hbp copied locate cycles lengths specify cycles length hbp satisfied call active cycles let fda number active cycles vns spanning compute number ugasts hbsc using following formula fsc lfsa fda hbp count number active cycles involved give weight number active cycles vns spanning map counts step hbp sort list descendingly according counts pick subset top list change circulant powers associated using interim new powers steps fsc reduced maintaining cycles length hbsc update fsc circulant powers step otherwise return step iterate target fsc achieved note step performed heuristically hbp fig partitioning code example entries circles squares assigned hbp circulant power arrangement circulants example suppose want design code using partitioning cpo solving optimization problem yields optimal vector gives cycles length graph hbp fig shows partitioning applied hbp next applying cpo results ugasts unlabeled graph code graph hbsc fig shows final circulant power arrangement circulants technique designing hbsc based solving set equations applying heuristic program two replicas optimize circulant powers moreover partitioning orders magnitude fewer number partitioning choices compared partitioning see lemma even use choice partitioning choices without compare performances explicitly reasons demonstrate technique better performance see section details also much faster technique wcm ramework emoval gast applying technique optimize unlabeled graph code optimize edge weights particular use wcm framework remove gasts labeled graph code edge weight processing multiple parameters control difficulty removal certain gast tanner graph code number distinct wcms associated ugast minimum number edge weight changes needed remove gast denoted egast min among parameters third parameter number sets edge weight changes cardinality egast min candidates gast removal process first two parameters studied discuss third parameter section number candidate sets cardinality egast min increases difficulty gast removal decreases section unless otherwise stated say nodes connected mean directly connected neighbors applies conceptually say edge connected node vice versa remark gast removed performing egast min edge weight changes edges connected cns see also whether candidate set edge weight changes indeed results gast removal determined checking null spaces wcms minimize number edge weight changes performed remove gast need work vns connected maximum number unsatisfied cns egast min bvm see bvm maximum number existing unsatisfied cns per gast define emu topological upper bound egast min maximum number existing cns per gast thus emu egast min note follows bvm section study gasts means upper bound achieved egast min emu moreover simplicity assume vns connected cns connected cns degree theorem consider gast code defined column weight cycles length number sets edge weight changes cardinality egast min emu candidates gast removal process given follows smu avm emu emu avm number vns connected cns smu avm nco nco number cns connecting two avm vns proof whether minimize number edge weight changes need target vns connected maximum number unsatisfied cns definition since number vns type avm connected unsatisfied cns case general case avm pertinent vns different emu ways selecting emu satisfied cns connected cns edges change weights simultaneously moreover edge different new weights excluding current weight thus number candidate sets emu emu smu avm emu rephrased version case emu gast removed single edge weight change moreover substituting emu gives number candidate sets follows inequality smu avm equality achieved shared cns vns unsatisfied cns otherwise nco subtracted proves avm note subtraction nco needed reason emu multiple edges connected exist candidate set additionally since codes girth least exist one connecting two vns gast fig gast gast appropriate edge weights assumed example consider gast fig gast avm egast min emu moreover nco one shared two vns two unsatisfied cns thus number candidate sets cardinality smu contrarily gast fig avm egast min emu thus general relation number candidate sets cardinality smu ode esign teps imulation esults section present code design approach flash memories experimental results demonstrating effectiveness steps approach specify code parameters solve optimization problem optimal vector overlap parameters using hbp apply circulant power optimizer reach powers circulants hbsc binary image hbsc designed assign edge weights generate next partition using couple components construct hsc using initial simulations practical flash channel combinatorial techniques determine set gasts removed graph hsc use wcm framework see algorithm remove many possible gasts section results proposed best achieved two techniques table umber ugast codes designed using different techniques design technique uncoupled best number ugasts start experimental results table comparing number ugasts codes designed using various techniques codes codes used underlying block codes code design techniques comparing proposed technique table demonstrates reductions number ugasts achieved technique technique technique ranges intriguingly table shows technique provides lower number ugasts best achieved underlying block codes used note best reached using exhaustive search reason could provide counts next provide simulation results verifying performance gains achieved code design approach flash memories flash channel use practical flash channel mixture nlm flash channel use reads sector size bytes define rber raw bit error rate uber uncorrectable bit error rate one formulation uber recommended industry frame error rate fer divided sector size bits simulations done software high speed cluster machines codes simulated defined block length bits rate code uncoupled code designed using technique code designed using technique cpo applied underlying block codes codes codes code designed using technique edge weights codes selected randomly code code result applying wcm framework code code optimize edge weights code code code ugasts additionally code code code ugasts ugast second common substructure dominant gasts codes simulated flash channels uber uncoupled rand weights uncoupled opt weights rand weights rand weights rand weights opt weights rber fig simulation results nlm flash channel codes designed using different techniques fig demonstrates performance gains achieved stage code design approach code outperforms code order magnitude gain first stage code outperforms code order magnitude gain second stage cpo code outperforms code orders magnitude gain third stage wcm moreover figure shows code designed using approach code achieves rber gain compared code code practical flash channel intriguing observation encountered performing simulations change error floor properties code code particular gast dominant object case code encountered gasts error profile code vii onclusion proposed combinatorial approach design codes optimized practical flash channels oocpo technique efficiently optimizes underlying topology code wcm framework optimizes edge weights codes designed using approach reduced number detrimental gasts thus outperforming existing codes flash channels proposed approach help increase reliability ultra dense storage devices emerging flash devices acknowledgement research supported part grant astcidema nsf eferences wang vakilinia chen courtade dong zhang shankar wesel enhanced precision multiple reads ldpc decoding flash memories ieee sel areas vol may maeda kaneko error control coding multilevel cell flash memories using nonbinary codes proc ieee dfts chicago usa hareedy lanka dolecek general ldpc code optimization framework suitable dense flash memory magnetic storage ieee sel areas vol parnell papandreou mittelholzer pozidis modelling threshold voltage distributions nand flash memory proc ieee globecom austin usa hareedy lanka guo dolecek combinatorial methodology optimizing codes theoretical analysis applications data storage jun online available http felstrom zigangirov periodic convolutional codes matrix ieee trans inf theory vol kudekar richardson urbanke spatially coupled ensembles universally achieve capacity belief propagation ieee trans inf theory vol pusane smarandache vontobel costello deriving good ldpc convolutional codes ldpc block codes ieee trans inf theory vol mitchell dolecek costello absorbing set characterization spatially coupled ldpc codes proc ieee isit honolulu jun iyengar papaleo siegel wolf corazza windowed decoding ldpc convolutional codes erasure channels ieee trans inf theory vol apr esfahanizadeh hareedy dolecek codes optimized magnetic recording applications ieee trans vol esfahanizadeh hareedy dolecek novel combinatorial framework construct codes minimum overlap partitioning proc ieee isit aachen germany jun bazarsky presman litsyn design quasicyclic ldpc codes ace optimization proc ieee itw sevilla spain fossorier codes circulant permutation matrices ieee trans inf theory vol

jan characters finite simple groups gunter malle alexandre zalesski abstract let finite group prime let sylow character called sylp restriction character regular representation addition vanishes elements order divisible said every finite simple group determine primes admits character except alternating groups characteristic moreover determine primes projective dimension algebraically closed field characteristic introduction let finite group prime let sylow character called sylp every additionally say sylp whenever divisible called additionally say sylp characters chevalley groups defining characteristic studied specifically simple groups lie type characteristic except characters prime determined main motivation study kind characters connection characters projective indecomposable modules study projective indecomposable modules dimension initiated malle weigel obtained full classification modules arbitrary finite simple groups assuming character module trivial character constituent restriction removed simple groups lie type defining characteristic parts proofs valid characters projective modules also even sylp characters paper complete classification projective indecomposable modules dimension simple groups first main result classification characters simple groups sole exception alternating groups prime theorem let finite simple group prime dividing let character respect one following holds irreducible triple proposition sylow cyclic proposition lie type characteristic see date january mathematics subject classification key words phrases characters projective indecomposable modules characters first author gratefully acknowledges financial support sfb trr gunter malle alexandre zalesski fact many instances even classify sylp characters examples case presented corollaries aware examples second main result determines reducible projective modules simple groups minimal possible dimension theorem let finite simple group prime dividing sylow reducible projective dimension one following holds psln odd prime note irreducible projective dimension bijection irreducible characters defect degree listed proposition simple groups paper built follows preliminaries recall classification irreducible characters section proposition section classify sylp characters case cyclic sylow proposition section treat sporadic groups theorem alternating groups handled section theorem odd section partial results see theorems exceptional groups lie type considered section theorem rest paper deals classical groups lie type start section ruling remaining possibilities defining characteristic case large sylow psubgroups primes settled section section discuss small cases proof main theorems achieved section treating case preliminaries start fixing notation let finite field elements algebraic closure cardinality set denoted greatest common divisor integers denoted prime write finite group irr set irreducible characters set linear characters degree denote trivial character regular character write sylp mean sylow group order coprime called denote center derived subgroup respectively subgroup denote centraliser normaliser respectively character write restriction maximal integer proper character prime fixed sylp groups cyclic sylow irreducible characters studied characters finite simple groups respectively inner product characters denoted sometimes character induced character denoted let finite groups normal subgroup let field becomes called generalised brauer ordinary restriction denoted brauer ordinary character afforded character also write let minimal integer divisible odd set next two lemmas follow definitions finite group sylp lemma let sylp character every linear character occurs multiplicity particular abelian multiplicity free reg proof follows corresponding properties lemma let direct product let irreducible characters respectively product lemma let normalised let faithful character abelian proof let every character character projective module lemma module question indecomposable induced irreducible character say thm follows let derived subgroup normal therefore lies kernel since hence faithful abelian claimed note take abelian follows representation afforded consists scalar matrices faithful required thus simple group necessary condition character remark normalised sylow called theory finite groups thus lemma tells admits faithful character every abelian lemma let finite group subgroup normal let let sylow respectively let character reg words particular sylp character character let sylp character character gunter malle alexandre zalesski proof assume reg follows coincides whence claim show vanishespat elements let afforded observe projection thenp thus element follows assumption whence claim obvious lemma let direct product suppose every sylp character every sylp character proof let sylp set let sylp generalised restriction lemma character let sylp character assumption whence result lemma let let character irr distinct characters addition character proof write irr distinct characters reducible general let let characters linearly independent follows every addition let whence required corollary let lemma sylow let irreducible constituents suppose every character proof let lemma assumption reg reg subcharacter therefore subcharacter multiple result follows proposition let finite group normal subgroup cyclic let character character conjugacy classes coincide proof let irr linear character generates irr elements vanishes follows thus write characters finite simple groups linear combination irreducible characters irr constant orbits multiplication clearly suffices show claim single orbit say irr minimal set ker irreducible note thus follows generates irr vanishes hence induced irr claimed define character gxg well know sum characters suitable assumption hence whence hence follows whence result remark let proposition necessarily indeed let hci cyclic group order let square root define irr character let regular dihedral group order normal subgroup one observes hence however character corollary let proposition let character suppose every irreducible character degree character particular characters neither proof proposition character clearly assumption every irreducible constituent therefore claim follows proposition lemma let finite group normal subgroup index suppose integer every character every character proof suppose contrary let character induced character contradiction following fact well known lemma let finite group normal subgroup index let algebraically closed field characteristic let projective indecomposable projective indecomposable proof induction sends projective modules projective modules furthermore green indecomposability theorem thm induction normal subgroups index preserves indecomposability indecomposable direct summand projective projective indecomposable statement also follows assumption gunter malle alexandre zalesski irreducible characters simple groups complete list irreducible characters simple groups degree suffices extract characters degree list irreducible characters degree obtained thm list already appeared prop case inadvertently omitted note irreducible character sylp proposition let simple group suppose irreducible sylp character one following holds simple group lie type characteristic steinberg character even odd psln odd prime psun odd prime odd prime prime problem determining minimal degree irreducible characters looks much complicated remark let point following cases explicitly mentioned proposition cyclic sylow section determine reducible characters simple groups cyclic sylow proposition let finite group cyclic sylow assume abelian irr proof let assumption abelian order prime irreducible characters degree hence corresponding pims dimension since brauer tree star pims uniserial vii cor thm indecomposable characters finite simple groups sufficiently large field characteristic quotient pim dimension strictly smaller projective let irr zero multiple claim follows else lies block full defect exists indecomposable lift thm projective hence dimension divisible green correspondent indecomposable nonprojective vii lem thus dim said claim follows lemma let simple group let prime sylow psubgroup cyclic let denote minimal degree irreducible character except case mod proof values every simple group either known explicitly good lower bound sporadic simple groups one inspect alternating groups simple groups lie type values listed lemma follows comparison data proposition let prime let simple group cyclic sylow let sylp character one following holds irreducible degree irreducible mod distinct irreducible characters degree proof suppose reducible result easily follows computation character table group suppose let irreducible constituent lemma therefore constituent lemma proposition let prime let simple group cyclic sylow reducible sylp character one following holds even psln odd prime psun odd prime furthermore case addition irreducible character unless possibly holds may sum two irreducible constituents equal degree proof additional statement follows proposition reducible case proposition may assume irreducible thus irreducible characters level determined thm belongs list thm drop list characters degree gunter malle alexandre zalesski remaining cases given statement proposition note list thm includes groups one first needs delete representations center instance odd hence even however every irreducible representation even degree faithful words irreducible representation even degree contrast exist irreducible representations psln psun odd degree prove converse show case sylp let representations afforded let hsi cor multiplicity every eigenvalue det follows eigenvalue therefore required next show cases follows inspection cases trivial cases one take preimage say sln sun respectively irreducible natural module groups described huppert easily follows unless order consideration sylow let contradiction sylow follows every element either therefore lemma assumptions notation proposition unique unless holds character projective module holds proper character minimal degree character either holds holds proof let irreducible unless holds show irreducible character degree unique unless holds follows character table group well known psln psun observed table case number characters equals number irreducible characters degree mod otherwise three characters see recall principal projective indecomposable module pim whose character contains constituent characters proposition contain constituent therefore character projective module say indecomposable principal compare list characters proposition main result comparison rules case proposition furthermore admits least two characters one character projective module leaves cases cases unique must character principal projective indecomposable module listed follows inspection table characters finite simple groups remark group several sylp characters one projective sporadic groups theorem let sporadic simple group reducible sylp character unless one following holds four characters constituents degrees six characters none proof groups primes conjugacy class taking strictly positive value irreducible characters degree cases like one solve little linear system equations integral solutions cases solutions exist listed statement note cases occur also proposition alternating groups section consider characters alternating groups alternating groups odd primes give short proof using recent result giannelli law replaces earlier direct proof lemma let irr addition fact also observed proof first part proposition addition irreducible character degree implies claim lemma let let let character equivalently commuting proof lemma trivial let let subgroups set characters distinct irreducible characters lemma induction lemma hence either unique irreducible character degree may see suppose lemma false rearrange get characters follows well every irreducible constituent contains irreducible constituent distinct well known easily follows branching rule implies recall single character degree therefore let order implies gunter malle alexandre zalesski suppose two irreducible characters degree let denote therefore assuming lemma false write let primitive root unity integers implies lemma follows unless argument applies lemma let odd let hook partition corresponding character takes positive value proof well known hook character mth exterior power irreducible reflection character constituent degree natural permutation character let young subgroup clearly thus min clearly positive since binomial coefficients strictly increasing middle observe suffices prove claim since restriction hook character contains hook characters done since symmetry may assume theorem let odd max sylp regular character every sylp character irreducible unless proof sylow cyclic claim proposition assume let sylow first assume main result restriction irreducible character contains trivial character moments thought shows true restriction irreducible character lemma sylp character irreducible assume thm irreducible characters whose restriction contain trivial character two characters degree irreducible character whose restriction contain trivial character degree hence sylp character form irr let rule irreducible character takes value particular reducible parametrised hook partition degree takes positive values lemma contradiction finally cases easily checked individually example irreducible characters degree class vanish positive either class character sylow also deals characters finite simple groups corollary let finite group suppose subgroup containing sylow max character proof follows lemma theorem alternating groups situation complicated case complete results part due existence infinite family examples construct set irreducible character corresponding partition young diagram hook leg length lemma let even fixes letters moved let irr correspond hook partition proof one observes restriction sum irreducible characters irreducible characters hook characters respective groups see lemma next use lemma states irr young diagram skew hook leg length case hook rim definition skew hook connected either row column case hence column hook length leg length odd even respectively proper diagram claimed proposition suppose even character integer proof let even order suppose first cycle length lemma suppose cycle length express product cycle even size say element fixing letters moved lemma gunter malle alexandre zalesski induction write induction statement follows corollary let even character character proof characters remain irreducible restriction follows therefore proposition elements even order last claim follows proposition suppose odd set observe provided therefore proposition suppose odd characters integer proof let even order cycle even size lemma proper diagram proper diagram set second sum written whence similarly proper diagram proper diagram set second sum written well characters finite simple groups see proof proposition proposition let odd irreducible whereas sum two irreducible constituents denote set set corollary let characters characters let characters none proof let even order let remains irreducible restriction coincide restriction follows hence character observe follows thus proposition therefore characters addition suppose let follows hence character addition never consider observe follows proposition therefore characters steinberglike lemma let addition character characters characters sum irreducible characters gunter malle alexandre zalesski irreducible character degree sum two irreducible characters degree proof easily checked know character tables use computer program possibilities one checks character exists right restriction similarly one considers restriction finally cases treated restricting theorem suppose character one constructed proposition characters unless latter case characters listed proposition proof let character argue induction steinberg like character unless power assume power write distinct exponents lemma may assume one summands say different young subgroup contains sylow lemma irr distinct characters character thus assumption character proposition particular well possible unless well latter case lemma conclude argument shows character particular hence also possibly interchanging may assume consider sum hooks branching rule character restricted contains character except excluded thus inductively contain constituent turn means constituents hooks thus induction observe rule lemma contains thus one hand side common constituent every second hook character occurs hand every second hook must indeed occur thus either defined claim follows proposition otherwise degree larger projective characters lemma let projective character degree projective character degree proof follows lemma theorem let reducible projective character degree proof one inspect decomposition matrix modulo observe projective character degree analysing character table characters finite simple groups one observes characters hence pim dimension one inspect decomposition matrix observe minimal dimension pim analysing character table one observes characters hence pim dimension let using known character table one finds unique regular character viz character constituents degrees recall principal pim one constituent however principal pim degree let subgroup normal indeed let partition parts size direct product copies sylow take subgroup question semidirect product latter permutes natural way one easily observes odd pim degree lemma generalised restriction pim dimension pim exist seen let write let index odd pim degree therefore direct product pim obviously dim contradiction pim degree result follows lemma exceptional groups lie type theorem let simple group lie type classical sylp character characteristic except group two reducible characters two irreducible characters proof proof thm compare maximal order sylow bounded order normaliser maximal torus smallest irreducible character degrees given example tab shows except small examples sylp characters closer inspection finitely many remaining cases shows two reducible characters two irreducible character cases arise groups lie type defining characteristic shown simple groups lie type sufficiently large rank characters respect defining characteristic apart irreducible steinberg character precisely characters classified except groups types deal remaining cases gunter malle alexandre zalesski proposition let odd reducible character respect proof freely use results methods first assume according prop suffices consider group coming algebraic group connected centre let reducible character linear constituent thm multiplying inverse character may assume trivial character occurs exactly lemma constituents belong principal may fact replace may assume adjoint type let parabolic subgroup lemma restriction rlh character show possibility compatible restriction levi subgroups clearly rlh also contains trivial character reducible reducible characters proper levi subgroups known lemmas particular must type rlh irr degree thus lies lusztig series regular semisimple element dual group centraliser maximal torus order thus contain constituent lying lusztig series easily seen centraliser either maximal torus type correspondingly first last bigger contains constituent apart principal series generalised restriction contradicting rlh next restriction levi subgroup type form rlh particular lies lusztig series regular semisimple element order dividing centraliser maximal torus order centraliser either maximal torus type correspondingly constituent lusztig series degree last one larger furthermore lemma contains least one regular constituent either degree one check known list character degrees found regular character small enough degree degree observe sum remaining character degrees note unipotent constituents since would lead unipotent constituents rlh cuspidal unipotent characters turns remaining candidates except one degree degree divisible mod mod follows would occur least times possible contradiction concludes proof case characters finite simple groups cases follow previous one application inductive argument proof thm proposition let reducible character respect proof first consider case previous proof prop lemma may work adjoint type let reducible character contains thm hence restriction rlh levi subgroup type lemma mod rlh cuspidal character labelled regular element torus order order dividing also regular constituent degree holds three conjugacy classes levi subgroups type comparison degrees shows possible case follows previous one application argument proof thm classical groups large rank application results obtained section show classical groups large rank steinberg like character provided defining characteristic throughout odd prime dividing set order modulo first illustrate method groups gln lemma let gln let sylow write exist subgroups abelian normal max character proof see contains subgroup contains sylow result follows lemma let suppose contrary let character lemma must character however false theorem classical groups argument similar involves technical details let order modulo equivalently odd mod equivalently note even lemma let gun sylow isomorphic odd gln mod lemma let gun let sylow suppose mod equivalently odd write exist subgroups abelian normal max character gunter malle alexandre zalesski proof suppose first let natural direct sum subspaces dimension let stabiliser decomposition xvi semidirect product factors let sylow normal abelian well known contains sylow therefore satisfies statement let odd embedding gum gumd see hilfssatz note gumd isomorphic subgroup result follows similar proof lemma lemma let even gum even resp odd isomorphic subgroup resp isomorphic subgroup well addition contains sylow respective group proof follows lemma well second case follows groups contain subgroups isomorphic additional statement read orders groups question cases considered lemmas similar lemma let let one following groups gun odd even even either either even odd let even odd even let denote sylow exist subgroups abelian normal proof case mod handled lemma remaining cases result follows lemmas isomorphic subgroup lemma result follows lemma follows lemmas note contains subgroups isomorphic one contains sylow similar note contains subgroups isomorphic one contains sylow case subgroup isomorphic respectively contains sylow one easily check result follows characters finite simple groups result alternating groups corollary implies following proposition let let max let one following groups psln psun odd odd otherwise odd otherwise odd otherwise character remains true group normal abelian proof suppose first lemma let sylp subgroups normal max perfect let derived subgroup set sylp perfect similar statement true quotient central subgroup result follows theorem using lemma minimal characters sylow section show simple classical group satisfying assumptions proposition defining characteristic sylow cyclic sylp character hence character observe cyclic abelian lemma case cyclic dealt section group let denote sequence integers irreducible character degree irreducible character universal covering groups finite classical groups values determined analysis three values play significant role mainly classical centerless groups pgln pgun groups mainly observe sometimes latter case immediate conclude sylp character cases observe exists element order irreducible character degree use different method recall denotes minimal integer divisible groups gln set let sln minimal degrees projective irreducible representations psln given table table obtained omitting representations realisable ordinary representations sln lemma let glen suppose suppose either sylp character proof coprime statement obvious case gunter malle alexandre zalesski table minimal degrees irreducible characters sln let exceptions table ignored except two cases trivial less remark lemma extend case case leaves deal next lemma let glep sylp character proof table odd exceptional cases let permutation character associated action vectors natural character degree unique irreducible character degree table hence coincides let sylp character follows irreducible constituents either every get contradiction soon show equivalent showing easily verified lemma suppose let sln gln sylp character proof let sylp character neither lemma groups gun section consider case sylow gun abelian implies order modulo equivalently odd mod equivalently note even characters finite simple groups lemma let odd let sylow gun abelian cyclic proof gln sylow abelian let sylow use lemma mod abelian odd abelian equivalently abelian equivalently similarly cyclic lemma follows lemma let sudp gudp sylp character remains true sun gun proof note means divide suffices prove lemma sudp first assume odd mod case similarly odd finally assume implies sylp character proof additional statement similar thus left primes first consider case lemma let sun gun sylp character proof let det element sylow follows lemma result follows show sylp character turn follows result sun coprime deal suppose contrary let sylp character first let table let single irreducible constituent table sylp lemma therefore false case read character table let let natural let orthogonal basis let subspace dimension set hbi hbi every element acts scalarly let derived subgroup sylp character contains sylow result follows lemma lemma let character proof lemma suffices prove character suppose contrary let character groups gunter malle alexandre zalesski let irreducible constituent characters degree positive class whereas degree vanish class follows positive class contradiction lemma let sup slp primitive root unity let diag irreducible character whose kernel order prime proof element written orthogonal basis underlying vector space unitary case extraspecial group order restriction direct sum irreducible representations well known easily checked character every representation vanishes claim follows let gun gln weil representations groups studied howe authors many applications mainly due fact irreducible constituents call irreducible weil representations essentially exhaust irreducible representations degree details given odd let underlying space weil representation general irreducible except case glp case sum irreducible subspace irreducible weil representations dimension greater parameterised restriction remains irreducible restriction sun sln every irreducible representation degree irreducible weil representation moreover every irreducible representation degree obtained irreducible weil representation tensoring onedimensional representation lemma let gup glp let irr let character irreducible constituent weil representation labeled let lemma except case glp addition element order proof consider case gup case glp similar let irr irreducible constituent labeled means let let character hhi fixed pth root unity multiplicity eigenvalue equals recall multiplicity eigenvalue matrix gup therefore zhk also characters finite simple groups compute note integer let subgroup order hzp first show second sum equals note equivalent therefore fixed partial sum claimed next compute observe sum simplifies note therefore let last sum equals let last sum equals therefore particular eigenvalue module fixed multiplicities trace equals similarly trace question equals words character otherwise lemma let gup glp sylp character true sup slp groups proof set lemma suffices prove lemma place suppose contrary let sylp character let irreducible constituent first observe hence gunter malle alexandre zalesski indeed note unitary case respectively linear case table value greater let unless former case settled lemma let case implies character degree positive class contradiction mentioned prior lemma either seen character obtained irreducible weil character tensoring linear character let lemma tensoring ignored assume irreducible weil character lemma unitary case linear case every irreducible constituent false trivial definition hence lemma contradiction irreducible weil representations remain irreducible upon restriction argument works intermediate groups lemma let let group sln gln sun gun sylp character proof unitary case result stated lemma case dealt lemma remaining case examined lemma let gln result follows lemma proved lemma result stated lemma case examined lemma statement follows lemma lemma let gln sln gun sun let group sylp character unless proof suppose contrary let sylp character suppose first note subgroup say isomorphic slep sudp let sylow let slep sudp lemma sylp character slep sudp contradicts lemma unless possibly sun case occur next suppose refine argument set glp gup set normal hence follows hence normal subgroup lemma sylp character however lemma characters finite simple groups sylp character unless left case gun group excluded assumption consider first suffices deal suppose contrary let character let irreducible constituent element class unless constituent say pick class unless must constituent say degree hence follows contradiction constituents degree particular constituents implies must occur multiplicity whence contradiction let lemma suffices deal set irreducible characters degree less positive class addition irreducible characters degree less equal positive class let character irreducible character degree constituent note irr sum constituents degree trivial character occur multiplicity greater must multiple false let irr note multiplicity occurs multiplicity sum constituents irreducible characters degree degrees well positive class contradiction suppose occurs sum constituent values class follows constituents may degrees inspecting one observes well positive class contradiction multiplicity must sum constituent values class therefore degrees constituents may let character degree sum constituent values class trivial character one whose value occur twice get contradiction therefore contradicts completes analysis case let contains subgroup isomorphic contains sylow result case follows addition done lemma similarly result follows remark group irreducible projective character degree hence projective character degree theorem let group sln gln sun gun suppose sylow cyclic character unless gunter malle alexandre zalesski proof linear case unitary case result follows proposition place view lemma linear case unitary case result follows lemma linear case unitary case result follows lemma linear case unitary case sylow cyclic remark proposition gives better bound yield essential advantage cases covered proposition use lemma anyway symplectic orthogonal groups lemma let even odd suppose sylow abelian proof let sylp abelian even see table greater odd whence result cases similar see thm proposition let odd let suppose sylp character proof let sylp lemma conjugate sylow subgroup gln lemmas gln sylp character unless possibly let even greater whence result odd proposition let even let odd suppose sylp character proof write integer contains subgroup spke respectively suffices prove lemma let lemma sylow contained subgroup isomorphic guk lemma lemma place group guk sylp character whence claim exceptional case considered let even greater whence result odd whence result characters finite simple groups similar argument works well odd even except irreducible characters let degree less degrees let sylp character degree irreducible constituent whenever contradiction let result follows case examined suppose even odd sylow contained subgroup isomorphic respectively note groups result proven except cases however sylow hence cyclic case examined propositions let even otherwise unless let irreducible characters degree less degrees therefore characters occur irreducible constituents character however values characters element class particular positive contradiction suppose let irreducible constituent let belong conjugacy class notation inspection character table one observes whenever therefore every irreducible constituent implies see however character takes positive values elements class case ruled remark let universal covering group one observes character characters steinberglike characters reducible irreducible classical groups section investigate sylp characters simple classical groups fields odd order prime linear unitary groups first deal smallest case proposition let odd let reducible character let reducible character proof mod else smallest character degree first case second follows characters first case unless second case follows gunter malle alexandre zalesski character table sum trivial steinberg character power cases otherwise two reducible characters degree let reducible character let lemma character irreducible proposition also true reducible irreducible characters degree indeed using character table one observes exist irreducible characters vanish follows reducible character recall denotes third smallest degree irreducible representation lemma let group psln psun odd proof let first odd table mod respectively mod thus unless latter case claim follows next let odd larger assume psln see table odd mod mod claim follows let odd table mod respectively mod thus unless latter case claim follows let odd suppose first whereas suppose less assume psun odd mod respectively mod claim follows finally assume psun even bounded given conclude proposition let psln psun odd character character proof lemma suffices prove result case assume equals order sylow let first psln odd let character lemma hence irreducible constituents degree characters finite simple groups see table known irreducible characters degree induced characters character stabiliser line underlying space gln irreducible character degree unique constituent permutation character let let sln matrix corresponding primitive element determinant corresponding element order since eigenvalue conjugate contained induced characters vanish particular note image even order write sum induced characters degree suitable evaluating see divisible odd prime equal proves part psln assume easy estimate shows may assume addition either power let order observe conjugate element thus may argue conclude candidate characters degrees clearly character consider case proof lemma shows unless latter case possible constituents degrees clearly one degrees contribute necessarily case character table shows character completes proof psln let psun odd let character according lemma hence irreducible constituents degree see table first semisimple characters lying lusztig series element order dual group pgun centraliser second unipotent character say corresponding character weyl group parametrised partition let sun regular element even order maximal torus order see lemma conjugate dual maximal torus contains characters vanish see prop unipotent value sign irr labelled permutation cycle shape see prop remark prop rule gives gunter malle alexandre zalesski may argue first part conclude thus completing proof next assume odd power assume power hence particular mod three smallest character degrees trivial character occurring easily seen integral solution possible decomposition three smallest degrees three smallest degrees neither case characters either finally proof lemma shows must possible constituents degrees easy consideration shows case needs special attention existence characters ruled known character table treat case considerably delicate lemma let reducible characters proof irreducible characters degree less take values class characters irreducible characters degree large since smallest character degree degrees constituents character thus need consider characters degrees clearly degree occur either mod see character degree appear least three times values elements order give contradiction remark irreducible characters see proposition lemma let let character proof suppose contrary note reducible let irreducible constituent maximal degree numerical data see let otherwise false let irr unless unless let indeed otherwise irreducible constituents degree implies contradiction follows let irr irreducible characters degree distinct addition follows contradiction characters finite simple groups let irr irreducible characters degree degree contradiction suppose obtain positive value class consideration rules suppose occurs otherwise occurs times false contradiction option exists irreducible characters degree less positive lemma let let character proof lemma proper character inspection character table see easily checked conjugacy class element order lemma every means lemma lemma let let character let irreducible constituents disregarding multiplicities proof let thus suppose contrary assume note otherwise hence character multiple character degree property note irreducible characters degree extend except degree degree degree degree corresponding characters degrees respectively let aip integers suppose computing get contradiction suppose whence computing get contradiction computing get contradiction unless case computing gives contradiction violates iii let whence conflicts computing yields contradiction let hence yields contradiction let violates let irreducible character degree less negative value follows note follows character degree occurs implies get contradiction gunter malle alexandre zalesski let computing leads contradiction lemma let character let irreducible constituents disregarding multiplicities proof let suppose contrary assume note every linear character therefore constituent let implies follows must contain least representations degree contradicts either equals unless violates unless contradiction lemma let let character proof lemma claim holds induction assume true lemma irr characters character induction lemmas applied lemma proposition let let character proof let direct product copies let character lemmas let semidirect product acts permuting factors contains sylow let index odd note see proof proposition result follows groups theorem let one glm slm pslm gum sum psum character moreover character proof let first glm gum mod result stated proposition let let sylow gll gul set contains sylow therefore lemma character character proposition result follows groups slm sum result follows lemma pslm psum statement follows lemma orthogonal symplectic groups let natural module odd let dimension fixed point subspace let characters finite simple groups denote weil character howe prop let irr lemma let semisimple fix vector let subspace dimension let element gvi fix vector let constant elements proof therefore prop whence claim prop whence claim choose element stabilising coincides matrix similar element satisfying hence let images elements kernel viewed character greater follows proposition let odd characters proof divides divides let minimal hand thm larger unless let set aside cases moment otherwise constituents either weil characters trivial character note weil character degree centre kernel degree odd constituents degree mod odd else according lemma trivial character occurs never power odd consider zsigmondy prime divisor trivial character must occur exactly let denote two weil characters interchanged outer diagonal automorphism observe induced element thus fixes involution classes let involution write number constituents necessarily compare degrees see gunter malle alexandre zalesski discuss two exceptions irreducible characters degree take values class character irreducible characters degree take positive values class except one degree takes value one degree takes value one latter two characters could occur character proposition let odd characters proof according thm larger unless either character degree less semisimple character degree see since trivial character occur regular character see example arise character degree less semisimple character degree see lead example character degree less character degree conclude proposition let odd characters proof second smallest character degree given see thm larger unless leaving cases aside moment see character multiple smallest character degree plus possibly trivial character arguing case symplectic groups see characters take value involutions constituents character could degree integral linear combination three degrees appearing adds second smallest character degree see thm larger conclude left case lemma let character proof irreducible characters degree take positive value class see let irreducible characters degree positive elements conjugacy class let irreducible characters degree take positive values class result follows implies result lemma let let character irreducible constituents disregarding multiplicities set characters finite simple groups proof suppose first note suppose contrary use notation characters degree less maximal degree among one irreducible character degree less negative degree positive must constituent character extends constituents degrees follows fact indeed degree characters positive well degree violates thus hence furthermore computing character table program computer package gap one observes distinct irreducible characters degree one irreducible character degree follows characters differ multiplication linear character one observes every linear character therefore must constituent constituents degree contradicts inequality thus irreducible characters hence positive contradicts let note suppose contrary irreducible characters degree less characters positive violates pgo psp lemma let let character let let character proof result contained lemma induction assume true hnp lemma irr characters character induction lemma applied lemma follows lemma lemma let character proof let direct product copies let character lemma let semidirect product acts permuting factors contains sylow let index odd note see proof proposition result follows groups result follows lemma statement follows lemma proposition let character gunter malle alexandre zalesski proof mod result stated lemma let let sylow set contains sylow let character therefore lemma character character lemma result follows proposition let character proof let let contains subgroup isomorphic one concludes contains sylow let sylow set contains sylow lemma character character lemma result follows proposition let character proof case dealt lemma assume let let contains subgroup isomorphic contains sylow set sylow contains sylow lemma character character lemma result follows theorem let let derived group let group character proof statement follows lemma propositions using lemma lemma collect results prove main theorems introduction proof theorem assume finite simple group possessing steinberglike character respect prime cases irreducible recalled proposition sylow cyclic proposition may assume sylow alternating odd cases theorem except characters sporadic groups listed theorem thus lie type case defining prime handled propositions respectively assume defining prime groups exceptional lie type handled theorem classical groups large rank odd result contained proposition cases psln psun completed theorem classical groups propositions finally cases covered proposition proposition psln psun theorem psln psun propositions classical groups theorem case characters finite simple groups proof theorem characters projective dimension particular order prove result need list given theorem sylow cyclic possibilities given lemma lie type characteristic see thm case theorem subsumed statement finally alternating groups discussed theorem references conway curtis norton parker wilson atlas finite groups clarendon press oxford curtis reiner methods representation theory applications finite groups orders wiley new york emmett zalesski regular orbits elements classical groups permutation representations comm algebra feit representation theory finite groups amsterdam giannelli law permutation characters sylow shoke zalesski groups fusion systems algebra howe character weil representation trans amer math soc huppert klassischen gruppen math james representation theory symmetric groups berlin lassueur malle simple endotrivial modules linear unitary exceptional groups math lassueur malle schulte simple endotrivial modules groups reine angew math character degrees multiplicities groups lie type rank available http malle weigel finite groups minimal manuscripta math malle zalesski prime power degree representations groups archiv math navarro characters blocks finite groups cambridge univ press cambridge hung ngoc nguyen low dimensional complex characters symplectic orthogonal groups comm algebra pellegrini zalesski characters chevalley groups vanishing elements internat algebra comput rudloff zalesski multiplicity eigenvalues elements irreducible representations finite groups group theory tiep zalesskii minimal characters finite classical groups comm algebra weir sylow classical groups finite fields characteristic coprime proc amer math soc zalesski minimal polynomials eigenvalues representations groups cyclic sylow london math soc zalesski low dimensional projective indecomposable modules chevalley groups defining characteristic algebra zalesski remarks characters simple groups archiv math gunter malle alexandre zalesski mathematik kaiserslautern postfach kaiserslautern germany address malle department physics informatics mathematics national academy sciences belarus minsk belarus address

stability integral delay equations stabilization models iasson karafyllis miroslav krstic dept mathematics national technical university athens zografou campus athens greece email iasonkar dept mechanical aerospace university california san diego jolla email krstic abstract present bounded dynamic output feedback laws achieve global stabilization equilibrium profiles partial differential equation pde model simplified chemostat model chemostat pde state means global stabilization established positive orthant particular function rather situation develop tools feedback laws employ distributed parametric knowledge model moreover provide family highly unconventional control lyapunov functionals clfs chemostat pde model two kinds feedback stabilizers provided stabilizers continuously adjusted input stabilizers results based transformation hyperbolic partial differential equation ordinary differential equation integral delay equation novel stability results integral delay equations also provided results independent interest allow explicit construction clf chemostat model keywords hyperbolic partial differential equation models chemostat integral delay equations nonlinear feedback control introduction models described foerster equation see references therein first order hyperbolic partial differential equation pde boundary condition models natural extensions standard chemostat models see optimal control problems agestructured models studied see references therein ergodic theorem see similar results asymptotic similarity proved important tool study dynamics age structured models see also study existence limit cycles work initiates study global stabilization problem means feedback control models specifically design explicit output feedback stabilizers sought global stabilization equilibrium age profile chemostat model chemostat feedback control problems described ordinary differential equations odes see dilution rate selected control input output weighted integral age distribution function assumed output functional form chosen appropriate form expression measurement total concentration microorganism bioreactor expression measured variable light absorption depends amount size distribution microorganism bioreactor main idea solution feedback control problem transformation first order hyperbolic pde integral delay equation ide see application strong ergodic theorem feature differentiates present work recent works feedback control problems first order hyperbolic pdes see present work studies global stabilization problem equilibrium age profile agestructured chemostat model means two kinds feedback stabilizers continuously applied feedback stabilizer feedback stabilizer entire model assumed unknown two cases considered equilibrium value dilution rate case equilibrium value dilution rate unknown absolutely nothing known model case equilibrium value dilution rate priori known first case family dynamic output feedback laws continuously adjusted dilution rate proposed equilibrium value dilution rate estimated observer second case output feedback law proposed arbitrarily sparse sampling schedule cases dilution rate control input takes values bounded interval consequently input constraints taken account main idea solution feedback control problem transformation pde ode ide preliminary results case extended present work given however instead simply designing dynamic output feedback laws guarantee global asymptotic stability equilibrium age profile present work additional goal explicit construction family control lyapunov functionals clfs chemostat model order achieve goal present work novel stability results linear ides independent interest newly developed results provide proof scalar strong ergodic theorem special cases integral kernel stability results linear ides similar studied work also studied since state chemostat model population density particular age given time state chemostat pde valued accordingly desired equilibrium profile function age variable state space pde system positive orthant particular function space pursue global stabilization positive equilibrium profile state space requires novel approach even novel formulation stability estimates norm state desired equilibrium zero takes infinite value population density age infinite also zero infinitely penalize population death washout main idea development particular logarithmic transformation state penalizes overpopulated underpopulated conditions infinite penalty washout condition structure paper described next section describe chemostat stabilization problem precise way provide statements main results paper theorem theorem section provides useful existing results uncontrolled pde section devoted presentation stability results ides allow construct clfs chemostat problem proofs main results provided section section presents result similar theorem uses reduced order observer instead observer simulations illustrate application obtained results given section concluding remarks paper given section finally appendix provides proofs certain auxiliary results notation throughout paper adopt following notation real number denotes integer part denotes interval let open subset metric space set denote class continuous mappings take values denote class continuously differentiable functions take values denotes functions continuous satisfy lim lim denotes functions continuously differentiable satisfy lim lim class strictly increasing unbounded functions see subset denotes class functions exists finite empty set derivative exists every continuous function meaningful right left limits tends point exist finite let function given constant use notation denote profile certain let function given constant use notation denote history certain let dmin dmax given constants saturation function sat interval dmin dmax defined sat min problem description main results model consider chemostat model dmin dmax dilution rate dmax dmin constants constant continuous functions system continuous model microbial population chemostat function called mortality function function denotes density population age time function birth modulus population boundary condition renewal condition determines number newborn individuals finally maximum reproductive age physically meaningful solutions solutions solutions satisfying chemostat model derived neglecting dependence growth microorganism concentration limiting substrate accurate model would involve enlarged system one pde age distribution coupled one ode substrate proposed context studying limit cycles constant dilution rates however approach neglecting nutrient equation chemostat new see example assume exists dmin dmax assumption necessary existence equilibrium point control system different identically zero function function form arbitrary constant equilibrium point control system notice continuum equilibria measured output control system given equation continuous function notice case corresponds total concentration microorganism chemostat feedback control continuously adjusted input let arbitrary constant set point let equilibrium age profile given consider dynamic feedback law given sat constants next consider solutions problem initial condition set solution problem initial condition mean pair mappings satisfies following properties finite possibly empty set derivative defined continuous see notation iii equations hold equation holds mapping called solution system initial condition defined define functional means equation assume following technical assumption holds function satisfies recall exists constant ready state first main result present work provides stabilizers continuously adjusted input theorem continuously adjusted input unknown equilibrium value dilution rate consider chemostat model assumption every exists unique solution initial condition furthermore exist constant function every unique solution initial condition defined satisfies following estimate max max moreover let pair constants satisfying continuous functional defined arbitrary constant max exp min min sufficiently small constant sufficiently large constants lyapunov functional system sense every solution system satisfies inequality lim sup remarked introduction theorem provide formulas dynamic output feedback stabilizers guarantee global asymptotic stability selected equilibrium age profile also provides explicit formulas family clfs system indeed continuous functional defined clf system remark family feedback laws parameterized guarantees global asymptotic stabilization every selected equilibrium age profile moreover feedback law achieves global exponential convergence rate see estimate sense estimate holds physically meaningful initial conditions indicated introduction logarithmic penalty penalizes overpopulated underpopulated conditions infinite penalty zero density age state converges desired equilibrium profiles positive initial conditions initial condition equilibrium population develop dead initial state feedback law dynamic output feedback law subsystem observer primarily estimates equilibrium value dilution rate observer highly reduced order since estimates two variables constant scalar functional state introduced remaining infinitely many states estimated key achievement stabilization without estimation nearly entire state proving result appropriately constructed transformed representation unmeasured infinitedimensional state iii family feedback laws require knowledge mortality function population birth modulus population maximum reproductive age population accordingly require knowledge equilibrium value dilution rate either instead estimated observer state see estimate feedback law work arbitrary input constraints condition needs satisfied equilibrium value dilution rate must satisfy input constraints dmin dmax reasonable requirement otherwise selected equilibrium age profile feasible parameters used control practitioner tuning controller selection values parameters affects value constant determines exponential convergence rate since proof theorem constructive useful formulas showing dependence constant parameters established proof theorem noted every pair constants possible find constants satisfying indeed every matrix hurwitz matrix consequently exists positive definite matrix matrix negative definite implies inequalities vii main idea construction feedback law transformation pde problem system consists ode ide along transformations presented figure logarithmic output transformation exploited rigorously proof theorem figure also shows observer actually observer system checking assumption theorem assumes birth modulus population satisfies assumption assumption needed establishment exponential estimate estimate could established without assumption means strong ergodic theorem see section role assumption crucial establishment clf given however since assumption demands specific property function exp involves unknown equilibrium value dilution rate verification validity assumption becomes issue following proposition provides useful sufficient conditions assumption proof provided appendix proposition means checking assumption let function satisfies following assumption moreover function satisfies exists constant set lebesgue measure every holds proposition shows assumption valid function satisfies assumption hand know assumption holds every function satisfying finite number zeros interval since sure assumption necessarily holds birth moduli population finite number zeros interval matter equilibrium value dilution rate matter mortality function exp dsda figure transformation pde boundary condition given ide ode inverse transformation control hand equilibrium value dilution rate priori known position achieve stabilization let sampling period consider system feedback law sat integers solution problem initial condition set mean mapping satisfies following properties finite possibly empty set derivative defined continuous iii equations hold equation holds mapping called solution system initial condition defined ready state second main result present work theorem feedback known equilibrium value dilution rate consider chemostat model every exists unique solution initial condition furthermore exist constant function every unique solution initial condition defined satisfies following estimate max max differences theorem theorem theorem applies feedback theorem applies continuously adjusted feedback theorem assumes knowledge equilibrium value dilution rate iii theorem assume property provide clf system explained assumption needed explicit construction clf finally reader notice constraint sampling period arbitrarily large values allowed arbitrarily sparse sampling case output given instead proof theorem works minor changes proof omitted case considered ideas behind proofs main results basic tool proofs main results present work transformation shown figure main idea comes recent work transformation hyperbolic pde ide however applied results straightforward way would end following ide exp however ide dependent instead would like describe effect control input convenient way achieved introducing one state given evolution described ode position obtain transformation decomposes dynamics dynamics ide dsda evolving subspace described equation ode achieving objective next step stability analysis zero solution ide exactly point strong ergodic theorem results linear ides used uncontrolled pde present section aims give reader background mathematical knowledge used study pdes specifically aim make reader familiar strong ergodic theorem pdes show relation pdes linear ides let constant let continuous functions consider initial value pde problem initial condition following existence uniqueness result follows directly proposition theorems pages lemma absolutely continuous function exists unique function satisfies function defined absolutely continuous satisfies mapping continuously differentiable equation holds almost moreover function called solution additional regularity properties hold solution satisfies properties shown following lemma lemma ides every satisfying function lemma finite empty set derivative defined continuous satisfies equation also unique solution integral delay equation ide initial condition lemma obtained integration characteristic lines solution ide obtained solution delay differential equation differential equation obtained formal differentiation ide solution satisfies verification requires integration parts straightforward show function strictly decreasing lim lim therefore exists unique holds equation condition following strong ergodicity result follows results section proposition theorem scalar strong ergodic theorem let unique solution exist constants every absolutely continuous function corresponding solution satisfies exp exp exp linear continuous functional defined results linear integral delay equations since previous sections demonstrated relation pdes linear ides next focus study linear ides present section provides stability results system described following linear ide constant results present section allow construction lyapunov functionals linear ides provide formulas lyapunov functionals pdes since zero dynamics controlled model described linear ides proofs results present section provided appendix notion every exists unique function satisfies function called solution initial condition solution obtained solution neutral delay equation theorem page guarantees existence unique function satisfies therefore ide defines dynamical system state see notation basic estimate consequences first result section provides useful bounds solution kernel notice following lemma allows discontinuous solutions well discontinuous initial conditions lemma basic estimate solution linear ides let given function consider ide let arbitrary constant exists unique function every satisfies moreover satisfies following inequality min inf inf inf sup sup sup min direct consequence lemma lemma every satisfying corresponding solution satisfies see notice hand may apply lemma lemma directly ide define exp follows sufficiently large another direct consequence lemma lemma quantity appearing right hand side transformation function thus valid transformation indeed straightforward verify every piecewise continuous function dmin dmax every solution corresponding input dmin dmax satisfies solution initial condition using equation get exp using equation definition get exp exp since consequence conclusion previous paragraph equation implies combining two equations get exp exp exp notice implies indeed exp exp exp exp exp using definition fact consequence get exp exp combined gives fact exp exp therefore using get exp exp function quantity consequently since exp strong ergodic theorem terms ides next state strong ergodic theorem theorem terms ide goal define operator every relation satisfy certain follows lemma theorem exist constants every satisfying unique solution ide initial condition satisfies following estimate exp exp property rephrased without reference pde every exp exist constants every unique solution ide initial condition satisfies using transformation obtain mapping solutions ide solutions ide moreover estimate implies following estimate exp functional defined means equation dsda found substituting functional functional defined therefore position conclude following property holds every exist constants every unique solution ide initial condition satisfies following estimate using property obtain following corollary restatement strong ergodic theorem theorem terms ides norm instead norm recall corollary strong ergodic theorem terms ides suppose exist constants every unique solution ide initial condition satisfies following estimate max max construction lyapunov functionals problem corollary provide functional allow derivation important property moreover provide information magnitude constant order construct functional obtain information magnitude constant need technical results first result deals exponential stability zero solution notice proof exponential stability property made means lyapunov functional lemma lyapunov functional general case suppose globally exponentially stable moreover functional defined max constant satisfies exp satisfies differential inequality lim sup every solution lemma useful next construct lyapunov functionals form used lemma however mostly interested kernels values satisfy show next even specific case possible construct lyapunov functional invariant subspace state space next introduce technical assumption function satisfies moreover exists following result provides construction lyapunov functional system assumption theorem lyapunov functional linear ides special kernels consider system satisfies assumption let real number define functional means equation max real number exp functional defined following relations hold lim sup every solution remark theorem version scalar strong ergodic theorem compare corollary kernels satisfy assumption corollary allow estimate magnitude constant determines convergence rate hand theorem allows estimate comparison lemma page differential inequality guarantee exp every solution using definition previous estimate guarantee max max exp max therefore bounds computed straightforward way using inequality exp allowable value moreover corollary provide functional equation however cost features loss generality corollary holds kernels satisfy theorem holds kernels satisfy assumption theorem allow guarantee exponential stability zero solution state evolves certain invariant subsets state space shown following result corollary lyapunov functional linear ides invariant sets consider system satisfies assumption let real number let functional defined define functional means equation max real number exp let positively invariant set system let constant open set continuous functional satisfies lim sup every every solution every every following hold solution initial condition lim remark differential inequality equivalent assumption mapping using assumption lemma guarantee mapping nondecreasing mapping every solution continuous functionals min max indeed lemma implies every solution holds consequently set form min max min max constants positively invariant set moreover using semigroup property solution get inequality shows mapping mapping every solution proofs main results next turn attention proof theorem throughout section use notation min dmin notice function satisfies equation sat equation fact dmin dmax imply inequality max dmax dmin also notice inequality min max min holds indeed inequality derived using definition distinguishing three cases dmin dmax dmax iii dmin case get since case case min dmax dmin get dmax min dmax dmin conclude holds dmax since conclude holds case proof similar case iii proof theorem based transformation shown figure following lemmas proofs found appendix lemma consider control system constant dmin dmax constant dmax dmin constants satisfies assumption control system defined set linear functional dsda measured output system given equation function satisfies consider system dynamic feedback law given sat constants let pair constants satisfying exist sufficiently large constants sufficiently small constants every constant every solution system initial condition unique exists satisfies differential inequality lim sup continuous functional defined max min lemma suppose exists constant continuous function satisfies lim sup following estimate holds ready provide proof theorem proof theorem define straightforward verify using definitions equation fact dsda define notice fact imply function satisfies next consider solution system initial condition lemma guarantees solution system exists solution ide since coincides solution delay differential equation initial condition therefore virtue function defined exp continuous finite possibly empty set derivative defined continuous since follows using conclude moreover using conclude equations hold equation holds finally virtue follows using conclude therefore differential inequality implies differential inequality lemma conjunction inequality implies following estimate holds since pair constants follows quadratic form positive definite therefore exist constants using previous inequality obtain following estimates max min min max estimates fact sufficiently large constants imply following estimate max min min using fact get estimate min exp max min min min implies following estimate exp max max min min combining obtain following estimate max exp taking account conclude validity relies showing exists function satisfies following inequality satisfying max order show taking account definitions suffices show exist functions following inequalities hold satisfying amax max min min max follows using repeatedly notation max fact max exp exp min inequality follows definition max following implications using get exp exp max min inequalities derived means definition directly implies max max min min moreover virtue since notice last equality used integration parts integral numerator combining using get consequently shows first inequality holds hand using get exp exp min using obtain exp exp exp following inequality direct consequence max min min exp exp consequently shows second inequality holds exp exp proof complete proof theorem based transformation shown figure following lemma proof found appendix lemma consider control system constant dmin dmax constant dmax dmin constants satisfies control system defined set dsda linear functional measured output system given equation function satisfies consider system dynamic feedback law given sat integers constant let operator defined relation every exist constant function every solution system initial condition unique exists satisfies following estimate max max min min min min ready provide proof theorem proof theorem define means straightforward verify using definitions equation fact dsda operator defined relation every define means notice fact exp imply function satisfies next consider solution system initial condition lemma guarantees solution system exists moreover exist constants function every solution system initial condition satisfies estimate solution ide since coincides solution delay differential equation initial condition therefore virtue function defined continuous finite possibly empty set derivative defined continuous since follows using conclude holds moreover using conclude equations hold equation holds finally virtue follows using fact min get estimate max max min min combining obtain following estimate max max min min estimate certain direct consequence inequalities certain proof complete using reduced order observer instead using observer system one think possibility using reduced order observer estimates equilibrium value dilution rate dynamic output feedback law given equations sat constants case solution problem initial condition means pair mappings satisfies following properties finite possibly empty set derivative defined continuous iii equations hold equation holds mapping called solution system initial condition defined observer case position prove exactly way proving theorem following result since proof almost identical proof theorem omitted theorem stabilization reduced order observer consider chemostat model assumption every exists unique solution initial condition furthermore exist constant function every unique solution initial condition defined satisfies following estimate max max moreover continuous functional defined arbitrary constant max exp min min given sufficiently small constant sufficiently large constants lyapunov functional system sense every solution system satisfies inequality lim sup family dynamic bounded output feedback laws presents features family difference lies dimension observer simulations demonstrate control design theorem three simulations carried simulation considered case birth modulus given constant condition holds model dimensionless dimensionless version obtained using appropriate scaling variables simple calculation found constant given output given equation words output total concentration microorganism chemostat chosen equilibrium profile stabilized profile given equation exp equilibrium value output given exp two feedback laws tested state feedback law given sat integers feedback law proposed output feedback law given sat integers feedback law given theorem feedback laws chose dmin dmax following family functions used initial conditions exp free parameters constant chosen condition holds simple calculations find however notice parameters used additional condition min must hold well simulations made generation uniform grid function values calculation integrals every made numerically however since wanted numerical integrator able evaluate exactly integrals every exponential function exp certain constants could use conventional numerical integration scheme like trapezoid rule simpson rule reason demand able evaluate exactly integrals every exponential function explained fact equilibrium profile given exponential function would like avoid error due error induced numerical integration end used following integration schemes derivation formulas based interpolation function exp points specifically obtain based interpolation exact integration formulas used example integral get exp exp exp exp exp ada combining formulas estimated values given obtain formula similarly derive formulas notice formulas allow numerical evaluation integrals every without knowledge since time step chosen equal discretization space step able use exact formula therefore position use following algorithm simulation system effect output feedback law algorithm given certain following calculate given calculate given integer set max dmin min dmax else set calculate using algorithm obvious modifications also used simulation system well simulation system effect output feedback law first simulation used parameter values initial conditions figure plot control values newborn individual values show values open loop feedback state output feedbacks simulation shows efficacy control design second simulation changed parameter values plotted values figure responses output feedback law output feedback law almost identical second simulation made initial condition close equilibrium profile sense initial condition large initial population difference performance feedback controllers distinguished final simulation tested robustness controller respect errors choice used controllers chose values instead applied following controllers state feedback law sat integers output feedback law given sat integers obtained cases lim lim error gives deviation desired value newborn individuals see figure notice constant error equivalent error set point since sat sat sat state feedback case sat sat sat output feedback case concluding remarks chemostats present challenging control problems hyperbolic pdes require novel results studied problem stabilizing equilibrium age profile chemostat using dilution rate control built family dynamic bounded output feedback laws continuously adjusted input ensures asymptotic stability arbitrary physically meaningful initial conditions require knowledge model also built bounded output feedback stabilizer guarantees asymptotic stability arbitrary physically meaningful initial conditions requires knowledge one parameter equilibrium value dilution rate addition provided family clfs chemostat model construction clf based novel stability results linear ides independent interest newly developed results provide proof scalar strong ergodic theorem special cases integral kernel since growth microorganism may sometimes depend concentration limiting substrate would useful solve stabilization problem enlarged system one pde age distribution coupled one ode substrate proposed context studying limit cycles constant dilution rates instead control going topic future research acknowledgements authors would like thank michael malisoff help initial stages writing process paper figure simulation initial condition given upper part figure shows response newborn individuals solid line bullets state feedback dashed line output feedback bulleted line system lower part figure shows applied control action solid line state feedback dashed line output feedback bulleted line shows equilibrium value dilution rate figure simulation initial condition given upper part figure shows response newborn individuals solid line state feedback output feedback identical bulleted line system lower part figure shows applied control action solid line state feedback output feedback bulleted line shows equilibrium value dilution rate figure control presence modeling errors error simulation initial condition given upper part figure shows response newborn individuals solid line state feedback dashed line output feedback dotted line shows equilibrium value newborn individuals lower part figure shows applied control action solid line state feedback dashed line output feedback bulleted line shows equilibrium value dilution rate references bastin coron boundary feedback stabilization linear hyperbolic systems bounded interval systems control letters bernard krstic adaptive stabilization hyperbolic pdes automatica boucekkine hritonenko yatsenko optimal control populations economy demography environment google ebook brauer mathematical models population biology epidemiology new york charlesworth evolution populations edition cambridge university press coron vazquez krstic bastin local exponential stabilization quasilinear hyperbolic system using backstepping siam journal control optimization meglio vazquez krstic stabilization system coupled firstorder hyperbolic linear pdes single boundary input ieee transactions automatic control feichtinger tragler veliov optimality conditions control systems journal mathematical analysis applications gouze robledo robust control uncertain chemostat model international journal robust nonlinear control hale lunel introduction functional differential equations springerverlag new york inaba semigroup approach strong ergodic theorem multistate stable population process mathematical population studies inaba asymptotic properties inhomogeneous foerster system mathematical population studies karafyllis kravaris syrou lyberatos vector lyapunov function characterization stability application robust global stabilization chemostat european journal control karafyllis kravaris kalogerakis relaxed lyapunov criteria robust global stabilization nonlinear systems international journal control karafyllis jiang new theorem application stabilization chemostat international journal robust nonlinear control karafyllis krstic relation delay equations hyperbolic partial differential equations esaim control optimisation calculus variations karafyllis malisoff krstic ergodic theorem stabilization hyperbolic pde inspired chemostat karafyllis malisoff krstic feedback stabilization agestructured chemostat models proceedings american control conference chicago khalil nonlinear systems edition krstic smyshlyaev backstepping boundary control hyperbolic pdes application systems actuator sensor delays systems control letters mazenc malisoff harmand stabilization robustness analysis chemostat model two species monod growth rates via lyapunov approach proceedings ieee conference decision control new orleans mazenc malisoff harmand results stabilization periodic trajectories chemostat two species ieee transactions automatic control exponential stability linear continuous time difference systems systems control letters pazy semigroups linear operators applications partial differential equations new york rao roxin controlled growth competing species siam journal applied mathematics rundnicki mackey asymptotic similarity malthusian growth autonomous nonautonomous populations journal mathematical analysis applications smith waltman theory chemostat dynamics microbial competition cambridge studies mathematical biology cambridge university press cambridge sun optimal control population dynamics spread universally fatal diseases applicable analysis international journal toth kot limit cycles chemostat model single species age structure mathematical biosciences appendix proof proposition define since dsda dsda dsda define lebesgue measurable sets integrals notice consequently equations conjunction fact implies since get moreover since follows therefore obtain equivalently definition fact implies combining previous inequality obtain desired inequality proof complete proof lemma local existence uniqueness every initial condition guaranteed theorem define solution exists sup inf let sufficiently small solution exists get definition equation sup sup sup sup sup sup sup sup sup max sup sup using fact assuming min obtain using fact constant fact distinguish following cases case implies case implies therefore case get similarly get definition equation fact min inf inf min inf inf min inf min inf min inf inf inf min inf inf min using fact constant fact obtain distinguishing cases min follows solution bounded min standard contradiction argument conjunction theorem implies solution exists indeed finite maximal existence time max solution conjunction theorem would imply lim sup max lim inf using induction position show max min integers max min moreover using fact max case max max max case max max arbitrary max get max max min sup sup max max contradicts assertion lim sup lim inf max max inequality direct consequence definitions fact inequalities proof complete proof corollary since holds since get let constants involved using get max max follows following estimate holds max max max get max max max max max max moreover using definition fact max imply get max max max max max two inequalities give max max max max max max max combining conclude estimate holds max proof complete proof lemma notice since follows mapping continuous max max max max exp max max exp max using obtain exp exp max exp max exp exp max exp max exp exp max consequently obtain max exp exp max since exp obtain max exp indeed proof follows exp exp max distinguishing cases exp exp max case leads contradiction since case conjunction max implies exp contradicts assumption exp exp max therefore obtain exp follows letting using obtain proof complete proof theorem notice since follows mappings continuous moreover definition implies dsda dsdw follows leibniz rule mapping continuously differentiable derivative satisfies notice derivation equality used fact using conclude holds next define using definition fact obtain moreover follows definition dsda since dsda obtain combining get real number therefore position apply lemma solution specifically get lim sup max real number exp finally notice definitions equality imply following equalities max max max differential inequality direct consequence equation inequality proof complete proof corollary working proof theorem show holds since continuous functional follows mappings continuous applying theorem taking account fact obtain lim sup let solution differential inequalities imply mappings consequently get implies virtue mean value theorem obtain existence using fact mapping combining relations obtain min sufficiently small differential inequality direct consequence inequalities fact mapping continuous proof complete proof lemma virtue remark corollary every solution ide exists unique satisfies specifically using lemma guarantee solution ide satisfies inf min indeed since follows assumptions lemma hold therefore get arbitrary inequality inf inf sup sup holds inequality direct consequence continuity inequality given facts satisfies solution ide satisfies position guarantee mapping defined continuous mapping follows every solution system differential equations sat sat exists locally unique moreover due fact right hand side differential equations satisfies linear growth condition follows solution system differential equations sat sat exists due definition equations position conclude constructed mappings coincide solution system initial condition uniqueness solution system direct consequence procedure construction solution define notice equations definition allow conclude following differential equations hold since inequalities hold follows quadratic forms positive definite recall remark follows exist constants using inequality holds conclude exist constants following differential inequality holds since satisfies assumption since mapping nona decreasing every solution ide continuous min functional min min recall remark follows mapping using remark corollary arbitrary constant conclude lim max max min min max min since min exp real number real number using facts obtain exp max exp min min using obtain following differential inequality lim sup max exp min max min selecting exp obtain definition lim sup exp min suppose since mapping continuous follows sufficiently small differential inequality implies mapping consequently virtue mean value theorem obtain sufficiently small therefore using obtain differential inequality lim hand using position conclude case using conclude unique solution therefore case implies finally facts imply definition allows conclude therefore differential inequality holds finally using get sat distinguish following cases case using fact function defined previous inequality obtain case using fact function defined previous inequality obtain max dmax dmin inequality max max min max max min combining three cases conclude implies max max dmin consequently obtain dmin combining using triangle inequality obtain every lim using definition obtain exp using obtain every lim exp therefore obtain definitions exp differential inequality min exp min dmax dmin proof complete proof lemma first notice differential inequality shows also make following claim claim claim holds virtue fact proof claim made contradiction suppose exists since follows consequently obtain lim sup using comparison lemma page obtain since obtain contradiction since obtain lim sup using comparison lemma page obtain using fact implies fact obtain estimate since satisfies conclude holds proof complete proof lemma virtue remark corollary every solution ide exists unique satisfies specifically using lemma guarantee solution ide satisfies working proof theorem also show given facts satisfies solution ide satisfies position guarantee mapping defined continuous mapping follows every solution differential equation sat integers initial condition exists locally unique moreover due fact right hand side differential equation bounded follows solution differential equation sat integers initial condition exists due definition equations position conclude constructed mappings coincide solution closedloop system initial condition uniqueness solution system direct consequence procedure construction solution using corollary conclude exist constants every unique solution ide initial condition satisfies following estimate max max follows following equation holds integers sat integers next show following claim claim following inequality holds integers min min dmax dmin proof claim distinguish following cases case dmin dmax definition implies using get directly implies case dmin definition implies dmin using get dmin inequality dmin implies dmin thus get min min min min inequality conjunction fact min dmax dmin implies holds case dmax definition implies dmax using get dmax inequality dmax implies dmax thus get max max max max inequality conjunction fact min dmax dmin implies holds proof claim complete claim following inequalities hold integers max min dmin min max proof claim proof inequalities made induction first notice inequalities hold next assume inequalities hold certain integer distinguish following cases case dmin dmax definition implies using get consequently get max max max dmax max directly implies second inequality place similarly obtain first inequality place case dmin definition implies dmin using get dmin consequently get min min min min min min min min min min min min first inequality place furthermore inequality dmin implies dmin consequently get dmin max max max dmax max second inequality place case dmax definition implies dmax using get dmax consequently get max max max max max max max max max max max max second inequality place furthermore inequality dmax implies dmax consequently get dmax min min min dmin min first inequality place proof claim complete next show following claim claim following inequalities hold min min max proof claim let arbitrary define notice definition implies inclusion distinguish following cases case dmin dmax definition implies using get equality conjunction facts inequality gives estimates specifically inequality conjunction facts implies since inequality shows holds case moreover equation gives conjunction facts implies hand inequality gives max dmax max max combining two inequalities get max since inequality shows right inequality holds case left inequality proved way case dmin definition implies dmin using get dmin therefore get combined gives min dmin min min min min min since inequality shows left inequality holds case furthermore inequality dmin implies dmin since dmin get dmin dmin equality conjunction facts shows right inequality holds exactly case finally notice following inequalities hold combined facts implies since inequality shows inequality holds case case dmax definition implies dmax using get dmax therefore get combined gives max dmax max max max since inequality shows right inequality holds case furthermore inequality dmax implies dmax since dmax get dmax dmax equality conjunction facts gives min min min min hand inequality gives min min dmin min min min combining two inequalities get min min since inequality shows left inequality holds case finally notice following inequalities hold min min combined facts implies since inequality shows inequality holds case proof claim complete using fact min get estimate exp max min max min min min min let integer next show following inequality holds exp indeed get directly implies hand get previous inequality conjunction fact gives exp exp consequently holds using induction position prove following inequality specifically inequality follows definition sequence fact inequality gives exp using induction prove formula directly implies using fact exp exp obtain following inequality exp exp exp max max exp exp min min min min min since follows thus obtain following inequality exp exp max max exp min min min min using implies max obtain max min min conjunction notice holds well consequently holds since integer follows may selected smallest integer conjunction max fact min satisfies min min fact exp exp integers get following inequality max exp min min exp using fact smallest integer satisfies max min min guarantee holds using fact obtain proof complete

pursuit single evader uncertain information saad aleem jan cameron nowzari george pappas department electrical systems engineering university pennsylvania philadelphia usa abstract paper studies problem involving single pursuer single evader interested developing pursuit strategy require continuous even periodic information position evader propose control strategy allows pursuer sample evader position autonomously satisfying desired performance metric evader capture work paper builds previously proposed pursuit strategy guarantees capture evader finite time finite number evader samples however algorithm relied unrealistic assumption evader exact position available pursuer instead extend previous framework develop algorithm allows uncertainties sampling information evader derive tolerable error pursuer guarantee capture evader addition outline advantages retaining evader history improving current estimate true location evader used capture evader even less samples approach sharp contrast existing works literature results ensure capture without sacrificing performance terms guaranteed compared classic algorithms assume continuous availability information key words control control analysis introduction paper study problem involving single pursuer single evader objective pursuer catch evader traditionally treatment problem assumes continuous periodic availability part agents entails numerous unwanted drawbacks like increased energy expenditure terms sensing requirement network congestion inefficient bandwidth utilization increased risk exposure adversarial detection etc contrast interested scenario relax sensing requirement pursuer replace triggered decision making pursuer autonomously decides needs sense evader update trajectory guarantee capture evader material paper partially presented ieee conference decision control december osaka japan corresponding author saad aleem tel email addresses aleems saad aleem cnowzari cameron nowzari pappasg george pappas preprint submitted automatica literature review two main areas related contents paper first popular problem garnered lot interest past engineering perspective problems studied extensively context differential games isaacs olsder sgall sufficient conditions derived pursuer capture evader agents equal maximum speeds constrained move within nonnegative quadrant alonso upper lower bounds discussed agents constrained circular environment pursuit strategies generalized extended kopparty ravishankar guarantee capture using multiple pursuers unbounded environment long evader initially located inside convex hull pursuers context robotic systems visibilitybased received lot interest past lavalle hinrichsen sachs isler problems pursuer visually searching unpredictable evader move arbitrarily fast simply connected polygonal environment similar problems studied suzuki yamashita gerkey march isler visibility limitations introduced pursuers agents actively sense communicate times related problem discussed bopardikar agents move agent limited range spatial sensing detailed review recent applications context search rescue missions motion planning involving adversarial elements found chung vast literature problems highlights multifaceted applications variety contexts however previous works usually assume continuous periodic availability sensing information especially part pursuer towards end want apply new ideas triggered control problem studied far contrast conventional approaches strategies based triggered control schemes study information could sampled control purposes agents act opportunistic fashion meet desired objective bernhardsson velasco triggered control allows analyze cost make less communication effort part agents achieving desired task guaranteed level performance system see heemels overview recent studies particular relevance paper works study subramanian fekri nowzari eqtami mazo tabuada implementations local agent strategies control focus detecting events intrinsic exogenous execution trigger agent actions control emphasis instead developing autonomous tests rely current information available individual agents schedule future actions context problems make use approach equips pursuer autonomous decision making order decrease required sensing effort tracking evader principle paper shares works aim trading increased decision making agent pursuer level less sensing effort still guaranteeing capture evader key result aleem relied receiving perfect information evader whenever pursuer decided sample reality exact position information evader may never available pursuer main contribution paper design robust policy allow noisy sensor measurements part pursuer still guaranteeing capture sporadic evader observations triggered control framework provides fresh insights dealing information uncertainties scenarios problem framework readily incorporates uncertainty sensing evader allows derive tolerable error bounds estimating evader position preserve capture guarantees theme pursuit policy quite similar existing works triggered control based latest current estimate evader pursuer computes certificate sleep duration follow current trajectory without sense evader addition discuss relative advantages retaining previous estimates leverage past information evader arrive better estimate evader true location show incorporating additional knowledge evader past improves update duration pursuer mitigates uncertainty detecting evader organization problem formulation mathematical model presented section section present design update duration pursuer derived aleem followed section allow uncertainty sampling evader position outline maximum tolerable error accommodated part pursuer without compromising strategy section discusses relative merits retaining previous estimates evader position hope increasing sleep durations pursuer readers encouraged detailed analysis problem appendix notation let sets positive real nonnegative real nonnegative integer numbers respectively denotes euclidean space euclidean distance contribution paper builds earlier work aleem applied framework triggered control design pursuit policy pursuer guarantees capture evader finite number observations work different existing methods literature analysis assume availability continuous periodic information evader instead framework guaranteed capture evader without sacrificing performance terms guaranteed compared classic algorithms assume continuous information available times problem statement consider system single pursuer single evader given time position evader given velocity given kue maximum speed evader similarly position velocity pursuer given kup maximum speed plane agents modelled single integrators note necessary assume agents particularly evader moving constant speeds times practical purposes upper bound evader speed vemax vemax analysis remain unchanged denote positions agents additionally pursuer moving along relative angle agents headings denoted see fig without loss generality normalize speed pursuer evader moves speed times dynamics pursuer evader given pursuer system evolves problem goal pursuer capture evader define capture evader instance pursuer within capture radius evader assuming pursuer exact information evader state times well known strategy pursuer move maximum speed direction evader isaacs strategy known classical pursuit given control law kre cos cos sin sin issue control law requires continuous access evader state times instantaneous updates control input instead want guarantee capture evader without tracking times updating controller sporadically pursuer decide opportunistic fashion sample evader position update control input framework pursuer knows position evader time last observation let sequence times pursuer receives information evader position updates pursuer implements hold control signal computed last time observation using given kre fig figure shows pursuer evader pursuer moving along relative angle agents headings denoted arrows indicate velocity vectors agents update policy pursuer suppose time pursuer observes evader distance agents krp notational brevity denote position agents instance observation rek rpk interested duration pursuer maintain course trajectory without observing evader specifically interested first instance separation agents possibly increase thus prompting pursuer sample evader state update trajectory let denote separation pursuer evader time consider objective function kre separation agents time goal design triggering function pursuer guaranteed capture evader also aware number samples evader required paper purpose identify function update duration pursuer determines next time updated information required words time pursuer receives updated information evader time want find duration next update note time becomes nonnegative time becomes nonnegative using derivative see appendix details given cos sin design update law study pursuit evasion problem consisting single pursuer single evader function evader parameters reachable set evader see separation strictly decreasing note dmax dmax maximum possible separation agents duration see appendix details given sfrag replacements fig figure shows plot normalized update time evader speed given thus given ball rek additionally fixed write explicitly function evader parameters denote let sup subject reachable capture time number samples using update policy pursuer guaranteed capture evader finite time finite number updates specifically find maximum number samples terms capture radius evader speed use guarantee finite summarized following theorem set evader rek dynamics denote update duration defined inf theorem capture finite samples let pursuer evader dynamics given agents initially separated given positive capture radius selftriggered update policy ensures capture finite observations finite time smallest duration exists evader state may increase separation agents agents modelled update duration obtained solving inf see appendix derivation given proof according proposition separation agents strictly decreasing successive updates fact new separation agents satisfies inequality given implies observations evader separation agents satisfies inequality initial separation agents using result maximum number samples calculated setting graph evader speed shown fig plot observe increasing evader speed decreases update duration pursuer evader moves faster law prescribes frequent updates guarantee capture underlying objective design selftriggered policy instance fresh observation separation agents must decreased following proposition characterizes result log nmax log expression shows positive capture radius pursuer guaranteed capture evader finite number samples completes first part proof selftriggered pursuit policy sequence times pursuer samples evader position denoted follows criteria means updates total duration pursuit denoted given without loss generality assume since pursuer guaranteed capture evader proposition decreasing separation let pursuer evader dynamics given agents separated time pursuer updates trajectory using update policy distance agents time strictly decreased proof given separation time new separation agents duration tcap max denotes max satisfies relationship psfrag replacements using inequality get max tcap thus tcap use relationship shows evader speed tcap finite remark performance note proving finite theorem showed strictly less however given capture radius shown satisfies inequality evader captured soon actual separation within capture radius time instance updates relationship upper bound classical pursuit strategy classical pursuit tcap bounded tcap allowable error evader estimate section study scenario pursuer acquires information position evader uncertainty interested analyzing effect imperfect observations selftriggered framework objective investigate maximum allowable error estimating evader position still allows catch evader using pursuit policy specifically want find maximum tolerable uncertainty estimating evader position instance observation function evader speed suppose pursuer estimates evader rbe observation imperfect corrupted associated noise instance observation true position evader notational brevity denote rbe rbek objective find allowable range error function evader speed pursuit policy still guarantee capture reachable set evader given rbek illustrated fig let applying previous framework analysis want maximize evader parameters subject constraint rek let sup last step made tcap fig figure shows variation maximum number samples evader speed given expression chosen max maximum number samples finite number maximum samples nmax denoted tcap bounded subject reachable set evader rek finding similar procedure finding outlined appendix difference reachable set evader increased whereas remains error estimated separation update duration defined initial separation capture radius isaacs worstcase occurs scenario evader actively moving away times shows pursuit policy guarantees capture performance classical case finite number evader samples expression guarantees capture finite samples evader state fig shows graph maximum number samples required guarantee capture evader speed capture radius number samples increase quite sharply approaches makes intuitive sense maximum number evader observations increase evader approaches maximum speed pursuer inf solving inf yields rpk rpk dmax fig time pursuer measures evader indicates rbek duration outlines boundary reachable set evader dmax denotes maximum possible separation agents simplify analysis select error scaled version current estimate separation parametrization allow study effect changing update duration also tell maximum tolerable error relative psfrag replacements setting current estimate separation get fig figure shows pursuer rpk estimating observer time pursuer detects rbek separated evader uncertainty rbek indicates reachable set evader slight abuse notation used inf instead note chosen arbitrarily find feasible domain invoke criteria yields however imposing positivity duration sufficient come desired domain design update duration rests underlying performance objective evader capture requires strict decrease true separation updates instance update new dmax estimate separation satisfies dmax given fig figure shows variation different values evader speeds satisfies condition evader speed increasing value decreases previous inequality allowed worstcase scenarios estimating evader position sufficient guarantee fore setting strict decrease actual separation updates results conservative set allowable values shown dmax illustrated fig thus thus evader speed maximum allowable error observation denoted satisfies inequality dmax let recall function deriving means note maximum allowable error dynamically changes decreases pursuer closes evader also means duration entire pursuit maximum allowable error rek sup case uncertainty access estimates instead true separation current separation tions denoted satisfies relationship samples presence uncertainty given initial agents preestimated separation maximum defined positive capture radius number evader updates calculated setting results capture radius incur zeno behavior using policy pursuer require infinitely many samples capture evader following theorem characterizes important result easy verify satisfies condition using similar analysis previous section estimate maximum number maximum number samples analysis memory previous section studied effect uncertainty sensing evader position provided bounds maximum tolerable error function evader speed allowed capture evader sporadic updates absence uncertainty outlined section pursuer employs memoryless pursuit policy catch evader pursuer needs current sample evader true position rek order calculate update duration current reachable set evader rek always subset reachable set evader introduce uncertainty estimating evader position statement longer true idea behind retaining evader estimated history potentially reduce actual reachable set evader combine current reachable set evader previous reachable sets thus improving current estimate true position evader allows improve increase update duration compared important consequence theorem presence uncertainty framework guarantees capture finite estimates evader means find maximum number samples guarantee capture design satisfies condition guarantees strict decrease measured separation successive updates general write fig figure shows plot maximum number samples different values evader speeds satisfies condition capture radius taken positive monotonically decreasing function thus evader speed duration positive constant observations suffices show duration incur zeno behavior evader speed maximum number samples guaranteed capture increases parameter increased see fig shows need sample evader state frequently order allow uncertainty estimating evader position instance update proof suppose pursuer observes evader instance order show zeno behavior suffices prove duration lowerbounded positive constant observations according selfdition psfrag replacements triggered update policy note higher values result smaller duration see fig given capture radius following relationship satisfied observations log theorem zeno behavior pursuer updates trajectory using update policy satisfies condition pursuer guaranteed incur zeno behavior duration pursuit log memoryless case uncertainty derived section specifically leverage knowledge previous estimates evader improving current update duration pursuer mitigating effect uncertainty estimating evader position consider problem pursuer receives uncertain information evader keeping track previous estimates purpose illustration analyze case pursuer retains previous estimate evader position extending framework case one previous estimates similar straightforward additionally assume pursuer samples evader associated error suppose pursuer sampled evader time computed update duration observed evader instance current reachable set evader given rbek previous reachable set evader given rbe based information actual reachable set evader given intersection two one particular scenario illustrated fig actual reachable set current reachable set update duration defined inf equivalently fixed optimal value following optimization problem subject sup explained earlier incorporating evader previous estimate potentially reduce actual reachable set evader case noisy measurements keeping track previous estimate equivalently adding constraints feasible reachable set optimization problem want formalize benefit retaining evader history terms improvement increase update duration compared memoryless update duration given recall update duration derived using current reachable set evader rek based latest estimate let denote optimal value problem used obtain memoryless update duration sup rpk observe relaxation obtained removing constraint corresponding previous reachable set evader result notice monotonically increasing parameter increasing increases feasible set optimization problem thus yielding potentially greater maximum value using monotonicity along fact allows infer first instance approaches greater equal first instance approaches means rek rek fig figure shows reachable sets evader based current rek previous estimates current reachable set denoted rbek previous reachable set noted actual reachable set intersection two inf inf consequence using let sup subject actual reachable set leveraging memory uncertainty relationship shows using previous estimate evader position potentially increase update durations pursuer case fig demonstrates one particular instance leveraging evader history yields greatest improvement evader rbek rek ous current reachable set intersect point suppose measured separation units using memoryless pursuit policy results whereas obtaining update duration solving problem results observe shows certain cases knowing one previous estimate evader almost nullify uncertainty sensing evader position consequently allow greater update duration case case fig figure shows two possible cases instance fresh observation evader case shows extreme case incorporating evader history precisely determines true position case shows scenario current reachable set subset previous reachable set thus previous estimate provides additional information towards multiple estimates extending framework case multiple previous estimates relatively straightforward suppose observation evader information previous updates current sleep duration computed inf optimal value optimization problem value treated length sliding window retaining fixed number previous estimates compute current update duration pursuer current update duration extreme case instance observation current previous reachable sets evader intersect actual reachable set reduced point equivalently means know precisely evader result longer update duration compared memoryless case would incorporated uncertainty observation yield conservative update duration let denote improvement update duration comparing see greatest improvement denoted given sup subject rek bej bej current reachable set previous estimate evader position given sometimes adding information previous estimate advantageous important realize incorporating previous reachable set evader always results improvement increase update duration increase selftriggered update duration evader past provides information current true position scenario current reachable set evader subset previous reachable set get additional information evader true position hence improvement update duration compared memoryless case illustrated case fig drop forget previous estimate evader position subset previous reachable set yield improvement increasing current update duration bej remark forgetting previous estimates note might capability store previous estimates evader necessary use computing current sleep duration pursuer illustrated earlier retaining history improves update duration reduces current reachable set evader allows forget estimates whose reachable sets either completely contain current reachable set reachable set another previous estimate formalize notion instance observation evader construct set remark numerical example illustrate improvement update duration numerically let values based initial measurement pursuer finds first update duration using samples evader find rbek bei bel bei bei denotes denotes collection indices among previous samples estimates forget reduce computation complexity problem thus improved update duration computed solving problem problem guaranteed optimal value estimates belonging set effective contribution actual reachable set evader thus removing change optimal value sup subject rek bej section provide numerical results case pursuer retains past samples evader outlined section study potential benefit retaining previous estimate evader observation pursuer tries capture evader thus simulations selftriggered update duration obtained solving optimization problem model agents single integrators normalize speeds agents maximum speed evader evader restricted move directions right left chooses best direction actively move away maximum possible speed pursuer times initialize agents actual separation units every observation pursuer samples evader current position associated error initially true position evader rek rbek note arbitrarily set capture radius outlined section evader speed error ture radius must satisfy relationship guarantees capture finite updates observe beginning pursuit adding history results relatively better gains compared towards end agents nearby values indicate separation decreases potential benefit adding computational overhead retaining previous observation simulations table comparison memoryless update times conclusions robust framework paper extends previous results address practical issues related uncertainty information evader elaborate case sampling perfect design update duration along tolerable error bounds estimating evader state show analysis preserve selftriggered controller updates pursuer incurs zeno behavior catching evader without losing previous performance guarantees methodology offers fresh perspective dealing uncertain information problems besides contrast majority previous works assume continuous least periodic information evader available times additionally study merits retaining evader history show allow potentially longer update durations incorporating past observations evader pursuer autonomous decision making future interested extending methods scenarios involving multiple agents deriving conditions cooperative strategies setting satisfies aforementioned relationship table shows variation memoryless update duration improved update duration takes account previous estimate evader observation since different result different measures separation instance observation fair comparison need compare normalized update durations denote true tions instance evader observation memoryaware memoryless pursuit strategies respectively results table indicate presence uncertainty sampling evader incorporating previous estimate allows greater sleep durations references aleem nowzari pappas pursuit single evader proceedings ieee conference decision control pages osaka japan alonso goldstein reingold lion man upper lower bounds orsa journal computing bernhardsson comparison periodic event based sampling stochastic systems proceedings ifac world congress volume pages rek olsder dynamic noncooperative game theory volume siam bopardikar bullo hespanha cooperative pursuit sensing limitations proceedings ieee american control conference pages new york chung hollinger isler search pursuitevasion mobile robotics survey autonomous robots eqtami dimarogonas kyriakopoulos eventtriggered control systems proceedings ieee american control conference pages baltimore gerkey thrun gordon pursuitevasion limited field view international journal robotics research heemels johansson tabuada introduction control proceedings ieee conference decision control pages maui isaacs differential games mathematical theory applications warfare pursuit control optimization courier corporation isler kannan khanna randomized polygonal environment ieee transactions robotics isler kannan khanna randomized local visibility siam journal discrete mathematics kopparty ravishankar framework pursuit evasion games information processing letters lavalle hinrichsen pursuitevasion case curved environments ieee transactions robotics automation mazo tabuada decentralized control wireless networks ieee transactions automatic control nowzari coordination robotic networks optimal deployment automatica sachs lavalle rajko pursuitevasion unknown planar environment international journal robotics research sgall solution david gale lion man problem theoretical computer science subramanian fekri sleep scheduling lifetime maximization sensor networks fundamental limits optimal solutions symposium information processing sensor networks pages new york acm suzuki yamashita searching mobile intruder polygonal region siam journal computing velasco fuertes marti self triggered task model control systems proceedings ieee systems symposium volume pages rek rpk fig figure shows pursuer evader rpk rek respectively separated time rek ball centered rek radius indicates reachable set evader thus pursuit trajectory parallel pursuer observe evader till next update instance thus modified dynamics given cos sin thus separation cos sin using cos sin agent updates possibly become nonnegative rek explicitly denotes terms evader parameters fixed problem formulated sup subject rek denotes optimal value problem update duration defined inf rek constraint problem independent cos setting sin yields arctan note see suppose setting get contradiction due symmetry problem assume since substituting get derivation update duration agents modelled dynamics time pursuer observes evader distance krek rpk without loss generality make relative vector pursuer evader parallel additionally matter convenience assume elaborated fig simplifies problem sup subject rek shows satisfies constraints problem means instance update maximizer relaxed problem feasible solution original problem hence optimal solution thus result note continuous parameter constraint rek get means must lie boundary rek thus substituting get reduces problem sup subject maximum separation pursuer updates trajectory using selftriggered update policy described maximum distance agents successive updates given dmax note relax problem omitting constraint relaxation results unconstrained optimization problem ignoring constraints problem let sup inf see duration pursuer moves distance units evader evader anywhere inside ball radius centered rek shown fig maximum separation pursuer evader denoted dmax given result relaxation let argmax unconstrained problem perform unconstrained maximization tive given setting yields argmax given solving rpk rek rek inf yields dmax fig indicates new separation agents rek outlines boundary reachable set evader dmax denotes maximum possible separation pursuer evader recall obtained relaxing constraint problem claim instance update maximizer relaxed problem feasible solution problem see maximizer given

diffeomorphic random sampling using optimal information transport apr martin sarang klas department mathematics florida state university bauer department bioengineering scientific computing imaging institute university utah sjoshi department mathematical sciences chalmers university technology university gothenburg abstract article explore algorithm diffeomorphic random sampling nonuniform probability distributions riemannian manifolds algorithm based optimal information transport oit analogue optimal mass transport omt framework uses deep geometric connections metric space probability densities information metric group diffeomorphisms resulting sampling algorithm promising alternative omt particular formulation free nonlinear equation compared markov chain monte carlo methods expect algorithm stand well large number samples low dimensional nonuniform distribution needed keywords density matching information geometry metric optimal transport image registration diffeomorphism groups random sampling introduction construct algorithms random sampling addressing following problem problem let probability distribution manifold generate random samples classic approach sample probability distribution higher dimensional space use markov chain monte carlo mcmc methods example algorithm alternative idea use diffeomorphic density matching density standard density samples drawn easily standard samples transformed diffeomorphism generate samples bayesian inference example distribution would posterior distribution would prior distribution case prior hard sample uniform distribution used subset real line standard approach use cumulative distribution function define diffeomorphic transformation however dimension greater one obvious change variables transform samples distribution prior thus led following matching problem problem given probability distribution find diffeomorpism denotes standard distribution samples drawn acting densities jacobian determinant benefit methods traditional mcmc methods cheap computation additional samples amounts drawing uniform samples evaluating transformation hand methods scale poorly increasing dimensionality contrary mcmc action diffeomorphism group space smooth probability densities transitive moser lemma existence solution problem guaranteed however dimension greater one space solutions thus one needs select specific diffeomorphism within set solutions moselhy marzouk reich proposed use optimal mass transport omt construct desired diffeomorphism thereby enforcing convex function omt approach implies solving one form another heavily nonlinear equation survey omt approach random sampling given marzouk article pursue alternative approach diffeomorphic based random sampling replacing omt optimal information transport oit diffeomorphic transport based geometry building deep geometric connections metric space probability densities information metric group diffeomorphisms developed efficient numerical method density matching efficiency stems solution formula explicit inversion laplace operator thus avoiding solution nonlinear pde paper explore method random sampling initial motivation medical imaging although applications including random sampling also suggested resulting algorithm implemented short matlab code available mit license https density transport problems let orientable compact manifold equipped riemannian metric volume density induced denoted without loss ofrgenerality assume total volume respect one furthermore space smooth probability densities given prob denotes space smooth group smooth diffeomorphisms diff acts space probability densities via diff prob prob result moser action transitive introduce subgroup volume preserving diffeomorphisms sdiff diff note sdiff isotropy group respect action diff spaces prob diff sdiff structure smooth infinite dimensional manifold furthermore diff sdiff infinite dimensional lie groups careful treatment topologies found work hamilton following focus attention diffeomorphic density matching problem problem common approach overcome nonuniqueness solution add regularization term problem search minimum energy solution required matching property energy functional diffeomorphism group following ideas mathematical shape analysis natural approach define energy functional using geodesic distance function dist riemannian metric diffeomorphism group regularized diffeomorphic matching problem written follows problem given probability density prob want find diffeomorphism diff minimizes energy functional diffeomorphisms free variable matching problem choice riemannian distance group diffeomorphisms although formulated moselhy marzouk proposed use metric diff diff corresponds optimal mass transport omt induces wasserstein distance prob see example article use metric rham operator lifted vector fields orthonormal basis harmonic hodge decomposition theorem independent choice orthonormal basis harmonic vector fields construction related metric space probability density predominant field information geometry call information metric see information underlying geometry connection information metric metric allows construct almost explicit solutions formulas problem using explicit formulas geodesics metric theorem let prob smooth probability density diffeomorphism diff minimizing distgi constraint given obtained solution problem unique geodesic connecting sin sin cos sin sin algorithm diffeomorphic random sampling described following section directly based solving equations numerical algorithm section explain algorithm random sampling using optimal information transport direct adaptation algorithm algorithm oit based random sampling assume numerical way represent functions vector fields diffeomorphisms numerical methods composing functions vector fields diffeomorphisms computing gradient functions computing solutions poisson equation sampling standard distribution evaluating diffeomorphisms oit based algorithm problem given follows choose step size positive integer calculate geodesic derivative time points using equation initialize set compute solve poisson equation compute gradient vector field construct approximations exp example update set continue step unless draw random samples uniform distribution set algorithm generates random samples distribution one save repeat whenever additional samples needed computationally intensive part algorithm solution poisson equation time step notice however need solve nonlinear equations necessary omt example example consider distribution defined cartesian coordinates exp exp normalized ratio maximum mimimum resulting density depicted fig left draw samples distribution using matlab implementation algorithm available mit license needed one may also compute inverse https implementation summarized follows solve lifting equations discretize torus mesh use fast fourier transform fft invert laplacian use time steps resulting diffeomorphism shown mesh warp fig draw uniform samples apply diffeomorphism sample applying diffeomorphism corresponds interpolation warped mesh resulting random samples depicted fig right draw new samples efficient example another samples drawn less second standard laptop fig left probability density maximal density ratio right samples calculated using oit based random sampling algorithm conclusions paper explore random sampling based optimal information transport algorithm developed given nature algorithm expect efficient competitor existing methods especially drawing large number samples low dimensional manifold however detailed comparison methods including mcmc methods outside scope paper left future work provide example complicated distribution flat method straighforward extended elaborate manifolds using finite element methods efficient solution poisson equation manifolds manifolds importantly one might use standard techniques first transform required distribution compact domain fig computed diffeomorphism shown warp uniform mesh every shown notice warp periodic satisfies solves problem minimizing information metric ratio largest smallest warped volumes bibliography amari nagaoka methods information geometry amer math providence bauer bruveris michor uniqueness metric space smooth densities bull lond math soc bauer joshi modin diffeomorphic density matching optimal information transport siam imaging sci friedrich die und symplektische strukturen math nachr hamilton inverse function theorem nash moser bull amer math soc hastings monte carlo sampling methods using markov chains applications biometrika khesin lenells misiolek preston geometry diffeomorphism groups complete integrability geometric statistics geom funct anal khesin wendt geometry groups series modern surveys mathematics vol berlin marzouk moselhy parno spantini sampling via measure transport introduction ghanem higdon owhadi eds handbook uncertainty quantification springer international publishing cham miller younes metrics equations computational anatomy annu rev biomed eng modin generalized equations optimal information transport factorization diffeomorphisms geom anal moselhy marzouk bayesian inference optimal maps journal computational physics moser volume elements manifold trans amer math soc otto geometry dissipative evolution equations porous medium equation comm partial differential equations reich nonparametric ensemble transform method bayesian inference siam journal scientific computing villani optimal transport old new grundlehren der mathematischen wissenschaften vol berlin

may adaptive algebraic multiscale solver compressible flow heterogeneous porous media matei yixuan wang hadi hajibeygi department geoscience engineering faculty civil engineering geosciences delft university technology box delft netherlands department energy resources engineering stanford university panama stanford usa abstract paper presents development adaptive algebraic multiscale solver compressible flow heterogeneous porous media similar recently developed ams incompressible linear flows wang jcp operates defining primal blocks top grid coarse grids facilitate construction conservative finite volume coarsescale system computation local basis functions respectively however unlike incompressible elliptic case choice equations solve basis functions compressible problems trivial therefore several basis function formulations incompressible compressible without accumulation considered order construct efficient multiscale prolongation operator restriction operator allows multiscale finite volume msfv finite element msfe methods finally order resolve highfrequency errors smoother stages employed order reduce computational expense operators prolongation restriction smoothers updated adaptively addition linear system loop infrequently updated systematic numerical experiments performed determine effect various options outlined convergence behaviour efficient strategy heterogeneous compressible problems developed based overall cpu times finally compared algebraic multigrid amg solver results comparison illustrate quite efficient nonlinear solver even iterated machine accuracy key words multiscale methods compressible flows heterogeneous porous media scalable linear solvers multiscale finite volume method multiscale finite element method iterative multiscale methods algebraic multiscale methods preprint submitted elsevier science july introduction accurate efficient simulation multiphase flow heterogeneous natural formations crucial wide range applications including hydrocarbon production optimization risk management carbon capture storage water resource utilizations geothermal power extractions unfortunately considering size domain along high resolution heterogeneity geological properties numerical simulation often beyond computational capacity traditional reservoir simulators therefore multiscale finite element msfe finite volume msfv methods extensions developed resolve challenge comparison different multiscale methods based original descriptions studied literature msfv msfe methods map discrete system much coarser space multigrid terminology map considered special prolongation operator represented adaptively updated basis functions restriction operator defined based either finite element msfe finite volume msfv combination msfv applied wide range applications see thus recommending multiscale promising framework reservoir simulators however developments including algebraic multiscale formulation ams focused incompressible linear flow equations compressibility effects considered pressure equation becomes nonlinear solution requires iterative procedure involving parabolictype linear system equations therefore development efficient general algebraic formulation compressible nonlinear flows crucial order advance applicability multiscale methods towards realistic problems present study introduces first algebraic multiscale iterative solver compressible flows heterogeneous porous media along thorough study computational efficiency cpu time convergence behaviour number iterations contrast cases incompressible flows construction basis functions compressible flow problems straightforward past corresponding author yixuanw pressible elliptic compressible elliptic parabolic basis functions considered however literature lacks systematic study reveal benefit using one option especially combined smoother stage moreover study overall efficiency multiscale methods based cpu time measurements done far compressible problems order develop efficient prolongation operator work several formulations basis functions considered basis functions differ amount compressibility involved formulation ranging incompressible elliptic compressible parabolic types terms restriction operator msfe msfv considered along possibility mixing iterations former latter allowing benefit symmetric positive definite spd property msfe conservative physically correct solutions msfv errors resolved global multiscale stage cams errors tackled using smoother paper consider two options smoothing stage widely used local correction functions different types compressibility involved general specific operator well ilu best procedure determined among various strategies basis cpu time heterogeneous problems important note setup linear system population measured alongside solve time study far appeared previously published compressible multiscale works though conservative method iterations enough order obtain approximation solution benchmark studies work iterated machine accuracy reached thus performance exact solver compared algebraic multigrid amg method samg comparative study compressible problems first kind made possible presented algebraic formulation allows easy integration existing advanced simulation platforms numerical results presented wide range heterogeneous domains illustrate quite efficient simulation nonlinear compressible flow problems paper structured follows first compressible algebraic multiscale solver presented several options prolongation restriction operators well solver considered adaptive updating operators studied along possibility infrequent linear system updates loop numerical results subsequently presented wide range heterogeneous test cases aimed determining optimum strategy finally compared algebraic multigrid solver samg terms number iterations overall cpu time compressible flow heterogeneous porous media single phase compressible flow porous media using darcy law without gravity capillary effects stated porosity density source terms respectively moreover fluid mobility permeability tensor fluid viscosity form nonlinear flow equation using implicit eulerbackward time integration reads linearized superscripts denote old new iteration levels respectively converges nonlinear therefore coefficient linearization lemma plays role iterations fact opens possibility alter computing based either resulting corresponding pressure previous timestep choice potentially lead different convergence behaviour thus computational efficiency algebraically written unknown pressure vector diagonal matrix dvi cell diagonal entry dvi volume cell also convective compressible flow matrix transmissibilities computed basis finitevolume scheme entries moreover vector contains integrated source terms volumes dvi total rhs terms denoted vector compressible algebraic multiscale solver relies grids superimposed grid see fig coarse grid cells domain cells cell coarse cell fig multiscale grids imposed given fine grid block highlighted right left sides respectively transfer operators defined multiscale restriction prolongation former defined based either finite element msfe finite volume msfv corresponds integral blocks contained block otherwise columns basis functions computed cells see fig subject simplified boundary conditions localization assumption contrast incompressible ams formulated based different choices basis functions depending level compressibility involved first two types read pressure dependent different sense consideration accumulation term alternatively one also formulate basis functions using pressure independent since based pressure previous time step equations subject boundary conditions along cell boundaries one also obtain equations corresponding four types local correction functions substituting corresponding rhs term eqs mentioned work systematic studies basis cpu time well number iterations performed order find optimum formulation basis function prolongation operator basis functions assembled cells used correction functions also assembled snd fig illustrates basis functions form partition unity compressibility effects included intrinsic nature parabolic compressible equation choices formulated affect computational efficiency constructing updating multiscale operators precisely basis functions eqs depend pressure hence updated adaptively pressure changes eqs pressure independent thus need computed problems flows need adaptively updated local transmissibility changes beyond prescribed threshold value basis correction functions previously used options yet studied fig two choices multiscale basis functions reference block left column summation basis functions block right column partition unity check approximates solution using prolongation operator matrix size basis function column map coarse solution reads system obtained using restriction operator solution prolonged using residual form reads residual note different options basis functions considered construction prolongation operator employs global solver resolving errors addition solver efficient convergent multiscale solver needs include smoother fine scale smoother accounts errors arising simplified localization conditions nonlinearity operator complex rhs term among choices smoother solvers correction functions ilu considered work procedure finally summarized table convergence achieved see initialize update linear system components based update residual adaptively compute basis functions use either eqs stage used apply update residual multiscale stage solve stage smooth times using iterative solver ilu used obtaining update solution update error compute assign table iteration procedure converging tolerance next section numerical results heterogeneous test cases presented order provide thorough assessment applicability problems numerical results numerical experiments presented section divided finding proper iterative procedure multiscale components efficiently capturing nonlinearity within flow equation systematic performance study comparing commercial algebraic multigrid solver samg note second aspect mainly provide computational physics community accurate assessment convergence properties compressible multiscale solver advantage many advanced linear solvers allows construction locally conservative velocity msfv stage therefore multiphase flow scenarios iterations necessary obtain accurate solutions studied numerical experiments paper sets distributed permeability fields spherical variograms generated using sequential gaussian simulations variance mean natural logarithm permeability test cases respectively unless otherwise mentioned furthermore grid size dimensionless correlation lengths principle directions provided table set realizations sets orientation angle referred layered fields also grid aspect ratio unless otherwise specified permeability set grid angle direction patchy variance mean table permeability sets realizations used numerical experiments paper layered fields refer sets orientation angle direction phase properties simulation time described numbers pressure density introduced peast pwest peast respectively coefficient set subsequent test cases paper pwest peast values relative standard atmospheric condition considered correspond pressure values set dirichlet conditions west east boundaries respectively cases unless otherwise mentioned also surfaces subject neumann conditions time introduced pwest peast average permeability length scale domain values pressure difference viscosity value homogeneous cases problem size units implementation used obtain results presented paper consists code cpu times measured intel xeon system ram determining effective iterative procedure multistage multiscale components efficient capturing nonlinearity within iterations important designing efficient multiscale strategy purposes conclusive result section set patchy fields permeability set table considered one realizations corresponding solution shown fig fig permeability left pressure solution right corresponding one realization permeability set table nonlinear linear level updates formulating convergence criterion one express error approximate solution step basis either linear nonlinear expressions according nonlinear error grid cell reads assembled vector allows computation error norm hand error based linearized equation leads computation residual norm order determine suitable sequence linear nonlinear stages patchy domain grid cells considered fig pressure equation solved using following solution strategy reached update parameters linear system matrix rhs vector based solve linear system using richardson iterative scheme preconditioned one multigrid table solution strategy used determine suitable stopping criterion error residual norms recorded iteration richardson loop presented fig note reduction residual norm beyond first iterations contribute reduction nonlinear error norm therefore one could ideally speed solution scheme monitoring error norm updating linear system decrease starts stagnate however computational cost evaluating nonlinear equation roughly linear system update thus much expensive evaluation residual norm fig also reveals stagnation error norm happens roughly residual norm approximately reduced initial value immediately linear system update fig shows convergence behaviour implementing heuristic strategy deemed quite efficient since two norms agreement hence following experiments strategy employed linear level iteration inner linear loop nonlinear level iteration outer nonlinear loop set see table error norm residual norm error norm residual norm iteration iteration fig error residual norm histories one realizations permeability set table single time step shown left strategy nonlinear stage fully converged linear solution obtained shown right strategy outer nonlinear loop residual reduced one order magnitude adaptive updating multiscale operators previous study described first adaptive aspect considered work namely updating linear system residual norm drops order magnitude procedure optimized employing adaptive updates multiscale components basis considered correction functions end one monitor changes entries transmissibility matrix rhs iteration steps fig shows adaptive update basis functions leads significant terms cpu time furthermore two adaptivity methods linear system local function updates combined shown fig hence perform iterations exploits adaptivity within multiscale components nonlinearity within flow equation note case compressible variant used basis correction functions however incompressible eqs used basis functions require updates iterations finally following results unless otherwise stated coarsening ratio taken found efficient see subsection global stage choice basis functions aim study determine optimum choice type basis functions algorithm correction function computed based cases hence updated adaptively pressure iterations ilu used smoothing possibilities basis functions eqs considered finally single sec cpu time sec cpu time sec sec iteration sec cpu time sec cpu time sec sec iteration iteration multiscale solution smoother solution iteration lin sys construction basis functions correction function fig effect different types adaptivity performance permeability set table time step adaptivity linear system update adaptivity multiscale operator update adaptivity fully adaptive terms linear system multiscale operator updates time step simulation takes initial solution time everywhere solution time total cpu time spent stage solver well number iterations given top bar fig measured also success rate convergence given inside parentheses beside average number iterations results show including compressibility basis functions translate faster convergence thus additional cpu time required adaptively update justified fact efficient use incompressible pressure independent basis functions eqs also inclusion accumulation term type restriction msfe msfv play important role patchy test case note none choices results successful convergence even though ilu smoothing iterations employed iteration attributed use correction functions investigated next paragraph cpu time sec average cpu time ilu different types prolongation restriction multiscale solution smoother solution lin sys construction basis functions correction function fig effect choice basis function performance problem time step results averaged realizations number iterations shown top bar success percentage also shown parentheses note simulations employ correction functions smoothing stage choice correction function note none results previous test case fig success rate described seen independent stage inclusion seen option necessity convergence fig presents results rerunning previous experiment time varying type correction function plot confirms eliminating altogether leads overall addition convergence success rate described explained sensitivity heterogeneity permeability field leads solver instability therefore considered candidate stage efficient procedure instead ilu performed order resolve errors smoothing stage number smoother iterations another variable framework number smoothing steps ilu applied order obtain best convergence rate cpu time results several experiments optimum choices incompressible basis functions incorporation various numbers ilu applications illustrated fig clear setup optimum scenario would cpu time sec average cpu time ilu different types correction restriction multiscale solution smoother solution lin sys construction basis functions correction function rre rre fig effect choice correction function cpu time multiscale solution permeability set table time step number iterations shown top bar last bars right correspond runs correction function used ilu found ilu iterations per call note runs without correction functions converged successfully average cpu time different number smoothing steps cpu time sec multiscale solution smoother solution lin sys construction basis functions correction function smoothing steps fig effect number ilu smoothing steps performance permeability set table grid aspect ratio time step number iterations shown top bar convergence success rate inside parentheses note excluding leads success rate scenarios sensitivity coarsening ratio size coarse system local problem cost coarsening factors used paper found optimal careful study sensitivity coarsening ratio thorough study new solver important illustrate also sensitivity change system size thus coarsening ratio important fact studied shown figs patchy fields cases studied paper optimum overall cpu times obtained cells size approximately domain length direction reservoirs different coarsening ratios initialization lin sys construction solution cpu time sec fig patchy fields averaged cpu time realizations different coarsening ratios permeability set table results support use coarsening ratio similar behaviour observed restriction operator reservoirs different coarsening ratios initialization lin sys construction solution cpu time sec fig patchy fields averaged cpu time realizations comparison different coarsening ratios permeability set table results support use coarsening ratio similar behaviour observed restriction operator reservoirs different coarsening ratios initialization lin sys construction solution cpu time sec fig patchy fields averaged cpu time realizations comparison different coarsening ratios permeability set table coarsening ratio offers best balance initialization basis function computation solution time results expensive initialization faster convergence subsequent similar behaviour observed restriction operator benchmark versus samg basis previously presented studies optimal strategy includes global multiscale stage using incompressible basis functions accompanied iterations ilu subsection compared samg three sets different test cases heterogeneous domains different sizes table permeability set table stretched grids terms permeability set table different variances permeability contrasts presentd experiments samg called perform single repeatedly richardson loop adaptivity controlled manually beginning outer iteration samg allowed update galerkin operators hand linear iterations samg instructed reuse previous grids operators test cases considered approach found efficient factor excess automatic solver control described aspects samg used commercial solver test case heterogeneous domains different sizes table subsection compared samg algebraic multigrid solver patchy layered permeability fields table consecutive time steps pressure solution one patchy one layered sample shown fig illustrating propagation signal western face entire domain figs show number iterations cpu time consecutive times different problem sets table note restriction operator converge test cases variant achieved success rate due spd property therefore ideal solution strategy would use msfe converge desired level accuracy employ single msfv sweep order ensure mass conservation addition figs illustrate cpu time vertical axis total number iterations top column permeability sets table coarsening ratios note except first basis functions fully computed slight edge samg mainly due adaptivity relatively inexpensive iterations initialization cost particularly high case due large number linear systems solved direct solver needed basis functions clear fig larger blocks requires less setup time iterations converge note performance studies presented paper computations since reservoir simulators typically run many high initialization time outweighed efficiency gained subsequent steps moreover given local support basis functions initialization greatly improved parallel processing furthermore multiscale iteration may prove necessary obtain accurate approximation pressure solution time step flow problems fig pressure solution one realizations permeability sets left right table top bottom respectively patchy reservoir cpu time sec initialization lin sys construction solution layered reservoir initialization lin sys construction solution fig averaged cpu time realizations comparison samg solvers permeability sets table successive coarsening ratio moreover employs ilu smoothing steps per iteration number iterations given top bar layered reservoir patchy reservoir initialization lin sys construction solution cpu time sec initialization lin sys construction solution fig averaged cpu time realizations comparison samg solvers permeability sets table successive time steps employs coarsening ratio along ilu smoothing steps per iteration number iterations given top bar symbol signifies convergence success rate restriction operator employed patchy coarsening ratio layered coarsening ratio initialization lin sys construction solution cpu time sec cpu time sec initialization lin sys construction solution layered coarsening ratio initialization lin sys construction solution cpu time sec patchy coarsening ratio cpu time sec initialization lin sys construction solution fig averaged cpu time realizations comparison samg solvers permeability sets left column right column table successive different coarsening ratios top row bottom row considered moreover employs ilu smoothing steps per iteration number iterations given top bar test case stretched grids terms study effect anisotropic permeability fields along radial injection flow pattern permeability set table considered settings previous test cases except following items dirichlet boundary conditions set centers two vertical sets grid cells one values respectively addition grid aspect ratios considered note nondimensional time calculated using characteristic length figure illustrates pressure solutions one permeability realizations first time step fig converged pressure solution one realizations permeability sets grid aspect ratio respectively left right one time step dirichlet boundary conditions set centers two vertical sets grid cells one values respectively performance samg presented fig contrast shown lead convergence success however successful runs similar cpu times observed results shown fig obtained coarsening ratios cases respectively note shown fig anisotropic transmissibility caused stretched grid effect would motivate use enhanced geometries strategy well developed algebraic multigrid community subject future studies patchy cpu time sec cpu time sec initialization lin sys construction solution initialization lin sys construction solution samg layered samg patchy cpu time sec initialization lin sys construction solution cpu time sec layered initialization lin sys construction solution fig performance top samg bottom permeability set table different grid aspect rations three successive time steps pressure solutions first time step shown one realizations fig test case effect permeability contrast study effect permeability contrast permeability set table considered different variances note studied cases variance described table settings default test cases dirichlet conditions set east west faces condition everywhere else figure illustrates performances samg test case note requires iterations permeability contrast increased improve performance one consider enriched multiscale strategies based local spectral analysis modified permeability field less contrast calculation basis functions note success rates shown patchy patchy layered successful runs cpu times comparable patchy layered initialization lin sys construction solution cpu time sec cpu time sec initialization lin sys construction solution samg layered initialization lin sys construction solution initialization lin sys construction solution cpu time sec cpu time sec samg patchy fig averaged cpu time comparison top samg bottom permeability set table different variances conclusions algebraic multiscale solver compressible flows heterogeneous porous media introduced algebraic formulation benefits adaptivity terms infrequent updating linearized system selective update basis functions used construct prolongation operator extensive numerical experiments heterogeneous patchy layered reservoirs revealed efficient strategy use basis functions incompressible advection terms paired iterations ilu postsmoothing finally several benchmark studies presented developed cams research similator compared multigrid solver samg results show competitive solver especially experiments involve simulation large number time steps drawback relatively high initialization time reduced choosing appropriate coarsening strategy running basis function updates parallel moreover due conservative property requires iterations per time step obtain good quality approximation pressure solution practical purposes systematic error estimate analyses multiphase simulations subject ongoing research addition performance extended enrichment multiscale operators enriched coarse grid geometries basis underlying transmissibility subjects future studies acknowledgements would like acknowledge financial support intersect alliance technology schlumberger petroleum services matei scientific visit delft november february since march matei phd research assistant delft sponsored authors also thank hamdi tchelepi stanford university many helpful discussions references hou multiscale finite element method elliptic problems composite materials porous media comput hou cai convergence multiscale finite element method elliptic problems rapidly oscillating coefficients math efendiev hou multiscale finite element methods theory applications springer efendiev ginting hou ewing convergence nonconforming multiscale finite element method siam numer aarnes hou multiscale domain decomposition methods elliptic problems high aspect ratios acta math jenny lee tchelepi method elliptic problems subsurface flow simulation comput jenny lee tchelepi adaptive fully implicit finitevolume method flow transport 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decoupling schemes predicting compressible fluid flows petr vabishchevicha nuclear safety institute russian academy sciences tulskaya moscow russia friendship university russia rudn university moscow russia jan peoples abstract numerical simulation compressible fluid flows performed using euler equations include scalar advection equation density vector advection equation velocity given pressure dependence density approximate solution value problem calculated using finite element approximation space fully implicit scheme used discretization time numerical implementation based newton method main attention paid fulfilling conservation laws mass total mechanical energy discrete formulation schemes splitting physical processes employed numerical solving problems barotropic fluid flows transition one time level next one iterative process used iteration linearized scheme implemented via solving individual problems density velocity possibilities proposed schemes illustrated numerical results model problem density perturbations keywords compressible fluids euler system barotropic fluid finite element method conservation laws schemes decoupling scheme introduction applied models continuum mechanics based conservation laws mass momentum energy transport scalar vector quantities due advection determines mathematical form conservation laws addition parameters flow positivity property monotonicity important properties differential problem continuum mechanics must inherited discrete problem flows ideal fluids governed euler equations whereas equations applied describing viscous flows mathematical problems validation models considered example corresponding author email address vabishchevich petr vabishchevich preprint submitted january books discussing existence solutions various sobolev spaces principal problems positivity fluid density also highlighted consideration also carried see instance discrete level various approximations time space computational fluid dynamics important problems associated two contradictory requirements namely necessary construct monotone approximations advective terms fulfil conservation laws construction monotone approximations discussed many papers see standard linear approximations considered basic problems continuum mechanics problems discretization space conservative approximations constructed basis using conservative divergent formulation continuum mechanics equations approach naturally implemented using interpolation method balance method regular irregular grids control method volume nowadays main numerical technique solve applied problems finite element method widely used computational fluid dynamics discretizations time computational fluid dynamics often constructed using explicit schemes strong restrictions time step sense stability moreover explicit schemes similar restrictions monotonicity approximate solution natural focus implicit schemes solve boundary value problems partial differential equations schemes widely used schemes weights linear problems study discretizations time based general theory stability schemes particular possible apply unimprovable coinciding necessary sufficient stability conditions formulated operator inequalities hilbert spaces present work value problem considered euler equations describing barotropic fluid flows section conservation laws mass momentum total mechanical energy discretization space performed section using standard lagrange elements density cartesian velocity components evaluate approximate solution new time level fully implicit scheme employed approximate solution mass conservation law holds estimate dissipation total mechanical energy fulfilled fully implicit scheme convenient numerical implementation solution new time level determined system coupled nonlinear equations density velocity decoupling scheme proposed section refers class linearized schemes splitting physical processes linearization carried field advective transport way time level solve individual problems density velocity possibilities proposed schemes illustrated results numerical solving model problem perturbation fluid density initially rest section solve numerically nonlinear discrete problem new time level newton method used calculations small number iterations two three sufficient process convergence influence grid size space time investigated observed decreasing time step results monotonization numerical solution main result paper proof robustness linearized decoupling scheme scheme involves separate solving standard advection problems density velocity demonstrates high iteration convergence approach used problems continuum mechanics numerical solving value problems equations mathematical models value problem considered describing barotropic fluid flows system equations includes scalar advection equation density vector advection equation velocity given pressure dependence density conservation laws mass momentum total mechanical energy discussed barotropic fluid continuity equation bounded domain form div density velocity momentum equation written conservative form div grad pressure considered fluid assumed barotropic known dependence pressure density assume domain boundaries rigid impermeability condition imposed initial conditions density velocity also specified value problem describes transient flows ideal barotropic fluid direct integration continuity equation domain taking account boundary condition results mass conservation law hilbert space define scalar product norm standard way kwk similar way space vector functions defined density conservation law mass written relation treated priori estimate equation directly expresses conservation law momentum integrating equation obtain ndx thus ndx udx multiplying taking account equation rewrite equation div div div integration domain view leads div second term expressed renormalized equation continuity define pressure potential equation particular ideal fluid const continuity equation div div integration renormalized equation continuity results expression div adding equality get arrive conservation law total mechanical energy equations basic conservation laws problem formulation convenience consideration introduce operators advective convective transport system euler equations advection operator divergent form written follows div assuming boundary condition satisfied velocity obtain continuity equation written form differential equation notation used similarly equation written form grad system equations prescribed dependence consider cauchy problem initial conditions see form considered problem key point property advection operator written divergent form implicit scheme solve numerically value problem euler equations use fully implicit backward euler scheme finite element discretizations space problems fulfilment conservation laws discrete level discussed discretization space solve numerically problem employ finite element discretizations space see define bilinear form div define subspace finite elements discrete operator similarly vector quantities representation employed simple specification boundary conditions assume separate parts boundary computational domain parallel coordinate axes finite element approximation used individual components vector constructing discretizations space arrive cauchy problem system operator equations corresponding space namely cauchy problem system ordinary differential equations instance put correspondence problem grad denoting onto solution problem satisfies system conservation laws solution problem see discretization time let step uniform simplicity grid time construct study schemes main attention given fulfillment corresponding conservation laws priori estimates important problem positivity density time level requires study considered present work solve numerically problem fully implicit scheme applied case approximate solution new time level determined using prescribed see value basic properties approximate solution related fulfillment conservation laws mass total energy simplify investigation assume density positive assumption time level view integration equation domain leads equality discrete analog mass conservation law momentum conservation law put correspondence equality ndx obtained integrating equation estimate total mechanical energy established following work multiplying equation integrating arrive grad first term view obtain grad definition operator makes possible rewrite inequality div estimate second term side inequality apply discrete analogue renormalized equation continuity multiply continuity equation following equality takes place min max assumption assume natural assumptions get second term taking account div grad div div view integration equation results div combining obtain inequality comparing conclude discrete level instead fulfillment conservation law total energy observe decrease energy noted property established additional assumption result consideration expressed following statement proposition fully implicit scheme produces approximate solution problem satisfies mass conservation law form ref momentum conservation law moreover assumptions hold estimate total mechanical energy also fulfilled decoupling schemes linearized scheme used solution new time level evaluated advective transport taken previous time level using linearization iterative process constructed numerical implementation fully implicit scheme approximate solution new time level determined sequential solving first linear problem advection density secondly linear problem velocity linearized scheme focus use techniques demonstrate following properties transition new time level implemented solving linear problems splitting respect physical processes employed namely problems density velocity solved separately individual problems velocity components example simplest decoupling scheme euler equations system linearized scheme advective transport involves velocity previous time level instead employ scheme grad first linear transport equation evaluate density new time level next linear decoupled system velocity components calculate velocity remark system equations given density general case coupled individual cartesian velocity components case parts boundary computational domain parallel axes cartesian coordinate system system equations decoupled evaluate independently individual components velocity linearized scheme discrete analogs mass conservation law see momentum conservation law see hold iterative decoupling scheme basis linearized scheme possible construct iterative algorithm numerical implementation fully implicit scheme approximate solution iteration denoted initial approximation previous time level assume new approximation new time level calculated previous iterations done similarly use system equations thus iteration firstly solve linear problem density calculate linear problem velocity numerical results possibilities fully implicit scheme decoupling schemes illustrated numerical results model problem density perturbation initially resting fluid test problem present results numerical solving model problem obtained using different techniques problem considered square assume dependence density pressure form simulate motion initially resting fluid initial density see specified form exp fully implicit scheme problem solved using standard uniform triangulation segments direction finite elements employed discretization space implement fully implicit scheme nonlinear discrete problem new time level newton method direct solver applied corresponding system linear algebraic equations compressible fluid flow shown fig presents density various time moments calculation used spatial grid time step density center computational domain maximum max minimal min values density entire domain given fig newton iterative method solving discrete problem new time moment converges quickly two three iterations enough table demonstrates convergence iterative process first step time present relative error first three iterations model problem obtained various time steps table convergence newton method iteration accuracy approximate solution test problem illustrated data density section solution calculated grid using various grids time shown fig similar data shown fig respectively easy see good accuracy reconstruction leading edge wave spatial grid refined also observe effect smoothing namely elimination increasing time step decoupling scheme using decoupling scheme greatest interest related convergence rate iterative process time step calculations equal present numerical results model problem figure density various time moments figure density central maximal minimal values figure solution problem various time moments calculated grid dotted line dashed solid figure solution problem various time moments calculated grid dotted line dashed solid figure solution problem various time moments calculated grid dotted line dashed solid figure solution problem various time moments calculated grid dashed line dotted solid consideration obtained different grids space dependence solution number iterations grid shown fig figure presents similar data grids respectively easy see see finest grid see fig linearized scheme yields substantially solution monotonized subsequent iterations main conclusion study demonstration high computational efficiency iterative decoupling scheme namely problems consideration sufficient two iterations using methods key point violation conservation law total energy fully implicit scheme instead conservation energy see estimate decreasing total energy observed dynamics total mechanical energy using linearized scheme iterative decoupling schemes various grids shown fig according calculated time moment solution obtained using decoupling scheme practically coincides solution derived fully implicit scheme data indicate conservation law total energy satisfied good accuracy decreasing time step results increasing figure solution problem various time moments calculated grid dashed line dotted solid figure solution problem various time moments calculated gridm dashed line dotted solid figure total mechanical energy various time steps obtained grid dashed line solid accuracy conservation law fulfillment acknowledgements publication financially supported ministry education science russian federation agreement figure total mechanical energy various time steps obtained grid dashed line solid figure total mechanical energy various time steps obtained grid dashed line solid references batchelor introduction fluid dynamics cambridge university press landau lifshitz fluid mechanics godunov romenskii elements continuum mechanics conservation laws springer leveque finite volume methods hyperbolic problems cambridge university press anderson computational fluid dynamics basics applications wesseling principles computational fluid dynamics springer lions mathematical topics fluid mechanics incompressible models oxford university press lions mathematical topics fluid mechanics compressible models oxford university press feireisl karper mathematical theory compressible viscous fluids analysis numerics springer kulikovskii pogorelov semenov mathematical aspects numerical solution hyperbolic systems taylor francis hundsdorfer verwer numerical solution equations springer verlag kuzmin guide numerical methods transport equations university morton kellogg numerical solution problems chapman hall london samarskii vabishchevich numerical methods solving problems urss moscow russian samarskii theory difference schemes marcel dekker new york versteeg malalasekra introduction computational fluid dynamics finite volume method prentice hall ern guermond theory practice finite elements springer larson bengzon finite element method theory implementation applications springer donea huerta finite element methods flow problems wiley zienkiewicz taylor finite element method fluid dynamics ascher numerical methods evolutionary differential equations society industrial applied mathematics leveque finite difference methods ordinary partial differential equations problems society industrial applied mathematics samarskii matus vabishchevich difference schemes operator factors kluwer academic dordrecht marchuk splitting alternating direction methods ciarlet lions eds handbook numerical analysis vol northholland vabishchevich additive schemes splitting schemes gruyter berlin galerkin finite element methods parabolic problems springer verlag berlin brenner scott mathematical theory finite element methods springer new york

relational concept analysis alexandre jessie marianne giacomo bourgogne dijon france lirmm cnrs montpellier montpellier france limos clermont auvergne france contact jcarbonnel huchard mar abstract formal concept analysis associated conceptual structures used support exploratory search conceptual navigation relational concept analysis rca extension formal concept analysis process relational datasets rca multiple interconnected structures represent good candidates support exploratory search relational datasets enabling navigation within structure well connected structures however building entire structures present efficient solution explore small localised area dataset instance retrieve closest alternatives given query cases generating concept neighbour concepts navigation step appears less costly alternative paper propose algorithm compute concept neighbourhood extended concept lattices concepts generated directly relational context family possess formal relational attributes algorithm takes account two rca scaling operators illustrate example keywords relational concept analysis formal concept analysis ondemand generation introduction many datasets thematic areas like environment product lines comprise databases complying relational data model typical applications currently involved concern issues relative watercourse fresqueau project inventory use pesticidal antibacterial antifungal knomana project analysis representation product lines applications wide range question forms classical querying establishing correlations descriptions objects several categories case based reasoning questions addressed complementary approaches including conceptual classification building knowledge pattern rule extraction exploratory search knomana project http http bazin carbonnel huchard kahn example one main purpose ongoing inventory support farmers advisors local entrepreneurs researchers selecting plants immediate interest agricultural crop protection animal health users face large amounts data mainly formulate general potentially imprecise potentially inaccurate queries without prior knowledge data exploratory search suitable approach context previous work shown formal concept analysis may relevant support data exploration expect relational concept analysis rca beneficial well considering rca relational dataset exploration brings issues relative use scaling logical operators iterative process presence several concept lattices connected via relational attributes despite additional complexity rca helps user concentrate classification objects several categories object groups concepts described intrinsic attributes relations object groups categories besides relational attributes offer support navigate object groups different categories concept lattices offer navigation object groups category several complementary strategies explore datasets using rca one may consist exhaustively computing concept lattices related artefacts like implication rules several steps using several logical operators considering object categories relationships another strategy followed consists computation concept neighbourhood comprising upper lower relational covers next section presents main principles relational concept analysis section computation concept neighbourhood presented section section illustrates algorithm example introduced section related work exposed section conclude paper perspectives section relational concept analysis formal concept analysis fca allows structure set objects described attributes canonical structure called concept lattice based formal context set objects set attributes incidence relation stating objects possess attributes context application fca extracts finite set formal concepts concept extent concept intent concept lattice obtained ordering concepts order extents call resp attributeconcept lowest resp greatest concept lattice possessing object resp attribute relational concept analysis relational concept analysis rca adaptation fca process relational datasets relational dataset composed several sorts objects described attributes relationships objects input rca takes relational context family rcf gathering set formal contexts set relational contexts defining links objects different formal contexts definition relational context family relational context family pair set formal contexts relations set relational contexts relations sets objects respectively called source context target context three contexts table present example rcf taken software product line domain table top displays two formal contexts one side presents data modelling tools attributes representing compatible operating systems data models tools may manage table side describes database management systems dbms according data types may handle table bottom presents relational context stating data modelling tools support database management systems astah erwin magic draw mysql workbench enum set geometry spatial audio image video xml json period windows mac linux conceptual physical logical etl table top two formal contexts side data modelling tools side database management systems dbms bottom relational context stating support dbms dbms mysql oracle postgresql teradata support mysql oracle postgresql teradata astah erwin magic draw mysql workbench applying rca contexts builds first time one concept lattice per context objects without taking links account two concept lattices associated table top presented fig second time rca introduces links objects different lattices depending bazin carbonnel huchard kahn windows mac linux mysql workbench physical conceptual enum geometry logical set json astah erwin mysql postgresql etl period teradata xml magic draw spatial audio image video oracle fig left concept lattice right concept lattice dbms relations expressed links take form relational attributes introduce abstractions concepts target context source context specific relation specific scaling operator example may introduce relational attribute support characterise support least one dbms offering json xml generally given two formal contexts relational context application rca extends set attributes set relational attributes representing links concepts extended attribute set denoted incidence relation extended take account new attributes denoted associating object depending relation concept denoted involved relational attribute scaling operator relational attribute thus form paper focus two scaling operators existential operator denoted associating object relational attribute linked least one object extent universal strict operator denoted associating object objects linked included extent concept lattice associated formal context structures objects attributes relations sets objects relational attributes fig presents extended concept lattice corresponding extended formal context according relation support existential scaling operator relational concept analysis windows exists support exists support exists support mac linux physical mysql workbench astah magic draw conceptual exists support exists support logical exists support exists support erwin etl exists support fig concept lattice extended context way complex data models including one relation rca produces succession concept lattices extended step new abstractions obtained previous step step concept lattices set ones built initial formal contexts step formal contexts set extended depending concepts concept lattices relations expressed exploration algorithm section present algorithm taking step exploration considers rcf potentially extended previous steps starting concept context rcf exploration strategy consists choosing set relations rcf source provided scaling operators objective one step complete intent corresponding extent well compute upper lower relational covers meanwhile rcf updated relational attributes next step redefining derivation operators explicit knowledge relational attributes context requires computation concepts target contexts however afford amounts exhaustive computation relational concepts multiple contexts would prefer manipulate minimal number relational attributes allowing derive relational attributes bazin carbonnel huchard kahn object described attribute instead abuse notation also necessarily described attributes form intents represented without loss information relational attributes constructed attributeswise maximal concepts however problem arises representation set intersection used compute intent set objects anymore similarly maximal relational attributes explicitly present context extent set attributes computed simple test set inclusion remedy provide three algorithms use sets attributes intrinsic relational maximal relational attributes given explicitly intersect takes input two sets attributes represented maximal relational attributes outputs set maximal relational attributes intersection relational attribute intersection exists two attributes holds scaling operator intersecting intents concepts attributes keeping maximal ones results maximal relational attributes uses algorithm algorithm intersect input formal context attribute set intent object output relational intersection attribute set intent foreach foreach intersect intersect foreach foreach intersect intersect return uses intersect compute intent set objects described maximal relational attributes starts set explicitly known attributes intersects description object context computes extent set maximal relational attributes object attribute checks whether intersect correct way depending scaling operator relational concept analysis algorithm input formal context set objects output computes intent set objects foreach intent return algorithm input formal context set attributes output computes extent set attributes foreach foreach else foreach else foreach return computing closed neighbourhood redefined derivation operators implicitly known relational contexts able compute upper lower relational covers concept easiest relational covers concept relational cover concept maximal relational attribute upper covers easy candidates generated adding object set perfect knowledge current extent computing corresponding concept covers candidates smallest extent computing lower covers challenging could computed adding attributes intent full set relational attributes known implicitly chose instead remove objects lower covers concepts maximal extents contained contain minimal generators simple bazin carbonnel huchard kahn way compute would remove minimal transversals minimal generators algorithm computes closed neighbourhood concept takes input set formal contexts rcf strategy starting concept context goal compute complete intent corresponding extent well upper lower relational covers extended context first loop lines computes ocj ocj relation scaling operator give rise new relational attribute added context growcontext line intent concept extended relational attributes added previous loop next loop lines computes relational covers concept relational attribute intent corresponding concept target context added cover lines lower covers computed removing extent minimal transversal set minimal generators extent finally upper covers computed lines candidates created adding object extent minimal resulting concepts kept algorithm rca input strategy concept output completed concept closed relational neighbourhood foreach ocj foreach growcontext ocj foreach foreach mint rans mingen foreach return relational concept analysis algorithm growcontext ocj input formal context relational context scaling operator object ocj set output extends context adds crosses foreach ocj obj example section illustrate defined algorithms consider rcf dbm support presented section decide apply strategy support let imagine user wants select data modelling tool runs windows windows handles logical conceptual data models logical conceptual traditional fca may compute formal concept associated attributes side fig inform user corresponding tools erwin magic draw tools also handle physical let apply algorithms concept retrieve supported dbms relational cover find closest alternatives query lower upper covers rca support lines extend context relational attributes representing dbms support target context case one relation support visited line line ocj takes dbms concepts righthand side fig loop lines considers objects growcontext called object associated relational attributes representing concepts ocj extents least one object linked support astah ysql oracle mysql oracle added associated astah end line obtain extended context presented table line updates intent input concept take account relational attributes windows conceptual physical logical bazin carbonnel huchard kahn sup sup sup windows mac linux conceptual physical logical etl astah erwin magic draw mysql workbench sup table formal context extended according relation support cepts dbms corresponding relational attributes form relational cover input concept lines lines compute minimal generators extent erwin magic draw magic draw minimal transversals magic draw erwin two concepts magic draw erwin extent represent lower cover respectively fig finally lines consider objects extent mysql workbench astah one compute concept corresponding union extent obtain two concepts fig represent upper cover related work lattice structures among first structures used support exploratory search task later attracted lot attention formal concept analysis theory many works focus conceptual neighbourhood present information related query closest variants paper consider rca retrieve conceptual neighbourhood interconnected lattices structuring intrinsic relational attributes exponential growth concept lattices consequence main limitation exploratory search lies complexity computation structures many solutions proposed reduce complexity conceptual navigation authors propose prune concept lattice restrict explorable dataspace computing iceberg concept lattices applying constraints bound final structure ease navigation authors seek extract simplified browsable structures first extract tree concept lattice reduce obtained tree using clustering methods tool searchsleuth enables exploratory search web queries field domain entirely processed using fca concept lattices tackle issue generate new formal context specific query relational concept analysis navigation step previous work proposed compute conceptual neighbourhood query concept lattice restricted poset condensed alternative concept lattices step conceptual neighbourhood computed present work also generate conceptual neighbourhood time interconnected concept lattices mimouni use rca structure query browse collection legal documents first build interconnected lattices representing different types legal documents referring approach allows retrieval concept corresponding user query explore variations query navigation neighbour concepts approach compute lattices first step hermann propose faceted search implementation tool sewelis allows browse relational datasets form rdf files also propose rlca relational extension logical formal analysis adaptation fca describe objects formulas logics instead binary attributes rca computes connected yet separate concept lattices one per sort objects rlca gathers objects descriptions relations objects one structure conclusion paper proposed algorithms compute conceptual neighbourhood query connected concept lattices generated rca first redefined traditional fca derivation operators take account relational attributes presented way compute relational upper lower covers given concept extended lattices without computing structures two rca scaling operators existential universal strict may used illustrated algorithms work running example domain software product line engineering future plan study properties algorithm implement perform exploratory search relational datasets scalability study real datasets projects fresqueau knomana available product descriptions envisioned end generate random queries exploration paths also collecting concrete questions knomana project partners real exploration tasks domain qualitatively evaluate benefits approach references alam napoli latviz new practical tool performing interactive exploration concept lattices proc int conf concept lattices applications cla bazin carbonnel kahn generation reducing complexity conceptual navigation proc int symp foundations intelligent systems ismis bazin carbonnel huchard kahn ben nasr acher bosco sannier baudry davril automated extraction product comparison matrices informal product descriptions systems software carbonnel huchard nebut analyzing variability product families canonical feature diagrams proc int conf software engineering knowledge engineering seke carpineto romano exploiting potential concept lattices information retrieval credo universal comp sci codocedo napoli formal concept analysis information retrieval survey proc int conf formal concept analysis icfca ducrou eklund searchsleuth conceptual neighbourhood web query proc int conf concept lattices applications cla dunaiski greene fischer exploratory search academic publication citation data using interactive tag cloud visualizations scientometrics hermann reconciling faceted search query languages semantic web int metadata semantics ontologies ridoux sigonneau arbitrary relations formal concept analysis logical information systems proc int conf conceptual structures iccs springer reconciling expressivity usability information access filesystems semantic web habilitation thesis matisse univ rennes habilitation diriger des recherches hdr defended november ganter wille formal concept analysis mathematical foundations springer godin gecsei pichet design browsing interface information retrieval proc int conf research development information retrieval sigir godin saunders gecsei lattice model browsable data spaces inf sci hacene huchard napoli valtchev proposal combining formal concept analysis description logics mining relational data proc int conf formal concept analysis icfca huchard hacene roume valtchev relational concept discovery structured datasets ann math artif intell marchionini exploratory search finding understanding comm acm melo grand aufaure browsing large concept lattices tree extraction reduction methods int intelligent information technologies mimouni nazarenko salotti conceptual approach relational application legal collections proc int conf formal concept analysis icfca palagi gandon giboin troncy survey definitions models exploratory search acm workshop esida iui stumme taouil bastide pasquier lakhal computing iceberg concept lattices titanic data knowledge engineering

probabilistic integration role statistical computation chris mark michael dino department statistics university warwick mathematics imperial college london school mathematics statistics physics newcastle university alan turing institute data science department engineering science university oxford department statistics university oxford oct department october abstract research frontier emerged scientific computation wherein discretisation error regarded source epistemic uncertainty modelled raises several statistical challenges including design statistical methods enable coherent propagation probabilities possibly deterministic computational order assess impact discretisation error computer output paper examines case probabilistic numerical methods routine statistical computation focus numerical integration probabilistic integrator equipped full distribution output reflects fact integrand discretised main technical contribution establish first time rates posterior contraction one method several substantial applications provided illustration critical evaluation including examples statistical modelling computer graphics computer model oil reservoir introduction paper presents statistical perspective theoretical methodological issues pertinent probabilistic numerical methods aim stimulate feel important discussion methods use contemporary emerging scientific statistical applications background numerical methods tasks approximate solution linear system integration global optimisation discretisation schemes approximate solution differential equations core building blocks modern scientific statistical computation typically considered computational return point estimate deterministic quantity interest whose numerical error neglected numerical methods thus one part statistical analysis uncertainty routinely accounted although analysis errors bounds often available highly developed many situations numerical error negligible action required however numerical errors propagated computational pipeline allowed accumulate failure properly account errors could potentially drastic consequences subsequent statistical inferences mosbach turner oates study numerical algorithms statistical point view uncertainty formally due discretisation known probabilistic numerics philosophical foundations probabilistic numerics best knowledge first clearly exposed work larkin kadane diaconis hagan theoretical support comes field complexity traub continuous mathematical operations approximated discrete finite operations achieve prescribed accuracy level proponents claim approach provides three important benefits firstly provides principled approach quantify propagate numerical uncertainty computation allowing possibility errors complex statistical structure secondly enables user uncover key contributors numerical error using established statistical techniques analysis variance order better target computational resources thirdly dual perspective numerical analysis inference task enables new insights well potential critique refine existing numerical methods final point recent interest led several new effective numerical algorithms many areas including differential equations linear algebra optimisation extensive bibliography reader referred recent expositions hennig cockayne contributions aim stimulate discussion suitability probabilistic numerical methods statistical computation decision made focus numerical integration due central role computational statistics including frequentist approaches bootstrap estimators efron tibshirani bayesian approaches computing marginal distributions robert casella particular focus numerical integrals cost evaluating integrand forms computational bottleneck end let distribution state space task compute rather estimate integrals form integrand function interest motivation comes settings possess convenient closed form function actually evaluated input epistemic uncertainty actual value attained use probabilistic model epistemic uncertainty advocated far back larkin probabilistic integration method focus known bayesian cubature method operates evaluating integrand set states socalled discretisation returns distribution expresses belief true value computational cost associated general name suggests distribution based prior captures certain properties updated via bayes rule basis evaluations integrand maximum posteriori map value acts point estimate integral rest distribution captures uncertainty due fact evaluate integrand finite number inputs however theoretical investigation best knowledge contrast map estimator first contribution therefore investigate claim posterior provides coherent honest assessment uncertainty due discretisation integrand claim shown substantiated rigorous mathematical analysis building analogous results reproducing kernel hilbert spaces prior particular rates posterior contraction point mass centred true value established however check prior given integration problem second contribution explore potential use probabilistic integrators contemporary statistical context developed strategies model evidence evaluation via thermodynamic integration large number candidate models compared inverse problems arising partial differential equation models oil reservoirs iii logistic regression models involving latent random effects spherical integration used rendering virtual objects prescribed visual environments case results presented relative advantages disadvantages probabilistic approach integration presented critical assessment outline paper structured follows sec provides background outlines analytic framework method studied sec describes novel theoretical results sec devoted discussion practical issues including important issue prior elicitation sec presents several novel applications probabilistic integration critical sec concludes appraisal suitability probabilistic numerical methods applied statistical context background first provide reader relevant background sec provides formal description secs explain analysis dual minimax analysis nonparametric regression sec relates ideas established sampling methods let measurable space either subspace general manifold sphere case equipped borel let distribution integrand assumed integrable function whose integral object interest notation functional arguments write vector arguments denote functions write vector whose ith element notation max used relation taken mean exist computer code reproduce experiments reported paper downloaded http cubature rule describes functional form states weights term quadrature rule sometimes preferred domain integration notation motivated fact pexpression integral respect empirical measure atomic measure weights negative need satisfy bayesian cubature probabilistic integration begins defining probability space associated stochastic process belongs linear topological space larkin considered gaussian process stochastic process random variables gaussian topological dual bogachev paper avoid technical obfuscation assumed contains continuous functions let denote expectation taken characterised mean function covariance function write paper assume without loss generality note priors could also used process affords heavier tails values assumed integrand next step consider restriction set induce posterior measure fact contains continuous functions ensures moreover restriction set also shown denoted see chap rasmussen williams final step produce distribution projecting posterior defined integration operator sketch procedure presented figure relevant formulae provided denote expectation variance taken respect write vector values vector whose ith entry matrix entries proposition induced distribution gaussian mean variance denotes integral respect argument proofs paper reserved supplement seen computational cost obtaining full posterior much higher obtaining point estimate integral however certain combinations point sets covariance functions reduce cost several orders magnitude see karvonen would case instead since canonical space continuous processes polish space polish spaces borel spaces thus admit regular conditional laws theorem theorem kallenberg integrand posterior distribution solution integral figure sketch bayesian cubature top row shows approximation integrand red posterior mean blue number function evaluations increased dashed lines represent posterior credible intervals bottom row shows gaussian distribution mean variance dashed black line gives true value integral number states increased posterior distribution contracts onto true value integral formally associates stochastic process prior model integrand turn provides probabilistic model epistemic uncertainty value integral without loss generality assume remainder paper eqn takes form cubature rule wibc wbc furthermore eqn depend function values location states choice covariance function useful allows state locations weights however also means variance endogeneous driven choice prior valid quantification uncertainty thus relies prior consider issue sec mean eqn coincides classical cubature rules specific choices covariance function example one dimension brownian covariance function min leads posterior mean piecewise linear interpolant states trapezium rule suldin similarly constructed covariance function cubature recovered karvonen showed cubature rules recovered clearly point estimator eqn natural object also received attention kernel quadrature literature sommariva vianello empirical interpolation literature kristoffersen recent work computational focus includes kennedy minka rasmussen ghahramani huszar duvenaud gunter briol karvonen oettershagen present paper focuses full posterior opposed point estimator papers studied cubature rules hilbert spaces next review analysis approximation properties cubature rule carried terms reproducing kernel hilbert spaces rkhs berlinet consider hilbert space inner product associated norm said rkhs exists symmetric positive definite function called kernel satisfies two properties reproducing property shown every kernel defines rkhs every rkhs admits unique reproducing kernel berlinetr sec paperr kernels assumed satisfy particular guarantees define kernel mean exists consequence smola name justified reproducing property permits elegant theoretical analysis many quantities interest tractable language kernel means cubature rules form eqn written approximation kernel mean given fixed integration error associated expressed tight upper bound error obtained expression decouples magnitude integrand kernel mean approximation error following sections discuss cubature rules tailored target second term upper bound optimality cubature weights denote dual space denote corresponding norm performance cubature rule quantified error wce rkhs sup wce characterised error estimating kernel mean integral inner product commute due existence bochner integral steinwart christmann sometimes called inequality hickernell fact minimisation wce natural corresponds solving problem feature space induced kernel let denote vector weights vector matrix entries obtain following fact several optimality properties integration rkhs collated sec novak relevant work optimal estimate without loss generality take form cubature rule form eqn precise adaptive estimator terms asymptotic wce cubature rule defined relate ideas consider challenge deriving optimal cubature rule conditional fixed states minimises wce rkhs weights fact solution convex problem shows reproducing kernel equal covariance function map identical optimal cubature rule rkhs kadane wasilkowski furthermore expression wce fact shows cubature rule based states regarding optimality problem thus reduced selection states selection states earlier work hagan considered states employed gaussian cubature methods rasmussen ghahramani generated states using monte carlo calling approach bayesian bmc recent work gunter briol selected states using experimental design target variance approaches briefly recalled monte carlo methods method cubature rule based uniform weights wimc random states simplest methods consists sampling states xmc independently densities markov chain monte carlo mcmc methods proceed similarly induce dependence structure among xmcmc denote rani mcmc uniformly dom estimators xmc weighted estimators many challenging integration problems since provide convergence rate wce widely applicable analyse instance central limit theorem clt gives convergence distribution however clt may measure epistemic uncertainty explicit model numerical error since valid asymptotically unknown depending integral estimated quasi monte carlo qmc methods exploit knowledge rkhs spread states efficient deterministic way domain hickernell qmc also approximates course adaptive cubature may provide superior performance single fixed function minimax result true general outside rkhs framework integrals using cubature rule uniform weights wiqmc cases optimal convergence rates well sound statistical properties qmc recently led interest within statistics gerber chopin buchholz chopin related method weights explored stein experimental design methods optimal obc rule selects states globally minimise variance obc corresponds classical cubature rules specific choices kernels karvonen however obc general implemented problem optimising states general smola sec pragmatic approach select states use experimental design methods greedy algorithm sequentially minimises method called sequential sbc straightforward implement using numerical optimisation probabilistic integration method often used osborne gunter sophisticated optimisation algorithms also used example empirical interpolation literature eftang stamm proposed adaptive procedures iteratively divide domain integration literature briol used conditional gradient algorithms task similar approach recently considered oettershagen present experimental design schemes possess computational efficiency come expect mcmc qmc moreover scale well highdimensional settings due need repeatedly solve optimisation problems established theoretical guarantees reasons focus next mcmc qmc methods section presents novel theoretical results probabilistic integration methods states generated mcmc qmc sec provides formal definitions sec establishes theoretical results probabilistic integration sampling methods mcmc lesser extent qmc widely used statistical computation pursue idea using mcmc qmc generate states aim exploit account possible impact numerical integration error inferences made statistical applications mcmc possible two states identical prevent kernel matrix kpfrom becoming singular duplicate states mcmc define xqmc procedure requires modification existing mcmc qmc sampling methods estimator associated full posterior distribution described sec moment taken emphasise apparently simple act mcmc samples dramatic improvement convergence rates integration sufficiently smooth integrand whilst main interest suitability statistical model justified since information contained function evaluations lost introduce additional bias methods contrast methods discretisation integral highlight efficient point estimation comes date aware previous use bmcmc presumably due analytic intractability kernel mean bqmc described hickernell marques best knowledge theoretical analysis posterior distributions associated either method goal next section establish fundamental results theoretical properties section present novel theoretical results bmc bmcmc bqmc setting consider assumes true integrand belongs rkhs prior based covariance function identical kernel supported rather hilbert scaler viewed technical detail indeed constructed via theoretical analysis similar could carried lemma cialenco bayesian markov chain monte carlo baseline begin noting general result estimation requires slight strengthening assumption kernel kmax implies bounded estimators lemma song show kmax wce converges probability classical rate turning bmcmc bmc special case consider compact manifold distribution assumed admit density respect lebesgue measure denoted byp define sobolev space consist measurable functions order rkhs derivative counting hence principled approach practitioners choose suitable rkhs results apply rkhs permitting flexibility choice kernel specific examples kernels provided sec analysis based scattered data approximation literature wendland minor technical assumption enables simplify presentation results set may augmented finite set increase clearly bearing asymptotics measurable write indicator function event theorem bmcmc suppose bounded away zero let suppose states generated reversible uniformly ergodic markov chain targets moreover exp depends arbitrarily small two norms vector space equivalent exists constants khk khk result shows posterior distribution posterior distribution concentrates open neighbourhood true integral result address frequentist coverage posterior assessed empirically sec although focus point estimation brief comment warranted lower bound wce attained randomised algorithms setting novak thus result shows point estimate one rate away bach obtained similar result fixed specific importance sampling distribution analysis directly imply asymptotic results vice versa completion work similar results point estimation appeared oettershagen bauer thm generalised several directions firstly consider general domains specifically scattered data approximation bounds used proof apply compact domain satisfies interior cone condition wendland technical results direction established oates second consider spaces example slight extension thm shows certain infinitely differentiable kernels lead exponential rates wce rates posterior contraction brevity details omitted bayesian quasi monte carlo previous section focused bmcmc sobolev space avoid repetition consider interesting spaces functions whose mixed partial derivatives exist even faster convergence rates obtained using bqmc formulate bqmc must posit rkhs priori consider collections states xqmc constitute qmc point set tailored rkhs consider uniform define thepsobolev space dominating mixed smoothness consist functions order space rkhs build intuition note normequivalent rkhs generated tensor product kernels sickel ullrich indeed tensor product univariate sobolev space kernel specific space seek appropriate qmc point set digital construction example qmc point set details refer reader dick pillichshammer details theorem bqmc let suppose states chosen according digital net prime exp depends arbitrarily small result shows posterior indeed rate contraction much faster compared terms point estimation optimal rate deterministic algorithm integration functions novak results understood hold qmc methods control variate trick bakhvalov used achieve optimal randomised wce steps outside bayesian framework general give guarantees clear far result generalised terms compared result bmcmc since would require use different qmc point sets summary section established rates posterior contraction bmc bmcmc bqmc general sobolev space context results essential since establish sound properties posterior shown contract truth evaluations made integrand course higher computational cost may restrict applicability method regimes however emphasise motivation quantify uncertainty induced numerical integration important task often justifies higher computational cost implementation far established sound theoretical properties bmcmc bqmc assumption prior unfortunately prior specification complicates situation practice since given test function infinitude rkhs belongs specific choice space impact upon performance method particular scale posterior driven scale prior uncertainty quantification provided endogenous prior could mitigate advantages probabilistic numerical framework important point discussed important highlight distinction bqmc former choice states depend rkhs allows possibility specification kernel evaluations integrand obtained whereas alternative methods kernel must stated discussion therefore centres prior specification relation several statistical techniques applied prior specification theoretical results address important issue whether scale posterior uncertainty provides accurate reflection actual numerical error closely related problem prior specification kriging literature stein stein consider parametric kernel distinction drawn scale parameters smoothness parameters former defined parametrising norm whereas latter affect set selection based data successful absence acute sensitivity parameters scale parameters wide body evidence demonstrates usually concern stein however selection smoothness parameters active area theoretical research cases possible elicit smoothness parameter physical mathematical considerations known number derivatives integrand attention instead restricted scale parameters several approaches discussed relation suitability marginalisation natural approach bayesian perspective set prior parameters marginalise obtain posterior recent results certain infinitely differentiable kernel establish minimax optimal rates approach including practically relevant setting supported ambient space yang dunson however act marginalisation involves intractable integral computational cost evaluating integral often dwarfed integral interest marginalisation nevertheless introduces additional undesirable computational challenge might require several approximations osborne however possible analytically marginalise certain types scale parameters amplitude parameters proposition suppose covariance function takes form reproducing kernel amplitude parameter consider improper prior posterior marginal distribution mean variance degrees freedom another approach kernel choice however perform poorly number data small since data needs reduced training test sets performance estimates also known large variance cases chap rasmussen williams since small scenario one primary settings interest felt unsuitable use applications empirical bayes alternative approaches empirical bayes selection scale parameters choosing maximise likelihood data sec rasmussen williams advantage providing objective function easier optimise relative however also note lead small since full irregularity integrand yet uncovered addition shown estimates need converge supported infinitely differentiable functions stein remainder chose focus combination marginalisation approach amplitude parameters approach remaining scale parameters empirical results support use approach though claim strategy optimal tractable intractable kernel means requires kernel mean available case several pairs subset pairs recorded table arbitrary arbitrary arbitrary arbitrary unif unif unif mixt gaussians unif unif mixt gauss unif known moments known log wendland weighted exponentiated quadratic exponentiated quadratic gegenbauer trigonometric splines polynomial kernel reference oates sec use error function kennedy sec integration parts wahba briol oates table list distribution kernel pairs provide expression kernel mean initial error refers tensor product kernels event pair interest lead kernel mean sometimes possible determine another pair available absolutely continuous respect derivative exists one construct importance sampling estimator proceed hagan one side contribution research novel generic approach accommodate intractability kernel mean described detail supplement used case studies presented sec results aims following section validate preceding theoretical analysis explore use probabilistic integrators range problems arising contemporary statistical applications assessment uncertainty quantification focus uncertainty quantification provided particular performance hybrid approach kernel parameters clear concerned accurate point estimation low computational cost wellstudied problem reaches far beyond methods paper rather aiming assess suitability probabilistic description integration error provided motivation expensive integrands perform assessment controlled environment considered inexpensive test functions varying degrees irregularity whose integrals accurately approximated included test function exp sin easy setting hard setting hard test function variable hence difficult approximate see fig one realisation states generated independently uniformly initially estimated integrals estimated integrals figure evaluation uncertainty quantification provided used empirical bayes marginalised left test functions top bottom dimension right solutions provided monte carlo black bayesian bmc red one typical realisation credible regions shown bmc green horizontal line gives true value integral blue curve gives corresponding lengthscale parameter selected used estimate work rkhs characterised tensor products kernels modified bessel function second kind kernel means exist case whenever used select lengthscale parameters kernel amplitude parameter marginalised prop smoothness parameter fixed note test functions space degree arbitrariness choice prior results shown fig used denote posterior credible regions value integral also display values length scale selected appear converge rapidly encouraging emphasise term credible used loosely since estimated rather marginalised figure evaluation uncertainty quantification provided used empirical bayes marginalised dimensions top bottom coverage frequencies computed top bottom realisations compared notional bayesian credible regions varying level number observations quadrant represents conservative credible intervals whilst quadrant represents intervals left easy test function right hard test function provide theoretical guarantees work negative side possible small values indeed posterior liable since absence evidence contrary selects large values correspond regular functions evident hard case next computed coverage frequencies credible regions sample size process repeated many realisations states shown fig may seen large enough uncertainty quantification provided easier function whilst complicated functions expected observed coverage small values performance subsequently investigated selected general performed worse marginalised results contained supplement finally understand whether theoretical results asymptotic behaviour realised practice note absence variance independent integrand may plotted function results supplement demonstrate theoretical rates observed practice bqmc however large values data required achieve accurate estimation increased numerical instability observed results test functions provided section illustrate extent uncertainty quantification possible using particular examples observed reasonable frequentist coverage number samples small remainder explore possible roles bmcmc bqmc statistical applications four case studies carefully chosen highlight strengths weaknesses presented brief critiques study contained full details found supplement case study model selection via thermodynamic integration consider problem selecting single best model among set based data assumed arise true model set bayesian solution assuming uniform prior models select map model focus case uniform prior models problem hence reduces finding largest marginal likelihood usually intractable integrals parameters associated model one approach model selection estimate turn say take maximum particular thermodynamic integration one approach approximation marginal likelihoods individual models gelman meng friel pettitt many contemporary applications map model example variable selection many candidate models map becomes sensitive numerical error since incorrect model assigned overly large value due numerical error case could selected place true map model explore potential exploit probabilistic integration surmount problem thermodynamic integration simplify notation consider computation single suppress dependence index corresponding model denote parameter space inverse temperature define power posterior distribution density thermodynamic identity formulated double integral log log thermodynamic integral log log standard practice discretise outer integral estimate inner integral using mcmc letting denote fixed temperature schedule thus using trapezium rule log log mcmc samples several improvements proposed including use numerical quadrature outer integral friel hug use control variates inner integral oates date probabilistic integration explored context probabilistic thermodynamic integration proposal apply inner outer integrals instructive since nested integrals prone propagation accumulation numerical error several features method highlighted transfer learning probabilistic approach two integrands assigned prior probability models inner integral assign prior data vector estimating times much data estimator eqn makes use function evaluations information transfer across temperatures made possible explicit model underpinning posterior gaussian random vector mean covariance defined obvious notation kernel matrix defined inclusion prior information outer integral known discretisation error substantial friel proposed correction trapezium rule mitigate bias hug pursued use simpson rule attacking problem probabilistic perspective want place stationary prior since known extensive empirical work vary smaller values indeed commonly used calderhead girolami would like encode information prior proceed importance sampling step log implies importance distribution small renders function approximately stationary made precise supplement stationary prior transformed integrand provides encoding prior knowledge used propagation uncertainty construction posterior log gaussian post prob std thermo int prob thermo int std thermo int candidate models post prob prob thermo int candidate models candidate models candidate models figure probabilistic thermodynamic integration illustration variable selection logistic regression true model standard probabilistic thermodynamic integration used approximate marginal likelihoods hence posterior models row represents independent realisation mcmc data fixed left standard monte carlo point estimates marginal likelihood assumed associated numerical error right probabilistic integration model numerical error integral propagated posterior models probabilistic approach produces probability distribution probability distribution numerical uncertainty modelled top usual uncertainty associated model selection mean covariance defined log log kernel matrix defined term arises outer integral term arises propagating numerical uncertainty inner integral outer integral simulation study experiment conducted elicit map model collection candidate logistic regression models variable selection setting could achieved many ways aim compare accuracy point estimates rather explore probability model unlike standard methods provided full details supplement results shown fig compared approximations model posterior obtained using standard method versus probabilistic method two realisations mcmc data fixed make observations probabilistic approach models numerical uncertainty top usual statistical uncertainty computation associated required less time total time taken afforded mcmc iii model always selected map numerical error ignored depended mcmc random seed contrast probabilistic approach either could feasibly map mcmc realisations numerical uncertainty top row fig shows large posterior uncertainty marginal likelihood could used indicator computational effort expended particular integral posterior variance dominated uncertainty due discretisation error outer integral rather inner integral suggests numerical uncertainty could reduced allocating computational resources outer integral rather inner integral case study uncertainty quantification computer experiments consider industrial scale computer model teal south oil field new orleans hajizadeh conditional field data posterior inference facilitated using mcmc lan oil reservoir models generally challenging mcmc first simulating models making cost individual mcmc samples minutes several hours second posterior distribution often exhibit strongly concentration measure computed statistics interest using bmcmc uncertainty quantification afforded aims enable valid inferences presence relatively mcmc samples full details provided supplement quantification uncertainty associated predictions major topic ongoing research field mohamed hajizadeh park due economic consequences associated inaccurate predictions quantities future oil production rate probabilistic model numerical error integrals associated prediction could provide complete uncertainty assessment particular integrals considered posterior means model parameter compared empirical benchmark obtained brute force mcmc bmcmc employed kernel whose selected using estimates posterior means obtained using standard mcmc bmcmc shown fig example posterior distribution provides sensible uncertainty quantification integrals integrals point accuracy bmcmc estimator matched standard mcmc estimator lack faster convergence bmcmc appears due inaccurate estimation kernel mean conjecture alternative exact approaches oates may provide improved performance context however standard confidence intervals obtained clt mcmc estimate asymptotic variance parameters parameter parameter parameter estimated integrals estimated integrals estimated integrals parameter parameter parameter estimated integrals estimated integrals estimated integrals parameter parameter parameter estimated integrals estimated integrals estimated integrals figure numerical estimation parameter posterior means teal south oil field model centered around true values green line gives exact value integral mcmc black line bmcmc point estimates red line provided similar performance mcmc confidence intervals based estimated asymptotic variance black dotted lines poorly calibrated whereas bmcmc credible intervals red dotted lines provide honest uncertainty assessment case study random effects aim explore whether flexible representations afforded weighted combinations hilbert spaces enable probabilistic integration focus bqmc methodology could applied probabilistic integrators weighted spaces formulation high infinite qmc achieved construction known weighted hilbert space spaces defined motivated tion many integrands encountered applications seem vary lower dimensional projections compared higher dimensional projections presentation follows sec dick pillichshammer idea goes back least wahba chap usual qmc work uniform let subset define weight denote collection weights consider space functions form belongs rkhs kernel denotes components indexed restrictive since function written form considering turn hilbert space defining inner product hfu constructed way rkhs kernel intuitively weights taken small whenever function depend heavily interaction states thus small function thatpis effectively measure effective dimension function given extreme case could even infinite provided sum remains bounded dick canonical weighted sobolev space dominating mixed smoothness defined taking component spaces finite dimensions bqmc rules based digital net attain optimal wce rates rkhs see supplement full details random effects regression illustration considered generalised linear models focus poisson random effects regression model studied kuo example context inference parameters following model log independent knots took equally spaced knots min inference requires multiple evaluations observed data likelihood therefore candidate probabilistic integration methods order model cumulative uncertainty estimating multiple numerical integrals order transform integration problem unit cube change perform variables wish evaluate denotes standard gaussian inverse cdf applied component probabilistic integration proceeds hypothesis integrand belongs least well approximated functions smoothness parameter weights intuitively integrand increase value knot compensated decrease value neighbouring knot changing values remote knots therefore expect exhibit strong individual pairwise dependence expect dependency weaker motivates weighted space assumption sinescu provides theoretical analysis choice weights weights order two used qmc bqmc bqmc true integral estimate figure application random effects regression dimensions based samples digital net error bars show credible regions improve visibility results shown error bars symmetric linear scale qmc estimate used approximate true value integral value parameter dmax dmax otherwise corresponds assumption interaction terms though still depend arguments full details provided supplement results fig showed posterior credible regions cover truth problem suggesting uncertainty estimates appropriate negative side bqmc method encode integrand consequently posterior mass placed negative values integral meaningful understand effect weighted space construction compared bqmc point estimate interactions interesting observation point estimates closely followed produced qmc case study spherical integration computer graphics probabilistic integration methods defined arbitrary manifolds formulations spaces suggested far back diaconis recently exploited context computer graphics brouillat marques forms setting final case study global illumination integrals analyse bqmc order estimate integrals form spherical measure uniform probabilistic integration applied compute global illumination integrals used rendering surfaces pharr humphreys therefore focus case uncertainty quantification motivated inverse global illumination integral estimate integral estimate red channel blue channel green channel number states number states number states figure probabilistic integration sphere employed estimate rgb colour intensities california lake environment error bars bmc blue bqmc green represent credible intervals estimates black qmc estimates red shown reference task make inferences noisy observation object via computerbased image synthesis measure numerical uncertainty could naturally propagated context limit scope restrict attention uncertainty quantification forward problem models involved global illumination based three main factors geometric model objects present scene model reflectivity surface object description light sources provided environment map light emitted environment interact objects scene reflection formulated illumination integral outgoing radiance outgoing light direction represents amount light emitted object assume known light hitting object direction term bidirectional reflectance distribution function brdf models fraction light arriving surface point direction reflected towards direction unit vector normal surface object investigation motivated strong empirical results bqmc context obtained marques assess performance bqmc consider typical illumination integration problem based california lake environment goal compute intensities three rgb colour channels corresponding observing virtual object fixed direction consider case object directly facing camera brdf took exp integrand modelled sobolev space low smoothness specific function space consider sobolev space formally defined supplement results bmc bqmc tested example ensure fair comparison identical kernels taken basis methods bqmc employed using spherical tdesign bondarenko shown bqmc point set used see supplement fig shows performance particular test function bqmc point estimate almost identical qmc estimate values overall bmc bqmc provided sensible quantification uncertainty value integral values considered conclusion increasing sophistication computational models numerical integration one component demands improved understanding numerical error accumulates propagates computation common settings integrands computationally intensive many numerical integrals required effective methods required make full use information available problem hand evidenced recent success qmc leverages smoothness properties integrands probabilistic numerics puts statistician centre stage aims model integrand approach eloquently summarised kadane proposed following vision future computation statistics thought set tools used making decisions inferences face uncertainty algorithms typically operate environment perhaps statisticians might join teams scholars addressing algorithmic paper explored probabilistic integration perspective statistician results highlight advantages disadvantages approach positive side general methodology described unified framework existing mcmc qmc methods associated probability distribution models discretisation error posterior contraction rates first time established negative side remain many substantial open questions terms philosophical foundations theoretical analysis practical application discussed philosophy several issues concerning interpretation first whose epistemic uncertainty modelled hennig argued uncertainty modelled hypothetical agent get design statistician selects priors loss functions agent best achieves statistician goals goals typically involve combination relatively behaviour perform well diverse range problems low computational overhead interpretation posterior subtle subjective inference many points contention objective inference also appear framework methodology options part numerical method modelled paper integrand considered uncertain distribution considered known however one could alternatively suppose unknown pursued oates regardless endogenous nature uncertainty quantification means practice one reliant effective methods estimation kernel parameters interaction standard methods empirical bayes task numerical uncertainty quantification demands theoretical research stein theory probabilistic integration theoretical work required results address coverage finite sample size interaction coverage methods kernel parameter estimation particularly important question recently addressed kanagawa behaviour integrand belong posited rkhs prior specification broad discussion required prior information included information ignored indeed practical considerations essentially always demand aspects prior information ignored competing computational statistical philosophical considerations play must balanced example rkhs framework studied paper advantage providing flexible way encode prior knowledge integrand allowing specify properties smoothness periodicity effective hand several important properties including boundedness less easily encoded possibility importance sampling eqn element arbitrariness appears preclude pursuit default prior even within rkhs framework issue integrands usually belong infinitude rkhs selecting appropriate kernel arguably central open challenge qmc research present practical perspective elicitation priors infinitedimensional spaces hard problem adequate choice prior informative numerical scheme significantly improve convergence rates method methods choosing kernel automatically could useful duvenaud would need considered suitability providing uncertainty quantification integral list meant exhaustive highlights many areas research yet explored acknowledgements authors grateful expert feedback received associate editor reviewers well barp cockayne dick duvenaud gelman hennig kanagawa kronander meng owen robert schwab simpson skilling sullivan tan teckentrup zhu authors thank lan marques providing code used case studies fxb supported epsrc grant cjo supported arc centre excellence mathematical statistical frontiers acems supported epsrc grant epsrc established career fellowship grant royal society wolfson research merit award work also supported alan turing institute epsrc grant programme engineering finally material also based upon work partially supported national science foundation nsf grant statistical applied mathematical sciences institute opinions findings conclusions recommendations expressed material author necessarily reflect views nsf references bach equivalence quadrature rules random features mach learn bakhvalov approximate calculation multiple integrals russian vestnik mgu ser math mech astron phys bauer devroye kohler krzyzak walk nonparametric estimation function noiseless observations random points multivariate berlinet reproducing kernel hilbert spaces probability statistics springer science business media new york bogachev gaussian measures mathematical surveys monographs american mathematical society bondarenko radchenko viazovska optimal asymptotic bounds spherical designs ann briol oates girolami osborne bayesian quadrature probabilistic integration theoretical guarantees proc adv neur nips brouillat bouville loos hansen bouatouch bayesian monte carlo approach global illumination comp graph forum buchholz chopin improving approximate bayesian computation via quasi monte carlo calderhead girolami estimating bayes factors via thermodynamic integration population mcmc comput statist data cialenco fasshauer approximation stochastic partial differential equations collocation method int comput cockayne oates sullivan girolami bayesian probabilistic numerical methods diaconis bayesian numerical analysis statist decis theory rel top dick pillichshammer digital nets sequences discrepancy theory carlo integration cambridge university press dick kuo sloan integration carlo way acta duvenaud automatic model construction gaussian processes phd thesis university cambridge efron tibshirani introduction bootstrap crc press eftang stamm parameter empirical interpolation int numer methods friel pettitt marginal likelihood estimation via power posteriors stat soc ser stat friel hurn wyse improving power posterior estimation statistical evidence stat gelman meng simulating normalizing constants importance sampling bridge sampling path sampling statist gerber chopin sequential carlo stat soc ser stat girolami 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predicting integrals random fields using observations lattice ann stein locally lattice sampling designs isotropic random fields ann steinwart christmann support vector machines springer science business media suldin wiener measure applications approximation methods izvestiya vysshikh uchebnykh zavedenii matematika van der vaart van zanten frequentist coverage adaptive nonparametric bayesian credible sets ann traub wasilkowski complexity academic press wahba spline models observational data regional conference series applied mathematics wendland scattered data approximation cambridge university press stein maximum likelihood estimation smooth gaussian random field model siam uncertainty quantification yang dunson bayesian manifold regression ann debevec malik hawkins inverse global illumination recovering reflectance models real scenes photographs proc ann conf comput graph int supplement supplement provides complete proofs theoretical results extended numerics full details reproduce experiments presented paper proof theoretical results proof fact prior data standard conjugacy results gps lead posterior mean covariance see chap rasmussen williams repeated application fubini theorem produces zzz dpn dpn proof completed substituting expressions two equations result main text additionally sets proof fact eqn main text converse inequality consider specific integrand supremum definition dual norm use reproducing property completes proof proof fact combining fact direct calculation gives required following lemma shows probabilistic integrators provide point estimate least good counterparts lemma bayesian let consider cubature rule corresponding rule wibc proof immediate fact shows weights wibc optimal choice space convergence controlled quality approximation lemma regression bound let fix states jensen inequality rproof application required note regression bound sharp general ritter prop consequence thm quite optimal lemmas refer point estimators provided however aim quantify change probability mass number samples increases lemma contraction assume suppose define interval radius centred true value integral vanishes rate exp proof assume without loss generality posterior distribution gaussian mean variance since posterior probability mass given distribution definition get upper bound definition wce terms bounded asymptotically erfc result follows fact erfc exp sufficiently small result demonstrates posterior distribution probability mass concentrates neighbourhood hence prior well calibrated see sec posterior provides uncertainty quantification solution integral result performing finite number integrand evaluations define fill distance set sup min scaling fill distance described following special case lemma oates lemma let continuous monotone increasing satisfy exp suppose bounded away zero samples uniformly ergodic markov chain targeting arbitrarily small proof thm initially consider fixed states fixing random seed standard result functional approximation due schaback see also wendland thm exists kernels alternative bounds wendland table augment finite number states ensure always holds regression bound lemma follows taking expectation sample path markov chain cex cex lemma scaling relationship arbitrarily small markov inequality convergence mean implies convergence probability thus using eqn completes proof generally result follows fact proof thm theorem dick pillichshammer assumes qmc rule based digital net prime satisfies log sobolev space dominating mixed smoothness order constant depends result follows immediately norm equivalence lemma contraction rate follows lemma proof prop denote posterior distribution integral conditional value following prop gaussian distribution mean variance given furthermore posterior amplitude parameter satisfies exp corresponds distribution parameters therefore distributed marginal distribution distribution claimed kernel means section propose approximate bayesian cubature weights wbc approximation optimal weights based approximation kernel mean see also prop sommariva vianello following lemma demonstrates bound contribution error inflate posterior reflect additional uncertainty due approximation uncertainty quantification still provided lemma approximate kernel mean consider approximation form performed analytically respect denote estimator moreover nka proof define let write consider wbc wbc wbc use denote tensor product rkhs since fact remains show second term equal indeed completes proof method posterior variance computed computable obtained used propagate numerical uncertainty remainder statistical task idea make use triangle inequality first term rhs available analytically fact square second term explicit upper bounds exist case states independent random samples instance song thm radial kernel uniform independent log sup probability least dependent eqn replaced estimate effective sample size write credible interval defined conservative upper bound described eqns conclude credible interval probability least note even though credible region inflated still contracts truth since first term rhs lemma bounded sum vanish resulting conservative posterior viewed updating beliefs based approximation likelihood function statistical foundations approach made clear recent work bissiri additional numerics section presents additional numerical results concerning calibration uncertainty multiple parameters higher dimensions calibration fig top row study quantification uncertainty provided setup main text optimising parameter magnitude parameter easy hard test functions notice led inferences low regime attains approximately correct frequentist coverage larger calibration experiments sec based bmc repeated dimension results shown fig bottom row clearly integrand evaluations required attain good frequentist coverage credible intervals due curse dimension however frequentist coverage reasonable large task empirical convergence assessment convergence bqmc studied based digital nets theoretical rates provided sec method figure gives results obtained left right one dimensional case theoretical convergence rate attained method cases considered however case rates observed number evaluations considered helps demonstrate important point addition numerical conditioning rates provide asymptotic may require large values observed figure evaluation uncertainty quantification provided results shown top bottom coverage frequencies computed top bottom realisations compared notional bayesian credible regions varying level left easy test function right hard test function posterior standard deviation posterior standard deviation figure empirical investigation bqmc left right dimensions sobolev space mixed dominating smoothness results obtained using tensor product kernels smoothness red green blue dotted lines represent theoretical convergence rates established kernel black line represents standard qmc kernel parameters fixed left right supplemental information case studies case study mcmc paper used manifold langevin algorithm girolami calderhead combination population mcmc population mcmc shares information across temperatures sampling yet previous work leveraged evaluation one inform estimates derived contrast occurs naturally probabilistic integration framework described main text mcmc used generate small number samples basis order simulate scenario numerical error computation marginal likelihood temperature ladder rungs employed reason according recommendation calderhead girolami convergence issues experienced mcmc previously successfully used oates prior elicitation motivate prior unknown function based work calderhead girolami advocated use schedule based extensive empirical comparison possible schedules good temperature schedule approximately satisfies criterion basis allocates equal area portions curve lie controlling bias trapezium rule substituting optimality criterion produces letting obtain formally treating continuous taking limit produces conclude transformed function approximately stationary reasonably assigned stationary prior however importance sampling transformation require support reason took experiment variance computation covariance matrix obtained due intractability kernel mean therefore explored approximation plugging place provides approximation posterior variance log likelihood took form empirical distribution employed based first samples remaining samples reserved kernel computation heuristic approach becomes exact sense underestimates covariance finite kernel choice experiments taken gaussian covariance functions example exp parametrised choice made capture smoothness integrands involved application found parameters possible learn data using parameters required large number data pin therefore experiments fixed mean mean cases remaining kernel parameters selected using data generation captures salient properties model selection discussed main text considered variable selection logistic regression logit model specifies active variables via binary vector model prior employed given model active parameters endowed independent priors single dataset size generated model parameter problem principle different models true model selected model thus sensitive numerical error computation pmarginal likelihood practice limited model space consider models speeds computation particular case rules models much lower posterior probability actual map model thus models compared case study background model teal south model pde computer model oil reservoir model studied grid layers parameters representing physical quantities interest include horizontal permeabilities layers vertical horizontal permeability ratio aquifer strength rock compressibility porosity experiments used emulator likelihood model documented lan order speed mcmc however might undesirable general due additional uncertainty associated approximation results obtained kernel choice numerical sec results obtained using kernel given exp corresponds sobolev space note satisfied used parameter fixed amplitude parameter variance computation due intractability posterior distribution kernel mean unavailable closed form overcome methodology supplement employed obtain empirical estimate kernel mean half mcmc samples used weights approximate integral half weights approximate kernel mean eqn used upper bound intractable posterior variance upper bound hold states must independent samples whereas obtained using mcmc therefore independent order ensure mcmc samples independent possible employed sophisticated mcmc methodology developed lan nevertheless emphasise gap theory practice hope fill future research results paper fixed eqn essentially credible interval formal investigation theoretical properties uncertainty quantification studied methods provided paper case study kernel choice canonical weighted sobolev space defined taking component spaces sobolev spaces dominating mixed smoothness space tensor product sobolev spaces smoothness parameter constructed way rkhs kernel bernoulli polynomials theoretical results finite dimensions construct digital net attains optimal qmc rates weighted sobolev spaces theorem let rkhs bqmc based digital attains optimal rate proof follows combining thm dick pillichshammer lemma qmc rules theorem explicitly take account values weights algorithm tailors qmc states specific weights known component component cbc algorithm details found kuo principle cbc algorithm lead improved rate constants high dimensions effort wasted directions varies little computational overheads also greater consider cbc algorithms bqmc paper note weighted hilbert space framework allows bound wce independently dimension providing sloan justifies use context details provided sec dick case study kernel choice function spaces consider sobolev spaces obtained using reproducing kernel normalised gegenbauer polynomials brauchart particularly simple expression kernel sobolev space obtained taking along pochhammer symbol specifically choices produce kernel associated tractable kernel mean hence initial error also available mmd bmc qmc bqmc number states figure application global illumination integrals computer graphics left spherical right wce monte carlo bayesian bmc quasi qmc bayesian qmc bqmc theoretical results states could generated case analogous results obtained sec obtained specifically thm brauchart bayesian lemma classical leads slow convergence regression bound argument lemma together functional approximation result gia thm gives faster rate bmc dimension rather focus methods present results based spherical qmc point sets briefly introduce conceptr spherical bondarenko define set satisfying polynomials degree restriction polynomial usual euclidean sense theorem exists exists spherical states moreover use spherical leads rate proof property spherical follows combining hesse sloan bondarenko lemma rate thm deterministic method brauchart although explicit spherical currently known approximately optimal point sets numerically high accuracy additional theoretical results point estimates found fuselier particular consider conditioning associated linear systems must solved obtain weights numerical results fig value wce four methods considered qmc bmc bqmc number states increases bmc bqmc appear attain rate although bqmc provides constant experiments based point sets provided womersley website http accessed environment map used example freely available http html accessed may factor improvement bmc note shown brauchart deterministic method space references bissiri holmes walker general framework updating belief distributions stat soc ser stat brauchart saff sloan womersley qmc designs optimal order quasi monte carlo integration schemes sphere math fuselier hangelbroek narcowich ward wright kernel based quadrature spheres homogeneous spaces numer hesse sloan errors sobolev space setting cubature sphere bull aust math kuo constructions achieve optimal rate convergence multivariate integration weighted korobov sobolev spaces complexity gia sloan wendland multiscale approximation functions arbitrary sobolev spaces scaled radial basis functions unit sphere appl comput harmon sloan carlo algorithms efficient high dimensional integrals complexity schaback local error estimates radial basis function interpolation scattered data ima numer

cointegration daily electric power system load weather stefanov eso ead veslets sofia bulgaria szstefanov paper makes thermal predictive analysis electric power system security day ahead predictive analysis set thermal computation expected computation obtained cointegrating daily electric power system load weather finding daily electric power system thermodynamics introducing tests thermodynamics predictive analysis made shows electricity consumers wisdom keywords predictive analysis security thermodynamics cointegration wisdom introduction electric power system affected weather changes exchanges eps eps load unpredictable load eps dynamically unpredictable model load weather cointegrated according fezzi made cointegration daily load wholesale price electricity therefore cointegration daily eps load weather load eps thermodynamically unpredictable modelling eps field possible internal model principle system theory model eps viewed open system model evolution eps behaviour thermodynamic unpredictability diminishes predicting rare events cooperative competitive phenomena eps type events phenomena predicted eps dispatchers intelligent system viewed dissipative model brain dynamics intelligent system field computation field realization sense able make predictive analysis eps wehenkel abed made predictive analysis security networkmodelled eps data latter analyses incomplete predict change eps security caused evolution eps behaviour aim paper thermal predictive analysis eps security day ahead analysis sought means cointegration daily eps load weather cointegration daily eps load modelled one descriptive two rescriptive models models constructed time whose moments calendar days descriptive load model regression indicators two load peaks distributed lag represents load variability flow integrator load regression indicators pat denotes impulse hour day load morning equal afternoon load afternoon yesterday morning load distributed lag last two regressors flow integrator leads substitution regressors rescriptive load models regressions cointegration links distributed lag represents load variability flow integrator load regressions pbt pct distributed lag last two regressors respectively flow integrator leads substitution regressors regressor meaning model descriptive models rescriptive sense regression models normal load behaviour regressions evolution load behaviour predicting daily load three models predicting ensemble models according three models data treated sequentially parallel data treated way sequential models daily load forecast better forecast parallel ones flow integration reduces data regressions give accurate load forecast flow data presents eps exchanges load models daily eps load econometrically modelled regressions modelling changes load environment regressions models dynamic unpredictability load daily eps thermodynamics cointegration distance regressions pat pbt respectively pct pbt angle respectively angle arctan pat pbt pat pat pbt pbt arctan pct pbt pct pct pbt pbt denotes mean time time series pat pbt pct obtained time series pat pbt pct subtracting mean eps entropy environment entropy arcos exp arcos exp eps recoherence recoherence positive quantity entropy monotonically increasing environment decoherence decoherence negative quantity entropy monotonically decreasing decoherence recoherence related inverse temperature viewing decoherence recoherence forward reverse process gives let quantities set follows min pat pbt pct max pat pbt pct min pat pbt pct max pat pbt pct eps work day work obtained transient fluctuation inverse temperature testing daily thermodynamics time test daily eps thermodynamics maximum likelihood seasonal cointegration tests daily let following times integer integer integer integer times obtained eps evolution set parity violation eps evolution follows spiral equivalent spiral obtained imel baev coarsening system loops time maximum likelihood statistic cointegration critical value critical value acceptance region first level seasonal cointegration sample size basic regression model constant seasonal dummies trend critical value time value among values compared time smaller time maximum likelihood statistic cointegration critical value critical value acceptance region second level seasonal cointegration sample size basic regression model constant seasonal dummies trend critical value time value among values compared time smaller time maximum likelihood statistic cointegration critical value critical value acceptance region first level seasonal cointegration sample size basic regression model constant seasonal dummies trend critical value time value among values compared time smaller time maximum likelihood statistic cointegration critical value critical value acceptance region third level seasonal cointegration sample size basic regression model constant seasonal dummies trend critical value time value among values compared time smaller time maximum likelihood statistic cointegration critical value critical value acceptance region second level seasonal cointegration sample size basic regression model constant seasonal dummies trend critical value time value among values compared time smaller time test daily eps thermodynamics consists critical value check times energy test daily eps thermodynamics test energy reserve exp exp test follows hypergeometric function presentation energy hypergeometric relations energy test daily eps thermodynamics consists checking positiveness reserve thermal computation evolution behaviour presented statistical submanifold evolution surface using reversible entropic mean standard deviation eps evolution behaviour cafaro cosh sinh cosh cosh angles work diffusion eps evolution behaviour gives following expected daily prices electricity inverse temperature reserve daily prices electricity found prices market uncertainty pennock multiplication ten photographic enlargement made price market uncertainty price electricity market day ahead three prices set following prices min max expected eps reliability respect rare event competitive phenomenon reliability found jordan curve descriptor introduced zuliani expected eps reliability respect cooperative phenomenon reliability found realization descriptor introduced daneev expected eps droop set inverse temperature eps scheme viewed euclidean bannai expected daily price electricity respect eps reliability daily price electricity minimizes eps lifetime variance accordance computation thermalisation reduces daily mean error relative daily peak load load forecast normalization euclidean bannai quantity determined computational potential introduced anders multiplication ten photographic enlargement made computational potential eps potential daily artificial dispatcher daily artificial dispatcher dad field intelligent system makes thermal predictive analysis set thermal predictive analysis predictive analysis eps security gives expected eps load expected electricity price expected reserve expected droop expected eps reliability predicting times predicting synchronization eps predicting energy reserve predicting stability eps dad predicting synchronization eps shows dad perceives cointegration dad predicting stability eps shows dad interprets cointegration dad intelligent system sense intelligence wisdom consists evasion prediction indeed falling synchronization evasion stability prediction thermalisation finding eps security field computation according therefore dad indeed field intelligent system dad resource heat dad logic logic resources girard linear logic conclusion natural linear logic dad wisdom electricity consumers dad presents average belief consumers evolution eps behaviour based expected warming weather consumers average belief eps reliability respect cooperative phenomenon consumers average belief eps reliability respect rare event competitive phenomenon dad stakes following part resources eps reliability respect rare event competitive phenomenon prpv pvpr prpv pvpr prpv dad stakes following part resources eps reliability respect cooperative phenomenon pvpr prpv pvpr pvpr prpv thus dad rational forecast gambler set sufficient conditions given wolfers prediction market prices coincide average beliefs among traders quantity obtained following equation standard deviation quantity set greatest positive root pro equation pro pro pro pro quantity obtained following equation standard deviation quantity set greatest positive root pvo equation pvo pvo dad checks expected eps synchronization time test daily eps thermodynamics dad checks expected eps stability energy test daily eps thermodynamics thus dad verifies forecasts results dad dad operates help dispatchers bulgarian eps regressions estimated hourly sampled values daily load dry bulb temperature preceding nine days well hourly sampled daily forecast dry bulb estimate regressions use made load data supplied bulgarian electric power system operator eso ead accuweather weather forecast used eso ead estimate three regressions obtained exact maximum likelihood method dynamic regression estimation given pesaran estimate correct sample small size mean error relative daily peak load daily load forecast bulgarian eps obtained dynamic regression dad reduces error table gives monthly average daily mean error relative daily peak load bulgarian eps load forecast made dad error given two months year choosing months error maximal table monthly average daily mean error relative daily peak load bulgarian eps load forecast made dad percent year month mmre dad reduces mean error relative daily peak load bulgarian eps daily load forecast reduction corresponds hypothetic increase average daily temperature assumed increase average daily temperature leads change accordance results crowley hypothetic warming weather day ahead results correct prediction expected eps security dad example expected zero reliability cooperative phenomenon gives true prediction eps decoupling conclusion aim paper thermal predictive analysis eps security day ahead aim achieved follows one descriptive two rescriptive dynamic models prediction daily load constructed models obtained cointegration daily eps load weather daily eps thermodynamics found eps inverse temperature eps work time test energy test daily eps thermodynamics obtained thermal computation expected eps security day ahead made predictive analysis eps security day ahead presented field intelligent system shows eps dispatchers electricity consumers wisdom shown proposed predictive analysis enhances eps security accurate prediction eps load prediction critical phenomena references hendry unpredictability foundations economic forecasting econometric society australasian meetings fezzi bunn structural analysis high frequency electricity demand supply interactions london business school working paper freeman vitiello brain dynamics dissipation spontaneous breakdown symmetry arxiv maclennan field computation 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published conference paper iclr raining wide residual networks deployment using single bit weight feb mark mcdonnell computational learning systems laboratory school information technology mathematical sciences university south australia mawson lakes australia bstract fast deployment trained deep neural networks embedded hardware learned weight parameter ideally represented stored using single bit usually increase requirement imposed report large improvements error rates multiple datasets deep convolutional neural networks deployed using wide residual networks main baseline approach simplifies existing methods binarize weights applying sign function training apply scaling factors layer constant unlearned values equal standard deviations used initialization imagenet models requiring less parameter memory achieve error rates respectively also considered mnist svhn achieving test results respectively cifar error rates halve previously reported values within errorrates network weights networks overfit also show significant improvements error rate learning batch normalization scale offset parameters applies full precision networks using schedule found training fast networks better accuracy standard schedules achieved peak performance training epochs full training code trained models matlab keras pytorch see https ntroduction fast parallel computing resources namely gpus integral resurgence deep neural networks ascendancy becoming methodologies many computer vision tasks however gpus expensive wasteful terms energy requirements typically compute using floating point bits recognized providing far precision needed deep neural networks moreover training deployment require availability large amounts memory storage trained models operational ram methods become embedded resourceconstrained sensors devices intelligent systems ranging robotics cars reliance computing resources need reduced end increasing interest finding methods drive resource burden modern deep neural networks existing methods typically exhibit good performance work conducted part hosted visit institute neural computation university california san diego part sabbatical period consilium technology adelaide australia published conference paper iclr ideal case parameters processing still fall error rates important benchmarks paper report significant reduction gap see figure results convolutional neural networks cnns deployed using weights stored applied using standard precision floating point networks deployed using weights represented process developing methods also obtained significant improvements obtained versions cnns used addition application custom hardware deploying deep networks networks deployed using previously shown pedersoli enable significant speedups regular gpus although yet possible using standard popular libraries aspects work first communicated subset material workshop abstract talk mcdonnell svhn mnist imagenet single crop imagenet bwn imagenet test error rate gap test error rate figure gaps using points except black crosses data best results reported paper dataset black points results full imagenet dataset comparison results rastegari black crosses notation corresponds network width see section elated ork ets new form cnn called deep residual network resnet developed used set many new accuracy records benchmarks comparison older cnns alexnet krizhevsky vggnet simonyan zisserman resnets achieve higher accuracy far fewer learned parameters flops operations per image processed key reducing number parameters resnets replace layers nets layers learned parameters lin springenberg simultaneously training much deeper network previously key new idea enabled deeper network trained effectively introduction many variations resnets since proposed resnets offer virtue simplicity given motivation deployment custom hardware chosen primary focus despite increased efficiency parameter usage similar cnns accuracy resnets still tends increase total number parameters unlike cnns increased accuracy result either deeper wider networks zagoruyko komodakis published conference paper iclr paper use wide residual networks zagoruyko komodakis demonstrated produce better accuracy less training time deeper networks educing memory burden trained neural networks achieving best accuracy speed possible deploying resnets similar networks mobile devices require minimising total number bits transferred memory processors given number parameters motivated considerations lot recent attention directed towards compressing learned parameters model compression reducing precision computations carried neural hubara detailed literature review recently published strategies model compression include reducing precision number bits used numerical representation weights deployed networks training courbariaux hubara merolla rastegari reducing number weights trained neural networks pruning han iandola quantizing compressing weights following training han zhou reducing precision computations performed inference courbariaux hubara merolla rastegari modifying neural network architectures howard theoretical analysis various methods proved results convergence variety methods range strategies focused approach simultaneously contributes two desirable attributes simplicity sense deployment trained models immediately follows training without extra processing implementation convolution operations achieved without multipliers demonstrated rastegari overall approach summary contributions strategy improving methods enable inference threefold baseline sought begin baseline deep cnn variant close error rates time commencement held wide residual networks zagoruyko komodakis starting point subsequent approaches exceeded accuracy resnets offer superior simplicity conforms third strategy list make minimal changes training aimed ensure training could achieved minimal changes baseline training simplicity desirable custom hardware custom hardware implementations mind sought simplify design baseline network hence version weights much possible without sacrificing accuracy ontributions ull recision ide ets although paper chiefly exceeded objectives fullprecision baseline network surpassed reported error rates using wide resnets zagoruyko komodakis achieved using convolutional layers prior work demonstrated best wide resnet performance using layers innovation achieves significant drop wide resnets simply learn scale offset factors layers retaining remaining attributes layers important done conjunction exchanging ordering final weight layer global average pooling layer see figure observed effect pronounced gaining around rate method advantageous networks overfit overfitting issue imagenet removing learning parameters detrimental published conference paper iclr ontributions deep cnn single bit weights inference first study aware consider gap compared weights changes accuracy across diverse range image classification datasets figure approach surpasses large margin previously reported error rates error rates halved networks constrained run inference time one reason achieved lower error rates case previously start superior baseline network previous studies namely wide resnets however approach also results smaller error rate increases relative error rates previously training requires number epochs case weights main innovation introduce simple fixed scaling method convolutional layer permits activations gradients flow network minimum change standard deviation accordance principle underlying popular initialization methods combine use method loshchilov hutter enables report results case far fewer epochs training reported previously earning model convolution weights sign weights propagate full precision weights updated follow approach courbariaux rastegari merolla find good results using inference time training apply sign function weights purpose forward backward propagation update weights using sgd gradients calculated using however previously reported methods training using sign weights either need train many hundreds epochs courbariaux merolla use computationallycostly normalization scaling channel layer changes minibatch training bwn method rastegari obtained results using simple alternative approach describe scale output conv layers using constant layer begin noting standard deviation sign weights convolutional layer kernels size close assuming mean zero contrast standard deviation layer networks initialized method number input channels convolutional layer number convolutional layers rgb inputs applying sign function alone mismatch principled approach controlling gradient activation scaling deep network although use still enable learning convergence empirically slow less effective address problem training using sign weights use initialization method weights updated also introduce scaling applied sign weights scaling constant unlearned value equal initial standard deviation method enables standard deviation information equal value would initially networks implementation training multiply sign weights layer value inference multiplication using scaling layer following weight layer weights network stored using deployed using see https hence custom hardware implementations would able perform model convolutions without multipliers rastegari significant gpu speedups also possible pedersoli published conference paper iclr fact scale weights explicitly training important although forward backward propagation equivalent scale input output feature maps convolutional layer also scales calculated gradients respect weights since calculated convolving input output feature maps consequence learning dramatically slower unless learning rates introduced cancel scaling approach similar bwn method rastegari constant scaling method faster less complex summary differences make comparison training follows let tensor convolutional weights convolutional layer weights processed following way forward propagation backward propagation weight updates sign spatial size convolutional kernel layer see figure full precision weights relu conv relu bit conv scale figure difference networks bit conv scale layers equivalent operations shown eqn ethods common baseline single bit networks etwork rchitecture resnets use identity mapping approach residual connections imagenet use design datasets mainly use network also report results layers residual block includes two convolutional layers preceded batch normalization rectified linear unit relu layers rather train deep resnets use wide residual networks wide resnets zagoruyko komodakis although zagoruyko komodakis others reported networks result better test accuracy networks found layers typically produces best results possibly due approach learning scale offset parameters baseline resnet design used experiments see figures several differences comparison zagoruyko komodakis details articulated appendix mostly simplicity little impact accuracy exception approach learning parameters raining trained models following aspects standard stochastic gradient descent methods used zagoruyko komodakis wide resnets specifically use loss minibatches size momentum learning weights situations learn scales offsets svhn mnist overfitting evident wide resnets use larger weight decay full imagenet use weight decay apart one set experiments added simple extra approach called cutout use standard light data augmentation including randomly flipping image horizontally probability two extra layers counted downsampling residual paths learn convolutional projections published conference paper iclr residual block repeat times input relu conv relu conv relu conv relu conv gap figure wide resnet architecture design mostly standard resnet zagoruyko komodakis first convolutional layer conv first residual blocks imagenet datasets output channels next blocks output channels widening parameter final convolutional layer convolutional layer gives output channels number classes importantly final convolutional layer followed prior gap softmax blocks number channels double downsampling blocks details depicted figure reduce spatial dimension feature map factor two relu layer closest input optional included best learn scale offset subsequent layer downsampling residual blocks avg pool double channels stride using zero padding relu stride conv relu conv double channels figure downsampling blocks wide resnet architecture standard resnet zagoruyko komodakis downsampling convolution used convolutional layers number output channels increases corresponding downsampling skip connections done residual block unlike standard resnets use average pooling layer avg pool residual path downsampling datasets plus svhn pad pixels sides using random values crop patch random location resulting image full imagenet scale crop flip use subtract mean initial layer network performs role use whitening augment using color brightness use initialization method describe important differences training approach compared usually reported batch norm scales offsets learned svhn training svhn layers except first one input relu used learn scale offset factor instead initializating channels keeping values training note also learn biases convolutional layers usual approach setting moments use layers inference mode keep running average training learning parameters found small benefit calculating moments used inference training finished simply form many minibatches possible full data augmentation used training applied pass trained network averaging returned moments batch published conference paper iclr best results using method using matconvnet found necessary ensure parameter used avoid divisions zero set different way used keras libraries network final weight layer convolutional layer significant difference resnets zagoruyko komodakis exchange ordering global average pooling layer final weight layer final weight layer becomes convolutional layer many channels classes training set design new seem new resnets corresponds architecture lin originated global average pooling concept also used springenberg convolutional layers follow final layer layer benefits conjunction learning scale offset described discussion section use warm restart learning rate schedule use warm restarts learning rate schedule reported wide resnet results loshchilov hutter whilst also speeding convergence method constantly reduces learning rate according cosine decay across certain number epochs repeats across twice many epochs restricted attention maximum epochs often epochs using method total number epochs reducing learning rate maximum minimum epochs epochs typically found could achieve test error rates epochs within error rates achievable epochs xperiments cutout literature experiments use simple standard data augmentation consisting randomly flipping training image probability padding image sides pixels cropping version image random location use augmentation although minor modification pad uniform random integers rather additionally experimented cutout devries taylor involves randomly selecting patch raw training image remove method shown combine methods set latest results see table found better results using larger cutout patches reported devries taylor hence choose patches size following method devries taylor ensured pixels chosen included patch equally frequently throughout training ensuring chosen patch location near image border patch impacts image part patch inside image differently devries taylor padding use uniform random integers replace image pixel values location patches apply cutout datasets esults conducted experiments six databases four databases rgb svhn full ilsvrc imagenet database russakovsky well mnist lecun details first three datasets found many papers zagoruyko komodakis downsampled version imagenet training validation images cropped using annotated bounding boxes downsampled chrabaszcz see published conference paper iclr http experiments carried single using matlab gpu acceleration matconvnet cudnn report results wide resnets except applied imagenet wider baseline resnets use terminology zagoruyko komodakis baseline channels layers first spatial scale use notation form denote wide resnets convolutional layers channels first spatial scale hence width full imagenet dataset use wide resnets channels first spatial scale given standard resnet baseline width corresponds width dataset zagoruyko komodakis denote networks table lists error rates indicates indicates superscript indicates standard crop flip augmentation indicates use cutout table lists error rates svhn full imagenet use cutout datasets results tabulated imagenet imagenet provide results testing also testing latter decision test image obtained averaging softmax output passing network times corresponding crops obtained rescaling scales described random positions scale imagenet error rates slightly lower expected wide resnet according results zagoruyko komodakis probably due fact use color augmentation table approach applied weights resnet epochs params table approach applied svhn imagenet weights resnet epochs params svhn imagenet single crop single crop table shows comparison results original work wide resnets subsequent papers reduced error rates datasets also show results knowledge current svhn without augmentation huang cutout augmentation devries taylor result svhn little short even though used resnet less million parameters training epochs table shows comparison results previous work trains models using sign weights training additional results appear hubara activations quantized error rates much larger published conference paper iclr table test error rates networks less parameters sorted method wrn zagoruyko komodakis weights wrn paper wrn chrabaszcz full precision wrn paper weights wrn cutout paper wrn cutout devries taylor wrn dropout zagoruyko komodakis xie full precision wrn cutout paper densenets huang regularization gastaldi cutout devries taylor params table test error rates using test time propagation training method courbariaux weight binarization merolla bwn googlenet rastegari cai resnet adam vgg cai paper single center crop svhn paper scales random crops imagenet full imagenet full imagenet full imagenet full imagenet full imagenet inspection tables reveals baseline networks trained cutout surpass performance deeper wide resnets trained dropout even without use cutout network surpasses error rate reported essentially network zagoruyko komodakis also better previous wide resnet results elaborated section improved accuracy due approach learning scale offset parameters networks observe always accuracy gap compared full precision networks discussed section using cutout reduces error rates expected comparison training wide resnets shown table effective use cutout augmentation network reduce error rate using quarter weights figure illustrates convergence overfitting trends wide resnets comparison use cutout wide resnets clearly even resnets gap error rate full precision weights small also noticeable method enables convergence good solutions epochs training longer epochs reduces test error rates also observed network powerful enough model training sets well accuracy modelling power reduced version particularly reduced modelling capacity weights consistent gap rate performance cases using cutout training longer gives improved error rates using cutout epochs suffices peak performance finally mnist wide resnet without data augmentation method achieved epoch training epochs whereas method achieved epochs comparison reported case courbariaux hubara published conference paper iclr error rate error rate test test test test train train train train training epoch training epoch figure convergence training left marker shows error rates test set training set end cycle training method resnets less million parameters right marker shows test error rate resnets without cutout indicates indicates blation studies wide resnets versions benefit method learning scale offset parameters accuracy case also benefits use training schedule loshchilov hutter demonstrate influence two aspects figure show test error rate changes training either methods used use cutout purpose figure comparison drops learning rate epochs clear methods lower final error rate around absolute learning parameters method enables faster convergence case significant reducing error rate however case clear best results best use learn parameters warm restart learn paper warm restart learn warm restart learn warm restart learn test error rate test error rate warm restart learn paper warm restart learn warm restart learn warm restart learn training epoch training epoch figure influence learning gain offset left case right case published conference paper iclr iscussion accuracy gap bit bit weights expected smaller error rate gap result network using fullprecision test error rate full precision network gets smaller indeed tables quantify gap error rate cases tends grow network grows illustrate trend approach plotted figure gap error rates error rate full precision case best performing networks six datasets used strong conclusions data made relative alternative methods first study knowledge consider two datasets nevertheless also plotted figure error rate gap reported rastegari two different networks using bwn weight method reasons larger gaps points unclear clear better accuracy results smaller gap cases challenge work derive theoretical bounds predict gap magnitude gap changes full precision error rate dependent many factors including method used generate models rate cases loss function throughout training much higher case case hence network able fit training set well one whether loss precision weights due mismatch gradients inherent propagating weights updating weights training open question latter case possible principled refinements weight update method used reduce gap however also interesting networks applied gap much smaller despite benefits full precision case extra depth also warrants investigation omparison bwn method approach differs bwn method rastegari two reasons first need calculate mean absolute weight values underlying full precision weights output channel layer following minibatch enables faster training second need adjust gradient term corresponding appearance weight mean absolute value found overall two methods work equally effectively two advantages faster training fewer overall parameters note found method rastegari also works equally effectively basis rather also note focus rastegari much case combines activations solely remains tested scaling method compares case also interesting understand whether use renders scaling sign weights robust different scalings whether networks use might sensitive precise method used influence learning batch normalization parameters unusual design choice learning batch normalization parameters made svhn mnist wide resnets overfitting evident datasets see figure end training typically loss function becomes close zero corresponding severe overfitting inspired regularization szegedy aims reduce overconfidence following softmax layer hypothesized imposing control standard deviation inputs softmax might similar regularizing effect removed final layer resnets replaced convolutional layer followed layer prior global average pooling layer turn learning scale offset layer ensures batches flowing softmax layer standard deviations grow throughout training tends increase entropy predictions following softmax equivalent lower confidence guo published conference paper iclr observing success methods wide resnets observed learning parameters layers also led increased overfitting increased test error rates removed learning layers except first one applied input relu used shown figure significant benefits approach networks table results surpass zagoruyko komodakis effectively wide resnet network essentially comparison network extra layers appear due use learned convolutional projections downsampling residual paths whereas use average pooling instead expected motivation found method appropriate datasets overfitting evident case learning batch normalization parameters significantly reduces test error rates omparison queeze compare approach squeezenet iandola reported significant memory savings trained model relative alexnet squeezenet approach uses two strategies achieve replacing many kernels kernels deep compression han regarding squeezenet strategy note squeezenet network closely resembles resnets used experimented briefly approach many plain springenberg squeezenet iandola mobilenet howard resnext xie found effectiveness relative baselines comparable variants also observed many experiments total number learned parameters correlates well classification accuracy applied squeezenet variant found obtain accuracy resnets depth increase width squeezenet approximately number learned parameters resnet conclude method therefore reduces model size baseline squeezenet architecture deep compression used factor albeit accuracy gap regarding squeezenet strategy squeezenet paper reports deep compression han able reduce model size approximately factor accuracy loss method reduces model size factor small accuracy loss gets larger accuracy gets smaller would interesting explore whether deep compression might applied models focus methods minimally alter training leave investigation complex methods future work regarding squeezenet performance best accuracy reported squeezenet paper imagenet error requiring model weights models achieve better error require model weights imitations urther ork paper focus reducing precision weights benefits model compression enabling inference multiplications also interesting desirable reduce computational load inference using trained model carrying layer computations numbers bits hubara rastegari cai facilitating requires modifying activations network relus quantized relus extreme case binary step functions use calculations expected combining methods reduced precision processing inevitably increase error rates addressed extension forthcoming submission acknowledgments work supported discovery project funded australian research council project number discussions visit hosting gert cauwenberghs hesham mostafa ucsd van schaik runchun wang western sydney university gratefully published conference paper iclr acknowledged discussions victor stamatescu muhammad abul hasan university south australia eferences cai sun vasconcelos deep learning low precision gaussian quantization corr url http chrabaszcz loshchilov hutter downsampled variant imagenet alternative cifar datasets corr url http courbariaux bengio david binaryconnect training deep neural networks binary weights propagations corr url http devries taylor improved regularization convolutional neural networks cutout corr url http gastaldi regularization corr url http guo pleiss sun weinberger calibration modern neural networks corr url http han mao dally deep compression compressing deep neural networks pruning trained quantization huffman coding corr url http zhang ren sun delving deep rectifiers surpassing performance imagenet classification proc ieee international conference computer vision iccv see zhang ren sun deep residual learning image recognition technical report microsoft research zhang ren sun identity mappings deep residual networks technical report microsoft research howard zhu chen kalenichenko wang weyand andreetto adam mobilenets efficient convolutional neural networks mobile vision applications corr url http huang liu weinberger van der maaten densely connected convolutional networks corr url http hubara courbariaux soudry bengio quantized neural networks training neural networks low precision weights activations corr url http iandola moskewicz ashraf han dally keutzer squeezenet accuracy fewer parameters model size corr url http krizhevsky sutskever hinton imagenet classification deep convolutional neural networks nips neural information processing systems lake tahoe nevada lecun bottou bengio haffner learning applied document recognition proceedings ieee studer samet goldstein training quantized nets deeper understanding corr url http published conference paper iclr lin chen yan network network corr url http loshchilov hutter sgdr stochastic gradient descent restarts corr url http mcdonnell wang van schaik training deployment deep residual networks stochastic binary quantization abstract talk neuro inspired computational elements workshop nice held ibm almaden march https merolla appuswamy arthur esser modha deep neural networks robust weight binarization distortions corr url http pedersoli tzanetakis tagliasacchi espresso efficient forward propagation bcnns corr url http rastegari ordonez redmon farhadi imagenet classification using binary convolutional neural networks corr url http russakovsky deng krause satheesh huang karpathy khosla bernstein berg imagenet large scale visual recognition challenge international journal computer vision ijcv doi simonyan zisserman deep convolutional networks image recognition corr url http springenberg dosovitskiy brox riedmiller striving simplicity convolutional net corr url http szegedy vanhoucke ioffe shlens wojna rethinking inception architecture computer vision corr url http xie girshick aggregated residual transformations deep neural networks corr url http zagoruyko komodakis wide residual networks zhou yao guo chen incremental network quantization towards lossless cnns weights corr url http ifferences tandard ets baseline network downsamples residual path using average pooling skip connections feature maps different sizes use increasing number channels per option however residual pathway use average pooling using kernel size stride downsampling instead typical lossy downsampling discards pixel values samples published conference paper iclr use batch normalization applied input layer literature reported various options optimal ordering usage placement relu layers residual networks following zagoruyko komodakis precede convolutional layers combination followed relu however different zagoruyko komodakis also insert optional relu immediately input layer first convolutional layer optional relu used unlike layers enable learning scale offset factors first layer enables avoid inputs network since layer provides necessary normalization optional relu included found learned offset ensures input first relu never negative accordance strategy simplicity weight layers thought block three operations sequence indicated figure conceptually followed relu thought single layer consisting relu adaptively changes centre point positive slope channel relative also precede global average pooling layer layer use relu point since nonlinear activation provided softmax layer found including relu leads differences early training completion training first conv layer many channels first residual block wide resnet zagoruyko komodakis specified first convolutional layer always constant number output channels even number output channels layers increases found need impose constraint instead always allow first layer share number output channels blocks first spatial scale increase total number parameters small relative total number parameters since number input channels first layer benefit change increased simplicity network definition ensuring one fewer change dimensionality residual pathway omparison esidual etworks lain cnn interested understanding whether good results achieved weights consequence skip connections residual networks therefore applied method plain networks identical residual networks except skip connections removed initially training indicated much slower convergence found altering initial weights standard deviations proportional instead helped change made change also applied equation summarized figure found convergence remained slower resnets small accuracy penalty comparison resnets epochs training consistent findings resnets deeper layers showed significant advantage plain networks experiment others done support view method particular resnets published conference paper iclr classification error rate epochs figure residual networks compared networks data figure networks width million learned parameters

sandboxing javascript technical report matthias keil peter thiemann university freiburg freiburg germany keilr thiemann jan abstract today javascript applications composed scripts different origins loaded run time origins equally trusted execution scripts isolated one another however scripts must access application state may allowed change preserving confidentiality integrity constraints application paper presents design implementation decentjs sandbox full javascript enables scripts run configurable degree isolation access control provides transactional scope effects logged review access control policy inspection log effects committed application state rolled back implementation relies javascript proxies guarantee full interposition full language code including dynamically loaded scripts code injected via eval restriction scripts must compliant javascript strict mode acm subject classification security protection keywords phrases javascript sandbox proxy introduction javascript used websites rely libraries connecting social networks feature extensions advertisement libraries packaged application others loaded run time origins different trustworthiness sometimes depending user input compensate different levels trust execution dynamically loaded code isolated application state today state art securing javascript applications include code different origins choice browsers apply protection mechanisms policy signed script policy scripts either run isolation gain full access script isolation guarantees noninterference working application well preservation data integrity confidentiality scripts must access part application state function meaningfully included scripts run authority application script exert control use data included script thus managing untrusted javascript code become one key challenges present research javascript existing approaches either based restricting javascript code statically verifiable language subset according http status march matthias keil peter thiemann licensed creative commons license leibniz international proceedings informatics schloss dagstuhl informatik dagstuhl publishing germany sandboxing javascript facebook fbjs yahoo adsafe enforcing execution model forwards selected resources otherwise isolated compartment filtering rewriting like google caja project implementing monitoring facilities inside javascript engine however approaches known deficiencies first two need restrict usage javascript dynamic features apply code generated run time require extra maintenance efforts analysis transformation needs kept sync evolution language implementing monitoring javascript engine fragile incomplete efficient solution works one engine hard maintain due high activity engine development optimization contributions present design implementation decentjs sandbox javascript enforces noninterference integrity confidentiality monitoring design inspired revocable references spidermonkey compartment concept compartments create separate memory heap website technique initially introduced optimize garbage collection objects created website allowed touch objects compartment proxies objects cross compartment boundaries used cross compartment wrappers make objects accessible compartments decentjs adapts spidermonkey compartment concept sandbox implements fresh scope run code isolation application state proxies implement membrane guarantee full interposition make objects accessible inside sandbox outline paper paper organized follows section introduces decentjs facilities programmer point view section recalls proxies membranes related work explains principles underlying implementation section discusses decentjs limitations section reports experiences applying sandboxing set benchmark programs finally section concludes appendix presents example demonstrating sandbox hosting library appendix shows example scenarios already use implemented system appendix shows operational semantics core calculus sandboxing appendix states technical results appendix discusses related work appendix reports experiences applying sandboxing set benchmark programs sandboxing primer section introduces sandboxing shows series examples explains sandboxing works transactional sandboxing inspired idea transaction processing database systems transactional memory sandbox implements transactional scope content examined committed rolled back central sandbox implementation membrane values cross sandbox boundary membrane supplies effect monitoring guarantees noninterference moreover features identity preservation handles shadow objects shadow objects allow modifications objects without effecting origins modified version keil thiemann function node value left right value left right function return function heightof node return heightof heightof node right function setvalue node node node setvalue setvalue figure implementation node node object consists value field left node right node prototype provides tostring method returns string representation function heightof computes height node function setvalue replaces value field node height recursively visible inside sandbox different sandbox environments may manipulate object sandboxing provides transactions unit effects represent set modifications write effects membrane effects enable check conflicts differences rollback particular modifications commit modification origin implementation system available access consider operations binary trees defined node figure along auxiliary functions example perform operations tree consisting one node two leaves value fields initially var root new node new node new node next create new empty sandbox calling constructor sandbox first parameter acts global object sandbox environment wrapped proxy mediate accesses placed top scope chain code executing inside sandbox second parameter configuration object sandbox first class value used several executions var sbx new sandbox parameters https sandboxing javascript one use sandbox wrap invocations function objects end sandbox api provides methods call apply bind analogous methods example may call setvalue root inside sbx setvalue root first argument call function object decompiled redefined inside sandbox step erases function free variable bindings builds new closure relative sandbox global object second argument receiver object call well actual arguments call wrapped proxies make objects accessible inside sandbox wrapper proxies mediate access target objects outside sandbox reads forwarded target unless local modifications return values wrapped proxies writes produce shadow value section represents modification object initially modification visible reads inside sandbox native objects like math object line also wrapped proxy methods decompiled exists string representation thus native methods must either trusted forbidden fortunately native methods side effects chose trust given wrapping sandboxing call line modify root object returns calling tostring inside sandbox shows effect root return effect monitoring execution sandbox records effects objects cross sandbox membrane resulting lists effect objects accessible contain effects read effects write effects respectively three lists offer query methods select effects particular object heightof root var rects print effects function print code snippet prints list effects performed global object executing heightof function root output shows resulting accesses heightof math effects get get first column shows timestamp second shows name effect last column shows name requested parameter list contain write accesses write effects value previous invocation setvalue keil thiemann var wectso root print write effects root function print write effects root set inspecting sandbox state inside outside sandbox may diverge different reasons distinguish changes differences conflicts change indicates value changed respect outside value difference indicates outside value modified sandbox concluded example difference previous execution setvalue arises replace left leaf element new subtree height outside sandbox new node new node new node changes differences examined using api similar effect api flags check whether sandbox changes differences well iterators conflict arises comparison different sandboxes two sandbox environments conflict least one sandbox modifies value accessed sandbox later consider conflicts demonstrate conflicts define function appendright adds new subtree right function appendright node node node node recapitulate global root still unmodified prints whereas root sbx prints let execute appendright new sandbox var new sandbox parameters appendright root calling tostring prints however sandboxes conflict following command show returns false sandboxes manipulate root manipulate different fields sbx recalculates field value whereas replaces field right neither reads data previously written sandbox however situation changes call setvalue also modifies right setvalue root var cofts returns list conflicts function print documents conflict confict get set right sandboxing javascript transaction processing commit operation applies select effects sandbox target effects may committed one time calling commit effect object calling commit sandbox object returns rollback operation undoes existing manipulation returns previous configuration effect rollbacks done basis sandbox whole however rollback remove shadow object thus rolling back values still shadow values sbx returns tostring root returns revert operation resets shadow object wrapped value following code snippet reverts root object sbx root root shadow object removed origin visible sandbox calling tostring inside sbx returns sandbox encapsulation implementation decentjs builds two foundations memory safety reachability memory safe programming language program access uninitialized memory memory outside range allocated datastructure object reference serves right access resources managed object along memory allocated javascript resources accessible via property read write operations objects thus controlling reads writes sufficient control resources decentjs ensures isolation actual program code intercepting operation attempts modify data visible outside sandbox achieve behavior functions objects crossing sandbox boundary wrapped membrane ensure sandboxed code modify way membrane implemented using javascript proxies precisely implementation sandboxing inspired revocable membranes access control based object capabilities identity preserving membranes keep sandbox apart normal program execution encapsulate objects passed membrane redirect write operations shadow objects section encapsulate code section withhold external bindings function section unprotected value passed inside sandbox proxies membranes proxy object intended used place target object proxy behavior controlled handler object typically mediates access target object target handler may proxy objects handler object contains trap functions called trapped operation performed proxy operations like property read property write function keil thiemann handler target proxy target proxy proxy target target target figure proxy operations operation invokes trap target proxy property get property set operation invokes target proxy proxya proxyb targeta targetb proxyc targetc figure property access identity preserving membrane dashed line around target objects property access wrapper returns wrapper property access returns wrapper application forwarded corresponding trap trap function may implement operation arbitrarily example forwarding operation target object latter default behavior trap specified figure illustrates situation handler forwards operations target membrane regulated communication channel object rest program membrane implemented proxy guards operations target result operation another object recursively wrapped membrane returned way objects accessed object behind membrane also behind membrane common use cases membranes revoking references object network enforcing write protection objects behind membrane figure shows membrane targeta implemented wrapper proxya property access wrapper returns wrapped object installing membrane new direct references target objects behind membrane become available identity preserving membrane guarantees target object one proxy thus proxy identity outside membrane reflects target object identity inside example refer object sandboxing javascript handler target proxy target proxy target proxy proxy target shadow target shadow shadow figure operations sandbox property get operation invokes trap handler get target proxy forwards operation proxy target property set operation invokes trap target proxy forwards operation local shadow object final property get operation also forwarded shadow object refer wrapper object shadow objects sandbox redefines semantics proxies implement expanders idea allows client side extension properties without modifying proxy target sandbox handler manages two objects target object local shadow object target object acts parent object proxy whereas shadow object gathers local modifications write operations always take place shadow object read operation first attempts obtain property shadow object fails read gets forwarded target object figure illustrates behavior similar javascript prototype chain sandboxed version object inherits everything outside cousin whereas modifications appear inside sandbox encapsulation extends functionality membrane object visible inside sandbox either object created inside wrapper outside object special proxy wraps sandbox internal values whenever committing value outside shown last example step mediates uses sandbox internal value outside form example required wrap arguments values passed committed sandbox function wrapping guarantees sandbox never gets access unprotected references outside sandbox scope apart access restrictions protecting global state modification membrane fundamental guarantee noninterference execute program code decentjs relies eval nested statement sbxglobal body getter setter functions require special treatment like functions decompiled applied shadow object see section keil thiemann var node new node var sbx new sandbox parameters sbxglobal function use strict function setvalue node node setvalue setvalue figure scope chain installed sandbox loading setvalue dark box represents global scope dashed line indicates sandbox boundary inner box shows program code nested inside statement places sandbox global top current environment scope chain executing body setup exploits eval dynamically rebinds free variables argument whatever scope call site construction related dynamic binding property defined sbxglobal shadows variable deeper scope chain employ proxy object place sbxglobal control variable accesses sandboxed code trapping sandbox global object hasownproperty method javascript traverses scope chain resolve variable access calls method hasownproperty objects scope chain starting top inside statement first object checked traversal proxied sandbox global hasownproperty method always returns true traversal stops javascript engine sends read write operations free variables sandbox global way obtain full interposition handler proxied sandbox global complete control free variables body figure visualizes nested scopes created execution setvalue example section sandbox global sbxglobal wrapper actual global object used access heightof library code nested empty closure provides fresh scope local functions variables step required javascript standalone block scopes blocks sandboxing javascript global scope jscode jscode jscode figure nested sandboxes application outer box represents global application state containing javascript global scope sandbox global object nested javascript code defined sandbox global java variables named functions created sandboxed code end fresh scope extra scope guarantees noninterference dynamically loaded scripts define global variables functions use strict declaration front closure puts javascript strict mode ensures code obtain unprotected references global object figure shows situation instantiating different sandboxes program execution every sandbox installs scope sandbox global top scope chain scripts nested inside defined respect sandbox global sandbox global mediates access javascript global object default implementation empty guarantee isolation however decentjs grant access making resources available sandbox global function recompilation javascript functions access variables functions lexical scope function defined mozilla says remembers environment calls wrapped functions may still cause side effects free variables modifying variable calling another function thus sandboxing either erase external bindings functions verify function free side effects former alternative default decentjs remove bindings functions passed membrane protection mechanism decompiles function recompiles inside sandbox environment decompilation relies standard implementation tostring method javascript function returns string containing source code function use external function sandbox first decompiles calling tostring method bypass potential tampering use private copy call next apply eval resulting string create fresh variant function explained section application eval nested statement supplies desired environment decompilation also places use strict statement front avoid frequent decompilation call eval respect code implementation caches compiled function applicable function created function name body strict mode requires use inside function valid either function called method receiver object specified explicitly using apply call https keil thiemann instead recompiling function may use string representation function verify function free side effects example checking function body however turns recompiling function lower impact execution time analyzing function body functions without string representation native functions like object array verified sanitized passing membrane either trust functions rule end decentjs may provided white list trusted function objects case functions remain wrapped sandbox proxy mediate property access addition normal function method calls access property bound getter setter function needs decompile verify getter setter execution dom updates document object model dom api manipulating html xml documents underlie rendering web page dom provides representation document content offers methods changing structure style content etc javascript api implemented using special objects reachable document object unfortunately document tree object programming language thus wrapped use inside sandbox possibility wrap interfaces particular document object grant access dom binding dom interfaces sandbox global instantiating new sandbox interfaces wrapped sandbox proxy mediate access number limitations default dom nodes accessed calling query methods like getelementbyid document object effect logging recognizes accesses method calls rather operations dom query functions special native functions string representation decompilation possible using query function must permitted explicitly white list query function must called method actual dom object implementing corresponding interface thus dom objects wrapped like objects require special wrapping calls method correct receiver object read operations managed way write operations must either forbidden affect original dom thus guest code modify original dom unless dom interface restricted operations unrestricted operations would possible insert new script elements document loads scripts internet executes normal application state without sandboxing however prohibiting write operation means majority guest codes executed sandbox overcome limitation decentjs provides guest code access emulated dom instead real one rely javascript library emulating full browser dom implement dom interface scripts running sandbox emulated dom merged global sandbox object executing scripts ses function cause side effects values passed argument https sandboxing javascript pseudo dom constructed inside sandbox accessed modified special treatment required however pseudo dom wrapped special membrane mediating operations performing effect logging dom elements sandbox owns direct reference sandbox internal dom provides following features user sandbox provides interface sandbox internal dom enables host program access aspects dom interface control data visible guest program host load page template evaluating guest code template arbitrary html document like host page blank web page libraries operate page documents reading writing particular element template used create environment guest code runs without restrictions example guest code introduce new script elements load library code internet libraries loaded executed inside sandbox well operations interface objects recorded example access window loading document effects examined using suitable api section host program perform inspection document tree search changes differences host recognizes newly created dom elements transfer content sandbox dom dom host program policies policy guideline prescribes whether operation allowed existing sandbox systems come facility define policies example policy may grant access certain resource may grant right perform operation cause side effect system provide access control policies manner known systems decentjs provides mechanism implement empty scope pass selected resources scope reference certain resource made available inside sandbox wrapped another proxy membrane enforces suitable policy example one may use work transactional membranes shadow write operations access permission contracts restrict access objects revocable references revoke access outside world discussion strict mode decentjs runs guest code javascript strict mode rule uncontrolled accesses global object restriction may lead dysfunctional guest code semantics subtly different mode javascript however assuming strict mode less restrictive restrictions imposed techniques restrict javascript dynamic features alternatively one could also provide program transformation guards uses may access global object keil thiemann scopes decentjs places every load scope hence variables functions declared one script visible execution another script sandbox indeed deliberately keep scopes apart avoid interference enable communication decentjs offers facility load mutually dependant scripts scope otherwise scripts may exchange data writing fields sandbox global object function decompilation top level closure wrapped sandbox free valriables declared sandbox bindings removed decompilation may change meaning function rebinds free variables pure functions decompiled without changing meaning however decompiling preserves semantics function free variables imported sandbox new closure formed within sandbox may closed variables defined sandbox task rightfully manual availability global bindings part policy conclusion decompilation unavoidable guarantee noninterference function defined another scope every property read operation may call getter function native functions decompilation requires string contains source code function calling standard tostring method work functions native function string representation trust native function regulated white list trusted functions method creates new function body first couple arguments bound arguments bind javascript provide string representation newly created function object array function initializer javascript objects initialized using literal notation initializer notation examples object literals using array objects using function objects using named unnamed function expression function using literal notation circumvents restrictions wrappings may imposed object array function constructors able intercept construction using literal notation enables unprotected read access prototype objects newly created object always inherits corresponding prototype however never get access prototype object able modify prototype writes created objects always effect object never forwarded prototype object pure function function maps input output without causing observable side effect sandboxing javascript even though elements contained native prototype objects uncritical default global sandboxed script could add sensitive data side effecting function one prototype objects thus bypass access unprotected data function constructor function constructor function creates new function object based definition given arguments contrast function statements function expressions function constructor ignores surrounding scope new function always created global scope calling enables access global variables prevent leakage sandbox never grants unwrapped access javascipt global function constructor even constructor safe native function special wrapping intercepts operations uses safe way construct new function respect sandbox noninterference execution sandboxed code interfere execution application code application run sandboxes present property called noninterference security community intuition sandboxed code runs lower level security application code sandbox code must able observe results computation global scope decentjs guarantees integrity confidentiality default empty sandbox guarantees run code full isolation rest application whereas sandbox global provide protected references sandbox summary sandboxed code may try write object visible application may throw exception may terminate membrane redirects write operations sandboxed code local replicas captures exceptions timeout could used transform executions exception alas timeout implemented evaluation evaluate implementation applied javascript benchmark programs google octane benchmark benchmarks measure javascript engine performance running selection complex demanding programs benchmark programs run times google claims octane measure performance javascript code found large web applications running modern mobile desktop browsers benchmark complex demanding expected run time increases executing benchmark sandbox programs like earleyboyer navierstrokes mandreel heavily affected others slightly affected richards crypto regexp code loading instance observed run time impact entirely depends number values cross membrane javascript timeout function schedules function run currently running javascript event interrupt running function https keil thiemann running times find sandbox causes average slowdown benchmarks acceptable compared systems numbers also show sandboxing effect logging enabled causes average slowdown additional factor top pure sandboxing execution program code inside sandbox nothing else normal program execution inside statement one additional call eval instantiating execution impact influenced number wrap operations values cross membrane number decompile operations functions iii number effects wrapped objects readouts internal counters indicate heavily affected benchmarks raytrace mandreel perform large number effects raytrace benchmark example performs million effects overall average slowdown acceptable compared languageembedded systems octane intended measure engine performance benchmark programs run times claim heaviest kind benchmark every library jquery less demanding runs without measurable runtime impact appendix also contains score values obtained running benchmark suite lists readouts internal counters conclusion decentjs runs javascript code configurable degree isolation access control rather disallowing access application state provides full browser compatibility browsers work without modifications long proxy api supported better performance systems additionally decentjs comes following features sandbox decentjs javascript library aspects accessible sandbox api library deployed language extension requires changes javascript system full interposition decentjs implemented using javascript proxies proxybased implementation guarantees full interposition full javascript language including dynamic features eval decentjs works code regardless origin including dynamically loaded code code injected via eval source code transformation avoidance javascript dynamic features required sandboxing decentjs sandbox provides transactional scope logs effects wrapper proxies make external objects accessible inside sandbox enable sandbox internal modifications object hence sandboxed code runs usual without noticing sandbox effects reveal conflicts differences changes respect another sandbox global state inspection log effects committed application state rolled back acknowledgments work benefited discussions participants dagstuhl seminar scripting languages frameworks analysis verification particular tom van cutsem provided helpful advice internals javascript proxies sandboxing javascript references adsafe making javascript safe advertising http agten acker brondsema phung desmet piessens jsand complete sandboxing javascript without browser modifications zakon editor annual computer security applications conference acsac orlando usa december pages acm arnaud denker ducasse pollet bergel suen execution dynamic languages vitek editor objects models components patterns international conference tools spain june july proceedings volume lecture notes computer science pages spain june springer semantics felleisen gardner editors esop volume lecture notes computer science pages rome italy mar springer dewald holz freiling adsandbox sandboxing javascript fight malicious websites shin ossowski schumacher palakal hung editors proceedings acm symposium applied computing sac sierre switzerland march pages sierre switzerland acm dhawan ganapathy analyzing information flow browser extensions annual computer security applications conference acsac honolulu hawaii december pages ieee computer society dhawan ganapathy enhancing javascript transactions noble editor ecoop programming european conference beijing china june proceedings volume lecture notes computer science pages springer facebook sdk javascript https felt hooimeijer evans weimer talking strangers without taking candy isolating proxied content stein mislove editors proceedings workshop social network systems sns glasgow scotland april pages acm goguen meseguer security policies security models ieee symposium security privacy pages translator securing web content http guarnieri livshits gatekeeper mostly static enforcement security reliability policies javascript code monrose editor usenix security symposium montreal canada august proceedings pages usenix association guha saftoiu krishnamurthi essence javascript hondt editor ecoop volume lecture notes computer science pages maribor slovenia june springer hanson proebsting dynamic variables proceedings conference programming language design implementation pages snowbird usa june acm press new york usa hedin jsflow tracking information flow javascript apis acm symposium applied computing sac gyeongju korea mar heidegger bieniusa thiemann access permission contracts scripting languages field hicks editors proceedings annual acm symposium keil thiemann principles programming languages pages philadelphia usa acm press heidegger thiemann heuristic approach computing effects bishop vallecillo editors proceedings international conference objects models components patterns volume lecture notes computer science pages zurich switzerland june springer keil thiemann efficient dynamic access analysis using javascript proxies proceedings symposium dynamic languages dls pages new york usa acm keil thiemann efficient dynamic access analysis using javascript proxies proceedings symposium dynamic languages dls pages indianapolis indiana usa acm keil thiemann treatjs contracts javascript boyland editor european conference programming ecoop july prague czech republic volume lipics pages prague czech repulic july schloss dagstuhl fuer informatik keil thiemann treatjs contracts javascript technical report institute computer science university freiburg keil thiemann treatjs online http maffeis taly isolation untrusted javascript proceedings ieee computer security foundations symposium csf port jefferson new york usa july pages ieee computer society magazinius phung sands safe wrappers sane policies self protecting javascript aura editor nordic conference secure systems lecture notes computer science springer verlag meyerovich livshits conscript specifying enforcing security policies javascript browser ieee symposium security privacy pages california usa may ieee computer society miller robust composition towards unified approach access control concurrency control phd thesis johns hopkins university baltimore usa miller samuel laurie awad stay caja safe active content sanitized javascript http google white paper miller shapiro paradigm regained abstraction mechanisms access control saraswat editor advances computing science asian programming languages distributed computation asian computing science conference mumbai india december proceedings volume lecture notes computer science pages springer patil dong liang jiang towards access control javascript contexts international conference distributed computing systems icdcs minneapolis minnesota usa june pages ieee computer society phung sands chudnov lightweight javascript susilo tupakula varadharajan editors asiaccs pages sydney australia mar acm politz eliopoulos guha krishnamurthi adsafety verification javascript sandboxing usenix security symposium san francisco usa august proceedings usenix association sandboxing javascript richards hammer nardelli jagannathan vitek flexible access control javascript hosking eugster lopes editors proceedings acm sigplan international conference object oriented programming systems languages applications oopsla part splash indianapolis usa october pages acm policy https secureecmascript ses https shavit touitou software transactional memory proceedings acm sigplan symposium principles distributed computing pages ottowa ontario canada acm press new york usa signed scripts mozilla http strickland findler flatt chaperones impersonators support reasonable interposition leavens dwyer editors oopsla pages acm terrace beard katta javascript javascript sandboxing scripts maximilien editor usenix conference web application development webapps boston usa june pages usenix association van cutsem miller proxies design principles robust intercession apis clinger editor dls pages acm van cutsem miller trustworthy proxies virtualizing objects invariants castagna editor ecoop volume lecture notes computer science pages montpellier france july springer wagner gal wimmer eich franz compartmental memory management modern web browser boehm bacon editors proceedings international symposium memory management ismm san jose usa june pages acm warth stanojevic millstein statically scoped object adaptation expanders proceedings acm sigplan conference object oriented programming systems languages applications pages portland usa acm press new york weikum vossen transactional information systems theory algorithms practice concurrency control recovery morgan kaufmann publishers san francisco usa keil thiemann doctype html html head libraries script script script body body page headline headline script headline changed headline figure motivating example listing shows snippet file script tags load libraries application state executing body within body tag uses jquery modify dom motivation javascript important client side language web pages javascript developers rely heavily libraries calenders maps social networking feature extensions thus code web page usually composed dynamically loaded fragments different origins however javascript language provision namespaces encapsulation management global scope variables functions every loaded script authority one hand javascript developers benefit javascript flexibility enables extend application state easily hand included script ability access manipulate every value reachable global object makes difficult enforce security policy javascript consequence program understanding maintenance becomes difficult side effects may cause unexpected behavior also number security concerns library code may access sensitive data example may read user input browser dom browsers normally provide isolation mechanisms however isolation always possible scrips key challenges javascript developer manage untrusted code control use data included scrips reason effects included code sandboxing javascript doctype html html head decentjs script runs datejs fresh sandbox script var sbx new sandbox new date function sbx effect return effect instanceof date body figure execution library code sandbox first script tag loads sandbox implementation body second script tag instantiates new sandbox loads executes datejs inside sandbox later commits intended effects native date object javascript issues example consider web application relies scripts various sources figure shows extract page first includes datejs library extending javascript native date object additional methods parsing formatting processing dates next loads jquery jquery plugin also simplifies formatting javascript date objects point want ensure loading code datejs jquery influence application state unintended way encapsulating library code sandbox enables scrutinize modifications foreign code may attempt commit acceptable modifications isolating javascript transactional sandboxing inspired idea transaction processing database systems software transactional memory sandbox implements transactional scope content examined committed rolled back isolation code decentjs sandbox run javascript code isolation https https https keil thiemann application state proxies make external values visible inside sandbox handle sandbox internal write operations internal dom simulates browser dom needed setup guarantees isolated code runs without noticing sandbox providing transactional features decentjs sandbox provides transactional scope effects logged inspection policy rules specified effects adhere rules committed application state others rolled back appendix gives detailed introduction decentjs api provides series examples explaining facilities figure shows modify figure load code sandbox first focus datejs consider jquery later section comment placeholder unmodified code considered initially create fresh sandbox line first parameter sandboxinternal global object scripts running sandbox whereas second parameter configuration sandbox global object acts mediator sandbox contents external world section placed top scope chain code executing inside sandbox used make outside values available inside sandbox wrapped proxy membrane mediate accesses host program next instruct sandbox load execute datejs library line inside sandbox afterwards proxy javascript native date object modified several ways among others library adds new methods date object extends additional properties write operations proxy wrapper produce shadow value section represents modification object initially modification visible reads inside sandbox reads forwarded target unless local modifications case shadow value returned return values wrapped proxies committing intended modifications execution sandbox records effects objects cross sandbox sandbox api offers access resulting lists inspection provides query methods select effects particular object loading datejs effect log reports reads writes three different however manual inspection effects impractical requires lot effort decentjs allows register rules sandbox apply automatically rule combines sandbox operation predicate specifying state operation allowed performed example consider extension date object intended modification install rule automatically commits new properties date object line general rule commiton takes target object date predicate predicate function gets invoked sandbox object sbx effect object describing effect target object example predicate checks effect property appendix shows full html code predefined configuration object standard use sandbox consists simple pairs verbose false lists contain effects values created inside sandbox appendix shows readout effect lists sandboxing javascript write operation extending javascript native date object property name already present construct function inside sandbox function written committed outside object free variables function contain objects inside sandbox arguments call function also wrapped calling function outside causes effects inside sandbox furthermore committing object way wraps object proxy writing outside target measures required guarantee sandbox never gets access unwrapped references outside world point mention data structure committed functions constructed inside sbx bound references functions still point objects inside sandbox thus using causes effect inside sandbox furthermore committing object wraps object proxy writing target intercepts use committed object wrap arguments committed function invoking function illustration figure shows extract membrane arising javascript native date object appendix executing datejs sbx shown left first box creates proxy element accessed date date date wrapped proxies created demand proxies forward read target structure visible inside sandbox identical structure visible outside extending native date object sbx yields state shown second box modifications visible inside sandbox new elements wrapped exist inside sandbox however special proxy wraps sandbox internal values whenever committing value outside shown last box step mediates uses sandbox internal value example wrapping value arguments calling function defined sandbox wrapping guarantees sandbox never gets access unprotected references outside shadowing dom operations example figure omits inclusion jquery simplification purposes however initial objective sandbox code reason modifications done loading code prevent application state unintended modifications isolating library like jquery challenging needs access browser dom calls native dom interface expose mixture public confidential information access neither fully trusted completely forbidden address issue decentjs provides internal dom serves shadow actual dom running web library sandbox figure demonstrates loading jquery library web sandbox extend first example create new empty sandbox line initialize sandbox internal dom loading html template line using configuration activates shadow dom instructing sandbox create dom interface merge interface global object shadow dom initially see section detailed discussion keil thiemann contains empty document instantiated actual html body html template shown line figure shows template extract original containing script tags jquery library selected parts html body loading template also loads executes code inside sandbox afterwards internal effect log reports two write operations fields jquery global window object one write operation htmlbodyelement interface child node part dom interface automatically commit intended modifications global window object install suitable rule lines jquery instantiated sandbox internal dom using modifies sandbox internal dom instead browser dom modifications must committed browser dom become visible line alternatively decentjs allows grant access browser dom white listing window document objects however white listing expose entire objects restrict access certain parts document model using transactions wrapped objects decentjs supports mechanism first examples figure prevent application state unintended modification loading untrusted code commit intended ones however datejs might modify javascript native date object avoid undesired overwrites decentjs allows inspect effects libraries check conflicts committing date predicate line checks conflicts arise comparison different sandboxes conflict flagged least one sandbox modifies value accessed sandbox later furthermore prescribe case conflicts methods datejs used end second rule discards modifications date second sandbox detecting conflicts rollback operation undoes existing manipulation returns previous configuration partial rollback result inconsistent state delete objects references inside sandbox remain unchanged full html example figure shows full html code example section effects lists sections shows resulting effect logs recorded sandboxes section see appendix detailed explanation output effects sbx read effects consider conflicts conflicts handled hazard represents problem occurs concurrent executions sandboxing javascript function print get get get get function print none read effects date date function print get get get get get get write effects write effects date date function print set set set set set set set set set set set keil thiemann set set set set set set set set set set set set set read effects function print get write effects function print set set set set set set set set set set set set set set set set set set set set set set sandboxing javascript set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set set keil thiemann set set set set set set set set set set set set set set set set set set set set set set set set set set set set set effects read effects function print get get get get get get sandboxing javascript get get get get function print none read effects window window function print get getownpropertydescriptor getownpropertydescriptor get getownpropertydescriptor getownpropertydescriptor getownpropertydescriptor getownpropertydescriptor getownpropertydescriptor getownpropertydescriptor get get get getownpropertydescriptor get getownpropertydescriptor get get write effects write effects window window function print set set keil thiemann date date prototype tostring date prototype date prototype tostring isweekday date tostring prototype tostring isweekday date prototype tostring prototype tostring isweekday figure shadow objects sandbox loading datejs section structure javascrip native date object shown solid lines left shadow values enclosed dashed line solid lines direct references objects whereas dashed lines indirect references proxy objects dotted lines connect target object first box shows sandbox reading whereas second box shows sandbox modifying structure date third box shows situation committing modifications date sandboxing javascript doctype html html head decentjs script runs jquery fresh sandbox script var new sandbox new jquery new body body page headline headline script headline changed headline headline getelementbyid headline figure execution web library sandbox first script tag loads sandbox implementation body second script tag instantiates new sandbox initializes sandbox predefined html template see figure later commits intended effects application state copies data sandbox internal dom doctype html html head script script body body page headline headline figure file contains script tags loading jquery code figure keil thiemann doctype html html head checks conflicts datejs script new date function sbx effect return date new date function sbx effect return date body figure checking conflicts html code first checks conflicts datejs jquery commits modification library rolls back sandboxing javascript doctype html html head decentjs script runs datejs fresh sandbox script var sbx new sandbox new date function sbx effect return effect instanceof date runs jquery fresh sandbox script var new sandbox new jquery new checks conflicts datejs script new date function sbx effect return date new date function sbx effect return date body body page headline headline script headline changed headline headline getelementbyid headline figure execution library code sandbox section paper first script tag loads sandbox implementation second script tag instantiates new sandbox sbx loads executes datejs inside sandbox later commits intended effects native date object third script tag instantiates sandbox initializes sandbox predefined html template see figure paper later commits intended modifications application state last script tag checks conflicts datejs jquery commits modification date rolls back script tag included body performs modification sandbox internal dom copies changes global dom keil thiemann application scenarios section considers example scenarios use implemented system examples drawn projects use work sandboxing mechanism guarantee noninterference treatjs treatjs contract system javascript enforces contracts monitoring treatjs implemented library aspects contract specified using full javascript language example base contract typenumber checks argument number var typenumber function arg return typeof arg number asserting base contracts value causes predicate checked applying predicate value javascript function used return value converted typenumber accepted treatjs relies sandbox presented work guarantee execution contract code interfere contract abiding execution host program access objects functions safe useful many contracts treatjs facilitates making external references visible inside sandbox example isarray contract references global object array var isarray array array function arg return arg instanceof array however treatjs forbids write accesses traps unintended write global variable type following code var typenumberbroken function arg type typeof arg return type number treatjs online web frontend experimentation treatjs contract system enables user enter code fragments run combination treatjs code aspects treatjs accessible user code however user code neither able compromise contract system website functioning writing browser document window objects without precaution code snippet like javascript programmers speak truthy falsy values convert true false http sandboxing javascript function observer target handler var sbx new sandbox parameters omitted var controller get function target name receiver var trap var result trap trap target name receiver var raw target name return observerof raw result result raw return new proxy target controller figure implementation observer proxy excerpt get trap evaluates user specific trap sandbox guarantee noninterference afterwards performs usual operation compares outcomes executions traps implemented way function arg return arg could change contract objects influence subsequent executions avoid issues website creates fresh sandbox environment builds function closure user input executes user code sandbox sandbox grants access treatjs api javascript objects like object function array provide access browser objects like document window new invocation reverts sandbox initial state observer proxies observer restricted version javascript proxy change behavior proxy target arbitrarily implements projection either implements behavior target object raises exception similar feature provided racket chaperones observer cause program fail often case fail would behave way observer present figure contains getter part javascript implementation observer constructor observer proxy accepts arguments constructor normal proxy object returns proxy interposes different handler controller wraps execution user provided traps sandbox controller get trap evaluates user get trap one exists sandbox next performs normal property access target value produce side effects obtain baseline value compare results observerof checks whether sandboxed result suitably related baseline value observer proxy subsection confused observer proxy mention paper observer mentions section normal proxy implementing membrane keil thiemann constant variable expression new figure syntax value closure dictionary object environment store figure semantic domains semantics sandboxing section first introduces untyped lambda calculus objects object proxies serves core calculus javascript inspired previous work defines syntax describes semantics informally later extends new calculus adds sandbox core calculus core syntax figure defines syntax expression either constant variable operation primitive values lambds abstraction application creation empty object property read property assignment variables drawn denumerable sets symbols constants include javascript primitive values like numbers strings booleans well undefined null syntax make proxies available user offers internal method wrap objects sandbox expression fresh term fresh wrap object values figure extensions sandboxing javascript term new figure intermediate terms semantic domains figure defines semantic domains main component store maps location object native object object represented triple consisting dictionary potential function closure value acting prototype dictionary models properties object maps constant value object may function case closure consists lambda expression environment binds free variables maps variable value object indicated place value either constant location evaluation semantics introduces intermediate terms model partially evaluated expressions figure intermediate term thus expression zero subexpressions replaced outcomes evaluation judgment similar standard evaluation judgment except input ranges intermediate terms states evaluation term initial store environment results final store value figure defines standard evaluation rules expressions inference rules expressions mostly standard rule composite expression evaluates exactly one subexpression recursively invokes evaluation judgment continue subexpressions evaluated respective rule performs desired operation sandboxing section extends base calculus calculus adds sandboxing function expressions calculus describes core features illustrates principles sandbox features application level implemented top calculus figure defines syntax semantics extension expressions contain sandbox abstraction sandbox construction fresh instantiates fresh sandbox terms extended fresh term new internal wrap term occour source programs wraps value sandbox environment objects contain object proxies proxy object single location controlled proxy handler mediates access target location simplification represent handler objects handler sandbox handler enforces viz secure location acts shadow object proxies target object single secure environment keil thiemann const var abs dom null app new new dom new new get dom dom dom undefined put figure inference rules intermediate terms sandboxing javascript fresh fresh fresh wrap figure sandbox abstraction application rules clarity write wrapped values imported sandbox sandbox environment contains wrapped values locations proxies shadow objects consequently values extended sandboxes represents sandbox expression wrapped sandbox environment executed value used application evaluation figure contains inference rules sandbox abstraction sandbox application formalization employs semantics model side effects keeping number evaluation rules manageable rule expression fresh rule evaluates subexpression invokes evaluation judgment continue rules show last step pretty big step evaluation subexpressions evaluated respective rule performs desired operation sandbox execution happens context secure sandbox environment preserve noninterference sandbox definition abstraction evaluate sandbox closure containing sandbox expression abstraction together empty environment rule sandbox executions starts fresh environment guarantees unwrapped values reachable sandbox sandbox abstraction rule proceeds secure environments either empty set environment contains secure wrapped values sandbox execution rule applies first expression evaluates sandbox closure second expression evaluates value wraps given value triggers evaluation expressions sandbox environment binding wrapped value value acts global object sandbox used make values visible inside sandbox sandbox encapsulation sandbox encapsulation figure distinguishes several cases primitive value sandbox closure wrapped wrap location points object location packed fresh proxy along fresh shadow object current sandbox environment packaging ensures access wrapped location use current environment keil thiemann wrap wrap dom compile dom wrap wrap wrap figure inference rules sandbox encapsulation null compile dom dom dom null compile compile compile compile figure inference rules object case location already wrapped sandbox proxy location sandbox proxy gets wrapped location existing proxy returned rule ensures object wrapped thus preserves object identity inside sandbox shadow object build recompiling figure target object shadow objects new empty object may carry sandboxed replication closure part object recompiling returns empty object later acts sink property assignments wrapped objects function object recompiling extracts function body closure redefines body respect current sandbox environment new closure put new empty object step erases external bindings function closure ensures application wrapped function happens context secure sandbox environment case function already recompiled function recompilation returns existing replication application read assignment function application property read property assignment distinguish two cases either operation applies directly object applies proxy target operation proxy object usual rules apply figure contains inference rules function application property access sandboxing javascript wrap dom dom wrap figure inference rules function application property read property assignment cases application wrapped function proceeds unwrapping function evaluating sandbox environment contained proxy function argument result known wrapped case property read wrapped object two cases depending accessed property written sandbox notation dom defined shortcut dictionary lookup dom property read affected field reads property shadow location otherwise continues operation target wraps resulting value assignment wrapped object continues operation shadow location javascript write operations change properties object dictionary affect object prototype therefor shadow object contain prototype informations acts shadow absorbs write operations keil thiemann technical results javascript memory safe programming language reference seen right modify underlying object expressions body shown contain unprotected references objects modify data prove soundness sandbox show termination insensitive noninterference requires show initial store sandbox application observational equivalent final store remains application detail sandbox application may introduce new objects even write shadow objects reachable inside sandbox every value reachable outside remains unmodified calculus appendix support variable updates environments way make changes persistent modify objects thus proving noninterference relates different stores looks differences store respect reachable values observational equivalence stores first introduce equivalence relation stores respect semantic elements definition two stores equivalent constants constants equal definition two stores equivalent locations equivalent location target definition two stores equivalent objects equivalent objects constituents definition two stores equivalent dictionaries equivalent dictionary content dom dom dom definition two stores equivalent closures equivalent closure environment abstractions equal definition two stores equivalent environments equivalent environment content dom dom dom sandboxing javascript definition two stores equivalent proxy objects equivalent objects constituents definition two stores equivalent sandbox closures equivalent sandbox environment abstractions equal observational equivalence stores states follows definition two stores observational equivalent environment equivalent values dom dom lemma suppose proof proof induction derivation noninterference theorem fresh holds proof proof induction derivation keil thiemann related work plethora literature securing javascript focus distinguishing features sandbox related work already discussed body paper sandboxing javascript closely related work sandbox mechanism design access control wrappers revocable references membranes language function cause effects objects reachable references parameters global variables revocable reference instructed detach objects longer reachable safe effects however membranes handle side effects every property access call getter implement sandbox way agten implement javascript sandbox using proxies membranes work place wrappers around sensitive data dom elements enforce policies prevent application state unprotected script inclusion however instead encapsulating untrusted code require scripts compliant ses subset javascript strict mode prohibits features either unsafe grant uncontrolled access use execute scripts javascript parser transforms scripts run time restricts handling dynamic code compared approach treatjs javascript contract system uses sandboxing mechanism similar sandbox presented work guarantee execution predicate interfere execution contract abiding host program work use javascript dynamic facilities traverse scope chain evaluating predicates use javascript proxies make external references visible evaluating predicates arnaud provide features similar sandboxing mechanism treatjs approaches focus access restriction guarantee free contract assertion however neither implements sandbox writing completely forbidden always leads exception sandbox works similar way guarantees access target objects redirects write operations shadow objects local modifications visible inside sandbox however access restrictions tree approaches affect values cross border two execution environments values defined used inside local values restricted write access values fine patil present jcshadow reference monitor implemented firefox extension tool provides access control javascript resources similar decentjs implement shadow scopes isolate scripts regulate granularity object access unlike decentjs jcshadow achieves better runtime performance efficient approach tied specific engine requires active maintenance keep development enigine decentjs contrast javascript library based reflection api part standard approaches implement restrictions filtering rewriting untrusted code removing features either unsafe grant uncontrolled access exampe caja compiles javascript code sanitized javascript subset safely executed normal engines static guarantees apply code created run time using eval mechanisms caja restricts dynamic features sandboxing javascript rewrites code cajoled version additional checks prevent access unsafe function objects static approaches come number drawbacks shown number papers first either restrict dynamic features javascript guarantees simply apply code generated run time second maintenance requires lot effort implementation becomes obsolete language evolves thus dynamic effect monitoring dynamic access restriction plays important role context javascript security shown number authors effect monitoring richards provide webkit implementation monitor javascript programs run time rather performing syntactic checks look effects access control revoke effects violate policies implemented transcript firefox extension dhawan extends javascript support transactions speculative dom updates similar decentjs builds transactional scope permits execution unrestricted guest code effects within transaction logged inspection host program also provide features commit updates recover effects malicious guest code jscontest framework helps investigate effects unfamiliar javascript code monitoring execution summarizing observed access traces access permission contracts comes algorithm infers concise effect description set access paths enables programmer specify effects function using access permission contracts jscontest implemented offline source code transformation javascrip flexibility requires lot effort construct offline transformation guarantees full interposition covers full javascript language implementation jscontest known omissions support prototypes apply code created run time using eval mechanisms redesign reimplementation jscontest using javascript proxies new implementation addresses shortcomings previous version guarantees full interposition full language code regardless origin including dynamically loaded code code injected via eval monitors read write operations objects access permission contracts specify allowed effects contract restricts effects defining set permitted access paths starting anchor object however approach works differently encapsulate sensitive data instead encapsulating dubious functions systems jsflow full javascript interpreter enforces information flow policies run time like decentjs jsflow implemented javascript compared decentjs jsflow interpreter causes significantly higher impact sandbox reimplements javascript semantics membrane similar slowdown reported another javascript interpreter conceived execute untrusted javascript code implementation provides wealth powerful features similar decentjs access control support full javascript language full browser compatibility however average slowdown range significantly higher decentjs keil thiemann evaluation results section reports experience applying sandbox select programs focus influence sandboxing execution time use google octane benchmark measure performance sandbox implementation octane measures javascript engine performance running selection complex demanding programs benchmark programs run times google claims octane measure performance javascript code found large web applications running modern mobile desktop browsers benchmark complex demanding use octane intended measure engine performance benchmark programs run times claim heaviest kind benchmark every library jquery less demanding runs without measurable runtime impact octane consists range performance tests web applications figure kernel simulation portable pdf viewer program focuses special purpose example function method calls arithmetic bit operations array manipulation javascript parsing compilation etc testing procedure benchmarks run machine two amd opteron processors ghz memory example runs measurements reported paper obtained spidermonkey javascript engine benchmarking wrote new start script loads executes benchmark program fresh sandbox setting sandbox global standard global object ensure benchmark program refer properties global object needed sandboxing wraps global object membrane mediates interaction benchmark program global application state run time measurements taken deterministic run requires predefined number using run results figure figure contains statistics benchmark programs two different configurations explained figure caption lists readouts internal counters multiple read effects field object counted one effect expected run time increases executing benchmark sandbox programs like earleyboyer navierstrokes mandreel heavily affected others slightly affected richards crypto regexp code loading instance unfortunately deltablue zlib run sandbox deltablue attempts add new property global object sandbox prevents unintended modifications new property visible inside https https programs run either one second predefined number iterations iterations one second runs another second sandboxing javascript benchmark richards deltablue crypto raytrace earleyboyer regexp splay splaylatency navierstokes mandreel mandreellatency gameboy emulator code loading zlib typescript total baseline time sec sandbox effects time sec slowdown sandbox effects time sec slowdown sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec figure timings running google octane benchmark suite first column baseline gives baseline execution times without sandboxing column sandbox effects shows time required complete sandbox run without effect logging relative slowdown sandbox time column sandbox effects shows time slowdown baseline run effect logging current sandbox objects created new object created using object literals zlib benchmark uses indirect eval write objects global scope allowed ecmascript specification another benchmark code loading also uses indirect call eval small modification makes program compatible normal eval safely used sandbox first experiment turn effect logging whereas second one remains enabled separates performance impact sandbox system proxies shadow objects impact caused effect system running times find sandbox causes average slowdown benchmarks experimental setup wraps global object membrane mediates interaction benchmark program global application state benchmark program contains every source required run benchmark separation except global objects global functions thing influences execution time access global elements absolute times raw sandboxing causes run time deterioration per sandbox operation effects effect logging enabled example benchmark requires seconds complete performs effects membrane baseline requires seconds thus sandboxing takes additional seconds hence overhead per operation effect logging enabled indirect call invokes eval function using name eval keil thiemann benchmark objects effects size effect list reads writes calls richards deltablue crypto raytrace earleyboyer regexp splay splaylatency navierstokes mandreel mandreellatency gameboy emulator code loading zlib typescript total figure numbers internal counters column objects shows numbers wrap objects column effects gives total numbers effects column size effect list lists numbers different effects running benchmark column reads shows number read effects distinguished number write effects column writes distinguished number call effects column calls multiple effects field object counted one effect results tests also indicate garbage collector runs frequently significant increase memory consumption benchmark find virtual memory size increases raw run full run effect logging however looking benchmarks difference virtual memory size compared baseline run ranges raw sandbox run without effect logging full run fine grained effect logging appendix shows memory usage different benchmark programs difference compared baseline google octane scores values octane reports result terms score octane explains score follows nutshell bigger better octane measures time test takes complete assigns score inversely proportional run constants computation chosen current overall score geometric mean individual scores matches overall score earlier releases octane new benchmarks integrated choosing constants geometric mean remains rationale maintain comparability https sandboxing javascript benchmark richards deltablue crypto raytrace earleyboyer regexp splay splaylatency navierstokes mandreel mandreellatency gameboy emulator code loading zlib typescript isolation effects baseline figure scores google octane benchmark suite bigger better block isolation contains score values raw sandbox run without effect logging whereas block effects contains score values full run effect logging last column baseline gives baseline scores without sandboxing figure contains scores benchmark programs different configurations explained figure caption scores taken deterministic run requires predefined number using run expected scores drop executing benchmark sandbox first experiment turn effect logging whereas second run effect logging splits performance impact impact caused sandbox system proxies shadow objects impact caused effect system memory consumption figure figure figure shows memory consumption recorded running google octane benchmark suite numbers indicate significant increase memory consumed example difference virtual memory size ranges raw sandbox run full run fine grained effect logging programs run either one second predefined number iterations iterations one second runs another second keil thiemann benchmark virtual size richards deltablue crypto raytrace earleyboyer regexp splay splaylatency navierstokes mandreel mandreellatency gameboy emulator code loading zlib typescript baseline resident size size shared size figure memory usage running google octane benchmark suite without sandboxing column virtual shows virtual memory size column resident shows resident set size column shows segment size column shows segment size values mbyte benchmark richards deltablue crypto raytrace earleyboyer regexp splay splaylatency navierstokes mandreel mandreellatency gameboy emulator code loading zlib typescript virtual size diff sandbox effects resident size diff size diff shared size diff figure memory usage raw sandbox run without effect logging column virtual shows virtual memory size column resident shows resident set size column shows segment size column shows segment size size shows size mbyte diff shows difference baseline sandbox size baseline size mbyte sandboxing javascript benchmark richards deltablue crypto raytrace earleyboyer regexp splay splaylatency navierstokes mandreel mandreellatency gameboy emulator code loading zlib typescript virtual size diff sandbox effects resident size diff size diff shared size diff figure memory usage full run effect logging column virtual shows virtual memory size column resident shows resident set size column shows segment size column shows segment size size shows size mbyte diff shows difference baseline sandbox size baseline size mbyte

enumeration groups whose order factorises primes feb bettina eick february abstract let denote number isomorphism types groups order consider integers products necessarily distinct primes exhibit formulas introduction construction isomorphism groups given order old fundamental problem group theory initiated cayley determined groups order many publications followed cayley work history problem found enumeration isomorphism types groups order related problem number isomorphism types groups order known see almost see asymptotic estimates determined however closed formula known function many details group enumeration problem found higman considered porc conjecture suggests function porc polynomial residue classes proved see bagnera girnat newman brien brien exhibit flavour results recall explicit porc polynomials follows theorem bagnera primes primes primes primes gcd gcd primes determined formula let denote number different primes dividing prime let denote number prime divisors mod following also proved prop theorem let square free pcm aim determine formulas product primes formulas follow results cited hence remains consider numbers factorise different primes cases determine explicit formula see theorems formulas polynomial residue classes finitely many sets conditions involved primes polynomial involved primes summarise following theorem theorem see theorems let different primes polynomial residue classes enumerations obtained paper overlap various known results example considered groups order western order vavasseur lin order glenn order moreover laue considered orders form well orders dividing notes written two reasons first provide uniform reasonably compact proof considered group numbers exhibit resulting group numbers closed formula case distinctions laue work also contains unified approach towards determination considered groups approach similar easy read extract results second aim notes provide uniform reliable source considered group enumerations reliability results based proofs well detailed comparision small groups library acknowledgements give details results available literature overlap results discussions theorems details provided mike newman author thanks mike newman also various discussions notes divisibility define function via otherwise following remark exhibits relation underlying gcd remark follows gcd counting subgroups linear groups group denote number conjugacy classes subgroups order recall following result remark let prime irreducible cyclic subgroup order exists irreducible cyclic subgroup order unique conjugacy next theorem counts conjugacy classes subgroups certain orders crs denote direct product cyclic groups order theorem let different primes let follows follows sqr follows proof let preliminary step proof investigate number conjugacy classes cyclic subgroups order remark irreducible subgroup form exists conjugacy class unique case reducible cyclic subgroup order thus embeds group diagonal matrices note reducible cyclic subgroup order divides exponent subgroup exists unique subgroup contains every cyclic subgroup order group acts permutation diagonal entries element group prime order cyclic irreducible subgroup order exists since gcd reducible subgroup order exists case number conjugacy classes reducible cyclic subgroups order enumerated otherwise subgroups order subgroups diagonals form orbits length two action permutation diagonal entries first consider cyclic subgroups order irreducible case note either reducible case note conjugacy classes reducible cyclic subgroups order subgroups diagonal form orbits length two thus number conjugacy classes cyclic subgroups order remains consider subgroups type subgroup reducible exist case exists unique conjugacy class subgroups first consider cyclic subgroups order thus previous cases yields cyclic subgroups order number cyclic subgroups order case using arguments remains consider case subgroups subgroup irreducible satisfies imprimitive diagonalisable one possibility primitive irreducible one possibility extend theorem following remark let different primes let subgroup order number groups order satisfying proof consider groups order irreducible subgroup singer cycle satisfies reducible subgroup group diagonal matrices group satisfies normalizer acts permutation diagonal entries proof theorem group diagonal form satisfies theorem let prime let let follows proof first consider diagonalisable subgroups order exist case group diagonal matrices subgroup form contains subgroups order group subgroups order fall permutation action diagonal entries orbits orbits orbits next consider groups diagonalisable arise irreducible subgroups first case exists one class theorem second case remark exists one class note two cases mutually exclusive summary exists irreducible subgroup order note gcd prime thus follows simplifies formula theorem counting split extensions two groups let denote set group homomorphisms aut direct product aut aut acts set via aut aut conjugation aut action written short form given stabilizer aut called group compatible pairs denoted comp abelian via case comp acts induced action via comp constructions used solve isomorphism problem extensions two different settings recall following extensions abelian kernel suppose abelian fully invariant extension example case coprime maps onto fitting subgroup extension following theorem seems folklore theorem let finite abelian finite group fully invariant extension let complete set representatives aut aut orbits let denote number orbits comp number isomorphism types extensions following theorem proved exploits situation special case denotes cyclic group order theorem dietrich eick let prime let let finite sylow either one two isomorphism types extensions two isomorphism types extensions isomorphic special type split extensions section recall variation theorem taunt preliminary step introduce notation let finite solvable groups coprime order let denote set representatives conjugacy classes subgroups aut let denote set representatives aut normal subgroups let let denote fixed isomorphism let denote subgroup aut induced action stabaut denote subgroup aut induced action naut thus via double cosets subgroups aut denoted aut theorem let finite solvable groups coprime order number isomorphism types split extensions proof taunt theorem claims number isomorphism types split extensions correspond orbits aut aut turn orbits correspond union orbits naut set isomorphisms latter translate double cosets apply theorem two special cases following let denote cyclic group order theorem let let cqk let finite group order coprime denote gcd let set representatives aut normal subgroups let indk aut odd number isomorphism types split extensions indk proof apply theorem group aut cyclic order hence exists unique subgroup order aut subgroup cyclic next aut abelian follows naut aut aut acts trivially conjugation hence trivial group indk theorem let let cqk let finite group order coprime let set conjugacy class representatives cyclic subgroups order dividing aut number isomorphism types split extensions equals proof use theorem divisor exists unique normal subgroup cyclic order note aut hence cases groups order groups order considered lin laue various places results lin laue agree lin results harmless typos also refer prop alternative description proof following result theorem let different primes proof classification groups order yields two nilpotent groups order groups remains consider groups desired order note every group order solvable burnside theorem groups normal subgroup groups form use theorem count number split extensions thus count number conjugacy classes subgroups order aut aut number conjugacy classes subgroups order aut exhibited theorem aut cyclic thus one subgroup order aut exists groups normal subgroup groups form use theorem count number split extensions purpose consider aut proper normal subgroups cyclic two proper normal subgroups cyclic quotient case arises case arises cases indk exists one aut normal subgroups cyclic quotient form yields indk case arises groups without normal sylow subgroup let group let fitting subgroup solvable follows normal sylow subgroup obtain thus option next acts faithfully conjugation since fitting subgroup hence contradiction thus case arise groups order groups order determined western laue western paper essentially correct final summary table groups group missing case mod missing group appears western analysis section minor issues section western paper disagreements results results laue case case tried track origin laue work theorem let different primes two special cases odd embark proof theorem note formula theorem simplified distinguishing cases odd follows thus theorem odd reads odd holds used simplify formula theorem proof proof follows strategy proof theorem burnside theorem asserts every group order solvable easy see five nilpotent groups order groups remains consider groups desired order groups normal subgroup groups form using theorem correspond conjugacy classes subgroups order aut five isomorphism types groups order groups aut aut subgroups order coprime aut one conjugacy class subgroups order one conjugacy class subgroups order leads special cases shows cases type group exist remains consider case aut cyclic order thus aut one subgroup order exists adds case aut solvable normal sylow thus aut contains subgroup order form case subgroups translate conjugacy classes subgroups aut adds extraspecial exponent aut aut surjective thus conjugacy classes subgroups order aut correspond conjugacy classes subgroups order counted theorem hence adds extraspecial exponent aut solvable normal sylow form thus one subgroup order aut exists adds conjugacy classes subgroups order aut counted theorem hence adds groups normal subgroup groups form using theorem determine aut proper normal subgroups cyclic order dividing determine indk case options indk hence adds case two normal subgroups aut one normal subgroup indk thus adds extraspecial exponent case one aut normal subgroups indk hence adds extraspecial exponent two aut normal subgroups one indk indk thus case adds case one aut normal subgroups indk thus adds groups without normal sylow subgroup let group let fittingsubgroup since solvable follows trivial construction thus options recall acts faithfully cyclic order group order act faithfully hence remaining possibility case order embeds aut thus fitting subgroup order follows normal subgroup order hence possible thus covered special case groups order except special cases remains sum values yields formulas exhibited theorem groups order groups order determined lin vavasseur laue lin work essentially correct minor mistakes agrees work laue results lin seems unaware work vavasseur compared results vavasseur theorem let different primes one special case proof first groups order solvable burnside theorem obvious isomorphism types nilpotent groups order next consider sylow theorems let denote number sylow group order recall mod thus impossible thus either impossible unless thus normal sylow groups normal sylow groups form consider arising cases aut thus one subgroup order aut use theorem determine case adds case similar first case adds let aut denote ker cases isomorphims type one aut normal subgroups order satisfies aut hence case remains count number conjugacy classes subgroups order see theorem thus adds clearly one aut normal subgroups satisfies aut hence case remains count number conjugacy classes subgroups order see theorem thus adds number groups order read yields formulae groups order groups order considered glenn laue glenn work several problems groups missing summary tables duplications invariants correct affects particular summary table laue agree glenn compared results laue theorem let different primes one special case equal proof exists one group order group order two nilpotent groups order following consider solvable groups order let fitting subgroup acts faithfully normal subgroup stabilizes series yields following cases case case aut hence abelian normal subgroup isomorphic normal subgroup stabilizes series occur case adds case case aut thus impossible since thus case adds case case aut hence since aut cyclic one subgroup order exists theorem case adds case cqr cqr acts faithfully note aut cyclic order thus case arises case unique subgroup aut order theorem case adds case order embeds aut theorem number groups corresponds number conjugacy classes subgroups order theorem adds case case aut thus abelian hence note thus acts form acts subgroup acts subgroup implies contradiction hence case adds case case aut thus abelian hence follows two cases distinguish first suppose sylow acts sylow splits thus group form monomomorphism aut theorem number groups given number subgroups order aut image sylow uniquely determined remains evaluate number subgroups order aut act sylow number second case suppose acts trivially action uniquely determined remains determine number extensions exist two isomorphism types extensions case summary case adds case case form aut monomorphism theorem count number subgroups order aut note aut thus number subgroups form aut remains consider number cyclic subgroups order aut number depends gcd gcd case arise thus suppose yields group yields groups yields groups similar results holds dual case obtain adds summary case adds case pqr case form kernel order use theorem count number arising cases group either cyclic cases one aut normal subgroups order satsifies autk maps surjectively aut thus remains count number subgroups order aut aut case adds case theorem count number conjugacy class representatives subgroups order aut recall aut aut thus remains count number conjugacy classes subgroups order aut cyclic aut number aut number determined theorem summary case adds case case dual previous case exception bigger prime contained group type form nilpotent order consider two cases cyclic aut case use theorem count desired subgroups consider case case use slightly different approach note groups case form cyclic aut kernel order option thus form case kernel order use theorem count number arising cases cyclic sufficient count conjugacy classes subgroups order aut theorem adds consider theorem detail first suppose trivial uniquely defined thus remains consider case subgroup aut induced stabaut trivial group let set conjugacy class representatives subgroups order let denote subgroup aut induced normalizer theorem yields number groups arising case aut next note determined via remark aut follows otherwise summary case adds evaluated remains sum values different cases determine final result final comments enumerations paper translate group constructions would interesting make explicit thus obtain complete irredundant list isomorphism types groups orders considered references bagnera works rend circ mat palermo preface pasquale vetro edited guido zappa giovanni zacher besche eick construction finite groups symb besche eick brien smallgroups library groups small order gap package webpage available besche eick brien millenium project constructing small groups internat algebra blackburn neumann venkataraman enumeration finite groups cambridge university press cayley theory groups depending symbolic equation philos conway dietrich brien counting groups gnus moas exotica math intelligencer dietrich eick groups order algebra eick horn hulpke constructing groups small order recent results open problems submitted dfg proceedings girnat klassifikation der gruppen bis zur ordnung staatsexamensarbeit braunschweig glenn determination abstract groups order distinct primes trans amer math higman enumerating problems whose solution porc proc london math die gruppen der ordnungen pqr math die gruppen mit quadratfreier ordnungzahl pages nachr ges wiss laue zur konstruktion und klassifikation endlicher gruppen vavasseur les groupes ordre nombre premier plus grand que nombre premier acad sci paris vie vavasseur les groupes ordre nombre premier plus grand que nombre premier ann norm lin groups order tamkang newman brien groups nilpotent lie rings whose order sixth power prime brien groups order odd prime algebra pyber group enumeration leads european congress mathematics vol budapest volume progr pages basel robinson applications cohomology theory groups campbell robertson editors groups andrews number lms lecture note series pages cambridge university press taunt remarks isomorphism problem theories construction finite groups proc cambridge philos western groups order proc london mat

asymmetric dynamics outer automorphisms aug mark bell university illinois mcbell march abstract consider action irreducible outer automorphism closure outer space action dynamics iteration points converge exponentially give family outer automorphisms goes infinity rate convergence goes infinity rate convergence goes one even require rate convergence remain bounded away one family constructed family also provides explicit example property described handel mosher uniform upper bound distance axes automorphism inverse keywords outer automorphism rate convergence asymmetric mathematics subject classification irreducible outer automorphism acts closure outer space section dynamics theorem therefore action pair fixed points iteration points converge natural embedding rpc set conjugacy classes free group embedding coordinate marked graph given length shortest loop freely homotopic page coordinate system convergence exponential measure rate convergence recall two definitions definition definition suppose polynomial roots ordered spectral ratio definition suppose irreducible outer automorphism let denote minimal polynomial stretch factor spectral ratio described rate convergence determined thus state main result paper complete characterisation possible build outer automorphisms converge rapidly one direct slowly terms spectral ratio theorem family fully irreducible outer automorphisms gives another difference irreducible outer automorphisms mapping classes surfaces mapping class spectral ratio defined terms dilatation definition however dilatation proposition automatically straightforward prove forward direction theorem considering contrapositive nothing check automorphisms finite order irreducible outer automorphism geometric hence mapping class oncepunctured torus induces however spectral ratio mapping class torus least golden ratio section therefore bounded away one thus devote remainder paper constructing explicit family outer automorphisms fixed start considering polynomials yxn yxn since linear polynomials irreducible therefore hilbert irreducibility theorem chapter infinitely many integers irreducible lemma suppose polynomial roots inside unit circle one root similarly polynomial one root roots inside unit circle one root example see figure proof let therefore theorem must number roots inside unit circle hence roots inside unit circle furthermore since deg one root locate intermediate value theorem shows must lie applying argument shows must roots inside unit circle verify two roots lie using intermediate value theorem figure polynomials fact knowing positions roots allows show many polynomials irreducible directly lemma whenever polynomial irreducible proof first assume however would mean contradicts fact hence root unit circle assume reducible one factors must roots inside unit circle therefore constant term product roots must modulus less one however means constant term must zero zero root false similarly taking account symmetry roots argument also shows irreducible whenever even furthermore many low odd values find prime image irreducible follows irreducible whenever mod values shown table table irreducible whenever mod following page let outer automorphism given use family conclude remaining direction theorem theorem suppose irreducible outer automorphisms fully irreducible furthermore proof interpret homotopy equivalence petalled rose since positive automorphism train track map transition matrix map sign characteristic polynomial kxn chose polynomial irreducible lemma polynomial roots lie interior unit circle one root hence hand given consider homotopy equivalence petalled rose direct calculation turns involved shows train track map transition matrix sign characteristic polynomial kxn chose polynomial irreducible thus lemma polynomial roots lie interior unit circle one root thus required also follows computation automorphism suppose periodic nielsen paths page note local whitehead graph connected page frobenious matrix thus full irreducibility criterion lemma fully irreducible hand periodic nielsen path must parageometric page thus parageometric corollary periodic nielsen paths local whitehead graph connected matrix hence fully irreducible either case fully irreducible handel mosher described uniform upper bound depending distance axes section finish noting explicit family outer automorphisms theorem suppose axes separated least log proof ease notation let begin noting eigenvectors matrices proof theorem respectively let metric graphs outer space corresponding roses lengths assigned images lie axis similarly images lie axis note log log furthermore direct calculation lengths candidate loops shows every log log therefore suppose distance two axes realised points without loss generality may assume lies segment lies segment however shown figure right hand side inequality bounded log log log required log log log figure axes note turns illegal hence never lone axis automorphisms theorem similarly turns indices wrap illegal also never lone axis automorphisms question axes remain within uniformly bounded distance remark eigenvectors also see goes infinity axes enter thinner thinner parts outer space acknowledgements author wishes thank yael christopher leininger many helpful discussions regarding result references mark bell schleimer slow dynamics pml arxiv december mladen bestvina michael handel train tracks automorphisms free groups ann math benson farb dan margalit primer mapping class groups volume princeton mathematical series princeton university press princeton michael handel lee mosher parageometric outer automorphisms free groups trans amer math electronic michael handel lee mosher parageometric outer automorphisms free groups trans amer math electronic michael handel lee mosher axes outer space mem amer math serge lang fundamentals diophantine geometry new york gilbert levitt martin lustig irreducible automorphisms dynamics compactified outer space inst math jussieu mosher pfaff lone axes outer space arxiv november pfaff constructing classifying fully irreducible outer automorphisms free groups arxiv may karen vogtmann automorphisms free groups outer space proceedings conference geometric combinatorial group theory part haifa volume pages

topology estimation using graphical models power distribution grids mar deepjyoti deka michael chertkov scott backhaus los alamos national laboratory new mexico usa institute science technology moscow russia grid medium low voltage part large power system structurally majority distribution networks operate radially energized lines form collection trees forest substation root tree operational may change time time however tracking changes even though important distribution grid operation control hindered limited monitoring paper develops learning framework reconstruct radial operational structure distribution grid synchronized voltage measurements grid subject exogenous fluctuations nodal power consumption detect operational lines learning algorithm uses conditional independence tests continuous random variables applicable wide class probability distributions nodal consumption gaussian injections particular moreover algorithm applies practical case unbalanced power flow algorithm performance validated power flow simulations ieee distribution grid test cases networks power flow unbalanced threephase graphical models conditional independence computational complexity ntroduction operation large power grid separated different transmission grid consists high voltage lines connecting generators distribution substations distribution grid consisting medium low voltage lines connect distribution substations loads structure transmission grid made loopy reliable delivery electricity substations via multiple redundant paths hand typical distribution grids operationally radial substation root loads positioned along nodes tree radial topology selected subset power distribution lines overall structurally form loopy graph switching breakers illustration radial operational topology presented fig topology estimation distribution grid thus refers problem determining current set operational lines energized respective switch statuses deepjyoti deka scott backhaus los alamos national laboratory los alamos email deepjyoti corresponding author backhaus chertkov los alamos national laboratory los alamos skolkovo institute science technology russia email chertkov work supported department energys office electricity part doe grid modernization initiative changes may conducted way without proper timely reporting distribution system operator accurate topology estimation distribution grid necessary checking system failure detection also taking consecutive optimization control decisions topology estimation hindered limited presence measurements flow breaker statuses distribution grid reduced monitoring ability often legacy past still today distribution grids received much less attention compared system however practices review emerging effort providing better observability distribution level deployment advanced nodal measurement devices like phasor measurement units pmus frequency monitoring network fnet systems alike similarly smart devices like thermostats electric vehicles monitors performance regulation participation demand response programs devices smart meters capable providing real time measurements voltages frequency though often limited grid nodes goal paper utilize nodal measurements voltages captured smart meters efficiently estimate operational radial topology distribution grids worth mentioning given loopy set permissible edges one construct large set candidate radial topologies thus brute force search current operating topology computationally prohibitive work overcome combinatorial difficulty utilizing framework graphical models characterize nodal voltages operationally radial grid design topology learning algorithm based learning framework general particular applicable systems characterized possibly unbalanced power flows furthermore learning algorithm require explicit information individual nodes injections value line impedances reconstruction limited information practical significance line parameters distribution grids seldom calibrated may known precision sufficient topology estimation prior work topology learning power grids general distribution grids particular fast growing area research past research efforts area differ methodology edge detection available measurements types flow model used case loopy transmission grids uses maximum likelihood estimator sparsityenforcing regularizers estimate grid structure information availaible via electricity prices uses markov random field model bus phase angles build dependency graph detect faults grids linearized power flow models radial grids constant resistance reactance ratio presents topology identification algorithm using sign inverse covariance matrix present greedy structure learning algorithm using trends second order moments nodal voltage magnitudes linearized power flow models observations known subset grid nodes sets line flow measurements used topology estimation using maximum likelihood tests efforts identify topology phase distribution grids include machine learning schemes compare available observations smart meters database permissible signatures identify topology changes phase recovery parameter estimation mentioned literature considers measurements static power flow models measurements arising system dynamics swing equations graphical model based reconstruction scheme proposed identify operational topology radial grids loopy grids broader category random distribution graphical models provide important graphical tool model structure different variables used learning estimation prediction problems diverse fields like natural languages genetic networks social interactions decoding communication schemes etc group lasso graph lasso based approximate scheme topology identification loopy grids discussed contrast prior work majority distribution grids unbalanced voltages injections different nodes overarching goal work present efficient algorithm topology estimation using nodal voltages radial distribution grid using graphical model framework well additional edges relating neighbors gridgraph based factorization voltage distribution present learning algorithm uses conditional independence based tests nodes per test also call quartet identify operational edges special case nodal injection fluctuations modelled gaussian probability distributions conditional independence quartet test reduces test voltage covariances nodes quartet learning framework several computational practical advantages first framework independent exact probability distribution individual node power usage voltage profile hence applicable general distributions second require knowledge line impedances similar network parameters calibrated infrequently hence may known accurately third computational complexity edge detection test grow network size test local test considers set four nodes furthermore tests conducted distributed fashion results extend reported earlier conference version version paper results presented limited simulations best knowledge first work claiming provable topology learning algorithm distribution grids material manuscript organized follows next technical introduction section provides brief discussion distribution grid topology power flow models associated nomenclature linearized power flow model counterpart described section iii section analyzes graphical model power grid voltage measurements emphasize induced graphical model built different conditional independence properties voltage distribution introduced utilized section develop main learning algorithm section devoted discussion details conditional independence quartet test experimental results algorithm ieee radial networks test presented section viii last section reserved conclusions contribution paper consider power flows distribution grid aim estimate radial operational topology using measurements nodal voltages first contribution work development linearized coupled power flow model generalizes prior work power flow model also relates similar linearized models specific radial grids analyze probability distribution complex nodal voltages grid framework graphical models second contribution work consists proving standard assumptions fluctuations power consumption distribution nodal voltages described specific chordal graphical model show edges graphical model include actual operational istribution rid tructure ower lows structure radial structure distribution grid represented radial graph set graph set undirected operational edge set realized closing switches set functional ull see fig operational grid note difference functional gridgraph collection disjoint trees tree spans subset nodes vtk substation root node connected set operational edges etk denote nodes letters undirected ledge line connecting nodes denoted terminal nodes degree one called leaf nodes node connected leaf node called parent node nodes degree greater called fig schematic radial distribution grid substations represented large red nodes operational grid formed solid lines black load nodes within tree marked color nodes path denotes unique set nodes edges connect circumvent notational complications limit remainder manuscript topology learning problem grids containing one operational tree extension case operational forest layout straightforward however prior discussing learning first review next subsection power flow models general necessarily tree power flow models notations valued quantities represented case use notation vector variable components multiple phases per phase component described without hat particular phase superscript value variable specific grid location bus line marked subscript subscript mentioned refers vector values variable permissible locations example represents complex real variable location wia wai represents value node phase represents vector complex real values locations phase power flow voltages currents injections entire grid defined say phase skipped notation paragraph notations convenience according kirchhoff laws complex valued equations governing power flow leaving node given real valued scalars denote respectively voltage magnitude phase active reactive power injection node exp mark respectively nodal complex voltage injection impedance line stands line resistance reactance unbalanced power flow equations three phases described next unbalanced power flow setting power systems three unbalanced note even though discuss solely system analysis extends general phase system phases operate four wires one phase common ground consequently active reactive power injections bus scalars components similarly complex voltage magnitude phase bus power flows line sizes component phase using notation described earlier complex power injections complex voltages grid pbi vbi stands voltage phase angles node reference phase angles three phase system defined modulo impedance line represented symmetric matrix relates three phase current line voltage difference according iiaj iibj iicj ribbj ribcj note diagonal values denote impedances values denote interphase impedances line resistance reactances similar structure kirchoff laws three phases configuration given diag diag inverse three phase admittance matrix phase components described note generalization one three phases next section introduce linear approximation power flows done first case extended case three phase unbalanced grids approximation derived assumptions fluctuations voltage magnitudes phases connected nodes small assumptions realistic distribution grids functioning operationally stable regime iii inear ower flow odels section discuss linear coupled approximate model three phase unbalanced power flow show generalizes linear power flow model lcpf discussed previous work case network terms functionality model geared towards medium low voltage distribution grids rather high voltage transmission grids first describe linear coupled power flow model format helps transition case three phase model linear coupled power flow model model linearized jointly phase angle difference neighboring buses deviations voltage magnitude reference voltage considered small arrive following set equations one conveniently express linear equations lcpf model matrix form diagonal matrix nonzero elements complex conjugates respective line impedances edge node incidence matrix every edge represented row eti etj standard basis vector associated vertex note deriving ignore losses active reactive powers lines thus getting conservation net active reactive powers make invertible one fixes voltage magnitude unity phase zero bus injections reference bus equal negative sum injections buses standard manipulations effectively remove reference bus system without loss generality measure voltage magnitudes phases buses relative reference bus removing entries corresponding reference bus matrix vectors arrive invertible system resulting next describe linear approximations three phase power flow linear coupled three phase power flow consider three phase described reference phase angle consider small deviations voltage magnitude phase angle nominal bus small angle difference neighboring buses three phases assumption stated follows states node angles different phases roughly separated amount reference phase angles small deviation assumptions approximate phase derived similar eqs ignoring second order terms voltage magnitude phase angle differences linear coupled power flow model three phases given note equations reduce eqs number phases limited one linearization provides several important attributes first contributions voltage magnitudes angles additive second either quantity contribution expressed terms differences values neighboring nodes phase third lossless phases individually sum injections nodes given phase zero collect nodal voltage magnitudes angles phase vectors line admittances pair phases diagonal matrices express linear equation similar case combining expressions power injections three phases arrive diag square block matrix every block diagonal matrix constructed admittances respective pairs due specific structure inverse similar block sparse pattern fact following holds theorem let three phase admittance impedance matrices respectively edge define block diagonal matrix admittances lines phase pair define vai vbi vci node follows takes following form similarly inverse proof omitted reduces showing equal identity matrix next step consists inverting thus deriving expression via since blockdiagonal rank deficient resolve apparent difficulty follow logic single phase calculations specifically reduce system considering reference bus reference voltage magnitudes angles three phases furthermore lossless injections reference bus given negative sum injections nodes therefore removing entries corresponding reference bus phase invert express three phase voltages terms three phase injections reduced system follows note reduced defined general grids possibly loopy proves distribution grids equivalent first order approximation distflow model hand ignoring line resistances voltage magnitude deviations one reduces model model similarly model radial grid appears equivalent lossless approximation three phase distflow model shows different derivation ignores components corresponding line losses power flow equations see following learning algorithms naturally extend linear models including model aside lossless another characteristic aforementioned models represent nodal complex voltages invertible function nodal injections buses property fundamental determining graphical model nodal voltages discussed following section raphical odel ower lows describe probability distribution nodal voltages distribution see fig considered model model first make following assumptions regarding statistics power injections distribution assumption nodes modeled nodes instance node kept constant loads nodes assumed generated probability distribution modeling exogenous process loads different nodes statistically independent note latter important part assumption concerning independence formalization observation act switching devices independently short intervals similar assumptions independence reported literature note assumption require components node uncorrelated particular active reactive injections node allowed dependent individual many random processes one would expect according law large numbers fluctuations load well modeled gaussian process assumption continuous random vector injections nodes within described following probability distribution functions pdf injection node single phase three phases respectively note contain exact information relation governed phase specific constant rotation using invertible relations voltages injections models one arrives following pdf complex voltage vector nodes satisfy satisfy represent determinants jacobian matrices invertible linear transformation injections voltages models respectively note transformation linear jacobian determinants constant describe graphical model representation probability distribution nodal voltages use following section topology estimation graphical model dimensional random vector described undirected graphical model node set vgm edge set egm representing conditional dependence edge egm vgm represents random variables corresponding nodes set stated differently set neighbors node represented random variables conditionally dependent follows definition deletion set nodes separates graphical model two disjoint sets node conditionally independent node given nodes discuss models distribution analyze structure respective gms nodal voltages note node single phase voltages represents two scalar variables voltage magnitude phase corresponding node similarly node three phase corresponds complex voltage node three phases six scalar variables consider model tree pdf single phase voltages given using one derives constant note term product sign right hand side includes voltages corresponding node neighbors consider two nodes neighbors terms include voltages similarly two hops away share common neighbor voltages appear however three hops away term includes voltages voltages thus pdf product separated terms containing implies voltages two nodes conditionally independent given voltages nodes distance greater hops next consider pdf three phase voltages using pdf nodal voltages expanded via vic using analysis single phase case one observes pdf nodal voltages similar feature voltages nodes greater two hops conditionally independent using aforementioned definition conditional independence structure arrive following lemma lemma graphical models pdf single phase voltages pdf three phase voltages contains edges single two hops nodes fig shows example construction correspondent either aforementioned power flow models node represents single three phase voltage corresponding node properties voltages worth mentioning first unlike loopy due edges nodes separated two hops note node immediate neighbors form clique set shown fig shown chordal every cycle size greater chord based factorizations edges nodes separators use properties chordal graphs paper omit discussion properties finally note structure requires independence nodal injections agnostic exact distribution nodal injection fig load nodes edges distribution tree dotted lines represent two hop neighbors gaussian case prior work loads modeled independent gaussian random variables linear functions gaussian random variables gaussian random variables distribution nodal voltages gaussian assumption multivariate gaussian gaussian distribution two properties particularly useful structure given offdiagonal terms inverse covariance matrix random vector variables conditionally independent given variables set conditional covariance given set zero first property used validate lemma gaussian shown supplementary material next section use specific structure described section develop topology learning algorithm voltage measurements particular arising gaussian distributions opology earning using voltage onditional ndependence consider pdfs nodal voltages correspondent respectively let corresponding voltages single phase three phase described previous section includes edges true neighbors two hop neighbors topology learning algorithm based separability properties gms satisfied edges corresponding true neighbors learn topology without ambiguity make following mild assumption structure operational assumption depth length longest path excluding substation node greater three theorem operational edge nodes exists distinct nodes single phase three phase proof given supplementary material theorem enables detection edges nodes using voltage measurements let tnl see fig comprise connections nodes vnl grid estimated using theorem let set nodes degree tnl node note consists nodes grid neighbors one node next theorem provides results helping determine true parent leaf node proof presented supplementary material let emphasize first result theorem identifies connections leaves parents single nonleaf neighbor parent fig remaining leaves children nodes two neighbors node fig connections identified second result theorem theorems imply topology learning steps algorithm remainder section discuss execution complexity algorithm execution topology learning algorithm proceeds three steps first edges nodes identified based theorem steps radial network nonleaf nodes tnl constructed step next leaves connected nodes degree set tnl identified using theorem steps finally edges leaves nodes connected two nodes set identified using theorem steps worth mentioning learning algorithm require information voltage measurements grid nodes require information vnl ull vkin vlin viin jin vnl vnl end end tnl vnl nodes degree tnl vnl vnl ine ull pick vnl vkin vlin viin jin vnl vnl end end end vnl ull vkin vlin viin jin vnl vnl vnl end end end vnl theorem let tnl nodes respective edges removed let set nodes degree tnl following statement holds node parent leaf node nodes vnl edges single phase three phase let leaf node parent set vnl parent nodes edges single phase three phase input complex voltage observations jin single phase three phase nodes permissible edge set ull output operational edge set algorithm topology learning grid tree note assumption satisfied includes least two nodes two hops away thus restrictive majority distribution grids real world test cases long paths assumption next theorem lists conditional independence properties voltage distributions distinguish true edges use following notation conditional independence random variables given variables fig learning steps algorithm radial grid fig learning edges nodes determining children nodes neighbors one node determining children nodes one neighbors line impedances statistics nodal injections set permissible lines available node pairs considered permissible edges set ull example steps reconstruction radial grid fig shown fig complexity worst case computational complexity number nodes grid derivation included supplementary material next describe conditional independence test algorithm discuss specifically effect gaussianity loads gaussian voltage distribution discuss special case gaussian nodal voltages detail used numerical simulations voltages gaussian distributed gaussian relations loads voltages linearwithin noted section conditional independence gaussian random variables equivalent vanishing conditional covariance consider following real covariance matrices xkl xkl xkl xkl xkl vak val refer real imaginary parts vector thus voltages conditionally complex independent given voltages following hold entry inverse note inverse size size conditional independence test per edge requires inversion matrix task single phase three phase complexity mentioned already previous section important feature test independence size network thresholding note due numerical errors empirical estimates true covariances may zero thus use fig layout radial distribution grid red circle marks bus black lines mark operational edges additional permissible edges available algorithm represented dotted green lines following thresholding test empirical conditional covariance make decision conditional independence voltages algorithm vkin vlin viin jin call condabs positive threshold however voltages nodes far apart may low correlation appear uncorrelated given even pair thus consider relative test termed condrel threshold vkin vlin viin jin graphical model clear removing single node make conditionally independent even one exists path empirically however covariance conditioning one two nodes may significantly reduced despite edge thus consider hybrid test conditional covariance termed condmod also look effect conditioning relative one vkin vlin viin jin min onditional ndependence est algorithm performs edge detection test edge verifying complex voltages nodes conditionally independent given voltages two nodes reduce complexity check conditional independence voltage magnitudes one phase given complex voltages note complex voltage node single phase case consists two scalars voltage magnitude phase angle correspondingly scalars three phase case total number scalar variables per conditional independence test case case general voltage distributions voltage measurements continuous random variables testing conditional independence general distributions task among tests conditional independence distances estimated conditional densities characteristic functions proposed one also bin domain continuous values use discrete valued conditional independence test another line work focuses conditional independence tests schemes conditional independence characterized using vanishing norm covariance operators reproducing kernel hilbert spaces rkhs advantage relative tests absolute test feature thresholds used less affected network parameters nodal injection covariances features vary within network vii imulations esults test algorithm extracting operational edge set tree grid loopy original edge set ull ull available node pairs considered potential edges first discuss choice conditional independence test used algorithm consider tree distribution network load nodes one substation shown fig simulate active reactive load profiles follow gaussian random variables uncorrelated across nodes covariance values around means generate nodal complex voltage samples relative errors topology estimation relative errors topology estimation thres thres thres rel abs mod number samples thres thres rel mod number samples relative errors topology estimation fig layouts modified ieee test distribution grids red circles represent substations network three phase network three phase network fig accuracy topology learning algorithm different conditional independence tests increasing number voltage samples bus test case fig ull edges ull edges node pairs considered legitimate cov cov cov cov number samples fig accuracy topology learning algorithm increasing number voltage samples generated bus test case fig injection covariances taken permissible edge set ull edges gaussian indepenedent lcpf model voltage measurements provided input together permissible edge set ull edges true edges additional edges shown fig test algorithm three conditional tests described eqs sample data set varying size average estimation errors relative number operational edges generated algorithm presented fig note increasing number samples leads lesser number errors three tests performance condmod best higher sample sizes demonstrated fig algorithm inputs node pairs total permissible set edges observe performance condmod better condrel tolerance values used three tests manually optimized trial error practice selected experiments conducted historical data remainder section use condmod conditional covariance estimation edge detection next discuss topology estimation using voltage samples generated single phase lossy equations consider radial modification system shown fig input set ull comprises edges true additional edges selected randomly consider gaussian load fluctuations generate samples using matpower show performance algorithm different sample sizes two distinct injection covariance tests fig comparison also present performance algorithm voltage samples generated injection samples model observe performance voltage samples similar one observed improves sample size increase thus confirming algorithm even though built linearization principles performance empirically well data generated power flow models finally discuss performance three phase power flow models first compare three phase voltages generated linearized model three phase power flow model consider three phase bus test cases modified ieee test cases modify nodal loads three phase remove shunts make line impedances three phases networks depicted figs fig show relative errors bus voltage magnitudes phase respect true values generated conventional sweep method results test network include two choices reference bus bus bus note maximum relative error less networks choice reference bus motivates evaluate performance topology identification using true three phase voltages generated compare consider bus reference node three phase test networks topology learning algorithm degree consider two different gaussian nodal injection covariances networks generate input voltage samples using three phase network include node pairs permissible edges ull performance algorithm different samples sizes case depicted fig phase ref bus phase ref bus phase ref bus phase ref bus phase ref bus phase ref bus relative error bus volt mag relative error bus volt mag relative errors topology estimation phase ref bus phase ref bus phase ref bus phase ref bus phase ref bus phase ref bus bus number bus number cov cov cov cov number samples bus network pick random additional edges added true edges input permissible edge set ull size algorithm along three phase voltage samples relative errors topology estimation different input sample sizes network shown fig note either three phase networks errors power flows less comparable errors observed linerized power flow model furthermore errors decrease increase sample size optimize values thresholds condmod used algorithm using trial error search practice determined historical simulated data viii onclusion paper develop algorithm allows estimate radial topology distribution grids particular derive linearized power flow model single unbalanced three phase cases develop graphical model based learning algorithm able estimate operational topology networks samples nodal voltages learning algorithm general require information nodal injection statistics line parameters best knowledge first approach develops algorithm guarantees topology estimation balanced effectively single phase unbalanced three phase networks learning algorithm uses conditional independence results voltages quartets nodes reduces conditional covariance test grids gaussian statistics computational complexity algorithm scales polynomially size networkprimarily due fact complexity conditional independence test independent network size demonstrate empirical efficacy algorithm number ieee test cases work number promising future extensions first realistic networks may portions three phase layout split three lines different lengths extension algorithm case straightforward second linear flow model based topology learning relative errors topology estimation fig accuracy voltages generated linear relative voltages model test systems fig base load selection reference buses cov cov cov cov number samples fig accuracy topology learning algorithm increasing number three phase voltage samples generated injection covariances bus test case fig node pairs considered permissible edges bus test case fig permissible edge set ull edges used jointly phase identification impendance estimation see example latter single phase case topology estimation presence missing nodes three phase layout another important research direction plan pursue near future finally plan extend empirical nonlinear approach towards establishing rigorous bounds errors linearized flow models shall enable extend theoretcial guarantees reconstruction algorithm linearized versions power flow models formulations eferences deka chertkov backhaus structure learning power distribution networks ieee transactions control network systems hoffman practical state estimation electric distribution networks power systems conference exposition psce ieee pes ieee phadke synchronized phasor measurements power systems computer von meier culler mceachern arghandeh microsynchrophasors distribution systems zhong billian zhang tsai conners centeno phadke liu power system frequency monitoring network fnet implementation power systems ieee transactions vol kekatos giannakis baldick grid topology identification using electricity prices arxiv preprint zhang dependency graph approach fault detection localization towards secure smart grid smart grid ieee transactions vol bolognani bof michelotti muraro schenato identification power distribution network topology via voltage correlation analysis decision control cdc ieee annual conference ieee deka backhaus chertkov learning topology power distribution grid without missing data control conference ecc european ieee tractable structure learning radial physical flow networks decision control cdc ieee conference ieee learning topology distribution grids using terminal node measurements ieee smartgridcomm sevlian rajagopal feeder topology identification arxiv preprint cavraro arghandeh von meier poolla approach distribution network topology detection arxiv preprint peppanen reno thakkar grijalva harley leveraging ami data distribution system model calibration situational awareness ieee transactions smart grid vol arya jayram pal kalyanaraman inferring connectivity model meter measurements distribution networks proceedings fourth international conference future energy systems acm talukdar deka materassi salapaka exact topology reconstruction radial dynamical systems applications distribution system power grid accepted american control conference acc talukdar deka lundstrom chertkov salapaka learning exact topology loopy power grid ambient dynamics proceedings eighth international conference future energy systems acm lokhov vuffray shemetov deka chertkov online learning power transmission ieee wainwright jordan graphical models exponential families variational inference foundations trends machine learning vol liao weng liu rajagopal urban distribution grid topology estimation via group lasso arxiv preprint deka talukdar salapaka topology estimation bulk power grids theoretical guarantees limits accepted bulk power systems dynamics control symposiumirep kersting distribution system modeling analysis electric power generation transmission distribution third edition crc press gan low convex relaxations linear approximation optimal power flow multiphase radial networks power systems computation conference pscc ieee chen chen hwang kotas chebli distribution system power flow rigid approach ieee transactions power delivery vol deka backhaus chertkov estimating distribution grid topologies graphical learning based approach power systems computation conference pscc ieee baran optimal sizing capacitors placed radial distribution system power delivery ieee transactions vol jan optimal capacitor placement radial distribution systems power delivery ieee transactions vol jan network reconfiguration distribution systems loss reduction load balancing power delivery ieee transactions vol apr abur exposito power system state estimation theory implementation crc press bolognani bof michelotti muraro schenato identification power distribution network topology via voltage correlation analysis ieee decision control cdc ieee lee baldick wind power ensemble prediction based gaussian processes neural networks ieee transactions smart grid vol dvorkin lubin backhaus chertkov uncertainty sets wind power generation ieee transactions power systems vol zhu giannakis sparse overcomplete representations efficient identification power line outages ieee transactions power systems vol white nonparametric hellinger metric test conditional independence econometric theory vol consistent test conditional independence margaritis learning bayesian network structure continuous domains fukumizu bach jordan dimensionality reduction supervised learning reproducing kernel hilbert spaces journal machine learning research vol gretton fukumizu teo song smola kernel statistical test independence advances neural information processing systems zhang peters janzing conditional independence test application causal discovery arxiv preprint eminoglu hocaoglu new power flow method radial distribution systems including voltage dependent load models electric power systems research vol online available http ieee standard interconnecting distributed resources electric power online available http kersting radial distribution test feeders power engineering society winter meeting ieee vol ieee garces linear load flow power distribution systems ieee transactions power systems vol park deka chertkov exact topology parameter estimation distribution grids minimal observability power systems computation conference pscc ieee upplementary aterial validation lemma gaussian injections mentioned section structure gaussian given entries inverse steps proof theorem voltages conditionally independent given voltages converse let true parent path pkl include node connected one node pkl include disconnected removing nodes otherwise pkl exists path containing two hop neighbors removing therefore relation hold parent let node vnl true parent leaf reduced weighted laplacian node edges exist using similar argument matrix tree weight edge given theorem conditional independence relation holds one arrives converse consider vnl consider path pik separated two hops voltages conditionally independent given voltages nodes next consider pik exactly two hops vnl otherwise neighbor pik assumption observe contains edges nodes violates relation edge belongs separated less three hops proposed therefore conditional independence relation satisfied inverse covariance matrix voltages model true parent vnl gaussian injections derived similar way covariance matrix consider vector injection profiles follows uncorrelated multivariate gaussian distribution diagonal covariance matrices denoting variance active reactive injections covariance active reactive injections respectively covariance matrix complex voltages satisfies proof theorem note single phase three phase includes edges node pairs one two hops away see lemma present proof extension straightforward part consider nodes voltages conditionally independent given voltages means removing nodes separates nodes disjoint groups prove edge nonleaf nodes contradiction let pkl unique path nodes separability least one nodes included pkl let node excluded pkl edge exists two hop neighbors thus removing disconnect one included pkl finally consider pkl least one node due edges two hop neighbors removing disconnect note leaf nodes part path two distinct nodes nonleaf nodes hence edge nodes contradiction part consider neighbors exits neighbor neighbor shown fig corresponding edges graphical model every path includes edge removing nodes thus disconnects nodes makes voltages conditionally independent proof theorem connected one node assumption exist nodes edges connected using computational complexity algorithm edge detection algorithm depends conditional independence tests test conducted voltages four nodes thus quartet unlike case general computational complexity test thus independent size network identifying edge pair nodes requires tests worst case combinations step considered therefore total complexity identifying network nodes determining nodes degree tnl complexity edge leaf degree one node tnl verified conditional independent test single neighbor two hop neighbor therefore complexity edge detection nodes tnl degree one leaves complexity number leaves tnl finally combinations neighbors two hop neighbors needed verify leaves vnl steps thus complexity overall complexity algorithm independent network size note assume prior information number edges node example set permissible edges ull given edge detection tests restricted set complexity reduce ull

jul groups extension philippe gille abstract let discretly henselian field whose residue field separably closed answering question raised prasad show semisimple group finite tamely ramified extension keywords linear algebraic groups galois cohomology theory msc introduction let discretly valued henselian field valuation ring residue field denote knr maximal unramified extension maximal tamely ramified extension semisimple simply connected groups theory available sense galois cohomology set knr computed terms galois cohomology special fibers group schemes permits compute residue field perfect hand perfect wild cohomology classes occur examples appear example study bad unipotent elements semisimple algebraic groups restrictions would like show vanishes see corollary related following result theorem let semisimple simply connected residue field separably closed knr date july author supported project anr geolie french national research agency gille theorem answers question raised gopal prasad found another proof reduction inner case type first observation result quite simple establish following additional hypothesis variety borel subgroups carries degree one point property holds away section open question holds groups type case actually strongly inner theorem proof galois cohomology argument using buildings section make stage remarks statement since knr discretly valued henselian field residue field observe implies also weak approximation argument prop reduces complete case residue field separably closed characteristic zero result follows steinberg theorem cor words main case address characteristic exponent acknowledgements grateful prasad raising interesting question also fruitful discussions variety borel subgroups degree one let field let separable closure let gal absolute galois group let nonsingular quadratic form celebrated result springer states witt index insensitive odd degree field extensions particular property maximal witt index insensible odd degree extensions rephrased saying algebraic group iff odd degree field extension fact generalizes semisimple groups without type theorem let semisimple algebraic without quotient type let finite field extensions coprime degrees gki proof far uniform hence gathers several contributions note split version absolutely almost simple case remind reader semisimple isomorphic inner twist group unique isomorphism denoting gqad adjoint quotient means exists galois cocycle gal gqad isomorphic denote gsc gqad simply connected cover gsc simply connected cover extension lemma following equivalent gqad furthermore gal gsc also equivalent iii gsc proof isomorphism class encoded image map gqad aut right handside map trivial kernel since exact sequence gqad aut split whence implication reverse inclusion obvious assume lifts implication iii obvious point map gsc trivial kernel whence implication iii proceed proof theorem proof let variety borel subgroups projective iff point thus prove degree one without loss generality assume simply connected according rlj absolutely almost simple simply connected group defined finite separable field extension notation rlj stands usual weilqrestriction variety borel subgroup isomorphic rlj borel subgroups reduction absolutely almost simple case assumption hence since separable follows carries degree one know prove case hence assume absolutely almost simple denote chevalley group twisted form reduction characteristic zero case characteristic let cohen ring residue field complete discrete valuation ring fraction field characteristic zero uniformizing parameter isomorphism class encoded galois cohomology class aut since aut smooth affine scheme use hensel lemma aut aut lifts semisimple simply connected group scheme let borel subgroups smooth projective let unramified field extension degree residue gille field denoting valuation ring consider maps left equality come projectivity right surjectivity hensel lemma follows degree one assuming result characteristic zero case follows whence may assume characteristic zero denote center tits class since tits class form zero classical restrictioncorestriction argument yields words strong inner form form means exists galois cocycle value twist inner conjugation lemma shows problem rephrased serre question triviality kernel map kernel indeed trivial case whence result remind reader one associate semisimple set torsion primes depends type since algebraic group splits extension degree whose primary factors belong get following refinement corollary let semisimple algebraic without quotient type let finite field extensions prime gki lemma together corollary implies following statement corollary let semisimple simply connected algebraic without factors type let finite field extensions prime maps gad gad trivial kernels proceed proof theorem away since theorem shows condition fullfilled case proof theorem assumption discretly valued henselian field given semisimple satisfying assumption extension becomes finite tamely ramified extension note prime denote borel subgroups want show reduced following cases perfect absolute galois group gal prime gal weak approximation prop may assume complete note operation change absolute galois group ibid case cdl cdl since perfect steinberg theorem cor yields case extension proper tamely ramified extension hence assumption implies remarks case proof need assume perfect prime different point gal separable cohomological dimension less equal see open question whether type split split coprime degree extensions positive answer question would imply serre vanishing conjecture groups type serre injectivity question positive answer arbitrary classical group simply connected adjoint holds certain exceptional cases cohomology buildings field introduction proposition assume separably closed let split semisimple connected proof reason finite level shall prove given finite tamely ramified extension put gal cyclic group whose order prime characteristic exponent let building comes equipped action let killing couple split defines apartment preserved action given galois cocycle defines section projection map provides action called twisted action respect cocycle fixed point theorem provides point fixed twisted action point belongs apartment since acts transitively set apartments exists suitable gille observe fixed pointwise standard action fixed consider equivalent cocycle compute fixed twisted action without loss generality may assume put stabg since fixed group preserved action let scheme attached know special fiber smooth connected quotient split unipotent radical split reductive important point action arises semilinear action explained beginning induces group since belongs carries natural maximal split maximal torus image still denoted maximal torus observe acts trivially aut idtx follows acts means group homomorphism int take generator denote image image cocycle relation yields generally observe fixed since get relation element order semisimple separably closed belongs maximal torus follows tad since belongs tad tad hence tad follows since map surjective assume without loss generality cocycle takes value trivial given homomorphism homomorphism lifts uniquely homomorphism fea main technical step claim fiber fea extension using claim fea image belongs image map hilbert theorem thus desired remains establish claim put ker group filtered decreasing filtration normal subgroups split unipotent equipped action page denote fea twisted cocycle fea surjection fiber map cor enough show fea happens fortunately filtration stable adjoint action image fea using lim lemma next subsection fea fea since fea maps onto kernel fiber fea cor conclude claim established permits complete proof theorem proof theorem usual reductions question boils semisimple simply connected case even absolutely almost semisimple simply connected case taking account cases established section remains deal case type denote split group type aut follows assumption gkt proposition states whence split record following cohomological application corollary let semisimple algebraic assume simply connected adjoint proof theorem permits assume denote gad adjoint quotient since map gad trivial kernel lem assume adjoint let consider twisted knr since isomorphic gkt theorem shows hence isomorphic means belongs kernel map aut int exact sequence aut splits kernel trivial thus knr appendix galois cohomology groups let separably closed field let algebraic equipped action finite group admits decreasing filtration gille normal pro unipotent stabilized unipotent algebraic lemma assume invertible smooth connected proof start algebraic case smooth connected unipotent according admits central characteristic filtration twisted form gna since smooth separably closed following exact sequence multiplication abelian group isomorphism exact sequence shows map onto induction follows maps onto whence consider case since smooth lim therefore successive approximations kernel map lim trivial according first case right handside trivial thus references lenstra forms odd degree extensions normal bases amer math black zero cycles degree one principal homogeneous spaces algebra bourbaki commutative berlin bruhat tits groupes sur corps local inst hautes etudes sci publ math bruhat tits groupes sur corps local existence une radicielle pub math ihes bruhat tits groupes sur corps local iii application cohomologie galoisienne fac sci univ tokyo gabber gille principaux sur les corps algebraic geometry garibaldi rost invariant trivial kernel groups low rank comment math helv gille sur les groupes sur corps global publications gille unipotent subgroups reductive groups characteristic duke math gille groupes sur corps dimension cohomologique monograph preparation extension knus merkurjev rost tignol book involutions ams colloq publ providence prasad new approach unramified descent theory preprint prasad finite group actions reductive groups descent bruhattits theory preprint groupes par demazure grothendieck lecture notes math springer serre cohomologie galoisienne new york serre cohomologie galoisienne bourbaki tits sur les des extensions corps les groupes simples acad sci paris math univ lyon claude bernard lyon cnrs umr institut camille jordan blvd novembre villeurbanne cedex france

submitted artificial life march simple model unbounded evolutionary versatility trend organismal evolution peter turney institute information technology national research council canada ottawa ontario canada phone fax abstract idea trends evolution biological organisms highly controversial commonly believed example trend evolution towards increasing complexity empirical theoretical arguments undermine belief natural selection results organisms well adapted local environments clear local adaptation produce global trend paper present simple computational model local adaptation randomly changing environment results global trend towards increasing evolutionary versatility model evolutionary versatility increase without bound environment must highly dynamic model also shows unbounded evolutionary versatility implies accelerating evolutionary pace believe unbounded increase evolutionary versatility trend evolution discuss testable predictions organismal evolution suggested model keywords evolutionary trends evolutionary progress trends evolutionary versatility evolvability baldwin effect running head unbounded evolutionary versatility national research council canada turney simple model unbounded evolutionary versatility trend organismal evolution introduction ruse argues almost evolutionary theorists including darwin believe progress evolution progress implies trend trend good example commonly believed layperson trend evolution towards increasing intelligence trend good several scientists suggested focus scientific question whether trends without regard question whether trends good mcshea presents excellent survey eight serious candidates live hypotheses trends evolution entropy energy intensiveness evolutionary versatility developmental depth structural depth adaptedness size complexity complexity appears popular candidate standard objection trends evolution natural selection local process results organisms well adapted local environments way local mechanism yield global trend hand seem complexity example increased steadily since life earth began seems suggest natural selection favours increasing complexity however many evolutionary theorists deny driving force natural selection behind apparent trends evolution gould presented extensive arguments driving force gould admits may trends evolution argues trends essence statistical artifacts example consider evolution life since first appearance prokaryotes mean level complexity would necessarily increase time organism significantly less complexity prokaryote would able live according gould apparent trend towards increasing unbounded evolutionary versatility plexity due random variation complexity plus existence minimum level complexity required sustain life selective pressure drives life towards increasing complexity discuss gould arguments detail section eight live hypotheses trends evolution paper focuses evolutionary versatility believe indeed selective advantage increasing evolutionary versatility evolutionary versatility number independent dimensions along variation occur evolution possible increasing evolutionary versatility may driving force behind apparent evolutionary trends increasing complexity discuss concept evolutionary versatility section section introduce simple computational model unbounded evolutionary versatility far know first computational model evolutionary mechanism one eight live hypotheses trends evolution model population evolves series eras era fitness landscape constant randomly changes one era next era model shows trend towards increasing evolutionary versatility spite random drift fitness landscape fact fitness landscape constant evolutionary versatility bounded model unbounded evolutionary versatility requires dynamic fitness landscape point model show possible principle natural selection drive evolution towards globally increasing evolutionary versatility without bound even though natural selection purely local process discuss related work section simple computational model evolutionary versatility related bedau seymour model adaptation mutation rates primary focus bedau seymour adaptation mutation rates primary focus paper evolutionary versatility bedau seymour model address evolutionary versatility core paper experimental evaluation model section turney first two experiments show parameter settings evolutionary versatility increase indefinitely third experiment show evolutionary versatility bounded fitness landscape static remaining experiments examine wide range settings parameters model experiments show behaviour model primarily determined parameters control amount change fitness landscape section discuss implications model one interesting implications model increasing evolutionary versatility implies accelerating evolutionary pace leads testable predictions organismal evolution discuss limitations future work section conclude section arguments trends evolution natural selection produces organisms well adapted local environments major objection trends evolution way local adaptation cause trend example although environments primates may favour increasing complexity environments parasites favour streamlining simplification generally accepted theoretical explanation natural selection could cause trend van valen red queen hypothesis attempts explain natural selection could cause trend towards increasing complexity based coevolution although van valen hypothesis criticized paper propose explanation natural selection could cause trend towards increasing evolutionary versatility attractive features proposal easily simulated computer leads testable principle predictions constant property shared environments would easy see could trend due adaptation constant property however computational model section shows largescale trend even fitness landscape changes completely randomly time unbounded evolutionary versatility aside theoretical difficulties trends question whether empirical evidence trend mcshea survey candidates trends address issue evidence candidates another paper finds solid evidence trend kinds complexity gould argues even empirical evidence trend imply driving force behind trend gould argues evolution performing random walk complexity space constraint minimum level complexity complexity organism drops certain level level prokaryotes longer live gould metaphor evolution drunkard random walk wall way bounded diffusion process wall minimum complexity causes random drift towards higher complexity random drift involve active selection complexity push drive towards increased complexity summary clear local selection produce global trend observation global trend imply driving force behind trend however model shows one way local selection produce global trend model makes testable principle predictions evolutionary versatility evolutionary versatility number independent dimensions along variation occur evolution species high evolutionary versatility wide range ways adapt environment vermeij argued selection increased evolutionary versatility lead organisms efficient better exploiting environments important point evolutionary versatility requires merely many dimensions along variation occur also dimensions independent turney ropy condition single gene affects two distinct traits appear unrelated traits appear vary dimensions linked pleiotropy effectively one dimension along variation occur several authors suggested would beneficial map modular since increasing modularity implies increasing independence traits mcshea points close connection evolutionary versatility modularity evolutionary versatility seems connected several seven live hypotheses increasing evolutionary versatility implies increasing complexity since organisms must new physical structures support new dimension variation dimensions supposed independent new physical structures must also least partially independent increasing accumulation many independent new physical structures implies increasingly complex organisms among live hypotheses developmental depth structural depth adaptedness perhaps energy intensiveness may connected evolutionary versatility evolutionary versatility also seems related evolvability evolvability capacity evolve increasing number independent dimensions along variation occur evolution implies increasing capacity evolve would seem increase evolutionary versatility must also increase evolvability hand properties increase evolvability may decrease evolutionary versatility example selection expected favour constraint produces symmetrical development humans sixth finger useful mutation would likely best new fingers appeared simultaneously hands instead requiring two separate mutations one left hand another right hand general selection favour constraint produces adaptive covariation constraints increase evolvability appear decrease evolutionary versatility unbounded evolutionary versatility increasing evolutionary versatility suggests increasing number independent dimensions adaptive covariation suggests decreasing number independent dimensions vermeij reconciles forces proposing increasing evolutionary versatility adds dimensions integrated adaptive covariation new dimensions added integrated ongoing cycle evolutionary versatility also appears related baldwin effect baldwin effect based phenotypic plasticity ability organism phenotype adapt local environment lifetime examples phenotypic plasticity include ability humans tan exposure sunlight ability many animals learn experience phenotypic plasticity facilitate evolution enabling organism benefit least survive partially successful mutation otherwise absence phenotypic plasticity might detrimental gives evolution opportunity complete partially successful mutation future generations evolution really free vary along given dimension variation along dimension leads death without children thus phenotypic plasticity increases effective number dimensions along variation occur evolution baldwin effect therefore seen mechanism increasing evolutionary versatility simple computational model evolutionary versatility following simple model unbounded evolutionary versatility three important features fitness function based shifting target demonstrate trend possible even optimal phenotype varies time fact model target must shift model display unbounded evolutionary versatility length genome change upper limit possible length genome necessary length bounded would finite number possible genotypes thus would bound evolutionary versatility mutation rate encoded genome mutation rate adapt turney environment allows model address claim mutation becomes increasingly harmful length genome increases authors argued natural selection tend drive mutation rates zero course mutation rate goes zero sets bound evolutionary versatility table shows parameters model baseline values experiments follow manipulate parameters determine effects behaviour model meaning parameters table become clear describe model table parameters model baseline values parameter name description pop size era length mutation length number individuals population number children born one run model number children born one era fraction target changes eras number individuals sampled selecting parents number bits genome encoding mutation rate run length change rate tournament baseline value figure description model evolutionary versatility model genome string bits model genetic algorithm opposed generational genetic algorithm children born generational genetic algorithm whole population updated simultaneously resulting sequence distinct generations parents selected using tournament selection tournament selection population randomly sampled two fittest individuals sample chosen parents see lines figure selective pressure controlled varying size sample new child created applying crossover parents lines new child undergoes mutation based mutation rate unbounded evolutionary versatility encoded child genome lines mutation flip bit genome add delete bit making bit string longer shorter initial section genome first bits encodes mutation rate genome remainder genome may null encodes phenotype phenotype bit string created genome simply copying bits genotype beginning plus one bit genotype continuing end genotype length genome exactly mutation code length simulation first starts running type null string fitness phenotype determined comparing target target random string bits fitness phenotype number matching bits phenotype target lines phenotype null fitness zero length target grows target always least long longest phenotype population lines new child born fitter least fit individual population replaces least fit individual lines target held constant interval time called era end era target randomly changed time target changes necessary fitness every individual lines instead dividing run series eras model could designed small continuous change target new child born special case current model target change rate small era one main motivation dividing run series eras increase computational efficiency model since computationally expensive fitness every individual time new child born actually would really necessary individuals time new child born could also argued organismal evolution characterized periods stasis followed rapid change punctuated equilibria turney set parameter values number individuals population number children born one run number children born one era fraction target changes eras number individuals sampled selecting parents number bits genome encoding mutation rate let pop array bit strings population let bit string pop string randomly generated bits generated equal probability pop individual pop let target empty string goal string determining fitness let fit array integers fitness pop let fit fit initial fitness pop childnum main loop randomly sample individuals bit strings pop sampling replacement take two fittest individuals parents randomly let mom one two parents let dad randomly pick crossover point cross falls inside bit strings mom dad parents may different lengths let child left side mom bit string cross followed right side dad bit string cross thus length child equals length dad let mutate set child mutation rate fraction interpreting first bits child encoded fraction example randomly flip bits child probability flipping bit mutate randomly add remove bit end child probability mutate adding removing equal probability remove bit length child minimum required length length child length target randomly add bit target equal probability let childfit number bits child match bits target first bit target aligned bit child child short childfit let worst oldest individual among least fit individuals pop let worstfit fitness worst childfit worstfit replace worst child replace worstfit childfit divides childnum remainder randomly flip bits target probability flipping bit fitness fit every individual pop end end figure description model evolutionary versatility model genetic algorithm crossover mutation mutation flip bit genome increase decrease genome length one bit mutation rate encoded genome parents chosen tournament selection unbounded evolutionary versatility feature model makes realistic recall evolutionary versatility number independent dimensions along variation occur evolution model evolutionary versatility genome length genome minus length part genome encodes phenotype first mutation bits independent directly affect phenotype shall ignore counting number independent dimensions along variation occur remaining bit genome independent dimension along variation occur dimensions independent fitness organism defined number matches phenotype target fitness sum fitnesses dimension fitness one dimension match one bit impact fitness another dimension match another bit note increasing evolutionary versatility increasing genome length necessarily imply increasing fitness additional bits necessarily match target mutation rate enables evolutionary versatility genome length increase also makes genome vulnerable disruptive fitness reducing mutations however design model implies increasing genome length tend correlated increasing fitness related models closely related work model bedau seymour bedau seymour model mutation rates allowed adapt demands environment find mutation rates adapt optimal level depends evolutionary demands environment novelty model similar mutation rates also allowed adapt work adaptive mutation rates includes bedau seymour model model distinct work share interest relationship adaptive mutation rates evolutionary demands turney environment novelty main difference paper previous work different objective none previous papers concerned trends evolution far know first model show possible evolutionary versatility increase without bound results experiments model section presents eight experiments model evolutionary versatility first experiment examines behaviour model baseline parameter settings second experiment runs model ten million births otherwise baseline case experiment gives lower resolution view behaviour model much longer time scale two experiments support claim model display unbounded evolutionary versatility given suitable parameter settings third experiment uses baseline parameter settings except target held constant constant target mutation rate eventually goes zero population becomes static results show model unbounded evolutionary versatility requires dynamically varying target remaining experiments vary parameters model one time experiments show model sensitive parameters determine pace change target comparison parameters affect target relatively little influence behaviour model experiment baseline parameter values figure shows results baseline parameter settings see table since model stochastic run different assuming random number seed different general behaviour runs assuming parameters experiment ran model times averaged results across runs experiment length era children start new era unbounded evolutionary versatility figure experiment baseline parameter values four plots show fitness genome length mutation rate fitness increase functions number children born target fitness function changes time one hundred children born fitness increase increase fitness since recent change target values averages whole population one hundred separate runs baseline configuration individuals times runs yields samples per value fitness drops however overall trend towards increasing fitness see first plot figure although probability mutation increase genome length equal probability mutation decrease genome length steady trend towards increasing genome length second plot figure mutation rate decreases steadily third plot although length era fixed era increase fitness turney since start era greater corresponding increase previous era fourth plot shows pace evolution accelerating evolutionary versatility mutation code length given genome length minus steady growth genome length second plot figure shows evolutionary versatility increasing least relatively short time span experiment experiment longer run length steady decrease mutation rate first experiment suggests mutation rate might zero mutation rate zero fitness longer increase without bound fitness would vary randomly target changed era fitness would always less genome length would become constant value second experiment ran model children born order see whether trends figure would continue longer time scale ran model times averaged results across runs first experiment population averages fitness genome length mutation rate fitness increase calculated time new child born second experiment increase speed model population averages calculated time children born figure shows results second experiment figure shows trends figure continue spite much longer time scale exception mutation rate quickly falls initial value hover indication mutation rate zero however since model stochastic always small probability mutation rate could zero figure fitness increase calculated average fitness population end era minus average fitness population start era unbounded evolutionary versatility figure experiment longer run length four plots show fitness genome length mutation rate fitness increase functions number children born first experiment target fitness function changes time one hundred children born values averages whole population ten separate runs baseline configuration values calculated ten thousand children born fitness increase calculated era average fitness increase calculated births since births era eras sample births value plot fitness increase average eras runs values three plots fitness genome length mutation rate averages runs turney experiment static target experiment investigated behaviour model target static second experiment population averages fitness genome length mutation rate calculated every births ran model times averaged results across runs used baseline parameter settings except length set length set zero figure shows results runs runs mutation rate zero every member population long children born longest run lasted births shortest run lasted births average run lasted births comparison second experiment runs ran children sign mutation rate would ever reach zero experiments support claim model unbounded evolutionary versatility requires dynamic target following two experiments investigate amount change target needed ensure unbounded evolutionary versatility experiment varying rate change target fourth experiment rate change target varied run length constant remaining parameters set baseline values figure shows behaviour model averaged ten separate runs time birth last novel child time mutation rate becomes zero every member population around birth child target change rate quickly rose around child target change rate approached see first plot figure could past conjecture threshold target change rate approximately average time birth last novel child approaches infinity approaches infinity unbounded evolutionary versatility figure experiment static target three plots show fitness genome length mutation rate functions number children born since target static fitness increase undefined values averages whole population ten separate runs model values calculated ten thousand children born ten runs made way birth mutation rate zero went second plot figure average fitness population time birth child final fitness rose steadily target change rate increased third plot could rise significantly limit set length conjecture would rise turney figure experiment varying rate change target experiment target change varies value experiment value experiments target change qualitative change behaviour model threshold appears separate bounded evolutionary versatility experiment left vertical dotted line unbounded evolutionary versatility experiment right vertical dotted line values plots based ten separate runs model ity rises infinity average mutation rate population time birth child final mutation rate increased steadily target change rate increased even past threshold experiment suggests relatively high amount change required ensure unbounded evolutionary versatility evolutionary versatility increase without bound target changed every hundred children era target must change least less environmental change mutation rate eventually drops zero experiment varying length era fifth experiment length era varied constant remaining parameters set baseline values figure shows behaviour model averaged ten separate runs like experiment experiment supports hypothesis relatively high amount change required ensure evolutionary versatility increase without bound target changes era length era children mutation rate stay zero experiment varying tournament size sixth experiment tournament size varied baseline value larger tournaments mean competition become parent higher selective pressure constant remaining parameters set baseline values figure shows behaviour model averaged ten separate runs results suggest model display unbounded evolutionary versatility long tournament compared era target change rate tournament size behaviour model relatively robust respect model displays unbounded evolutionary versatility relatively wide range values turney figure experiment varying length era experiment era length varies value experiments length experiment length length qualitative change behaviour model threshold appears separate unbounded evolutionary versatility experiment left vertical dotted line bounded evolutionary versatility experiment right vertical dotted line values plots based ten separate runs model experiment varying population size seventh experiment size population varied baseline value population run constant remaining parameters set baseline values figure shows behaviour unbounded evolutionary versatility figure experiment varying tournament size experiment varies run length tournament size larger tournaments mean greater selective pressure results suggest unbounded evolutionary versatility long size greater see first two plots third plot final fitness average fitness population time birth child continues rise even tournament greater runs reach child mutation rate see second plot suggests advantage higher selective pressure beyond needed obtain unbounded evolutionary versatility model averaged ten separate runs population small model susceptible random variations turney figure experiment varying population size experiment varied baseline value population size population size plots suggest runs unbounded evolutionary versatility even population size individuals however appears model becomes less stable population size smaller populations risk mutation rate could fall zero random chance large population model tend behave way every time runs figure suggests model becomes unstable population size less although sharp boundary unlike experiment sharp boundary experiment sharp boundary unbounded evolutionary versatility experiment varying number bits encoding mutation rate final experiment number bits genome used encode mutation rate varied baseline value run length constant remaining parameters set line values figure shows behaviour model averaged ten separate runs figure experiment varying number bits encoding mutation rate experiment number bits genome used encode mutation rate varied model displays unbounded evolutionary versatility number bits mutation length less seems quantization effects make model unstable encoding short ideal mutation rate may lie zero smallest value encoded genetic algorithm forced set mutation rate zero even though less ideal turney results suggest model display unbounded evolutionary versatility greater code length less bits model becomes susceptible quantization errors example bits encode values ideal mutation rate genome may forced set mutation rate zero although value less would better implications model claim model shows trend towards increasing evolutionary versatility organismal evolution claim model supports idea certain conditions possible evolutionary versatility increase without bound model active selection increased evolutionary versatility selective force drives increase merely statistical artifact due bounded diffusion process model shows purely local selection process yield global trend model also shows environment must highly dynamic target fitness function must change significantly repeatedly sustain increasing evolutionary versatility environment sufficiently dynamic disruptive effects mutation outweigh beneficial effects selection drive mutation rates zero mutation rate zero throughout population genome length longer increase evolutionary versatility bounded length longest genome population believe fact trend towards increasing evolutionary versatility organismal evolution although model prove belief model suggests way test belief model predicts increasing evolutionary versatility accelerating pace evolution see fourth plots figures therefore predict find evidence unbounded evolutionary versatility erating pace evolution biological organisms difficult objectively verify claim pace evolution accelerating natural measure pace evolution historical frequency innovations analysis complicated several factors one confounding factor record recent past superior record distant past may give illusion innovations recent past distant past another confounding factor population growth may expect innovations recent history simply innovators third factor difficulty counting innovations need objective threshold importance innovations vast number trivial innovations ignored suggest tests avoid objections predict fossil record show decreasing recovery time major catastrophes mass extinction events ice ages meteorite impacts volcanic eruptions also predict decrease average lifetimes species recent species accelerating rate two tests involve counting frequency innovations makes relatively objective limitations future work several limitations work one limitation run model infinity prove empirically evolutionary versatility grow infinity conjecture baseline settings parameters table expected mean average evolutionary versatility model rise infinity rises infinity conjecture supported empirical evidence figure proven theoretical argument yet developed theoretical argument another limitation model abstractness sophisticated model would include mapping internal implicit fitness function instead current external explicit fitness function turney ping fitness function allow varying degrees dependence independence among dimensions traits characteristics along variation occur evolution possibility covariation coevolution multiple species relationships however point exercise make model abstract possible order identify minimum elements needed display unbounded evolutionary versatility abstractness model intended make clear susceptible analysis might seem conflict model free lunch theorems informally free lunch theorems show universal optimization algorithm optimal fitness landscapes example one free lunch theorem theorem shows two optimization algorithms average fitness obtained equals average fitness obtained average calculated possible fitness landscapes sampled uniform probability model reach infinite fitness levels fitness landscapes violate free lunch theorem since average fitness must also infinite problem free lunch theorems concerned fitness finite number iterations fitness infinite number iterations case infinite number children model presented intended new superior form optimization algorithm intent model show possible certain conditions evolutionary versatility increase without bound furthermore model intended show local selection case local certain period time drive global trend global across periods time towards increasing evolutionary versatility model universal display unbounded increase evolutionary versatility certain parameter settings certain fitness landscapes fitness landscape defined parameters general design unbounded evolutionary versatility model figure experiments show model appears display unbounded evolutionary versatility baseline fitness landscape fitness landscape defined parameter settings table experiments show neighbouring fitness landscapes evolutionary versatility bounded experiments explored infinitely many possible fitness landscapes fitness landscapes explored appeared display unbounded evolutionary versatility conclusions paper introduces simple model unbounded evolutionary versatility model primarily intended address claim natural selection produce trend purely local process model shows local selection produce global trend towards increasing evolutionary versatility model suggests trend continue without bound sufficient ongoing change environment evolutionary versatility increase without bound must possible lengths genomes increase bound length genomes must bound evolutionary versatility model unbounded evolutionary versatility must therefore allow mutations occasionally change length genome seems possible genomes reach certain length benefit might obtained greater length would countered damage mutation useful genes found far point evolutionary versatility might stop increasing address issue model allows mutation rate adapt experiments show indeed little change environment damage mutation greater benefit mutation mutation rate goes zero evolutionary versatility stops increasing however sufficient change environment appears mutation rate reaches stable value evolutionary versatility continues increase indefinitely turney perhaps interesting observation fitness increase era grows time see fourth plots figures increasing evolutionary versatility leads accelerating pace evolution one interesting questions model whether plausible highly abstract model evolution life earth one test plausibility look signs pace organismal evolution accelerating example fossil record show decreasing recovery time major catastrophes decrease average lifetimes species evidence pace evolution accelerating evolutionary versatility may better able account seven live hypotheses clear hypotheses could used explain acceleration although seems natural consequence increasing evolutionary versatility acknowledgments thanks reviewers helpful comments earlier version paper thanks dan mcshea many constructive criticisms general encouragement references aboitiz lineage selection capacity evolve medical hypotheses altenberg evolution evolvability genetic programming advances genetic programming kinnear mit press anderson learning evolution quantitative genetics approach journal theoretical biology ayala concept biological progress studies philosophy biology ayala new york macmillan ayala progress defined biological concept evolutionary progress nitecki chicago university chicago press unbounded evolutionary versatility genetic algorithms varela bourgine eds towards practice autonomous systems mit press baldwin new factor evolution american naturalist bedau seymour adaptation mutation rates simple model evolution complexity international blickle thiele comparison selection schemes used genetic algorithms technical report gloriastrasse zurich swiss federal institute technology eth zurich computer engineering communications networks lab tik blickle thiele mathematical analysis tournament selection proceedings sixth international conference genetic algorithms eshelman morgan kaufmann san mateo davis adapting operator probabilities genetic search proceedings third international conference genetic algorithms morgan kaufmann san mateo dawkins evolution evolvability artificial life langton dawkins climbing mount improbable new york norton fogel fogel atmar programming chen proceedings asilomar conference signals systems computers california maple press gilinsky good probabilities origination persistence extinction families marine invertebrate life paleobiology gould trends changes variance new slant progress directionality evolution journal paleontology gould full house spread excellence plato darwin new turney york harmony hinton nowlan learning guide evolution complex systems lewin red queen runs trouble science mcshea metazoan complexity evolution trend evolution mcshea possible trends organismal evolution eight live hypotheses annual review ecology systematics nitecki evolutionary progress edited collection chicago university chicago press raup taxonomic survivorship curves van valen law paleobiology riedl approach phenomena quarterly review biology riedl order living organisms systems analysis evolution translated jefferies translation die ordnung des lebendigen new york wiley ruse monad man concept progress evolutionary biology massachusetts harvard university press simon architecture complexity proceedings american philosophical society syswerda uniform crossover genetic algorithms proceedings third international conference genetic algorithms california morgan kaufmann syswerda study reproduction generational genetic algorithms foundations genetic algorithms rawlins editor morgan kaufmann unbounded evolutionary versatility turney architecture complexity new blueprint synthese turney shift bias lessons baldwin effect evolutionary computation turney increasing evolvability considered trend evolution proceedings genetic evolutionary computation conference workshop program workshop evolvability van valen new evolutionary law evolutionary theory vermeij adaptive versatility skeleton construction american naturalist vermeij gastropod evolution morphological diversity relation shell geometry journal zoology vermeij biological versatility earth history proceedings national academy sciences united states america vermeij adaptation versatility evolution systematic zoology wagner altenberg complex adaptations evolution evolvability evolution whitley kauth genitor different genetic algorithm proceedings rocky mountain conference artificial intelligence denver whitley genitor algorithm selective pressure proceedings third international conference genetic algorithms california morgan kaufmann whitley dominic das anderson genetic reinforcement learning neurocontrol problems machine learning turney williams adaptation natural selection new jersey princeton university press wolpert macready free lunch theorems optimization ieee transactions evolutionary computation

apr crafting adversarial input sequences recurrent neural networks nicolas papernot patrick mcdaniel ananthram swami richard harang pennsylvania state university university park mcdaniel united states army research laboratory adelphi learning models frequently used solve complex security problems well make decisions sensitive situations like guiding autonomous vehicles predicting financial market behaviors previous efforts shown numerous machine learning models vulnerable adversarial manipulations inputs taking form adversarial samples inputs crafted adding carefully selected perturbations legitimate inputs force machine learning model misbehave instance outputting wrong class machine learning task interest classification fact best knowledge previous work adversarial samples crafting neural network considered models used solve classification tasks frequently computer vision applications paper contribute field adversarial machine learning investigating adversarial input sequences recurrent neural networks processing sequential data show classes algorithms introduced previously craft adversarial samples misclassified neural networks adapted recurrent neural networks experiment show adversaries craft adversarial sequences misleading categorical sequential recurrent neural networks ntroduction efforts machine learning security communities uncovered vulnerability machine learning models adversarial manipulations inputs specifically approximations made training algorithms well underlying linearity numerous machine learning models including neural networks allow adversaries compromise integrity output using crafted perturbations perturbations carefully selected often indistinguishable time yield important changes output machine learning model solutions making models robust adversarial perturbations proposed literature models remain largely vulnerable existence threat vector puts machine learning models risk deployed potentially adversarial settings taxonomy attacks deep learning classifiers introduced select perturbations changing class label assigned neural network classifier class different legitimate class specific target class chosen adversary two approaches followed fast gradient sign method forward derivative method approaches estimate model sensitivity differentiating functions defined architecture parameters approaches differ perturbation selection techniques primarily evaluated models trained solve image classification tasks tasks simplify adversarial sample crafting model inputs use linear differentiable images encoded numerical vectors thus perturbations found model input easily transposed corresponding raw image contrary study adversarial samples models mapping sequential inputs nondifferentiable manner categorical sequential outputs recurrent neural networks rnns machine learning models adapted neural networks suitable learning mappings sequential inputs outputs instance powerful models sentiment analysis serve intelligence community performing analysis communications terrorist networks furthermore rnns used malware classification predicting sequential data also finds applications stock analysis financial market trend prediction unlike neural networks rnns capable handling sequential data often rnns introduce cycles computational graph efficiently model influence time presence cyclical computations potentially presents challenges applicability existing adversarial sample algorithms based model differentiation cycles prevent computing gradients directly applying chain rule issue left future work previous work precisely question investigate paper study particular instance adversarial refer adversarial mislead rnns producing erroneous outputs show forward derivative adapted neural networks cyclical computational graphs using technique named computational graph unfolding experiment demonstrate using forward derivative model jacobian adversary produce adversarial input sequences manipulating sequences output sequential rnn classification predictions made categorical rnn manipulations require adversary alter part model training process data fact perturbations instantly manipulate model output test time trained deployed make predictions new inputs contributions paper following formalize adversarial sample optimization problem context sequential data adapt crafting algorithms using forward derivative specificities rnns includes showing compute forward derivative cyclical computational graphs investigate transposing adversarial perturbations model inputs raw inputs evaluate performance technique using rnns making categorical sequential predictions average changing words word movie review sufficient categorical rnn make wrong class predictions performing sentiment analysis reviews also show sequences crafted using jacobian perturb sequential outputs second rnn paper intended presentation initial efforts line research include discussion future work relevant advancement research topic bout ecurrent eural etworks facilitate discussion adversarial sample crafting techniques section iii provide overview neural networks specifically recurrent neural networks along examples machine learning applications tasks solved using models machine learning machine learning provides automated methods analysis large sets data tasks solved machine learning generally divided three broad types supervised learning unsupervised learning reinforcement learning method designed learn mapping association inputs outputs instantiation supervised learning settings output data nature characterizes varying problems like classification pattern recognition regression method given unlabeled inputs machine learning task falls unsupervised learning common applications include dimensionality reduction network finally reinforcement learning considers agents maximizing reward taking actions environment interested readers referred presentation machine learning neural networks neural networks class machine learning models useful across tasks supervised unsupervised reinforcement learning made computing activation functions inputs order produce outputs typically processed neurons computation performed neuron thus takes following formal form parameter referred weight vector whose role detailed neural network neurons typically grouped layers network always least two layers corresponding input output model one intermediate hidden layers inserted input output layers neuron link fig recurrent neural network sequential input processed time step value hidden neuron evaluates state time step adding result multiplying current input value weight result multiplying previous state weight bias finally applying hyperbolic tangent output multiplies hidden neuron state weight adds bias network possesses one hidden layer referred shallow neural network otherwise network said deep common interpretation hidden layers extract successive hierarchical representations input required produce output neural networks principally parameterized weights placed links neurons weight parameters hold model knowledge values learned training considering collections inputs corresponding labels context supervised learning recurrent neural networks recurrent neural networks rnns variant vanilla networks described adapted modeling sequential data without specificities vanilla neural networks offer scalability required modeling large sequential data specificities recurrent neural networks include importantly introduction cycles model computational graph results form parameter sharing responsible scalability large sequences words addition links neurons different layers recurrent neural networks allow links neurons layer results presence cycles network architecture cycles allow model share parameters links connecting neuron outputs throughout successive values given input value different time steps case rnns equation thus becomes following notation introduced neuron named time step input sequence note cycle allows activation function take account state neuron previous time step thus state used transfer aspects previous sequence time steps upcoming time steps example recurrent neural network throughout sections iii illustrated figure neuron iii rafting dversarial equences following formalize adversarial sequences build techniques designed craft adversarial samples neural network classifiers adapt problem crafting adversarial sequences recurrent neural networks adversarial samples sequences adversarial samples context machine learning classifier adversarial samples crafted legitimate sample selecting norm appropriate input results altered sample misclassified class different legitimate class adversarial target class chosen class class different legitimate class thus adversarial sample solves following optimization problem first formalized min case adversary interested target class different legitimate class finding exact solution problem always possible especially case deep neural networks due nonlinearity thus previous efforts introduced discussed find approximative solutions adversarial sequences consider rnns processing sequential data input output data sequences case one experiments equation hold output data categorical thus adversarial sample optimization problems needs generalized specify adversarial target vector matched closely possible model processing adversarial input stated min output sequence desired adversary norm appropriate compare vectors rnn input output domain acceptable error model output adversarial sequence adversarial target example norm compare input sequences number sequence steps perturbed detail approximative problem found computing model jacobian using fast gradient sign method fig unfolded recurrent neural network neural network identical one depicted figure exception recurrence cycle unfolded biases omitted clarity illustration cost function associated model parameter controlling perturbation magnitude increasing input variation parameter increases likeliness misclassified albeit simultaneously increases perturbation magnitude therefore distinguishability long model differentiable fast gradient sign method still one inserts recurrent connections computational graph model fact goodfellow used method craft adversarial samples deep boltzmann machine uses recurrent connections classify inputs fixed size adversarial sample crafting method described equation thus used recurrent neural networks long loss differentiable inputs however also interested solving equation model processing input sequence steps using forward derivative forward derivative introduced alternative means craft adversarial samples method design considers threat model adversaries interested misclassifying samples chosen adversarial targets nevertheless technique also used achieve weaker goal misclassification target class different original sample class forward derivative defined model jacobian fast gradient sign method approximates problem equation linearizing model cost function around input selecting perturbation using gradient cost function respect input gradient computed following steps typically used training instead computing gradients respect model parameters intent reducing prediction error normally case training gradients computed respect input yields following formulation adversarial samples sgn link ith component input component output precisely evaluates sensitivity output component input component gives quantified understanding input variations modify output value component pair leverage technique known computational graph unfolding compute forward derivative presence cycles case rnns looking back equation one observe compute neuronal state time step recursively apply formula decrementing time step yields following unfolded version equation thus unfolding recurrent components computational graph recurrent neural network made acyclic instance figure draws unfolded neural network corresponding rnn originally depicted figure using unfolded version graph compute recurrent neural network jacobian defined following matrix step input sequence step output sequence input output sequences length using definition unfolding recursively time step hidden neuron state reach write evaluated using demonstrated context neural networks craft adversarial sequences two types rnn forward derivative previous work introduced adversarial saliency maps select perturbations using forward derivative context classification neural networks due space constraints include overview saliency maps study binary classifier section thus simplifying perturbation selection indeed perturbing input reduce one class probability necessarily increases probability given second class thus adversarial sequences crafted solely considering jacobian column corresponding one output components consider crafting adversarial sequences models outputting sequences craft adversarial sequence legitimate input sequence need select perturbation within acceptable margin desired adversarial output hence approximatively solving equation consider output sequence jacobian column corresponds step output sequence identify subset input components high absolute values column comparably small absolute values columns jacobian matrix components large impact rnn output step limited impact output steps thus modify components direction indicated sgn output sequence step approach desired adversarial output component method evaluated second part section valuation craft adversarial sequences categorical sequential rnns categorical rnn performs sentiment analysis classify movie reviews lieu intelligence reports positive negative mislead classifier altering words review second rnn trained learn mapping synthetic input output sequences attack alters model output identifying contribution input sequence step recurrent neural networks categorical output rnn movie review classifier takes input sequence performs sentiment analysis classify negative outputs positive outputs able achieve error rate training set changing average words reviews average word long experimental setup experiment long short term memory lstm rnn architecture lstms prevent exploding vanishing gradients training introducing memory cell gives flexibility selfrecurrent connections compared vanilla rnn allowing remember forget previous states rnn composed four lstm mean pooling shown figure mean pooling layer averages representations extracted memory cells lstm layer softmax formats output probability vectors softmax layer mean pooling layer lst movie terrific lstm layer lst lst lst embeddings integers words fig lst rnn recurrent model classifies movie reviews rnn implemented python theano facilitate symbolic gradient computations train using little training testing reviews reviews sequences words dictionary includes words frequently used reviews special keyword words dictionary maps words integer keys convert integer sequences matrices row encodes word set word embeddings matrices used input rnn described trained architecture achieves accuracies respectively training testing tests jacobian tensor matrix word embedding vector also vector three dimensions indicated consider softmax layer instead output probabilities compute jacobian gradient computations stable results maximum logit index corresponds class assigned sentence experimental setup sequential rnn described figure train set synthetically generated input output sequence pairs inputs values per step outputs values per step sequences steps long values randomly sampled standard normal distribution inputs outputs random samples altered introduce strong correlation given step output sequence previous last previous step input sequence model trained epochs learning rate cost mean squared error model predictions targets figure shows example input sequence output sequence predicted input variable values adversarial sequences demonstrate adversaries craft adversarial sequences sentences misclassified model thus need identify dictionary words use modify sentence way switches predicted class positive negative turn attack described section iii based computing model jacobian evaluate jacobian respect embedding inputs gives precise mapping changes made word embeddings variations output pooling word input sequence sgn arg gives direction perturb word embedding components order reduce probability assigned current class thus change class assigned sentence unlike previous efforts describing adversarial samples context computer vision face difficulty set legitimate word embeddings finite thus set word embedding coordinates real value adversarial sequence overcome difficulty follow procedure detailed algorithm find word dictionary sign difference embeddings original input word closest sgn embedding takes direction closest one indicated jacobian impactful model prediction iteratively applying heuristic sequence words eventually find adversarial input sequence misclassified model achieved error rate training set changing average words training reviews reviews average word long instance change review rent one even dollar rental following misclassified adversarial sequence excellent rent one even dollar rental algorithm inserting words highly positive connotations input sequence mislead rnn model recurrent neural networks sequential output rnn predicts output sequences input sequences although use symthetic data models instance applied forecast financial market trends output variable values algorithm adversarial sequence crafting lstm model algorithm iteratively modifies words input sentence produce adversarial sequence misclassified lstm architecture illustrated figure require select word sequence arg sgn end return input sequence output sequence time step fig example input output sequences experimental setup input graph solid lines indicate legitimate input sequence dashed lines indicate crafted adversarial sequence output solid lines indicate training target output dotted lines indicated model predictions dashed lines prediction model made adversarial sequence adversarial sequences compute model jacobian quantifies contributions input sequence step output sequence craft adversarial sequences instance interested altering subset output steps simply alter subset input steps high jacobian values low jacobian values figure shows example inputs outputs solid lines correspond legitimate input sequence target output sequence small dotted lines output show model predictions closely matches target adversarial crafted modify value red step value blue step making important changes input sequence value black step value red step due space constraints completing qualitative results detailed quantitative evaluation left future work iscussion elated ork work part active line studies behavior machine learning models trained deployed adversarial settings theoretical approach described section iii applicable neural network model recurrent components independent output data type experiments performed lstm architecture categorical outputs vanilla rnn model sequential outputs preliminary validation approach albeit necessitating additional validation rnn model variants well datasets future work also address grammar adversarial sequences improve semantic meaning make sure indistinguishable humans paper considered threat model describing adversaries capability accessing model computational values parameters learned training realistic environments always possible adversaries without type access system hosting machine learning model acquire knowledge parameters limitation addressed context deep neural network classifiers authors introduced attack adversaries targeting classifier oracles targeted model queried labels inputs adversary choice used substitute model approximate decision boundaries unknown targeted model crafted adversarial samples using substitute samples also frequently misclassified targeted model due property known adversarial sample transferability samples crafted misclassified given model often also misclassified different models however adapting attack method rnns requires additional research efforts left future work onclusions models learned using rnns immune vulnerabilities exploited adversary carefully selecting perturbations model inputs uncovered context networks used computer vision classification paper formalized problem crafting adversarial sequences manipulating output rnn models demonstrated techniques previously introduced craft adversarial samples misclassified neural network classifiers adapted produce sequential adversarial inputs notably using computational graph unfolding experiment validated approach crafting adversarial samples evading models making classification predictions predictions future work investigate adversarial sequences different data types shown experiments switching computer vision natural language processing applications introduced difficulties unlike previous work consider data attack performing attacks weaker threat models also contribute better understanding vulnerabilities lead defenses acknowledgments research sponsored army research laboratory arl accomplished cooperative agreement number arl cyber security cra views conclusions contained document authors interpreted representing official policies either expressed implied arl government government authorized reproduce distribute reprints government purposes notwithstanding copyright notation eferences szegedy zaremba sutskever bruna erhan goodfellow fergus intriguing properties neural networks proceedings international conference learning representations computational biological learning society goodfellow explaining harnessing adversarial examples proceedings international conference learning representations computational biological learning society papernot limitations deep learning adversarial settings proceedings ieee european symposium security privacy ieee papernot mcdaniel goodfellow jha practical 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learning freedman statistical models theory practice cambridge university press goodfellow deep boltzmann machines advances neural information processing systems mozer focused algorithm temporal pattern recognition complex systems vol werbos generalization backpropagation application recurrent gas market model neural networks vol hochreiter schmidhuber long memory neural computation vol bergstra breuleux bastien lamblin theano cpu gpu math expression compiler proceedings python scientific computing conference scipy vol austin maas learning word vectors sentiment analysis proceedings annual meeting association computational linguistics human language technologies portland oregon usa hinton learning distributed representations concepts proceedings eighth annual conference cognitive science society vol amherst mesnil deng bengio investigation architectures learning methods spoken language interspeech barreno nelson machine learning secure proceedings acm symposium information computer 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characterization asymptotic dimension growth jan goulnara arzhantseva graham niblo nick wright jiawen zhang abstract give characterization asymptotic dimension growth apply cat cube complexes finite dimension giving alternative proof wright result finite asymptotic dimension also apply new characterization geodesic coarse median spaces finite rank establish subexponential asymptotic dimension growth strengthens recent result wright introduction concept asymptotic dimension first introduced gromov coarse analogue classical topological covering dimension started attract much attention proved novikov higher signature conjecture holds groups finite asymptotic dimension fad lot groups spaces known finite asymptotic dimension among instance finitely generated abelian groups free groups finite rank gromov hyperbolic groups mapping class groups cat cube complexes finite dimension see excellent survey results recently behrstock hagen sisto introduced powerful new notion hierarchically hyperbolic spaces showed finite asymptotic dimension recovering number results including notably mapping class groups number cat cube complexes hand many groups spaces infinite asymptotic dimension examples wreath product grigorchuk group thompson groups etc generalizing fad dranishnikov defined asymptotic dimension growth space asymptotic dimension growth function eventually constant space fad dranishnikov showed wreath product finitely generated nilpotent group finitely generated fad group polynomial asymptotic dimension growth also showed polynomial asymptotic dimension growth implies property hence coarse conjecture provided space bounded geometry later ozawa extended result spaces subexponential growth see also bell analyzed asymptotic dimension function affected various constructions mathematics subject classification key words phrases asymptotic dimension growth cat cube complex coarse median space mapping class group partially supported european research council erc grant goulnara arzhantseva fellowship trust royal society goulnara arzhantseva graham niblo nick wright jiawen zhang paper give alternative characterization asymptotic dimension growth function inspired brown ozawa proof property gromov hyperbolic groups theorem turn inspired use study two notable examples cat cube complexes finite dimension coarse median spaces finite rank techniques used study examples developments used wright establish property uniformly locally finite coarse median spaces finite rank byproduct obtain new proof finite asymptotic dimension cat cube complexes allows one explicitly construct required controlled covers compares wright original proof discussed cat cube complexes nice class curved spaces first studied gromov gave purely combinatorial condition recognizing curvature cube complexes many groups act properly cat cube complexes instance artin groups many small cancellation groups thompson groups admit actions makes possible deduce properties groups corresponding properties cat cube complexes wright proved asymptotic dimension cat cube complex bounded dimension proved constructing family cobounded maps cat cube complexes dimension indexed use characterization finite asymptotic dimension give direct proof result namely construct uniformly bounded covers suitable properties explicit proof loses however sharp bound asymptotic dimension thus give alternative proof following variant wright theorem theorem let cat cube complex finite dimension finite asymptotic dimension key point approach analyse normal cube path distance cube complex introduced niblo reeves consider ball respect normal cube path distance rather ordinary edgepath distance decompose ball intervals use induction dimension order construct separated net satisfying suitable consistency property process give detailed analysis normal balls normal spheres balls spheres respect normal cube path distance see section details second application coarse median spaces introduced bowditch coarse variant classical median spaces notion coarse median group leads unified viewpoint several interesting classes groups including gromov hyperbolic groups mapping class groups cat cubical groups bowditch showed hyperbolic spaces exactly coarse median spaces rank mapping class groups examples coarse characterization asymptotic dimension growth median spaces finite rank also established interesting properties coarse median spaces rapid decay property quadratic dehn function etc intuitively coarse median space metric space equipped ternary operator called coarse median every finite subset approximated finite median algebra approximations coarse median approximated actual median distortion controlled metric extends gromov observation space finite subsets well approximated finite trees recently wright proved coarse median space finite rank exponential volume growth property following proof using characterization asymptotic dimension growth obtain following result theorem let geodesic coarse median space finite rank exponential volume growth subexponential asymptotic dimension growth hierarchically hyperbolic spaces examples coarse median spaces see hence theorem broader scope though weaker conclusion finite asymptotic dimension result proven expect following general result conjecture every geodesic coarse median space finite rank finite asymptotic dimension result ozawa subexponential asymptotic dimension growth implies property thus theorem strengthens result paper organized follows section give preliminaries asymptotic dimension growth cat cube complexes coarse median spaces section provide characterization asymptotic dimension growth function special case give characterization finite asymptotic dimension sections deal cat cube complexes section study normal balls spheres essential approach prove theorem section section deals coarse median case prove theorem preliminaries asymptotic dimension notion asymptotic dimension first introduced gromov coarse analogue classical lebesgue topological covering dimension see also let metric space call family subsets inf write joint goulnara arzhantseva graham niblo nick wright jiawen zhang union family said uniformly bounded mesh sup diam finite let cover define denoted minimal integer ball intersects elements usual denotes multiplicity cover maximal number elements intersection number called lebesgue number every subset diameter exists element lebesgue number cover defined infimum lebesgue numbers definition say asymptotic dimension metric space exceed write asdim every space covered subspaces decomposed uniformly bounded subspaces disjoint xij sup diam xij say asdim asdim asdim less basic examples spaces groups finite asymptotic dimension example asdim group integers gromov spaces word hyperbolic groups finite asymptotic dimension definition easy see asymptotic dimension subspace ambient space equivalent definitions asymptotic dimension list one later use guide reader others proposition let metric space asdim exists uniformly bounded cover asymptotic dimension growth llet consider direct sum infinitely many copies integers since group contained mentioned results infinite asymptotic dimension order deal dranishnikov studied following concept generalization property finite asymptotic dimension definition let metric space define function adx min cover called asymptotic dimension function characterization asymptotic dimension growth note adx monotonic lim adx asdim like case volume function growth type asymptotic dimension function essential function recall write exists write clear equivalence relation define growth type class define asymptotic dimension growth growth type adx result bell dranishnikov growth type asymptotic dimension function invariant proposition let two discrete metric spaces bounded geometry adx ady particular asymptotic dimension growth finitely generated groups give alternative equivalent definition asymptotic dimension growth used characterization lemma let metric space define adx min cover proof given suppose cover define inner denotes usual set define since still cover definition obvious adx yields conversely suppose cover consider lebesgue number less easy show implies adx definition preceding lemma use either adx asymptotic dimension function recall exists polynomial subexponential function adx said polynomial subexponential asymptotic dimension growth dranishnikov shown polynomial asymptotic dimension growth implies property gave class groups property proposition let finitely generated nilpotent group finitely generated group finite asymptotic dimension wreath product goulnara arzhantseva graham niblo nick wright jiawen zhang polynomial asymptotic dimension growth particular polynomial asymptotic dimension growth cat cube complexes recall basic notions results structure cat cube complexes omit details proofs direct readers information cube complex polyhedral complex cell isometric euclidean cube gluing maps isometries dimension complex maximum dimensions cubes cube complex associate intrinsic dint minimal cube embeds isometrically finite dimension dint complete geodesic metric see general discussion polyhedral complex associated intrinsic metric also another metric associated let graph vertex set equip metric minimal number edges path connecting two given vertices clearly connected geodesic metric interval defined consists points geodesic geodesic metric space cat geodesic triangles slimmer comparative triangle euclidean space cube complex dint gromov given combinatorial characterization cat condition cat simply connected link vertex flag complex see also another characterization cat condition obtained chepoi see also cube complex cat intersection consists single point called median case call graph median graph equipped ternary operator indeed median algebra particular following equations hold permutation obviously lemma let implies proof since implies lemma proof chepoi result median graph hence weakly modular see implies characterization asymptotic dimension growth cat cubical complex equipped set hyperplanes hyperplane intersect divides space two halfspaces given two hyperplanes four possible intersections halfspaces nonempty say crosses denoted occurs cross common cube also denoted furthermore given maximal collection pairwise intersecting hyperplanes exists unique cube cross thus dimension maximal number pairwise intersecting hyperplanes also define intervals language hyperplanes consists points lie halfspaces containing call subset convex obviously halfspaces convex since geodesic crosses hyperplane also implies hyperplane separates coarse median spaces according gromov hyperbolic spaces considered locally coarse version trees sense every finite subset approximated finite tree controlled way one wants approximate space locally finite median algebras graphs would turn definition coarse median spaces introduced bowditch see details definition let metric space ternary operation say coarse median space coarse median following conditions hold exist constants exists function following property finite subset exists finite median algebra maps refer parameters furthermore exists always choose median algebra condition rank say coarse rank finitely generated group said coarse median cayley graph coarse median note definition coarse median group required equivariant group action remark according bowditch without loss generality may always assume satisfies median axioms goulnara arzhantseva graham niblo nick wright jiawen zhang large class groups spaces shown coarse median including gromov hyperbolic groups artin groups mapping class groups cat cube complexes etc bowditch proved coarse median groups property rapid decay quadratic dehn function etc yielded unified way prove properties groups recently wright proved coarse median spaces finite rank exponential volume growth property characterization asymptotic dimension growth section establish characterization asymptotic dimension growth obtain several interesting consequences main result instance get characterization group finite asymptotic dimension theorem let discrete metric space function following equivalent adx exists function growth type assign subset satisfying iii proof lemma assume exists function exists uniformly bounded cover suppose choose define let check four properties condition assume mesh mesh mesh words mesh immediate definition sets iii characterization asymptotic dimension growth assume implies hand suppose assume implies also define let define since take cover since know implies mesh finally let analyse consider take assumptions condition finally lemma adx taking preceding theorem constant function obtain characterization finite asymptotic dimension corollary let discrete metric space following equivalent asdim assign subset satisfying iii turn case graph obtain characterization finite asymptotic dimension easier check corollary given graph vertices edges equipped length metric following equivalent asdim goulnara arzhantseva graham niblo nick wright jiawen zhang assign subset satisfying iii connected edge remark distinction two corollaries corollary assumption required endpoints edge rather arbitrary pair points corollary point preceding corollaries generalized case arbitrary asymptotic dimension growth use generalization omit proof corollary implied directly corollary focus following proof proposition let define define since take cover since know implies mesh finally let analyse consider take definition length metric know exists sequence vertices hypothesis know implies normal cube path normal distance next two sections focus cat cube complexes prove theorem prove constructing uniformly bounded cover suitable properties construction relies deeply analysis normal balls spheres give section normal cube paths introduced niblo reeves play key role construction cover determine distance function vertices balls spheres defined terms distance essential proof theorem throughout section fix cat cube complex fixed vertex graph vertex set edge set give edge metric restriction metric normal cube paths given cube denote union cubes contain subface characterization asymptotic dimension growth definition let sequence cubes cube dimension least consists single point denoted call cube path unique cube minimal dimension containing diagonally opposite vertices define vertex diagonally opposite vertex diagonally opposite vertices called vertices cube path say cube path length cube path number cubes sequence cube path called normal normal cube paths cat cube complexes behave like geodesics trees precisely existence uniqueness normal cube paths connecting pair vertices established see also proposition two vertices exists unique normal cube path note order important since general normal cube paths reversible proposition intersection normal cube path hyperplane connected words normal cube path crosses hyperplane proposition let two normal cube paths let vertices normal cube paths omit proofs three propositions readers find original paper however let recall construction normal cube path follows consider hyperplanes separating adjacent key fact hyperplanes cross unique cube adjacent lying interval cube defined first cube normal cube path one proceeds inductively construct required normal cube path also need following lemma abstracted lemma let normal cube path hyperplane hyperplane intersect proof otherwise lemma know exists cube face moreover contains edge since contains contradiction definition normal cube path two vertices consider hyperplanes separating partial order inclusion explicitly let set hyperplanes separating let halfspace containing define note definition depends goulnara arzhantseva graham niblo nick wright jiawen zhang vertices choose may change circumstances still write abbreviation avoid ambiguity point vertices necessary write mean strict containment lemma intersect proof need show necessity let normal cube path assume since intersect assume obviously since proposition since intersect implies combining two lemmas following result existence chains proposition let normal cube path hyperplane exists chain hyperplanes proof lemma exists hyperplane intersect define inductively define sequence hyperplanes required conclusion follows lemma finally give lemma used proof consistency part main theorem lemma let let vertex normal cube path proof otherwise construction normal cube path know also vertex normal cube path since words contradiction assumption normal metric define new metric using normal cube paths definition define dnor length normal cube path call dnor normal metric one needs verify dnor indeed metric easy see dnor dnor note normal cube path one general symmetric relation obvious order show symmetric relation triangle inequality give following characterization lemma let relation defined dnor sup characterization asymptotic dimension growth proof suppose normal cube path dnor denote right hand side equality lemma chain proposition intersects one cube denoted obviously implies hand proposition chain hyperplanes implies proposition dnor indeed metric proof lemma posets carry opposite orders one thus deduce dnor dnor symmetric difference operation inclusions order preserving therefore lemma dnor dnor dnor normal balls normal spheres recall two points interval words set vertices union edge geodesics subset called convex let closed ball edge metric centre radius generally convex example take however see normal metric balls convex precisely define normal ball centre radius bnor dnor normal sphere centre radius snor dnor lemma bnor convex proof given bnor geodesic bnor assume first vertex bnor implies dnor let vertex preceding dnor since dnor since exists unique hyperplane separating according lemma exists chain since every geodesic intersects hyperplane see example implies also chain contradiction dnor lemma since intersection two convex sets still convex following corollary corollary set bnor convex goulnara arzhantseva graham niblo nick wright jiawen zhang well known convex subset cat cube complex point unique point closest see example statement true intrinsic cat metric cube complex edge metric vertex set similar statement normal distance proposition exists unique point bnor bnor point characterized max bnor furthermore dnor snor implies also unique point snor max snor proof exist bnor attains maximum consider median corollary bnor contradiction corollary bnor conversely bnor let corollary bnor implies choice lemma satisfying dnor bnor take geodesic let vertices since dnor implies since definition know bnor bnor however since dnor dnor dnor dnor contradiction use proposition flexibly give another characterization also viewed alternative definition rest subsection fix dnor normal cube path proposition let vertex normal cube path provided proposition prove result let focus subsets recall endowed relation defined prior lemma definition subset called closed characterization asymptotic dimension growth lemma let vertex normal cube path maximal following sense closed contains chains lengths proof proceed induction suppose lemma holds let vertex normal cube path given closed containing chains lengths maximal chain closed set contains chains lengths induction contained similarly implies proposition closeness get chain length greater contradiction proof proposition proposition bnor implies however closed contains chains lengths according lemma lemma implies implies finally characterize points lie intersection snor used next subsection decompose snor union intervals let cube normal cube path vertex cube path let set hyperplanes intersecting proposition following equivalent snor crosses last cube normal cube path separates proof proposition since dnor lemma maximum length chains take chain obviously intersects different cubes implies separates since separates must cross cube normal cube path since know chain also chain cross first cubes normal cube path cross last cube dnor however implies lemma dnor contradiction immediate lemma goulnara arzhantseva graham niblo nick wright jiawen zhang another description implied proposition directly lemma maximal length chains decomposition snor want decompose set snor proceed induction dimension proof theorem throughout subsection fix dnor let defined proposition end preceding subsection defined set hyperplanes intersecting normal cube path decompose union intervals dimensions lower number intervals controlled dimension make possible induction dimension definition define snor separates proposition immediately obtain following two lemmas lemma crosses last cube normal cube path separates lemma snor definition know bnor separates implies convex moreover show actually interval lemma let point minimising proof since convex hand let choice know implies proposition thus lemma proposition snor dim dim proof need show dim dim hyperplane crossing proposition dim dim give another characterization useful proof consistency condition theorem characterization asymptotic dimension growth lemma let closest point unique point bnor separates hyperplane separate proof since bnor separates hyperplane separates choose since separate however lemma contradiction remains show unique point satisfying conditions otherwise let another point satisfying hypothesis lemma let hyperplane separating assume obviously hypothesis separate well contradiction since separates cross implies lemma however lemma dnor dnor contradiction since dnor separates wright construction conclude section recent application normal cube paths invoked wright order provide new proof finite dimensional cat cube complexes property key proof construction family maps property interval neighbourhood endpoint interval maps push neighbourhood interval maps defined terms normal cube paths follows definition maps given define follows let vertex normal cube path dnor let dnor lemma let defined proof need show every halfspace containing contains also hyperplane one associated halfspaces say contains either former case need check case separates denote normal cube path denote vertices cube path shall argue hyperplane separating used within first steps cube path suppose cube cross hyperplane separating hence every hyperplane separates hyperplane separating necessarily separates hence crosses hyperplanes crossing contradicts maximality step normal cube path thus hyperplanes separating must crossed within first steps goulnara arzhantseva graham niblo nick wright jiawen zhang since vertex cube path hyperplanes separating must crossed thus actually also separates use remarkable properties maps construct sets defined characterization finite asymptotic dimension next section finite dimensional cat cube complexes throughout section fix cat cube complex finite dimension equipped basepoint make use characterization obtained corollary order prove theorem constructing sets corollary order prove finite asymptotic dimension need find constant assign subset satisfying iii define easy show satisfies iii satisfy need modification intuitively construct uniformly separated net precise require following lemma lemma exist two constants depending dimension subsets maps denotes power set satisfying postpone proof lemma first show use construct hence conclude proof theorem proof theorem let constants lemma let lemma know define thing left complete proof verify conditions corollary characterization asymptotic dimension growth definition know lemma know implies immediately definition iii lemma assume former let obviously use first part lemma second equation hand use first part lemma fourth equality know hence definition third part lemma last thing prove lemma use analysis section construct inductively recall section proposition snor dim dim order carry induction dimension require stronger version lemma flexible choice endpoints intervals explicitly lemma exist two constants depending dimension map satisfying obvious lemma implied lemma one needs take prove lemma proof lemma fix carry induction dim given dim define dnor goulnara arzhantseva graham niblo nick wright jiawen zhang since dim indeed isometric interval define follows consists single point distance function taking integer part obvious suppose dim defined map satisfying focus dim dnor proposition vxnl fxh snor vxnl farthest point hnl set hyperplanes crossing cube normal cube path also dim vxnl dim induction cxh vxnl pxh vxnl already defined define cxh vxnl let vertex normal cube path dnor implies vxnl snor define pxh vxnl hnl vxnl need verify requirements hold first suppose let hyperplane separating given snor proposition vxnl vertex normal cube path vnl vertex normal cube path due property proposition vxnl vnl proposition vnl snor snor vxnl characterization asymptotic dimension growth recall denotes set hyperplanes separating obviously implies hnl hnl hnl lemma lemma hnl proposition hand hnl lemma unique point bnor separates hyperplane separate implies since hnl hnl induction new base point vnl vnl since vxnl vnl vnl vxnl implies cxh vxnl vnl cxh since cxh vxnl vxnl cxh vxnl cxh vxnl vxnl cxh vxnl vxnl claim vxnl vnl indeed vxnl vnl holds naturally vxnl vnl lemma vxnl since vxnl vnl vxnl vnl vnl vxnl former hold since vnl vnl vxnl implies vnl vxnl vxnl vnl vxnl claim cxh vxnl cxh vxnl vnl cxh cxh vxnl cxh vxnl cxh cyh since snor one need show let vertex normal cube path analysis know hnl vxnl vnl vnl vxnl hnl hnl hnl vxnl hnl since vxnl vnl vnl goulnara arzhantseva graham niblo nick wright jiawen zhang first equation comes claim inductively know pxh vxnl definition pxh vxnl hnl vxnl pxh vxnl hnl vxnl hnl vnl second assume pxh vxnl hnl vxnl definition induction know since dim vxnl third consider suppose satisfying snor means exists hnl vnl let vxnl vxnl obviously vxnl vxnl induction cxh vxnl cxh vxnl exist values snor since exist hyperplanes vnl cxh vxnl take lemma holds constants coarse median spaces section discuss coarse median case prove theorem fix coarse median space geodesic metric coarse median parameters finite rank definitions notations section according remark also assume coarse median satisfies recall characterization asymptotic dimension growth theorem geodesic uniformly locally finite coarse median space finite rank exponential growth property result theorem says coarse median space subexponential asymptotic dimension growth thus combining ozawa result theorem yields strengthening theorem prove theorem use several notations lemmas use notation given coarse interval defined result bowditch exists constant depending parameter also recall median axiom holds coarse median case constant depending parameters actually take given denote need following lemmas lemma let coarse median space let lemma let geodesic coarse median space rank every exists exists lemma let coarse median space fix exist constants depending parameters coarse median structure following holds let satisfy satisfies proof theorem proof based construction used prove property readers convenience give sketch proof fact verify stronger conditions sets required apply theorem fix base point let constants lemma first apply lemma obtain sequence conclusion lemma holds furthermore choose inductively arrange sequence increasing goulnara arzhantseva graham niblo nick wright jiawen zhang fix lemma applied produces point define need verify sets satisfy condition statement proposition need show exists subexponential function satisfying iii construction iii hold naturally lemma know thing left find subexponential function condition holds following argument follows totally proof omit calculation readers turn original paper details take notation denote lemma one deduce since lemma implies point satisfies lemma implies consequently trt depends linearly proposition number possible points bounded polynomial depending uniform local finiteness since exponential growth follows crt constants take crt recall limit extend function setting completes 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mathematical society roller poc sets median algebras group actions extended study dunwoody construction sageev theorem southampton preprint archive sageev ends group pairs curved cube complexes proceedings london mathematical society smith asymptotic dimension first grigorchuk group infinity revista complutense wright coarse medians property wright finite asymptotic dimension cat cube complexes geometry topology novikov conjecture groups finite asymptotic dimension annals mathematics coarse conjecture spaces admit uniform embedding hilbert space inventiones mathematicae zeidler coarse median structures groups master thesis university vienna vienna austria wien mathematik wien austria address school mathematics university southampton highfield united kingdom address wright

normalizing flows riemannian manifolds nov mevlana gemici google deepmind mevlana danilo rezende google deepmind danilor shakir mohamed google deepmind shakir abstract consider problem density estimation riemannian manifolds density estimation manifolds many applications optics plasma physics appears often dealing angular variables used protein folding robot limbs general directional statistics spite multitude algorithms available density estimation euclidean spaces scale large normalizing flows kernel methods variational approximations methods immediately suitable density estimation general riemannian manifolds revisit techniques related homeomorphisms differential geometry projecting densities use generalize idea normalizing flows general riemannian manifolds resulting algorithm scalable simple implement suitable use automatic differentiation demonstrate concrete examples method recent years much interest applying variational inference techniques learning large scale probabilistic models various domains images text one main issues variational inference finding best approximation intractable posterior distribution interest searching class known probability distributions class approximations used often limited approximations implying solution ever able resemble true posterior distribution widely raised objection variational methods unlike mcmc true posterior distribution may recovered even asymptotic regime address problem recent work normalizing flows inverse autoregressive flows others referred collectively normalizing flows focused developing scalable methods constructing arbitrarily complex flexible approximate posteriors simple distributions using transformations parameterized neural networks gives models universal approximation capability asymptotic regime works distributions interest restricted defined high dimensional euclidean spaces many distributions defined special homeomorphisms euclidean spaces interest statistics beta dirichlet gaussian wrapped cauchy fisher find little applicability variational inference large scale probabilistic models due limitations related density complexity gradient computation many distributions unimodal generating complicated distributions would require creating mixture densities using auxiliary random variables mixture methods require knowledge tuning number mixture components necessary heavy computational burden gradient computation general quantile functions mode complexity increases linearly mixtures opposed exponential increase normalizing flows conditioning auxiliary variables hand constrains use created distribution due need integrating auxiliary factors certain scenarios methods computation gradients difficult due fact simulation random variables general reparameterized rejection sampling work present methods generalizes previous work improving variational inference using normalizing flows riemannian manifolds interest spheres tori product topologies like infinite cylinders figure left construction complex density first projecting manifold transforming density projecting back right illustration transformed densities corresponding uniform density sphere blue empirical density obtained monte carlo red analytical density equation green density computed ignoring intrinsic dimensionality special manifolds homeomorphic euclidean space corresponds dimensionality tangent space point homeomorphism continuous function topological spaces continuous inverse bijective bicontinuous maps point one space unique continuous manner example manifold unit surface unit ball embedded homeomorphic see figure normalizing flows main result differential geometry used computing density updates given represents relationship differentials infinitesimal volumes two equidimensional euclidean spaces using jacobian function transforms one space result applies transforms preserve dimensionality however transforms map embedded manifold intrinsic euclidean space preserve dimensionality points result become obsolete jacobian transforms rectangular infinitesimal cube maps infinitesimal parallelepiped manifold relation volumes given det metric induced embedding tangent space correct formula computing density becomes det det density update going manifold euclidian space given det det application method introduce inverse stereographic transform define maps bijective bicontinuous manner determinant metric associated transformation given det det using formulae left side figure map uniform density enrich density using normalizing flows map back onto obtain arbitrarily complex density original sphere qthe right side figure show density update based riemannian metric det red correct closely follows kernel density estimate based samples blue also show using generic volume transformation formulation dimensionality preserving transforms green leads erroneous density resemble empirical distributions samples transformation references rezende mohamed wierstra stochastic backpropagation approximate inference deep generative models icml kingma welling variational bayes iclr karol gregor ivo danihelka alex graves danilo jimenez rezende daan wierstra draw recurrent neural network image generation icml eslami nicolas heess theophane weber yuval tassa koray kavukcuoglu geoffrey hinton attend infer repeat fast scene understanding generative models arxiv preprint danilo jimenez rezende shakir mohamed ivo danihelka karol gregor daan wierstra generalization deep generative models icml matthew hoffman david blei chong wang john paisley stochastic variational inference journal machine learning research danilo jimenez rezende shakir mohamed variational inference normalizing flows arxiv preprint diederik kingma tim salimans max welling improving variational inference inverse autoregressive flow corr laurent dinh jascha samy bengio density estimation using real nvp tim salimans diederik kingma max welling markov chain monte carlo variational inference bridging gap francis bach david blei editors icml volume jmlr workshop conference proceedings pages arindam banerjee inderjit dhillon joydeep ghosh suvrit sra clustering unit hypersphere using von distributions mach learn december siddharth gopal yiming yang von clustering models tony jebara eric xing editors proceedings international conference machine learning pages jmlr workshop conference proceedings marco fraccaro ulrich paquet ole winther indexable probabilistic matrix factorization maximum inner product search proceedings thirtieth aaai conference artificial intelligence february phoenix arizona pages arindam banerjee inderjit dhillon joydeep ghosh suvrit sra generative clustering directional data proceedings ninth acm sigkdd international conference knowledge discovery data mining kdd pages new york usa acm alex graves stochastic backpropagation mixture density distributions corr lars maaloe casper kaae sonderby soren kaae sonderby ole winther auxiliary deep generative models corr scott linderman david blei christian naesseth francisco ruiz rejection sampling variational inference adi formula using matrix volume siam journal matrix analysis applications adi application matrix volume probability linear algebra applications marcel berger bernard gostiaux differential geometry manifolds curves surfaces manifolds curves surfaces volume springer science business media

sep annals statistics vol doi institute mathematical statistics estimation nonlinear regression harris recurrent markov degui dag jiti university york university bergen monash university paper study parametric nonlinear regression harris recurrent markov chain framework first consider nonlinear least squares estimators parameters homoskedastic case establish asymptotic theory proposed estimators results show convergence rates estimators rely properties nonlinear regression function also number regenerations harris recurrent markov chain furthermore discuss estimation parameter vector conditional volatility function apply results nonlinear regression processes derive asymptotic distribution theory comparable obtained park tribute paper process published heard professor peter hall one significant contributors areas nonlinear regression time series analysis sadly passed away fundamental work done professor peter hall area martingale theory represented book christopher heyde hall heyde martingale limit theory applications academic press enables authors paper using martingale theory important tool dealing different types estimation testing issues econometrics statistics related annals paper gao king ann statist theorem hall heyde plays essential role establishment important theorem short would like thank including paper dedicated issue honour professor peter hall fundamental contributions statistics theoretical econometrics received january revised august supported norwegian research council supported part australian research council discovery early career researcher award supported two australian research council discovery grants grant numbers ams subject classifications primary secondary key words phrases asymptotic distribution asymptotically homogeneous function recurrent markov chain harris recurrence integrable function least squares estimation nonlinear regression electronic reprint original article published institute mathematical statistics annals statistics vol reprint differs original pagination typographic detail gao phillips econometrica numerical studies including simulation empirical application provided examine finite sample performance proposed approaches results introduction paper consider parametric nonlinear regression model defined true value parameter vector assumed known throughout paper assume compact set lies interior standard assumption literature construct consistent estimator parameter vector derive asymptotic theory important issues modern statistics econometrics observations satisfy stationarity weak dependence conditions extensive literature theoretical analysis empirical application parametric nonlinear model extension see example jennrich malinvaud early references severini wong lai skouras nie recent relevant works pointed literature assuming stationarity restrictive unrealistic many practical applications tackling economic financial issues time perspective often deal nonstationary components instance neither consumer price index share price index exchange rates constitute stationary process traditional method handle data take difference eliminate possible stochastic deterministic trends involved data estimation stationary model however differencing may lead loss useful information thus development modeling technique takes nonstationary nonlinear phenomena account time series analysis crucial without taking differences park phillips hereafter study nonlinear regression regressor satisfying unit root structure prove rates convergence nonlinear least squares nls estimator depend properties integrable rate convergence slow asymptotically homogeneous rate convergence achieve even convergence recently chan wang consider model nonlinear nonstationary regression structure proposed paper establish corresponding results certain technical conditions weaker used paper also pointed recent paper myklebust karlsen null recurrent markov process nonlinear generalization linear unit root process thus provides flexible framework data analysis example gao yin show exchange rates british pound dollar time period january february nonstationary necessarily follow linear unit root process see also bec rahbek shephard similar discussion exchange rates french franc german mark time period december april hence gao yin suggest using nonlinear threshold autoregressive tar stationary unit root regimes proved recurrent markov process see example example section example empirical application section framework null recurrent markov chains extensive literature nonparametric semiparametric estimation karlsen karlsen myklebust lin chen schienle chen gao gao using technique split chain nummelin meyn tweedie generalized ergodic theorem functional limit theorem developed karlsen far know however virtually work parametric estimation nonlinear regression model regressor generated class harris recurrent markov processes includes stationary nonstationary cases paper aims fill gap function integrable directly use existing results functions harris recurrent markov processes develop asymptotic theory estimator case belongs class asymptotically homogeneous functions much challenging case function longer bounded nonparametric semiparametric estimation theory problems kernel function usually assumed bounded compact support unfortunately existing results asymptotic theory null recurrent markov process focus case bounded integrable chen hence paper first modify conventional nls estimator asymptotically homogeneous use novel method establish asymptotic distribution well rates convergence modified parametric estimator results show rates convergence parameter vector nonlinear cointegrating gao models rely properties function also magnitude regeneration number null recurrent markov chain addition also study two important issues closely related nonlinear mean regression harris recurrent markov chains first one study estimation parameter vector conditional volatility function asymptotic theory estimation method based rates convergence proposed estimator would depend property volatility function derivatives meanwhile also discuss nonlinear regression processes asymptotically homogeneous using theorem section obtain asymptotic normality parametric estimator stochastic normalized rate comparable theorem however derivation done markov perspective carries potential extending theory nonlinear nonstationary autoregressive processes seems hard approach rest paper organized follows preliminary results markov theory especially harris recurrent markov chain function classes introduced section main results paper extensions given sections respectively simulation studies carried section empirical application given section section concludes paper outline proofs main results given appendix supplemental document gao includes additional simulated examples detailed proofs main results proofs auxiliary results preliminary results make paper section first provide basic definitions preliminary results harris recurrent markov process define function classes way similar introduced markov theory let markov chain state space transition means pprobability set assume markov chain harris recurrent definition markov chain harris recurrent set given returns infinitely often probability one defined karlsen nonlinear nonstationary regression harris recurrence allows one construct split chain decomposes partial sum functions blocks independent identically distributed parts two asymptotically negligible remaining parts let regeneration times number observations number regenerations karlsen use notation instead process defining real function defined nummelin know sequence random variables converge zero almost surely divided number regenerations using lemma karlsen general harris recurrence yields stochastic rates convergence asymptotic theory parametric nonparametric estimators see theorems distribution size number regenerations priori known structure fully depend underlying process obtain specific rate asymptotic theory null recurrent process next impose restrictions tail behavior distribution recurrence times markov chain definition markov chain recurrent exist small nonnegative function initial measure constant slowly varying function stands expectation initial distribution gamma function parameter gao definition small function definition found existing literature page nummelin assuming recurrence restricts tail behavior recurrence time process regularly varying function fact small functions lemma karlsen find holds recurrent markov chain invariant measure markov chain small function minorization inequality karlsen letting following argument karlsen may show regeneration number recurrent markov chain following asymptotic distribution process parameter kasahara since null recurrent case rates convergence nonparametric kernel estimators slower stationary time series case karlsen myklebust gao however necessarily case parametric estimator model section show rate convergence null recurrent case slower stationary time series integrable may faster stationary time series case asymptotically homogeneous addition rates convergence also depend magnitude measures recurrence times markov chain examples recurrent markov chains stationary positive recurrent process next give several examples recurrent markov chains example recurrent markov chain process defined let random walk sequence random variables distribution absolutely continuous respect lebesgue measure density function satisfying inf compact sets existing papers including kallianpur robbins shown defined recurrent markov chain consider parametric tar model form nonlinear nonstationary regression compact subset complement satisfies corresponding conditions example recently gao yin shown tar process recurrent markov chain furthermore may generalize tar model parameter vector according granger autoregressive process also recurrent markov chain example recurrent markov chain let sequence random variables taking positive values defined positive constant myklebust karlsen prove recurrent integer function slowly varying positive function examples recurrent markov chain framework restricted linear processes see example furthermore null recurrent class invariance property recurrent transformation also recurrent granger invariance property hold processes examples recurrent markov chain refer example schienle general conditions diffusion processes ensure harris recurrence satisfied refer bandi phillips function classes similar park phillips consider two classes parametric nonlinear functions integrable functions asymptotically homogeneous functions include many commonlypq used functions nonlinear regression let kak aij aij kak euclidean norm vector function gao invariant measure harris recurrent markov chain differentiable integrable integrable invariant density function random walk case example reduces conventional integrability definition vector function said integrable exist neighborhood bounded definition comparable definition however definition need condition definition paper makes integrable function family paper slightly general next introduce class asymptotically homogeneous functions definition vector function let nonzero said asymptotically homogeneous following two conditions satisfied locally bounded uniformly continuous respect remainder term order smaller asymptotic order limit homogeneous function definition quite similar function except regularity condition definition replaced local boundness condition definition following definition order smaller either locally bounded bounded vanishes infinity note two definitions similarly generalized case matrix functions details omitted save space furthermore process positive recurrent asymptotically homogeneous function might also integrable long density function process integrable decreases zero sufficiently fast diverges infinity nonlinear nonstationary regression main results section establish asymptotic results parametric estimators derivatives belong two classes functions introduced section integrable function first consider estimating model nls approach also used unit root framework define loss function minimizing obtain resulting estimator arg min let derivabefore deriving asymptotic properties tives integrable give regularity conditions assumption harris recurrent markov chain invariant measure sequence random variables mean zero finite variance independent integrable assumption integrable matrix positive definite neighborhood remark assumption assumed harris recurrent includes positive null recurrent markov chains restriction assumption may replaced condition irreducible ergodic strongly mixing process mean zero certain restriction mixing coefficient moment conditions theorem karlsen myklebust hence mild conditions include gao arch processes special examples however case techniques used proofs theorems need modified noting compound process harris recurrent furthermore homoskedasticity error term also relaxed may allow existence certain heteroskedasticity structure conditional variance function satisfies assumption unit variance however property function would affect convergence rates given following asymptotic results example ensure validity theorem need assume indicates integrable literature karlsen myklebust need assume independence assumption quite standard similar corresponding conditions particular assumption key condition derive global consistency nls estimator following theorem next give asymptotic properties applicable stationary positive recurrent nonstationary null recurrent time series theorem let assumptions hold minimizes loss function solution consistent asymptotically normal distribution estimator form null vector asymptotically normal remark theorem shows stochastic convergence rate stationary nonstationary cases however usually unobservable specific rate depends recurrent see corollary next discuss link directly observable hitting time indeed support number times process visits time defined lemma karlsen nonlinear nonstationary regression possible estimator strongly consistent shown karlsen however usually somewhat limited practical use due slow convergence rate remark karlsen simulated example given appendix supplemental document discuss finite sample performance estimation method theorem obtain following corollary directly corollary suppose conditions theorem satisfied let support asymptotically normal distribution form estimator remark practice may choose compact set additional simulation study example given supplemental document two types recurrent markov processes choose positive constant carefully chosen works well setting continuous derivative function show density function estimated kernel method replacing nls estimated value obtain consistent estimate note observable estimated hence calculating variance residuals ebt inference purposes one may need estimate recurrent explicitly computed without knowing information theorem section following corollary corollary suppose conditions theorem satisfied furthermore recurrent markov chain gao defined section remark slowly varying positive function slower integrable case rate convergence rate convergence parametric nls estimator stationary time series case combining theorem result strengthened normal distribution mean zero covariance matrix identity matrix independent similar result also obtained chan wang corollary complement existing results rates convergence nonparametric estimators recurrent markov processes karlsen myklebust gao random walk case corresponds recurrent markov chain rate convergence similar result obtained processes type asymptotically homogeneous function next establish asymptotic theory parametric estimator derivatives belong class asymptotically homogeneous functions unit root process establish consistency limit distribution using local time technique method relies nls estimator linear framework unit root process functional limit theorem partial sum process continuous mapping theorem harris recurrent markov chain general process allows possibly nonlinear framework however particular null recurrent markov chain seen nonlinear generalization linear unit root process hence techniques used establishing asymptotic theory applicable possibly nonlinear markov chain framework meanwhile mentioned section methods used prove theorem applied directly asymptotically homogeneous functions usually bounded integrable leads violation conditions ergodic theorem process null recurrent fact existing limit theorems null recurrent markov process chen consider case bounded integrable hence quite challenging extend theorem case general null recurrent markov chains establish asymptotic theory nls estimator case asymptotically homogeneous functions nonlinear nonstationary regression address concerns modify nls estimator let positive increasing sequence satisfying dominated certain polynomial rate define modified loss function modified nls mnls estimator obtained minimizing arg min truncation technique enables develop limit theorems parametric estimate even function derivatives unbounded similar truncation idea also used ling estimate model second moment may exist called method ling however assumption ling indicates stationarity model harris recurrence considered paper general includes stationary nonstationary cases integrable case discussed section easily show asymptotic regularity conditions example distribution compare finite sample performance two estimators find quite similar furthermore positive recurrent mentioned last paragraph section although asymptotically homogeneous derivatives unbounded integrable may reasonable assume derivatives respect integrable case theorem corollary section still hold estimation role sample size implies consistency stationary time series case derived hence consider null recurrent remaining subsection let easy check disjoint define gao defined assumption additional assumptions introduced establish asymptotic properties assumption asymptotically homogeneous asymptotic orders limit homogeneous functions respectively furthermore asymptotic orders independent function continuous interval achieves unique exist continuous minimum sequence positive numbers lim neighborhood continuous interval exist continuous positive definite matrix sequence positive numbers lim bounded furthermore iii asymptotic orders positive nondecreasing small set invariant density function bounded away zero infinity nonlinear nonstationary regression remark assumption quite standard see example condition theorem restriction asymptotic orders independent relaxed cost complicated assumptions lengthy proofs example ensure global consistency need assume exist neighborhood inf inf establish asymptotic normality need impose additional technical conditions asymptotic orders limit homogeneous functions similar condition theorem assumption explicit forms derived special cases example generated random walk process implies derived similarly explicit forms two cases details thus omitted define next establish asymptotic theory null recurrent theorem let null recurrent markov process assumptions hold solution minimizes loss function consistent estimator asymptotically normal distribution identity matrix gao remark theorem asymptotic distribution asymptotically homogeneous regression function quite different integrable regression function process null recurrent finding comparable choice estimation method asymptotic theory discussed corollaries remark corollary modify inference purposes define satisfies conditions corollary show recurrent use asymptotically normal distribution theory conduct statistical inference without knowing information observable explicitly computed replacing estimated value theorem section following two corollaries rate convergence quite general recurrent markov processes convergence rate theorem corollary suppose conditions theorem satisfied furthermore let recurrent markov chain taking positive constant corollary suppose conditions theorem satisfied let random walk process positive constant mnls estimator furthermore nonlinear nonstationary regression remark simple linear regression model regressors generated random walk process imply existence super consistency corollaries show rates convergence parametric estimator nonlinear cointegrating models rely properties function also magnitude two corollaries give choice special cases fact random walk process defined example standard brownian motion denotes weak convergence furthermore continuous mapping theorem billingsley sup sup implies reasonable let may chosen sup second equality due reflection principle implies obtained given general recurrent markov process choice optimal remains open problem conjecture may defined chosen option take method study issue future research discussions extensions section discuss applications asymptotic results estimating nonlinear heteroskedastic regression nonlinear regression processes furthermore also discuss possible extensions model cases multivariate regressors nonlinear autoregression nonlinear heteroskedastic regression introduce estimation method parameter vector involved conditional variance function simplicity consider model defined satisfies assumption unit variance positive true value parameter vector gao involved conditional variance function estimation parametric nonlinear variance function defined important empirical applications many scientific studies depend understanding variability data covariates integrated han park study maximum likelihood estimation parameters arch garch models recent paper han kristensen considers quasi maximum likelihood estimation models stationary nonstationary covariates next consider general harris recurrent markov process use robust estimation method model letting positive number log log log log log log log log log log since main interest lies discussion asymptotic theory estimator first assume known simplify discussion model seen another nonlinear mean regression model parameter vector estimated logtransformation would make data less skewed thus resulting volatility estimator may robust terms dealing transformation commonly used estimate variability data stationary time series case peng yao gao chen cheng peng however extension harris recurrent markov chains may nonstationary done literature estimation method constructed based noting assumed known define log case derivatives integrable logb obtained transformed nonlinear least squares lnls estimator minimizing log letting assumptions satisfied replaced respectively asymptotic results developed section still hold nonlinear nonstationary regression case derivatives asymptotically homogeneous modified nonlinear least squares lmnls estimator obtained minimizing log defined section asymptotic results developed section still hold regularity conditions slightly modified version assumptions hence possible achieve result null recurrent practice however usually unknown needs estimated next briefly discuss issue case may define loss function log log estimators obtained minimizing simulated example example given appendix supplemental document examine finite sample performance lnls lmnls estimations considered cases respectively nonlinear regression processes mentioned consider nonlinear regression regressors generated sequence random variables satisfies summability conditions simplicity assume throughout subsection establish suite asymptotic results nls estimator parameter involved defined open problem establish results using recurrent markov chain framework quite challenging defined longer markov process except special cases example next consider solving open problem case asymptotically homogeneous derive asymptotic theory using theorem discussion integrable case gao complicated considered future study main idea approximate defined show asymptotically homogeneous function asymptotically equivalent function random walk process assumption see appendix supplemental document make use theorem define next give asymptotic results case unit root process proof provided appendix supplemental document theorem let assumptions appendix supplemental document hold defined assumption solution minimizes loss function consistent estimator asymptotically normal distribution number regenerations random walk remark theorem establishes asymptotic theory unit root process results comparable theorems however establish asymptotic ity stochastic rate establish asymptotic mixed normal distribution theory deterministic rate probability take corollary find rate convergence derived nonlinear nonstationary regression extensions multivariate regression nonlinear autoregression theoretical results developed section limited nonlinear regression univariate markov process natural question whether possible extend general case multivariate covariates unit root framework well known difficult derive limit theory case multivariate unit root processes vector brownian motion transient dimension larger equal contrast framework harris recurrent markov chains possible generalize theoretical theory multivariate case certain restrictions example possible extend theoretical results case one nonstationary regressor several stationary regressors next give example vector autoregressive var process may included framework certain conditions example defined consider var process matrix vector sequence random vectors mean zero eigenvalues matrix inside unit circle mild conditions theorem myklebust karlsen shows var process geometric ergodic belongs category positive recurrence hand matrix exactly one eigenvalue unit circle mild conditions theorem myklebust karlsen shows var process recurrent case asymptotic theory developed section applicable however two eigenvalues unit circle different restrictions might null recurrent recurrent transient three eigenvalues unit circle var process would transient indicates limit theory developed paper would applicable next briefly discuss nonlinear autoregressive model form autoregression case independent thus proof strategy developed paper needs modified following argument karlsen order develop gao asymptotic theory parameter estimation nonlinear autoregression may need process harris recurrent compound process also harris recurrent essentially consider sums products like general form treated karlsen verification harris recurrence discussed example given section establish asymptotic theory parameter estimation model studied future research simulated examples section provide simulation studies compare finite sample performance proposed parametric estimation methods illustrate developed asymptotic theory example consider generalized linear model defined exp generated one three markov processes process random walk process iii tar process sequence standard normal random variables three processes error process sequence random variables independent simulation study compare finite sample behavior nls estimator mnls estimator sample size chosen aim example illustrate asymptotic theory developed section regression function integrable following discussion section process defined positive recurrent random process defined tar process defined iii recurrent generate replicated samples simulation study calculate means standard errors parametric estimators simulations mnls estimation procedure choose defined case cases iii easy find simulation results reported table numbers parentheses standard errors nls mnls estimator replications table following interesting findings parametric estimators perform better stationary case nonstationary cases iii consistent asymptotic nonlinear nonstationary regression table means standard errors estimators example sample size nls mnls regressor generated case nls mnls regressor generated case nls mnls regressor generated case iii results obtained section theorem corollaries indicate convergence rates parametric estimators achieve stationary case recurrent case finite sample behavior mnls estimator nls estimator since means little sample information lost two parametric estimators improve sample size increases addition case ratio standard errors close theoretical ratio case iii ratio standard errors close theoretical ratio hence confirms asymptotic theory valid example consider quadratic regression model defined generated either one three markov processes introduced example unit root process sequence random variables error process defined example simulation study interested finite sample behavior mnls estimator illustrate asymptotic theory developed section regression function asymptotically homogeneous comparison purpose also investigate finite sample behavior nls estimation although establish related asymptotic theory framework null recurrent markov chains sample size chosen example replication number mnls estimation procedure previous example choose case cases gao table means standard errors estimators example sample size nls mnls regressor generated case nls mnls regressor generated case nls mnls regressor generated case iii nls mnls regressor generated case simulation results reported table following conclusions regression model asymptotically homogeneous regression function parametric estimators perform better nonstationary cases stationary case finding consistent asymptotic results obtained sections mnls estimator performs well nls estimator particular nonstationary cases nls mnls estimations improve sample size increases empirical application section give empirical application proposed parametric model estimation methodology example consider logarithm export import data data come website https spanning january august monthly sample size let defined log log puk log monthly average nominal exchange rate pit denotes consumption price index country example let denote logarithm either export import value data plotted figures respectively meanwhile real data application considered gao yin suggests may follow threshold autoregressive model proposed paper shown recurrent markov process furthermore application estimation method gives supports roughly follows recurrent markov chain nonlinear nonstationary regression fig plot real exchange rate avoid possible confusion let yex yim export import data respectively interested estimating parametric relationship yex yim order find suitable parametric relationship first estimate relationship nonparametrically based yex mex yim mim karlsen myklebust fig plot logarithm export import data gao mex mim estimated yex yim probability density function standard normal distribution bandwidth chosen conventional method parametric calibration procedure based preliminary nonparametric estimation suggests using polynomial relationship form yex eex export data estimated values using method section respectively yim eim import data estimated values respectively plots given figures respectively figures suggest relationship exchange rate either export import variable true relationship may fig plot polynomial model fitting nonlinear nonstationary regression fig plot polynomial model fitting also depend macroeconomic variables real interest rate period discussed section would like extend proposed models univariate case multivariate case future application able find accurate relationship among export import variable exchange rate macroeconomic variables conclusions paper systematically studied nonlinear regression general harris recurrent markov chain framework includes stationary nonstationary cases note nonstationary null recurrent process considered paper markov perspective unlike indicates methodology potential extended nonlinear autoregressive case paper develop asymptotic theory nls estimator integrable also propose using modified version conventional nls estimator asymptotically homogeneous adopt novel method establish asymptotic theory proposed modified parametric estimator furthermore using discuss estimation parameter vector conditional volatility function also apply results nonlinear regression processes may establish asymptotic distribution theory comparable obtained simulation studies empirical applications provided illustrate approaches results gao appendix outline main proofs appendix outline proofs main results section detailed proofs results given appendix supplemental document major difference proof strategy based unit root framework kristensen rahbek proofs rely limit theorems functions harris recurrent markov process lemmas whereas kristensen rahbek proofs use limit theorems integrated time series start two technical lemmas crucial proofs theorems proofs two lemmas given appendix supplemental document gao lemma let integrable function suppose assumption satisfied uniformly satisfies assumption uniformly positive definite furthermore lemma let hah asymptotically homogeneous function asymptotic order independent limit homogeneous function hah suppose null recurrent markov process invariant measure assumption satisfied hah continuous interval furthermore letting hah hah hah hah nonlinear nonstationary regression defined section exist continuous positive definite matrix sequence positive numbers bounded lim defined section uniformly jah hah jah satisfies assumption uniformly jah furthermore hah jah hah proof theorem theorem need verify following sufficient condition weak consistency jennrich sequence positive numbers uniformly continuous achieves unique minimum sufficient condition proved using lemma theorem thus proved combining device billingsley lemma complete proof asymptotically normal distribution details found appendix supplementary material proof corollary asymptotic distribution proved using theorem gao proof corollary convergence result proved using following proof lemma gao detailed proof given appendix supplementary material proof theorem proof similar proof theorem prove weak consistency similar need verify sufficient condition sequence positive numbers uniformly continuous achieves unique minimum using assumption following proofs lemma see appendix supplementary material may prove thus weak consistency result combining device lemma complete proof asymptotically normal distribution details given appendix supplementary material proof corollary using theorem following proof lemma gao directly prove proof corollary convergence result follows corollary proved using theorem acknowledgments authors grateful professor runze associate editor two referees valuable constructive comments suggestions substantially improved earlier version paper thanks also professor peter phillips colleagues commented paper participants various conferences seminars earlier versions paper presented work started first third authors visited second author department mathematics university bergen supplementary material supplement estimation nonlinear regression harris recurrent markov chains doi provide additional simulation studies detailed proofs main results section proofs lemmas theorem nonlinear nonstationary regression references bandi phillips nonstationary processes handbook financial econometrics hansen eds bec rahbek shephard acr model multivariate dynamic mixture autoregression oxf bull econ stat billingsley convergence probability measures wiley new york chan wang nonlinear cointegrating regressions nonstationary time series working paper school mathematics univ sydney chen often harris recurrent markov chain recur ann probab chen limit laws second order additive functionals harris recurrent markov chains probab theory related fields chen cheng peng conditional variance estimation heteroscedastic regression models statist plann inference chen gao estimation regression regressors bernoulli gao nonlinear time series semiparametric nonparametric methods monographs statistics applied probability chapman boca raton gao yin estimation threshold autoregressive models stationary unit root regime econometrics gao kanaya uniform consistency nonparametric estimators null recurrent time series econometric theory han kristensen asymptotic theory qmle garchx models stationary nonstationary covariates bus econom statist han park persistent covariate asymptotic theory mle econometrics limit theorems null recurrent markov processes mem amer math soc jennrich asymptotic properties least squares estimators ann math statist kallianpur robbins sequence sums independent random variables duke math karlsen myklebust nonparametric estimation nonlinear cointegration type model ann statist karlsen myklebust nonparametric regression estimation null recurrent time series statist plann inference karlsen nonparametric estimation null recurrent time series ann statist kasahara limit theorems processes poisson point processes applications brownian excursions math kyoto univ kristensen rahbek testing inference nonlinear cointegrating vector error correction models econometric theory gao lai asymptotic properties nonlinear least squares estimates stochastic regression models ann statist nie efficient statistical inference procedures partially nonlinear models applications biometrics gao supplement estimation nonlinear regression harris recurrent markov lin chen local linear null recurrent time series statist sinica ling local likelihood estimators models econometrics geometric ergodicity autoregressive model autoregressive conditional heteroscedastic term statist sinica malinvaud consistency nonlinear regressions ann math statist meyn tweedie markov chains stochastic stability cambridge univ press cambridge myklebust karlsen null recurrent unit root processes econometric theory nummelin general irreducible markov chains nonnegative operators cambridge tracts mathematics cambridge univ press cambridge park phillips asymptotics nonlinear transformations integrated time series econometric theory park phillips nonlinear regressions integrated time series econometrica peng yao least absolute deviations estimation arch garch models biometrika schienle nonparametric nonstationary regression many covariates working paper berlin severini wong profile likelihood conditionally parametric models ann statist skouras strong consistency nonlinear stochastic regression models ann statist granger modelling nonlinear economic time series oxford univ press oxford asymptotic theory nonlinear least squares estimation ann statist department mathematics university york heslington campus york united kingdom department mathematics university bergen post box bergen norway gao department econometrics business statistics monash university caulfield caulfield east victoria australia

memcapacitive neural networks yuriy pershin massimiliano ventra jul abstract show memcapacitive memory capacitive systems used synapses artificial neural networks example approach discuss architecture neural network based memcapacitive synapses moreover demonstrate plasticity simply realized devices memcapacitive synapses alternative memristive synapses neuromorphic computation pershin department physics astronomy usc nanocenter university south carolina columbia pershin ventra department physics university california san diego jolla california diventra memcapacitive neural networks electronic devices memory memristive memcapacitive meminductive systems promising components unconvential computing applications ability store process information space location moreover devices used memory computing elements neural networks density consumption easily achieved far memristive devices considered electronic synapses artificial neural networks article show instead memcapacitive systems could play similar role thus offering benefit low power dissipation instances full compatibility cmos technology added benefit electronic realizations smart electronics according definition memcapacitive systems described equations charge capacitor time applied voltage memcapacitance depends state system vary time set state variables describing internal state system continuous vector function currently well established biological neural networks synapse strength encodes memories electronic circuits memory feature memcapacitive systems provided internal states characterized play similar role figure show example memcapacitive neural network leaky network input neurons connected block output neuron help memcapacitive synapses memcapacitive synapse contains memcapacitive system two diodes assumed switching memcapacitive system involves voltage threshold voltage pulse amplitudes involved network subjected voltage pulse input neuron memcapacitive system charges integrating capacitor proportion capacitance soon voltage across reaches threshold nout nout fires forward voltage pulse uses controllable switch reset diodes connected ground discharge pulse disappearance replaced resistors fig presents simulations network ltspice environment assuming firing one input neuron compare circuit response subjected input pulse sequence periodic firing different values corresponding synaptic connection memcapacitance stronger synaptic connection fig results faster charging integrating capacitor higher rate output neuron nout firing result compatible operation excitatory synapses order model inhibitory synapses one use synaptic connection similar shown fig diodes connected opposite polarity integrating capacitor power supply voltage instead ground inset fig shows schematics inhibitory synapse explicitly moreover inhibitory synapse driven inverted pulses evaluate strengths weaknesses using memcapacitive systems synapses compare energy dissipation memcapacitive memristive neural networks indeed circuit depicted fig also operate memristive synapses replacing memcapacitive ones let estimate amount energy lost cases purpose consider situation single voltage pulse fired charges small amount charge case memcapacitive network dissipated energy energy stored due pulse namely case memristive network consequently application memcapacitive synapses two times energy efficient however memristive networks require less components diodes connected ground fig required memristive synapses used let consider realization plasticity stdp memcapacitive gnd gnd fig vout nout memcapacitive synapses neural network input electronic neurons connected output neuron nout using memcapacitive synapses inset schematics inhibitory synapse fig time vout voltage voltage vout time simulation memcapacitive network fig one spiking neuron regular spikes input neuron trigger output neuron nout voltage threshold shown horizontal dashed line different frequencies depending strength memcapacitive synapse plot obtained diode model voltage across lines shifted clarity synapses purpose consider bipolar memcapacitive system threshold also suitable network considered biological neural networks postsynaptic signal reaches synapse action potential presynaptic neuron synapse shows depression ltd namely strength decreases weaker connection neurons depending time difference postsynaptic presynaptic signals conversely postsynaptic action potential reaches synapse presynaptic action potential synapse undergoes longtime potentiation ltp namely signal transmission two neurons increases proportion time difference presynaptic postsynaptic signals electronic circuits stdp implemented using double voltage pulses shown fig see also ref case pulse overlap provides opposite voltage polarities across synapse depending timing presynaptic postsynaptic pulses using spice model memcapacitive system threshold simulate dynamics memcapacitive synapse subjected ltp ltd pulses bottom line fig clearly shows corresponding increase decrease synaptic strength memcapacitance conclusion presented alternative simulate synaptic behavior uses memcapacitive systems instead memristive systems corresponding memcapacitive neural networks operate presynaptic voltage postsynaptic fig memcapacitance ltp ltd time stdp memcapacitive synapses plot obtained using model bipolar memcapacitive system threshold clow chigh lines shifted clarity low energy consumption cases compatible cmos technology making promising candidates neuromorphic computing work partially supported nsf grant center magnetic recording research ucsd burroughs wellcome fund collaborative research travel grant eferences chua kang memristive devices systems proc ieee vol ventra pershin chua circuit elements memory memristors memcapacitors meminductors proc ieee vol ventra pershin parallel approach nature physics vol snider cortical computing memristive nanodevices scidac review vol chang ebong bhadviya mazumder nanoscale memristor device synapse neuromorphic systems nano vol pershin ventra experimental demonstration associative memory memristive neural networks neural networks vol pershin ventra neuromorphic digital quantum computation memory circuit elements proc ieee vol traversa bonani pershin ventra dynamic computing random access memory cowan sudhof stevens synapses johns hopkins university press biolek ventra pershin reliable spice simulations memristors memcapacitors meminductors

minimum cuts shortest cycles directed planar graphs via noncrossing shortest mar march abstract let simple directed planar graph nonnegative edge weights study fundamental problems computing global cut minimum weight cycle minimum weight best previously known algorithm former problem running time obtained algorithm nussbaum sankowski maximum flows best previously known result latter problem algorithm exploiting duality two problems planar graphs solve problems log log log time via algorithm finds shortest cycle kernel result log log algorithm computing noncrossing shortest paths among nodes well ordered common face directed plane graph extended algorithm italiano nussbaum sankowski undirected plane graph introduction let simple graph nonnegative edge weights unweighted weights edges identical let subgraph weight sum edge weights let denote graph obtained deleting edges paths allowed repeat nodes throughout paper nodes path subgraph global cut nodes cycle node problem seeks cut minimum weight instance consisting edge minimum cut graph figure best known algorithm directed due hao orlin runs log time undirected nagamochi ibaraki stoer wagner solved problem log time karger solved problem expected preliminary version paper appeared master thesis first author journal version appeared siam journal discrete mathematics graduate institute computer science information engineering national taiwan university email sirbatostar corresponding author department computer science information engineering national taiwan university author also holds joint appointments graduate institute networking multimedia graduate institute biomedical electronics bioinformatics national taiwan university address roosevelt road section taipei taiwan roc research author supported part grant email hil web figure simple planar graph simple bidirected plane graph obtained adding edges weights respectively seeking minimum cut respectively shortest cycle dual time kawarabayashi thorup recently announced first known algorithm undirected unweighted improving upon algorithm gabow designed twenty years ago problem seeks cycle minimum weight instance cycle weight shortest cycle graph figure since shortest directed cycle containing edge obtainable shortest problem directed graphs reduced computing shortest paths log time vassilevska williams williams argued finding truly subcubic algorithm problem might hard directed respectively undirected unweighted itai rodeh solved problem log respectively min time time multiplying two matrices undirected planar chalermsook fakcharoenphol nanongkai showed time complexity aforementioned problems log times finding minimum weight given nodes plugging log algorithms frederickson borradaile klein erickson reduction chalermsook solved problems time plugging log log time algorithm italiano nussbaum sankowski reduction chalermsook solved problems log log log time best known result problems log log algorithm sankowski relying upon oracle italiano paper addresses problems case directed planar problem thoroughly studied undirected planar graphs surprisingly prior work specifically directed planar graphs djidjev claimed technique unweighted undirected planar graphs solves problem unweighted directed planar time left open problem finding shortest cycle unweighted directed planar time weimann yuster gave algorithm problem adjustable solve problem also time via similar techniques proof lemma handle degeneracy shortest cycles reduced time problem unclear adjust algorithm solve problem without increasing required time much algorithm baum sankowski maximum flows solves problem directed planar time result theorem takes log log log time solve problems simple directed planar graph nonnegative edge weights pointed anonymous reviewers mozes nikolaev nussbaum weimann recently announced log log algorithms problem however unlike theorem algorithm requires condition unique shortest path two nodes general directed planar graphs nonnegative edge weights apply isolation lemma perturb edge weights meet condition high probability thus results monte carlo randomized algorithms related work known nontrivial algorithm problem due chang works undirected unweighted planar graphs undirected embedded orientable surface genus erickson fox nayyeri solved problem log log time based algorithm sankowski undirected planar graphs undirected unweighted embedded orientable surface genus djidjev solved problem log time undirected unweighted weimann yuster solved problem log time directed planar even unweighted theorem remains best known algorithm unweighted embedded surface technique djidjev solved problem time problem negative edge weights reduced one nonnegative edge weights using standard reweighting technique via tree cygan gabow sankowski studied problem graphs whose edge weights bounded integers yuster studied version undirected asking node shortest cycle containing node see algorithms compute shortest cycles prescribed topological properties see approximation algorithms problem closely related problem seeks minimum given nodes dual problem seeks maximum extensively studied even planar graphs see minimum obtained time maximum identifying edges nodes reachable nodes reachable residual graph respect efficient reductions direction known orlin gave known algorithms maximum problem general graphs integral edge weights undirected planar reif gave algorithm minimum problem frederickson improved time complexity reif algorithm log best known algorithms problems due italiano run log log time attempt janiga koubek generalize reif algorithm directed planar turned flawed borradaile klein erickson gave log time algorithms problems directed planar graphs directed planar unweighted brandes wagner eisenstat klein solved problems time algorithm kaplan nussbaum capable exploiting condition nodes close directed planar algorithm obtains minimum weights given nodes given node subsets directed planar algorithm borradaile klein mozes nussbaum computes subgraph minimum weight undirected planar borradaile sankowski gave algorithm compute gomoryhu tree compact representation minimum weights nodes kernel result log log algorithm computing noncrossing shortest paths among nodes well ordered common face directed plane graph extended algorithm italiano undirected plane graph closely related problem seeks noncrossing paths given terminal pairs faces minimum total weight plane graph takahashi suzuki nishizeki solved problem undirected plane graphs log time papadopoulou addressed geometric version problem terminal pairs boundaries polygonal obstacles plane complexity gave algorithm case erickson nayyeri generalized result takahashi solving problem undirected planar graphs log time also generalized result papadopoulou solve geometric version time algorithms computes implicit representation answers may total size polishchuk mitchell addressed problem finding noncrossing thick paths minimum total weight takahashi suzuki nishizeki also considered rectilinear version problem technical overview outline proof theorem consists series reductions based upon duality simple cycles minimal cuts plane graphs section gives reduction problems planar graph problem finding shortest cycle plane graph lemma let balanced separator corresponds fundamental cycle respect tree shortest cycle cross recursively computed subgraphs separated although afford compute shortest cycle crosses section reduces problem finding shortest cycle finding cycle cycle crosses property shortest shortest cycle cross shortest cycle lemma reduction recursive algorithm using balanced separator thus introduces log overhead running time cycle crosses shortest path shortcutted cycle crosses section reduces problem finding cycle finding cycle cycle whose weight cycle crosses shortest path exactly lemma technique reif incises along section reduces problem finding cycle finding shortest noncrossing paths among nodes well ordered boundary external face lemma matter fact problem solved log algorithm klein already yielding improved algorithms problems mozes also mentioned algorithms obtained plugging log minimum algorithm borradaile klein directed version reduction algorithm chalermsook achieve time complexity theorem section solves problem log log time extending algorithm italiano rected plane graph section concludes paper reduction finding shortest cycles directed graph bidirected two nodes edge edge graph figure bidirected degree node bidirected number neighbors degree bidirected maximum degree nodes bidirected plane graph bidirected planar graph equipped plane embedding edges two adjacent nodes bundled together figures show two bidirected plane graphs cycle passing node simple cycle degenerate node passes edges two nodes cycle simple respectively degenerate respectively cycle figure graph figure cycle degenerate simple cycle simple cycle degenerate shortest degenerate cycle weight shortest cycle weight theorem proved following lemma lemma takes log log log time compute shortest cycle simple bidirected plane graph nonnegative edge weights proof theorem adding edges weights respectively input graph affect weight minimum cuts respectively shortest cycles hence may assume without loss generality input graph least four nodes simple bidirected plane graph face triangle see figures examples let dual simple bidirected plane graph faces sharing set edges obtainable time follows two adjacent nodes directed edges two faces incident bundled edges face immediately succeeds face clockwise order around node see figure example observe minimal cut simple cycle nonnegativity edge weights shortest cycle minimum cut instance shortest cycle graph figure weight corresponds turn corresponds minimum cut although degenerate cycle shortest cycle correspond cut manner since node exactly three neighbors statement theorem problem follows applying lemma nonnegativity edge weights takes time obtain shortest degenerate cycle examining degenerate cycles exactly two nodes lemma statement theorem problem immediate following claim takes time obtain simple bidirected plane graph shortest cycle computed shortest cycle time let simple bidirected plane graphs nonnegative edge weights obtained following operation node neighbors clockwise order around plit figure bidirected plane graphs edge weights black solid lines shortest cycles blue dashed lines shortest cycles red dotted lines adds path new nodes replaces edge edge weight replaces edge vui edge weight deletes see figure example simple bidirected plane graph obtained time iteratively applying plit node degree prove claim suffices ensure following statement shortest cycle obtainable time shortest nondegenerate cycle least two edges internal nodes replace path vuj vuj since resulting obtainable cycle since may pass could see figure example remains show shortest simple cycle nonnegativity edge weights even let cycle obtained follows path vuj replace path vuj otherwise let since simple one path vuj since see figure example shortest nondegenerate cycle rest paper proves lemma via balanced separating cycles let simple cycle bidirected plane graph nonnegative edge weights let intg respectively extg denote subgraph consisting nodes edges boundary faces inside respectively outside nondegenerate cycle one shortest cycle figure bidirected plane graph edge weights black blue dashed cycle segmented cycle whose segments shortest paths shortest cycle intg weight shortest cycle extg weight red dotted cycle weight unique cycle unique shortest cycle bidirected plane graph respectively shortest cycle intg respectively extg say segmented consists following three paths order shortest path edge reverse shortest path one allowed node let shortest paths segments see figure example section proves lemma using lemmas section proves lemma lemma let simple bidirected plane graph nonnegative edge weights given segmented simple cycle together segments takes log log time compute cycle lemma henzinger klein rao subramanian takes time compute tree rooted given node connected simple directed planar graph nonnegative edge weights lemma lipton tarjan goodrich klein mozes sommer lemma let simple undirected plane triangulation nonnegative face weights summing weight face given spanning tree takes time obtain edge total weight faces inside respectively outside simple cycle proof lemma give recursive algorithm input graph assumed connected without loss generality node whose neighbors replace incident edges edges weights respectively resulting graph obtainable time remains simple connected bidirected plane graph see figure example let number faces since maximal simple path nodes edges nodes shortest cycle obtained time shortest cycle found time case figure bidirected plane graph faces edge weights black tree rooted blue dashed lines plane triangulation consists edges numbers weights faces undirected version consists black solid edges blue dashed edges undirected version blue dashed lines edges red dotted lines obtain shortest cycle case let obtainable tree rooted arbitrary node ensured lemma face simple undirected unweighted version size let triangulated faces via adding edges without introducing multiple edges let arbitrary one faces assigned weight let remaining faces assigned weights let resulting simple plane triangulation undirected version spanning tree see figure example lemma ensures edge obtainable time face weights inside respectively outside simple cycle sum instance edge example figure adjacent let otherwise let consist edges weights union obtain simple bidirected plane graph let least common ancestor let segmented simple cycle consisting edge reverse lemma takes log log time compute cycle let matter subgraphs recursively compute shortest cycle respectively respectively also shortest cycle respectively definition cycle minimum weight shortest cycle passes edge weight cycle otherwise return shortest cycle algorithm runs log log time without accounting time subsequent recursive calls number implying respectively faces respectively log levels recursion overall number faces recursion level implying overall number nodes recursion level algorithm runs log log log time figure tvs red dotted cycle blue dashed cycle minimum weight obtained incising along cycles cross separating cycle section proves lemma lemma proved lemmas section proves lemma graph let denote weight shortest let lemma let simple connected bidirected plane graph nonnegative edge weights let nodes boundary external face order takes overall log log time compute let simple bidirected plane graph simple path aligns subgraph reverse path simple path passing least one edge deviates subgraph edges internal nodes simple path cycle consists path aligning path deviating simple cycle path aligning first edge path deviating intg last edge path deviating extg instance figure whose path aligning node first edge respectively last edge path deviating extg respectively intg also cycle weight lemma let simple bidirected plane graph nonnegative edge weights let simple cycle given path aligning takes log log time compute cycle proof let minimum weight instance red blue cycles figure two minimum weight let shortest nondegenerate cycle passing least one endpoint obtained time via examining shortest lemma edges incident least one endpoint passes endpoint implying figure red dotted intersecting intg blue dashed cycle intersecting extg red dotted cycle consists order degenerate cycle obtained red dotted cycle replacing blue dashed green cycle cycle contained cycle ensured lemma rest proof assumes pass endpoint thus internal nodes let simple bidirected plane graph obtainable time follows suppose nodes order let incise along adding new nodes new path reverse letting edge vui respectively incident intg replaced vvi respectively weight letting weight edge reverse embedding resulting graph external face see figure example lemma takes overall log log time compute let respectively minimizes respectively lemma takes time obtain simple shortest simple shortest weight respectively minimum respectively let respectively cycle corresponding respectively let path deviates let first last nodes respectively first edge intg corresponds implying last edge intg corresponds implying instance red respectively blue cycle figure corresponds red respectively blue figure thus one minimum weight cycle ensured lemma proof lemma let intg extg let given segments let shortest cycle whose number edges minimized shortest cycles cycle cycle including one ensured lemma rest proof assumes neither contains lemma suffices ensure need following claim claim intersects else would cycle illustrated figure contradicting assumption since consists four paths order aligns deviates assumption thus see figure illustration remains prove claim assume contradiction intersects index nodes precedes succeeds deviate deviates let cycle obtained replacing path since shortest path since reverse thus even degenerate nondegenerate cycle see figure illustration nonnegativity edge weights shortest cycle whose number edges fewer number edges contradicting definition noncrossing shortest paths among nodes external face section proves lemma via extending techniques reif italiano undirected planar graphs algorithms lemma graphs lemma reviewed data structures algorithm lemma given data structures enables efficient partition boundary nodes via noncrossing paths lemma given tools involving noncrossing shortest paths lemma given lemma proved lemmas graph let simple bidirected plane graph division partition bidirected plane subgraphs piece multiplicity node number pieces containing node multiplicity two boundary node face piece hole piece face see division pieces nodes boundary nodes holes lemma klein mozes sommer takes time compute simple bidirected plane graph whose faces contains three nodes figure piece boundary nodes one hole boundary nodes hole let connected component piece let denote complete directed graph boundary nodes see figure example dense distance graph see simple directed graph boundary nodes simplified union connected components pieces keeping exactly one copy parallel edges minimum weight edge underlying connected component piece weight equal path underlying path consists underlying edge lemma klein given simple bidirected plane graph nonnegative edge weights takes log time compute data structure path first edges underlying path obtained log log time algorithm consider following equation distinct nodes simple directed graph edge weights monge unit complete equipped cyclic ordering nodes equation holds distinct nodes order monge unit complete bipartite equipped ordering two maximal independent sets equation holds distinct nodes one independent set order distinct nodes independent set order monge decomposition simple directed graph edge weights set monge units node subsets graph simplified union monge units multiplicity node number monge units contain size sum multiplicities nodes equivalent figure two graphs simplified union two monge units form following lemma proved mozes using algorithm klein used kaplan mozes nussbaum sharir specifically hole piece complete graph nodes nodes equipped cyclic ordering monge unit instance subgraphs figure induced equipped cyclic orders holes two monge units two holes piece mozes showed complete bipartite graph nodes nodes contains exactly one simplified union monge units instance subgraph figure consisting edges simplified union graphs figures edges graph figure respectively respectively form monge unit edges graph figure respectively respectively form monge unit lemma given simple bidirected plane graph nonnegative edge weights takes log time obtain monge decomposition multiplicity node times multiplicity summarized following lemma given monge decomposition graph obtainable data structures range minimum queries see kaplan gawrychowski mozes weimann algorithm fakcharoenphol rao outputs tree time lemma given monge decomposition simple strongly connected directed graph nonnegative edge weights takes time compute tree rooted given node lemma let given simple plane graph nonnegative edge weights takes log time compute data structure subset boundary nodes subgraph induced strongly connected takes time compute tree rooted given node sum multiplicities nodes proof let monge decomposition ensured lemma let consist subgraph induced monge unit remains figure noncrossing shortest paths monge unit induced cyclic ordering respectively orderings nodes first respectively second type thus monge decomposition preserving property multiplicity node times multiplicity implying size takes overall time obtain induced cyclic ordering two induced orderings nodes monge unit since weight edge obtained time weight implicit representation time lemma follows lemma noncrossing paths let simple connected bidirected plane graph let distinct nodes boundary external face connected plane graph order simple simple noncrossing empty path instance figure red blue noncrossing noncrossing let denote connected bidirected plane subgraph enclosed boundary external face following order see figure example let proof lemma needs data structure following property distinct nodes external face order disjoint simple simple noncrossing given takes log time obtain consists boundary nodes sum multiplicities nodes number edges see figure illustration lemma takes time compute data structure given simple connected bidirected plane graph proof given edge set takes time obtain nodes belongs let consist nodes rest proof assumes let respectively consist pieces contains nodes respectively edges since connected let obtainable undirected bipartite graph nodes pieces adjacent contains nodes connected component either belong belong since connected connected component contains node figure illustration definition blue solid green red dotted disjoint noncrossing boundary nodes form boundary nodes form boundary nodes form belongs piece takes overall time obtain hole piece since piece holes remains show defined hole piece takes log time determine nodes number nodes plus number edges assume without loss generality external face piece hole obtainable data structure consists cyclic ordering incident edges around node following items hole piece arbitrary simple path node node external face ordering indices nodes cyclic ordering indices nodes takes overall time obtain hole piece first part edge takes time determine whether second part subset piece hole takes time determine ordering indices nodes cyclic ordering indices nodes case illustrated figure takes overall log time via sorting ordering indices compute node first node traversal starting following order node preceding traversal determined time case illustrated figure let node preceding first node let boundary external face illustrated figure let node preceding first node traversal starting following order either way takes figure illustrations proof lemma figure illustration proof lemma time obtain determine whether otherwise noncrossing shortest paths lemma let simple connected bidirected plane graph nonnegative edge weights nodes boundary external face order shortest path shortest noncrossing proof illustrated figure suppose shortest let respectively first respectively last node let obtained replacing order boundary external face well defined shortest noncrossing lemma let simple connected bidirected plane graph nonnegative edge weights let distinct nodes boundary external face order let simple shortest noncrossing let number nodes given consider problem computing problem solved log time given set nodes least one shortest passes least one node problem solved time given problem solved time proof since given takes time obtain excluding edges internal nodes statements follow lemmas statement assumption simple shortest simple shortest given disjoint recursive algorithm easure solving computing indices let lemma takes time linear number nodes obtain simple shortest noncrossing call easure otherwise apply statement consisting arbitrary node call easure otherwise apply statement consisting arbitrary node algorithm statement obtains calls easure since computed lemma statement correctness holds trivially choice easure runs log levels recursion since holds call easure node appears two subgraphs level recursion thus overall running time level recursion algorithm runs log time proving lemma proof lemma let modification equals weight shortest resulting remains simple connected bidirected plane graph add new nodes external face edges edges contract strongly connected subgraph single node delete delete except one copy set multiple edges minimum weight thus rest proof assumes distinct cycles implying shortest paths simple let bidirected plane graph obtainable time identifying nodes new node triangulating face size larger let max lemma computed time let division induced piece obtained piece deleting edges added triangulate faces size larger piece nodes boundary nodes holes piece let consist indices indices least one boundary node since boundary node cardinality turn subroutine olve solve lemma return let median let respectively shortest whose first respectively last edges obtained log log time case let call abel olve olve return case call abel first node call abel last node case let index solve lemma return let implying solve lemma call olve return solve lemma call olve return case implying let let solve lemma call olve return solve lemma call olve return figure subroutine olve boundary nodes introduce new pieces form partition nodes let resulting division new piece nodes edges boundary nodes holes thus boundary node let simple bidirected plane graph edge weights obtained reversing direction edge let corresponding equation takes log log time compute data structures ensured lemmas nodes shortest path let denote need subroutine abel compute label node shortest path assumption one node let node node let arbitrary node let node precedes let node succeeds let subroutine abel runs time per node overwrite running abel nodes obtained time indices let set consist indices let shortest obtainable time lemma lemma follows lemma arbitrary node rest proof assumes algorithm proving lemma calls abel abel olve main subroutine olve defined figure elaborated solves computing indices condition figure illustrations proof lemma black red dots blue dashes shortest shortest disjoint node set boundary nodes given equation remains prove olve correctly solves log time boundary nodes since induce connected subgraph belong common piece implying solved lemma log time number nodes case afford directly compute shortest median lemma instead subgraph induced given set boundary nodes compute shortest respectively respectively first respectively last edges whose underlying path respectively obtained log log time lemma lemma contains least one shortest implying subgraph induced contains least one shortest therefore shortest intersect takes log log time per node obtain case figure subroutine lets calls abel olve olve intersects takes log log time per node obtain first node last node stated first two bullets case figure subroutine calls abel abel illustrated figure contains exactly one solved time lemma stated first bullet case figure illustrated figure let solved calling olve solved lemma time case similar second bullet case figure states two cases illustrated figure shortest disjoint solved calling olve since least one shortest passes subproblem solved time lemma case similar case figure states two cases correctness holds trivially since computed somewhere execution olve lemma since chosen median subroutine call olve log levels recursion executing olve let sum multiplicities nodes lemma time computing log order maintain condition given whenever olve called apply lemma obtain log time calling olve olve number edges since disjoint boundary node contained one two subgraphs recursion level since pieces piece boundary nodes sum subgraphs recursion level since edge appears one subroutine calls olve sum throughout execution olve equation overall time computing log log overall time finding paths log log since edges disjoint obtainable log log time per node therefore running time olve dominated sum log time solving subproblems lemmas bottom recursion since sum running time olve log lemma proved concluding remarks give first known log log log algorithms finding minimum cut shortest cycle simple directed planar graph nonnegative edge weights case restricted unweighted algorithm remains best known result problem best algorithm problem running log time obtained plugging minimum stcut algorithm brandes wagner eisenstat klein directed version reduction algorithm chalermsook thus interesting future direction reduce running time algorithms problems special case extending results graphs also interest acknowledgment thank anonymous reviewers helpful comments references borradaile klein log algorithm 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survey algorithms amgad walid faizan mohamed abdur saleh purdue university west lafayette usa umm university makkah ksa may may abstract algorithm finds path containing minimal cost two vertices graph plethora algorithms studied literature span across multiple disciplines paper presents survey algorithms based taxonomy introduced paper one dimension taxonomy various flavors problem one general algorithm capable solving variants problem due space time complexities associated algorithm important dimensions taxonomy include whether algorithm operates static dynamic graph whether algorithm produces exact approximate answers whether objective algorithm achieve goal directed survey studies classifies algorithms according proposed taxonomy survey also presents challenges proposed solutions associated category taxonomy introduction problem one topics computer science specifically graph theory optimal one minimum length criteria source destination surge research algorithms due problem numerous diverse applications applications include network routing protocols route planning traffic control path finding social networks computer games transportation systems count various graph types algorithms consider general graph mathematical object consisting vertices edges aspatial graph contains vertices positions interpreted locations space hand spatial graph contains vertices locations edge planar graph plotted two dimensions edges crossing continuous edges need straight also various settings identified example graph static vertices edges change time contrast graph dynamic vertices edges introduced updated deleted time graph contains either directed undirected edges weights edges either negative weights values real integer numbers relies type problem issued majority algorithms fall two broad categories first category singlesource sssp objective find vertex vertices second category apsp objective find pairs vertices graph computation generate either exact approximate solutions choice algorithm use depends characteristics graph required application example approximate algorithms objective taxonomy produce fast answers even presence large input graph special called spanner also created main graph approximates distances computed given large body literature algorithms computing objective survey present breakdown algorithms appropriate taxonomy taxonomy aims help researchers practitioners application developers understand algorithm works help decide type category algorithms use given specific scenario application domain figure illustrates proposed taxonomy branch describes specific category problem figure taxonomy algorithms taxonomy figure proposed taxonomy classifies various algorithms multiple highlevel branches static branch figure lists algorithms operate graphs fixed weights edge weights denote distance travel time cost weighting criteria given weights fixed static algorithms perform precomputations graph algorithms related work try achieve query time compared precomputation storage requirements static algorithms consists two classical algorithms fall two main categories sssp apsp sssp algorithms compute given vertex vertices apsp algorithms compute pairs vertices graph hierarchical algorithms break problem linear complexity problem lead enhanced performance computation orders magnitude algorithms optimize terms distance time toward target solution distance oracle algorithms include preprocessing step speed query time distance oracle algorithms either exact approximate dynamic branch figure lists algorithms process update query operations graph time update operation insert delete edges graph update edge weights query operation computes distance source destination vertices dynamic algorithms include apsp sssp algorithms algorithms target graphs change time predictable fashion stochastic algorithms capture uncertainty associated edges modeling random variables parametric algorithms compute solutions based values specific parameter replacement path algorithms computes solution avoids specified edge every edge source vertex destination vertex replacement paths algorithms achieve good performance reusing computations edge avoids hand alternative path algorithms also computes shortest path vertices avoids specified edge distinguishing factor categories replacement paths required indicate specific vertex edge hand alternative avoids specified edge problem finds approximate weighted planar divisions related work zwick survey adopts theoretical regards exact approximate shortest paths algorithms zwick survey addresses sssp pairs apsp spanners weighted graph variation distance oracles survey illustrates various variations category adopts handling negative edge weights well directed undirected graphs sen surveys approximate algorithms focus spanners distance oracles sen survey discusses spanners distance oracles algorithms constructed practical applicability static setting sommer surveys query processing algorithms index size query time sommer survey also introduce transportation network class algorithms include algorithms general graphs well planar complex graphs many surveys focus algorithms target traffic applications especially route planning methods related work network denotes graph holzer classify variations dijkstra algorithm according adopted speedup approaches survey emphasizes techniques guarantee correctness argues effectiveness techniques highly relies type data addition best speedup technique depends layout memory tolerable preprocessing time contrast optimal algorithms survey algorithms target heuristic algorithms quickly identify heuristic algorithms aim minimize computation time survey proposes main distinguishing features heuristic algorithms well computational costs goldberg investigates performance shortestpath algorithms road networks theoretical standpoint goldberg reviews algorithms dijkstra illustrates heuristic techniques computing given subset graph survey proves good bounds graph also discusses pruning illustrates algorithms altered compute reaches maintaining time bound original counterparts delling wagner survey route planning speedup techniques problems including dynamic timedependent variants example authors argue shortcuts used static networks work problem definition network essence investigate networks existing techniques adopted bast illustrates techniques fast routing road networks transportation networks bast survey argues algorithms networks different require specialized techniques also survey presents technique performs dijkstra algorithm moreover survey presents two open questions namely achieve despite lack hierarchy transportation networks efficiently compute local searches neighborhoods demetrescu italiano survey algorithms investigate fully dynamic directed graphs emphasis dynamic dynamic transitive closures survey focuses defining algebraic combinatorial properties well tools dynamic techniques survey tackles two important questions namely whether dynamic achieve space complexity whether shortest path algorithms setting solved efficiently general graphs nannicini liberti survey techniques dynamic graph weights dynamic graph topology list classical recent techniques finding trees large graphs dynamic weights target two versions problem namely refer cost updates weights dean survey focuses techniques dynamic setting surveys one special case namely fifo network exposes structural properties allow development efficient algorithms survey presents aspects different predecessors first presents taxonomy aid identifying appropriate algorithm use given specific setting second branch taxonomy algorithms presented chronological order captures evolution specific ideas algorithms time moreover survey comprehensive cover recent algorithms invented publication surveys problem definition given set vertices source vertex destination vertex set weighted edges set find minimum weight input algorithm graph consists set vertices edges graph defined edges directed undirected edges explicit weights weight defined unweighted implicit weight considered calculating algorithm complexity refer size set vertices size set edges static algorithms section review algorithms sssp shortestpath apsp problems sssp definition given graph source compute distances simplest case sssp graph unweighted cormen suggest breadthfirst search simply employed starting scan root vertex inspecting neighboring vertices neighboring vertex probes vertices path minimum number edges source destination vertex identified static algorithms dijkstra algorithm solves single source sssp problem given vertex vertices graph dijkstra algorithm used directed graphs weights algorithm identifies two types vertices solved unsolved vertices initially sets source vertex solved vertex checks edges unsolved vertices connected source vertex destination algorithm identifies shortest edge adds corresponding vertex list solved vertices algorithm iterates vertices solved dijkstra algorithm achieves time complexity one advantage algorithm need investigate edges particularly useful weights edges expensive disadvantage algorithm deals weighted edges also applies static graphs dijkstra algorithm performs search order find optimum known greedy algorithm dijkstra algorithm follows successive approximation procedure based bellman ford optimality principle implies dijkstra algorithm solve dynamic programming equation method called reaching method advantage dynamic programming avoids search process tackling dynamic programming algorithms probe exponentially large set solutions avoids examining explicitly possible solutions greedy dynamic programming versions dijkstra algorithm terms finding optimal solution however difference may get different paths optimal solutions fredman tarjan improve dijkstra algorithm using fibonnaci heap implementation achieves nlogn running time total incurred time heap operations log operations cost fredman willard introduce extension includes log variant dijkstra algorithm structure termed provides constant amortized costs heap operations log amortized cost deletion driscoll gabow propose heap termed relaxed fibonacci heap relaxed heap binomial queue allows heap order violated algorithm provides parallel implementation dijkstra algorithm another line optimization improved priority queue implementations boas boas implementations based stratified binary tree proposed algorithm enables online manipulation priority queue algorithm processing time complexity loglog storage complexity loglog study thorup indicates presence analogy sorting sssp problem sssp harder sorting edge weights thorup describes priority queue giving complexity loglog per operation loglog complexity sssp problem study examines complexity using priority queue given memory arbitrary word size following analogy han proposes deterministic integer sorting algorithm linear space achieves time complexity loglog logloglog sssp problem approach han illustrates sorting arbitrarily large numbers performed sorting small integers thorup proposes deterministic linear space time algorithm building hierarchical bucketing structure avoids sorting operation bucketing structure dynamic set element inserted deleted elements buckets picked unspecified manner list algorithm thorup works traversing component tree hagerup improves algorithm thorup achieving time complexity log width machine word done deterministic linear time space algorithm bellman ford moore develop sssp algorithm capable handling negative weights unlike dijkstra algorithm operates similar manner dijkstra attempts compute instead selecting shortest distance neighbor edges shortest distance selects neighbor edges proceeds cycles order guarantee changes propagated graph provides faster solution bellmanford algorithm dijkstra algorithm unable detect negative cycles operate negative weights static algorithms however negative cycle computed reason due lower total weight incurred due traversal cycle algorithm achieves complexity strong points include ability operate negative weights detect negative cycles however disadvantages include slower compared dijkstra algorithm also algorithm terminate iterations affect graph weights karp addresses issue whether graph contains negative cycle defines concept termed minimum cycle mean indicates finding minimum cycle mean similar finding negative cycle karp algorithm achieves time complexity yen proposes two performance modifications bellman ford moore first involves relaxation edges edge relaxed value vertex changes second modification dividing edges based linear ordering vertices set edges partitioned one subsets followed performing comparisons two sets according proposed partitioning scheme slight improvement yen proposes introduced bannister eppstein instead using arbitrary linear ordering use random ordering result fewer number iterations subsets apsp definition given graph compute distances source vertex destination elements set general case apsp graph edge weights case dijkstra algorithm computed separately vertex graph time complexity logn vast number algorithms proposed handle real shortestpath problem algorithm tries find pairs apsp weighted graph containing positive negative weighted edges algorithm detect existence cycles resolve cycles complexity algorithm number vertices detection cycle done probing diagonal path matrix algorithm find exact vertices pairs store intermediate vertices calculating however using simple update one store information within algorithm steps space complexity algorithm however space complexity reach using single displacement array strong point algorithm handle edges detect cycles main drawback though timing complexity running dijkstra algorithm vertices convert sssp apsp logn timing complexity lower sparse graph many studies proposed better running time algorithm edge weights notable enhancement proposed fredman relies approach approach relies theorem proposed aho hopcroft complexity matrix multiplication using multiplication approach similar shortestpaths shows comparisons suffices solve apsp problem algorithm achieves complexity loglogn table summarizes enhancements proposed edges date table algorithms complexities edges static algorithms time complexity loglogn logn loglogn loglogn logn loglogn logn author best result han takaoka achieve loglogn reduction factor compared result approach focuses distance product computation first nxn matrix divided dimensions determined based specific criterion algorithm proceeds series matrix manipulations index building encoding partitioning steps reaches proposed bound best edge weight complexity logn first algorithm sorts adjacency lists increasing weight fashion performs sssp computation times proceeds iterations first phase uses notion potential edges vertices selects labels edge minimum potential potential derived defined probability distribution complete directed graphs arbitrary edge lengths contain negative cycles algorithm runs two main phases specific invariant logn complexity best positive integer edge weight complexity exponent proposed coppersmith winograd proposed algorithm provides transition fastest exact approximate algorithms linear error rate algorithm focuses directed graphs small positive integer weights order obtain additive approximations approximations polynomial given actual distance pairs vertices distance oracles definition given graph distance oracle encompasses data structure index undergoes preprocessing query algorithm term distance oracle proposed thorup zwick proposes faster alternative sssp apsp algorithms achieved preprocessing graph creating auxiliary data structure answer queries distance oracle operates two phases namely preprocessing phase query phase preprocessing phase information data structures indexes computed contrast query processing phase processes queries efficiently using outcome preprocessing phase distance oracles may return exact approximate distances distance oracle provides efficient space terms data structure index storage query time exact distances fakcharoenphol rao propose algorithm planar graphs balances preprocessing query time preprocessing complexity space time complexity proposed approach creates graph given subset vertices followed computation tree first graph divided set static algorithms bipartite graphs distance matrices bipartite graph need comply condition referred monge condition proposed result holds long noncrossing condition enforced klein propose algorithm fast preprocessing complexity nlog directed planar graph graph include positive negative edges given planar directed graph source vertex algorithm finds curve known jordan curve jordan curve identified passes vertices boundary vertex one passes cutting graph duplicating boundary vertices creates subgraphs algorithm passes five stages recursively compute distances within graph arbitrary boundary vertex compute distances boundary vertices use variant compute graph distances boundary vertex boundary vertices use dijkstra algorithm compute graph distances boundary vertex vertices use dijkstra algorithm compute graph distances source vertix requires time nlogn djidjev proposes faster query time algorithm proves distance oracle space complexity preprocessing query time complexity djidjev objective algorithm product sssp apsp problems proposed algorithm provides complexity class directed graphs separator theorem holds cabello improves preprocessing time provides theoretical proof distance oracle preprocessing space complexity query time complexity slower algorithm proposed djidjev logarithmic factor still covers wider range proposed approach constructs data structure pair vertices answer queries algorithm queries data structure pairs proposes constant algorithm unweighted graphs proves distance oracle space complexity algorithm relies wiener index graph weiner index defines sum distances pairs vertices graph proposed technique shows existence subquadratic time algorithms computing wiener index computing wiener index complexity computing average vertex pairs distances henzinger propose sssp algorithm requiring log time absolute value edge smallest negative value proposed algorithm also achieves similar bound planar graphs planar bipartite graphs also propose parallel dynamic variant algorithm key component approach use based planar separators mozes sommer propose algorithm answer distance queries pairs vertices planar graphs edge weights prove nloglogn distance oracle time complexity space complexity distance queries answered graph preprocessed generated data structure size nloglogc query time cycle vertices approximate distances approximate distance oracles algorithms attempt compute querying distances important note algorithms deal finite metric spaces produce approximate answers algorithms create spanners spanner sparse approximates original graph regarded spanning tree maintains locality aspects graph locality aspects defines stretch stretch multiplicative factor static algorithms indicates amount distances increase graph stretch result utilizing spanner edges algorithms approximate distances triangulation using concept called landmark beacon selected random sampling vertex stores distances landmarks note given definition approximate distance oracles actual still guaranteed retrieved zwick presents apsp algorithm directed graphs utilizes matrix multiplication approximate distance computed log define stretch represents largest weighted edge identified graph aingworth propose apsp algorithm undirected graphs unweighted edges adopt matrix multiplication approach using fast matrix multiplication error propose two algorithms one achieves additive error time log also provide estimate graph paths distances log another algorithm achieves query time log dor improve previous surplus results proposing apsp algorithm computes surplus estimate also show surplus estimate takes computed work relies one main observation set vertices represent vertices high degree value words set vertices said represent set vertices neighbor cohen zwick improve work proposed dor weighted undirected graphs proposing algorithm computes surplus estimate distances estimate show finding estimated distances directed graphs hard problem similar boolean matrix multiplication makes proposed approximation algorithm valid undirected graphs algorithm relies two important aspects partitioning graph assumption directed use sssp algorithm dijkstra patrascu roditty improve stretch bound intermediate vertices expense increasing space requirements achieve approach defines notion balls defined balls around vertex grow geometrically stop based specific criteria given vertices happens balls intersect agarwal also propose estimate approach implemented distributed fashion approach mainly meant compact routing protocols aims characterize space time approximate distance queries sparse graphs approaches space versus query time depends number edges spanners elkin peleg propose general space complexity constant also constants claim stretch spanners minimized simultaneous evaluation fashion baswana sen propose spanner stretch computed size provide theoretical proof spanner stretch computed without distance computation linear time novel clustering technique proposed approach take rounds round explores adjacency vertex list order determine edges need removed advantage approach applicability various computational environments synchronous distributed model external memory model crcw pram model planar graphs thorup proposes distance oracle approach provides constant number separators contrast lipton vertex stores distances set landmarks per level process performed recursively logn levels static algorithms kawarabayashi propose planar graph algorithm provides tunable polylogarithmic query time achieved maintaining linear space requirement respect graph size proposed approach achieves preprocessing time complexity nlog query time log achieves faster running time thorup approach computes set connections covers vertices graph every vertex containing connections contrast subset vertices covered using kawarabayashi approach approach times number paths space complexity complex networks chen proposes distance oracle random graphs estimate space complexity approach adopts distance oracle proposed thorup zwick use vertices landmarks adaptation includes selecting vertices highest degree landmarks encodes vertex labels search algorithm based adding annotations vertices edges graph consist additional information information allows algorithm determine part graph prune search space simple search hart propose simple algorithm termed algorithm proposes heuristic approach finding unlike dijkstra algorithm informed algorithm searches routes lead final goal optimal greedy algorithm sets aside algorithms ability maintain distance traveled account always finds admissible heuristic function used strong point algorithm meant faster dijkstra since explores less number vertices downside use good heuristic method reach variants algorithm use landmarks techniques order achieve better performance various setups goldberg werneck propose preprocessing phase initially number landmarks selected followed computation stored vertices landmarks propose technique using computed distances addition triangle inequality property technique based algorithm landmark chosen triangle inequality gutman offers comparable solution problem work based concept reach gutman technique relies storing reach value euclidean coordinates vertices advantage gutman approach combined algorithm compared work goldberg werneck gutman outperforms proposed technique given one landmark performs worse given sixteen landmarks downside gutman approach depends assumptions longer preprocessing complexity inapplicability dynamic setting potamias propose approximate technique distance estimation large networks theoretical proof presented indicate problem propose heuristic solutions specific propose smart landmark selection technique yield higher accuracy reaching times less space selecting landmarks random among evaluated strategies centrality robust degree strategy also strategies based partitioning exhibit better computational cost across datasets kleinberg propose algorithm provable performance guarantees triangulation embedding algorithms basically designed triangulation static algorithms use triangle inequality deduce unmeasured distances indicate multiplicative error fraction distances achieved reconstruction given constant number beacons algorithm also achieves constant distortion distances claim dijkstra algorithm enhanced precomputing shortestpath distances propose partition graph clusters perform two operations store start end point store shortest connection pair clusters proposed algorithm achieves scaling factor contrast dijkstra algorithm advanced search edge labels approach relies precomputing information edge vertices superset represents vertices start edge graph first partitioned set regions size alongside precomputed set boundary vertices order compute edge flags sssp computation done regions boundary vertices various work kohler schulz lauther present variations propose algorithm sparse directed graphs edge weights termed approach approach preprocesses graph data generate information speeds queries dividing graph regions determining arc specific region lies given suitable partitioning scheme search approach times faster standard dijkstra algorithm large graph schilling present improvement searching region approach achieves subnetwork million vertices goldberg werneck propose based search landmarks alt algorithm uses triangle inequality show precomputing distances set landmarks bound shortest path computational cost propose average landmarks corners graph turn approach leads speed route planning bauer study systematically combine techniques proposed dijkstra algorithm adding approaches hierarchical approaches present generalized technique demonstrates performance improved results show highway vertex routing achieves best maintaining adequate preprocessing cost also present hierarchical search landmarks alt algorithm dense graphs delling present algorithm termed public transit router raptor raptor based dijkstra algorithm probes route graph raptor works fully dynamic scenarios extended handle example flexible departure times bauer delling uses hierarchical based techniques extend edge flag approach using contraction hierarchies preprocessing hence tackling main processing drawback edge flags proposed work termed shortcuts sharc short key observation sharc enough set edge flags edges focuses preprocessing important edges another observation sharc incorporates hierarchical aspects implicitly sharc also extends edge flag approach achieve fast unidirectional query algorithm maue propose algorithm utilizes precomputed cluster distances pcd proposed approach first partitions graph clusters followed precomputing shortest connections pairs clusters pcds produce bounding factors distances used prune search compared algorithm turn achieves static algorithms comparable alt using less space hierarchical hierarchical algorithms deal generating vertex hierarchy preprocessing stage hierarchical structure prominent areas road networks exhibits hierarchical properties ordering important streets motorways urban streets general methods using contraction hierarchies provide low space complexity contraction hierarchies contain many variants methods highway hierarchies vertex routing hand routing hub labels provide fast following sections discuss various algorithms follow hierarchical approach highway hierarchies highway hierarchies capture properties example highway edges exhibit better representation shortest paths although may located source destination vertices algorithm generates hierarchy graphs enables fast query time correctness guarantees sanders schultes propose static undirected highway hierarchies algorithm around notion correctly defining local search highway network appropriately define local search one visits tuning parameter closest vertices source target highway edge created lies path source vertex destination vertex edge within closest vertices source destination nannixini propose algorithm relies lengths extend original algorithm sanders schultes case directed graphs aim find fastest paths large dynamic road network quasi updates contraction hierarchies contraction hierarchy level vertex reaching levels hierarchical models improve query performance search conducted upwards manner graph reduced space complexity edges stored lower endpoints geisberger propose contraction hierarchies vertices initially ordered importance hierarchy generated contracting least important vertices iterative manner contracting process replacing passing vertex call shortcuts propose hierarchical algorithm utilizes bidirectional search technique batz propose version algorithm tackles road networks proposes fast exact route planning algorithm issue faces space complexity tackle problem using approximations functions lead significant space reduction preserving correctness proposed approach relies approximating shortcuts acquire edge weights weights used bidirectional search algorithm create corridor shortcuts searched kieritzcite propose distributed memory parallelization contraction hierarchies algorithm identifies vertices contracted every iteration parallelization achieved process contracts vertices independently vertices contractions overlap attempt approximate ordering sequential algorithms used static algorithms geisberger devise algorithm based contraction hierarchies calculate preprocessing step relies hierarchical properties road networks order add shortcut edges use modified version dijkstra algorithm visits hundred vertices turn makes suitable implement mobile devices graphs overlay graph set vertices lie specific level level use vertex upper levels turn method depends correct selection vertices act landmarks higher levels schulz propose decomposition method targets space reduction method precomputed replaces weights single edges weight equal length result subgraph smaller size compared original graph subgraph distances set vertices graph distance set vertices original graph holzer introduce several vertex selection criteria overlay graphs include criteria determine representative subset original graph investigate criteria effectiveness multilevel overlay graphs achieved computation transit vertex routing transit vertex routing precomputed shortest paths landmarks identified graph algorithm requires extensive preprocessing exhibits fast query time requires limited number landmarks located different locations bast propose transit vertex routing suggest vertical horizontal sweep sufficient compute set transit vertices also illustrate techniques make approach arz propose variant contraction hierarchies achieves order magnitude speeds similar time needed find contraction hierarchies propose locality filter affect query time hub labeling modeling road networks graph method used computing shortest paths one method used modeling process labeling algorithms labeling introduced distributed computing field labeling preprocessing stage vertex computed assigned forward label reverse label forward label encompasses set vertices vertex contains computed distance dist reverse label consists set vertices vertex contains computed distance dist labels later used query stage determine vertices minimize distance source destination label perceived set hubs vertex direct connection labeling algorithm ensures two vertices one hub common computing shortest path hub labeling starts preprocessing vertices vertex precomputes distance set landmarks vertex label query algorithm fast long number landmarks source destination vertices small storing labels consecutive manner allows algorithm exhibit good locality abraham delling propose labeling scheme given vertex considers dynamic algorithms sets vertices visited forward contraction hierarchy reverse contraction hierarchy contraction hierarchies algorithm computes intersection forward reverse sets contain vertex babenko propose approximation algorithm producing small labels main target reduce size maximum reduction process leads unbalanced solutions vertices skewed label sizes propose approximation algorithm maximum label size runs logn proposed approach reduces problem problem cohen propose data structure storing reachability label using cover paths graph vertex precomputes label lin lout pair least one vertex lout lin distance labeling query finds source totdestination finding minimum distance lout lin label lout lin size label guaranteed polynomial preprocessing time approximately logn finding cover invariant paths whose size larger set chang propose distance labeling size smaller another labeling approach preprocessing phase algorithm stores parent function assigns parent vertex vertex avoiding preprocessing proposed approach performs vertex separation graph divides multiple connected subgraphs graph decomposed minimal tree represents set vertices set edges approach uses distance query compute minimum distance time complexity query processing represents width represents height decomposed tree highway node routing motivation behind using highway node routing prominent vertices overlap various generate sparse overlay graphs result would faster query processing lower space overhead schultes sanders proposes dynamic algorithm allows query time thousand times faster compared dijkstra algorithm choice vertices achieved capitalizing previous results addition using required vertex sets defined highway hierarchies algorithms simplify complications computation prepreprocessing step also leads simplification query processing algorithm especially dynamic variants abraham suggests road networks necessarily significant highway dimension proposed algorithm relies realizing balls specific radius every exits sparse set length vertex set every ball radius contains less number vertices set sparse dynamic algorithms main requirement dynamic algorithms process updates query operations efficiently online fashion update operation edges inserted deleted graph query operation distance vertices computed fully dynamic algorithms process insertions deletions incremental algorithms process insert operations delete operations decremental algorithms process delete operations insert operations implies incremental decremental algorithms dynamic algorithms partially dynamic following section illustrates algorithms demonstrate aforementioned differences apsp algorithms reports distances two vertices graph algorithms attempt answer distance queries two vertices dynamically maintaining changes occur graph inserts deletes updates demetrescu italiano propose fully dynamic algorithm directed graphs edge weights every edge predefined number values algorithm achieves amortized time complexity log update operations achieving optimal query processing time proposed algorithm update operation inserts deletes vertex addition possible edges algorithm also maintains complete distance matrix updates thorup improves demetrescu italiano reducing graph problem smaller set decremental problems thorup adopts idea minimum spanning tree utilizing efficiency decremental algorithm solve shortestpaths problem bernstein presents algorithm apsp undirected graph positive edge weights bernstein algorithm achieves update time almost linear query time loglogn proposed query algorithm deterministic update procedure randomized algorithm behavior depends distance source vertex destination vertex since known beforehand algorithm relies guessing several different values roditty zwick propose fully dynamic apsp algorithm unweighted directed graphs algorithm randomized correctness returned results claimed high proposed algorithm passes set phases rely ideas decremental algorithm demonstrate incremental decremental versions sssp problems similar terms complexity static problem directed undirected graphs bernstein proposes approximate algorithm improves existing studies respect delete operation edge weight increase algorithm computes decremental weighted graphs approach achieves update time using randomized algorithm henzinger enhances fastest deterministic algorithm shiloach even achieving update time also achieves constant query time also propose deterministic algorithm update time query time loglogn introduce two techniques namely lazy tree algorithm proposed approach maintains tree bounded distance tree based technique algorithm reports distances given source vertex dynamic algorithm computes update query operations online fashion update operation inserts deletes modify edge weight query operation probes distance source vertex given target vertex fakcharoenphol rao propose algorithm planar graphs edge weights algorithms achieves time complexity nlog performs update query operations log amortized time proposed algorithm uses monge matrices combination dijkstra algorithms searching time bernstein roditty propose dynamic algorithm achieve update time better without sacrificing query time specific obtain total update time constant query time main type graphs achieve result moderately sparse graphs bernstein roditty propose two randomized decremental algorithms operate unweighted undirected graph two approximate problems henzinger improve update operation time bernstein roditty maintaining constant query time algorithm utilizes data structure given parameter constant maintains vertices referred centers main property data structure every vertex within specific distance tree termed tree proposed algorithm property data structure fastest moderately small algorithms algorithm processes graphs edges associated function known function function indicates much time needed travel one vertex another vertex query operation probes path source destination vertex graph returned result represents best departure time found given time interval algorithms kanoulas propose algorithm finds set fastest paths source destination given specified time interval specified interval defined user represents departure arrival time query algorithm finds partitioning scheme time interval creates set assigned set fastest paths unlike algorithm proposed algorithm probes graph instead multiple times ding propose algorithm finds departure time minimizes travel time road network also traffic conditions dynamically changing road network algorithm capable operating variety graphs george propose graph tag graph changes topology time tag vertices edges modeled time series apart time dependence also responsible managing edges vertices absent instance time propose two algorithms compute using network best best finds time given query using greedy algorithm hand best algorithm finds best earliest travel time entire period using tag time complexity best logt respectively represents edges represents vertices represents time instance ding propose algorithm problem large graph edge delay function denotes time taken source vertex destination vertex given time user queries least travel time ltt proposed algorithm achieves space complexity time complexity nlogn stochastic algorithms algorithms nannicini propose bidirectional algorithm restricts search set vertices defined algorithm bidirectional algorithm operates two modes first mode namely theforward search algorithm run graph weighted specific cost function second mode namely backward search run graph weighted function delling wagner reanalyzes various technique concluded techniques operate graphs guarantee correctness augmenting preprocessing query phases subroutines foschini study computational complexity problem timedependent graphs conclude linear functions causes shortest path destination changes logn times study complexity arrival time mapping problem parametric problem order analyzed correctly demiryurek propose technique computation timedependent spatial graphs propose technique based bidirectional algorithm operates two main stages first stage partitions graph set partitions overlap next calculate distance label vertices borders second state online probes fastest path utilizing heuristic function based computed distance labels results indicate proposed technique decreases computation time reduces storage complexity significantly stochastic algorithms stochastic attempts capture uncertainty associated edges modeling random variables objective becomes compute based minimum expected costs two notable lines research problem adaptive nonadaptive algorithms adaptive algorithms determine next best next hop would based current graph certain time instance algorithms focus minimizing length path adaptive algorithms mahmassani propose algorithm determine apriori source vertices single destination vertex computation done departure time busy time graph also propose apriori nikolova propose algorithm maximizes probability without exceeding specific threshold length define probabilistic model edge weights drawn known probability distribution optimal path one maximum probability indicating path pass specific threshold algorithms loui proposes using utility function length path utility function monotone utility function exhibits linear exponential behavior parametric algorithms becomes separable edge lengths allows utility function identified using classical algorithms via paths maximize utility function nikolova propose algorithm optimal route planning uncertainty define target function path length departure time starting source indicate path start time jointly optimizable due penalizing behavior exhibit late early arrivals also indicated joint optimization reducible classic shortestpath algorithms parametric algorithms parametric objective compute vertices based specific parameter probes parameter values known breakpoints tends change edge value varies based linear function parameter value mulmuley shah propose model computation variant parallel random access machine proof starts definition parametric complexity problem plotting weights function results optimal cost graph concave breakpoints defined fixed set linear weight functions fixed graph young propose model computed edge values makes tractable predecessors tractability allows obtaining polynomial time use algorithm proposed karp orlin modify use fibonacci heaps instead order improve performance erickson proposes algorithm computing maximum flow planar graphs algorithm maintains three structures namely edge spanning tree predecessor dual vertex set slack value dual edge set compute initial predecessor pointers slacks nlogn using dijkstra algorithm replacement algorithms consider graph set vertices set edges every edge source destination replacement path algorithm calculates avoids emek propose algorithm computes replacement path time algorithm requires nlog time preprocessing stage hloglogn time answer replacement path query number hops weighted planar directed graph roditty zwick propose randomized algorithm replacement path unweighted directed graph complexity algorithm monte carlo algorithm improves vickrey pricing problems factor bernstein proposes approximate algorithm computes paths log nlog nlogn mlog time largest smallest graph bernstein algorithm achieves running time applied simple problem alternative algorithms alternative algorithms alternative problem reports paths avoid given vertex edge termed unwanted vertex unwanted edge key difference alternative user required specify unwanted vertex edge replacement paths goal alternative path problem reusing previously computed results unwanted vertex edge turn achieves better performance existing algorithms dynamic solve alternative problem high complexity update operation xie propose storage schemed termed ispqf extension reduces number vertex space complexity forest spqf spqf algorithm find alternative single source source destination avoids vertex well pairs set sources set destinations avoid vertex weighted region algorithms mitchell papadimitriou define weighted region problem wrp generalization shortest path problem obstacles problem assumes plane subdivided weighted polygonal regions objective minimize cost according weighted euclidean metric study mitchell papadimitriou sheds light discriminating properties weighted region problem planar divisions proposes algorithm runs number vertices number bits required encode problem instance specific log maximum integer representing vertices triangulation error value tolerated mata mitchell propose algorithm compute approximate weighted planar subdivision problem constructing sparse graph termed approach uses snell law refraction divide vertices cones bound path vertex complexity build graph vertices number cones scanned produces paths within factor optimal solution conclusion paper devise taxonomy problem branch taxonomy illustrate discriminating features highlight research taxonomy provides investigators problem guideline required problem definition maps within current related work acknowledgements walid aref research supported part national science foundation grant iis references references abraham delling labeling algorithm shortest paths road networks experimental algorithms abraham delling goldberg werneck hierarchical hub labelings shortest paths algorithmsesa abraham fiat goldberg werneck highway dimension shortest paths provably efficient algorithms proceedings annual symposium discrete algorithms pages agarwal godfrey approximate distance queries compact routing sparse graphs ieee infocom pages aho hopcroft design analysis computer algorithms addisonwesley longman publishing boston usa edition aiello chung random graph model massive graphs stoc aingworth chekuri motwani fast estimation diameter shortest paths without matrix 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stabilization disturbed linear systems digital channels jan mohammad javad khojasteh mojtaba hedayatpour jorge massimo franceschetti present control strategy stabilizing scalar linear system digital communication channel bounded delay presence bounded system disturbance propose scheme determine lower bounds packet size information transmission rate sufficient stabilization show small values delay timing information implicit triggering events enough stabilize system positive rate contrast delay increases beyond critical threshold timing information alone enough stabilize system transmission rate begins increase finally large values delay require transmission rates higher prescribed classic theorem results numerically validated using linearized model inverted pendulum index control communication constraints control quantized control rate transmission apparently counterintuitive result explained noting act triggering essentially reveals state system perfectly tracked controller previous work quantifies information implicit timing triggering events function communication delay given triggering strategy showing phase transition behavior system disturbances delay communication channel small enough positive rate transmission needed achieve exponential stabilization delay communication channel larger critical threshold implicit information act triggering enough stabilization transmission rate must increase results compared implementation subject delay ntroduction literature however considered extent implicit information triggering events still valuable presence system disturbances disturbances add additional degree uncertainty state estimation process beside one due unknown delay effect properly accounted motivation consider stabilization linear timeinvariant system subject bounded disturbance communication channel bounded delay comparison consider weaker notion stability requiring state bounded times beyond fixed horizon without imposing exponential convergence guarantees allows simplify treatment derive simpler control strategy design scheme strategy show size packet transmitted channel every triggering event certain fixed value small values delay strategy achieves stabilization using implicit information transmitting rate arbitrarily close zero contrast values delay given threshold transmission rate must increase eventually surpasses one prescribed classic theorem follows small values delay successfully exploit implicit information triggering events compensate presence system disturbances hand large values delay imply information excessively aged corrupted disturbance increasingly higher communication rates required results numerically validated implementing strategy stabilize inverted pendulum linearized equilibrium point communication channel proofs omitted brevity appear full elsewhere networked control systems ncs feedback loop closed communication channel fundamental component systems cps context theorems state minimum communication rate achieve stabilization equal entropy rate system expressed sum logarithms unstable modes early examples datarate theorems appeared key later contributions appeared works consider communication channel capable noiseless transmission finite number bits per unit time evolution system extensions noisy communication channels considered stabilization channels including erasure channel special case studied additional formulations include stabilization systems random open loop gains channels stabilization switched linear systems systems uncertain parameters multiplicative noise optimal control stabilization using strategies paper focuses case stabilization using eventtriggered communication strategies context key observation made delay communication process system disturbances controller knowledge triggering strategy possible stabilize system positive khojasteh franceschetti department electrical computer engineering university california san diego hedayatpour faculty engineering applied science university regina canada department mechanical aerospace engineering university california san diego mkhojasteh massimo cortes hedayatm notation throughout paper represent set real natural numbers respectively also log represent base natural logarithms respectively function let denote limit namely addition resp denote nearest integer less resp greater equal denote modulo function mod whose value remainder division sign denotes sign roblem also define triggering interval tks referring generic triggering reception time convenience skip tkr tkc setting classical theorem states controller stabilize plant receives information least rate let number bits transmitted sensor time define information transmission rate formulation block diagram networked control system tuple represented figure plant described scalar lim sup since every triggering interval sensor sends bits tks lim sup controller estimated state represented evolves times fig system model linear model plant state control input respectively represents process disturbance latter upper bounded positive real number positive real number positive real number assume sensor measures system state exactly controller acts infinite precision without delay however measured state sent controller communication channel supports finite data rate subject bounded delay precisely sensor transmits packet via communication channel controller receive packet entirely without error unknown bounded delay sequence triggering times sensor transmits packet length tks bits denoted tks sequence times controller receives corresponding packet decodes denoted tkc communication delays uniformly finite real number follows tkc tks communication delay tkc starting assume sensor knowledge time actuator performs control action ensure sensor also compute time practice corresponds assuming instantaneous acknowledgment actuator sensor via control input discussed obtain causal knowledge one monitor output actuator provided control input changes reception time case sensor access system state one use narrowband signal control input excite specific frequency state signal time control action applied state estimation error defined use error determine triggering event occurs controller design ensure property similar practical stability system iii ontrol esign section proposes control strategy along quantization policy generate send packets every triggering event stabilize scalar continuoustime linear system described section along way also characterize sufficient information transmission rate accomplish assume triggering event occurs positive real number controller knows triggering time also knows follows may compute exact value transmitting one single bit every triggering time general however controller knowledge delay knows bound let estimate constructed controller knowing using decoded packet received communication channel define following updating procedure called jump strategy triggering time sensor encodes system state packet size consisting sign quantized version denote send controller using bound decoding received packet controller reconstructs quantized version finally controller estimate follows sign jea using quantization policy described theorem sensor causal knowledge delay communication channel sensor calculate time next show proposed scheme triggering intervals uniformly lower bounded consequently show zeno behavior namely infinitely many triggering events finite time interval lemma consider model plant dynamics estimator dynamics triggering strategy jump strategy packet size satisfies tks holds next propose quantization algorithm rely lemma lower bound packet size ensure theorem consider model plant dynamics estimator dynamics triggering strategy jump strategy control enough information state estimation error satisfies exists quantization policy achieves packet size max log next show using encoding decoding scheme sensor causal knowledge delay communication channel compute state estimated controller proposition consider model plant dynamics estimator dynamics triggering strategy jump strategy using lemma deduce rtr constant design parameter find lower bound size packet ensured next result bounds large difference triggering time quantized version lemma model plant dynamics estimator dynamics triggering strategy jump strategy using rtr lim sup sensor chooses packet size large enough satisfy following equation possible frequency transmission events triggered captured triggering rate noting jump strategy initial conditions possible delay process noise values combining bound theorem arrive following result corollary consider model plant dynamics estimator dynamics triggering strategy jump strategy control enough information state estimation exists error satisfies quantization policy achieves delay process noise realization information transmission rate max log figure shows sufficient transmission rate function bound channel delay expected rate starts zero increases goes theorem next result ensures property similar practical stability system theorem consider model plant dynamics estimator dynamics triggering strategy jump strategy assume pair stabilizable control enough information state estimation error satisfies sensor use quantization policy proposed theorem exists time real number provided packet size lower bounded addition add process noise linearized system model vector length four assume elements upper bounded also simple feedback control law derived hurwitz let follows note although theorem holds linear system delay linearizion valid sufficiently small values sufficient rate theorem rate channel delay upperbound sec fig illustration sufficient transmission rate function corollary follows transmission rate lower bounded sufficient ensure property similar practical stability stated theorem imulation implement proposed control scheme dynamical system linearized inverted pendulum section initially mathematical model inverted pendulum mounted cart presented nonlinear equations linearized equilibrium state system addition canonical transformation applied linear system decouple equations motion consider problem motion pendulum constrained plane position measured angle assume inverted pendulum mass length moment inertia also pendulum mounted top cart mass constrained move direction nonlinear equations governing motion cart pendulum written follows damping coefficient pendulum cart gravitational acceleration linearizion define equilibrium position pendulum small deviations derive linearized equations motion using small angle approximation let define state variable position velocity cart respectively assuming one write evolution time follows moreover design first three coordinates diagonalized system stable state estimation controller simply constructs follows cos diagonalization eigenvalues system hence three four modes system stable need actuation also gain system diagonalizable eigenvalues distinct result diagonalization matrix enables apply theorem unstable mode system consequently stabilize whole system using eigenvector matrix diagonalize system obtain starting unstable mode system follow using problem formulation section estimated state unstable mode evolves times tkc starting triggering occurs estate estimation error unstable mode let eigenvalue corresponding unstable mode equal using theorem choose exceeds triggering function depends random channel delay upper bound second row figure evolution unstable state state estimation presented finally last row figure represents evolution actual states linearized system time finally figure presents simulation information transmission rate versus delay communication channel stabilizing linearized model inverted pendulum simulations theorem rate channel delay upperbound sec fig information transmission rate simulations compared datarate theorem note rate calculated simulations start zero delay minimum channel delay upper bound equal one sampling time seconds example chosen simulations simulation time seconds size packet max log packet size simulation two differences lower bound provided theorem packet size integer used ceiling operator since least one bit send packet take maximum result ceiling operator simulation results following simulation parameters chosen system simulation time seconds sampling time seconds theorem developed based continuous system simulation environments digital tried make discrete model close continuous model choosing small sampling time however minimum upper bound channel delay equal one sampling time set three simulations carried follows simulation assumed process disturbance zero channel delay upper bounded sampling time simulation assumed process disturbance upper bounded channel delay upper bounded sampling time finally simulation assumed process disturbance upper bounded channel delay upper bounded simulation results simulation presented figure column represents different simulation first row shows triggering function absolute value state estimation error unstable coordinate soon absolute value error equal greater triggering function sensor transmit packet jumping strategy adjusts reception time practically stabilize system amount error onclusions presented control scheme stabilization noisy scalar continuous linear timeinvariant systems communication channel subject random bounded delay also developed algorithm quantized version estimated states leading characterization sufficient transmission rate stabilizing system illustrated results linearization inverted pendulum different channel delay bounds future work study identification necessary conditions transmission rate investigation effect delay nonlinear systems implementation proposed control strategies real systems acknowledgements research partially supported nsf award eferences hespanha naghshtabrizi survey recent results networked control systems proceedings ieee vol kim kumar systems perspective centennial proceedings ieee vol special centennial issue murray astrom boyd brockett stein future directions control world ieee control systems vol wong brockett systems finite communication bandwidth constraints stabilization limited information feedback ieee transactions automatic control vol baillieul feedback designs controlling device arrays communication channel bandwidth constraints aro workshop smart structures pennsylvania state univ tatikonda mitter control communication constraints ieee transactions automatic control vol nair evans stabilizability stochastic linear systems finite feedback data rates siam journal control optimization vol sahai mitter necessity sufficiency anytime capacity stabilization linear system noisy communication link part scalar systems ieee transactions information theory vol tatikonda mitter control noisy channels ieee transactions automatic control vol matveev savkin estimation control communication networks springer science business media khina halbawi hassibi almost practical tree codes ieee international symposium information theory isit july sukhavasi hassibi linear anytime codes control noisy channels ieee transactions automatic control vol sec bit sec bit triggering function sec bits triggering function triggering function time seconds time seconds time seconds time seconds time seconds time seconds time seconds time seconds time seconds fig simulation results first row represents absolute value state estimation error unstable mode system second row represents unstable mode state estimate finally last row represents evolution actual states real system time minero franceschetti dey nair data rate theorem stabilization feedback channels ieee transactions automatic control vol minero coviello franceschetti stabilization markov feedback channels general case ieee transactions automatic control vol kostina peres ranade control systems uncertain gain annual allerton conference communication control computing yang liberzon feedback stabilization switched linear systems unknown disturbances constraints ieee transactions automatic control ranade sahai control capacity information theory isit ieee international symposium ieee ding peres ranade zhai multiplicative noise stymies control arxiv preprint ranade sahai estimation control communication control computing allerton annual allerton conference ieee tatikonda sahai mitter stochastic linear control communication channel ieee transactions automatic control vol kostina hassibi tradeoffs control communication control computing allerton annual allerton conference ieee khina nakahira hassibi algorithms optimal control feedback ieee annual conference decision control cdc dec khina pettersson kostina hassibi control awgn channels via analog joint coding decision control cdc ieee conference ieee khojasteh tallapragada franceschetti value timing information control scalar case annual allerton conference communication control computing tallapragada stabilization linear systems bounded bit rates ieee transactions automatic control vol wang lemmon stabilizing quantized event triggered control systems proceedings acm international conference hybrid systems computation control acm pearson hespanha liberzon control minimal encoding encoders ieee transactions automatic control vol ling bit rate conditions stabilize scalar linear system based event triggering ieee transactions automatic control appear linsenmayer blind data rate bounds containability scalar systems vol kofman braslavsky level crossing sampling feedback stabilization constraints ieee conference decision control cdc ieee khojasteh tallapragada franceschetti value timing information control arxiv preprint khojasteh tallapragada franceschetti versus control communication channels ieee annual conference decision control cdc dec ling bit rate conditions stabilize linear system feedback dropouts ieee transactions automatic control vol lakshmikantham leela martynyuk practical stability nonlinear systems world scientific

mar numbers enochs estrada iacob abstract define invariants modules commutative noetherian local ring show periodicity invariants provided hypersurface case also gorenstein see finitely generated matlis dual numbers introduction consider commutative noetherian local ring known module complete projective resolution finite gorenstein projective dimension prove finitely generated finite gorenstein projective dimension construct complete projective resolution homotopically minimal complexes unique isomorphism modules free modules finite ranks ranks modules usually denoted called betti numbers boundedness sequence betti numbers module well interplay boundedness betti numbers eventual periodicity module studied intensively see example focus invariants rank module call invariants numbers see another way define invariants arbitrary module use analogous procedure construct complete injective resolution homotopically minimal complexes hence unique isomorphism define invariants prime ideal mathematics subject classification key words number complete projective resolution eventually periodic complex matlis duality enochs estrada iacob main results theorem theorem give sufficient conditions residue field guarantee periodicity numbers finitely generated finite gorenstein projective dimension respectively periodicity numbers prove theorem eventually periodic minimal projective resolution period every finitely generated invariants periodic period also prove theorem hypothesis invariants periodic period every module finite gorenstein injective dimension second part paper consider commutative local gorenstein ring finitely generated prove minimal complete projective resolution minimal complete injective resolution follows also see section definitions preliminaries recall module gorenstein projective exact hom roj exact complex projective modules ker definition module finite gorenstein projective dimension exists exact sequence gorenstein projective modules integer least property gorenstein projective dimension short exists infinite gorenstein projective dimension gorenstein injective modules gorenstein injective dimension defined dually tate cohomology modules defined means complete resolutions recall definition definition module complete projective resolution exists diagram projective resolution totally acyclic complex map complexes isomorphism numbers known module complete projective resolution finite gorenstein projective dimension particular gorenstein every complete projective resolution complete injective resolutions defined dually known module complete injective resolution finite gorenstein injective dimension gorenstein ring every module complete injective resolution numbers numbers let commutative local noetherian ring let rmodule finite gorenstein projective dimension complete projective resolution finitely generated choose minimal projective resolution recall complex said homologically minimal homology isomorphism isomorphism mod complex said homotopically minimal homotopy isomorphism isomorphism homologically minimal also homotopically minimal thus minimal projective resolution homotopically minimal fact homologically minimal see page chapter unique isomorphism show first finitely generated also get homotopically minimal also unique isomorphism use following lemma let finitely generated gorenstein projective reduced nontrivial projective direct summands exists exact hom roj exact complex finitely generated free module proof let finitely generated gorenstein projective reduced dual hom also also exact projective cover also finitely generated gorenstein projective reduced since dual module finitely generated gorenstein projective module also reduced exists short exact sequence enochs estrada iacob projective cover gorenstein projective finitely generated reduced gives exact sequence finitely generated projective finitely generated gorenstein projective reduced projective preenvelope therefore projective envelope injective map also since coker direct summand finitely generated gorenstein projective reduced projective envelope injection coker also finitely generated gorenstein projective reduced also exact projective cover proposition exists short exact sequence projective preenvelope finitely generated reduced gorenstein projective continuing obtain exact hom roj complex finitely generated free prove finitely generated finite gorenstein projective dimension construct complete projective resolution unique isomorphism see let partial minimal projective resolution gorenstein projective since resolution minimal also reduced nontrivial projective direct summands using lemma hard see exists exact hom roj exact complex finitely generated free module ker since complex minimal projective resolution follows homologically minimal complex page finitely generated construct complete projective resolution numbers homotopically minimal unique isomorphism call diagram minimal complete projective resolution arbitrary free modules finite rank usual ranks denoted denote ranks numbers called betti invariants call invariants invariants see another way define invariants arbitrary module use analogous procedure construct complete injective resolution homotopically minimal complexes hence unique isomorphism using matlis bass define bass invariants prime ideal invariants arbitrary prime ideal main results periodicity invariants recall first following definition complex said eventually periodic period saying periodic period obvious meaning remark minimal complete projective resolution finitely generated eventually periodic trivially also eventually periodic using minimality seen fact periodic case period see say invariants periodic period however may happen invariants periodic without periodic speak invariants periodic without associated complex periodic enochs estrada iacob prove residue field eventually periodic minimal projective resolution numbers finitely generated finite gorenstein projective dimension periodic prove hypothesis numbers periodic finite gorenstein injective dimension use deduce following balance result see also section theorem let commutative ring complete projective injective resolutions respectively equivalently finite homologies hom hom naturally isomorphic analogous result also holds proof without lost generality let assume hom hom hom hom ker ker follows corollary note case using fact class gorenstein injective modules closed cokernels monomorphisms second statement follows way prove main results theorem residue field eventually periodic minimal projective resolution period every finitely generated module finite gorenstein projective dimension invariants periodic period proof let minimal complete projective resolution since eventually periodic period complex periodic period consequently complex periodic period gives every numbers minimal complete projective resolution homologically since minimal complex follows proposition consider complex thus ker homology module ker vector space dimension since since see theorem residue field eventually periodic minimal projective resolution period module finite gorenstein injective dimension invariants periodic period proof let minimal complete projective resolution let minimal complete injective resolution hom hom naturally isomorphic homology modules theorem periodic period hom periodic period get hom hom bass see hom vector space whose dimension precisely remark eventually periodic betti sequence bounded converse true general counterexample given schulz proposition eisenbud proved converse hold group rings finite groups also holds commutative noetherian local setting rings considered complete intersections fact shown hypersurface complete intersection ring codimension one minimal free resolution eventually becomes periodic hypersurface ring theorem theorem hold remark main results hold provided eventually periodic minimal projective resolution betti numbers enochs estrada iacob bounded corollary betti numbers bounded hypersurface theorems hold hypersurface matlis duality let commutative local gorenstein ring let finitely generated exists diagram homotopically minimal unique isomorphism show minimal complete injective resolution module denotes matlis dual homr since injective exact complexes injective modules let ker injective preenvelope hom injective envelope direct summand proof corollary exact sequence rtj therefore rtj projective precover since projective cover follows direct summand rtj injective envelope since exact finitely generated free ker gorenstein projective follows exact complex injective modules ker gorenstein injective also gorenstein flat case injective module hom hom injective module follows gorenstein injective thus totally acyclic injective complex since minimal projective resolution follows minimal injective resolution similarly minimal injective resolution theorem gorenstein injective module reduced thus fact injective cover similarly injective cover numbers thus minimal left injective resolution complete injective resolution minimal follows also references bass ubiquity gorenstein rings math bergh complexity periodicity colloq christensen jorgensen tate homology via pinched complexes transactions ams eisenbud homological algebra complete intersection application group representations transactions ams enochs jenda gorenstein injective projective modules mathematische zeitschrift enochs estrada iacob balance unbounded complexes bull london math enochs jenda relative homological algebra walter gruyter gruyter exposition math enochs jenda relative homological algebra walter gruyter gruyter exposition math gasharov peeva boundedness versus periodicity commutative local rings transactions ams gulliksen proof existence minimal resolutions acta peeva exponential growth betti numbers journal pure applied schultz boundedness periodicity modules rings journal

function exponent sparse regression codes optimal encoding paper studies performance sparse regression codes lossy compression distortion criterion sparse regression code codewords linear combinations subsets columns design matrix shown encoding sparse regression codes achieve shannon function gaussian sources well optimal exponent completes previous result showed optimal exponent achievable distortions certain threshold proof result based second moment method popular technique show random variable strictly positive high probability context number codewords within target distortion source sequence first identify reason behind failure standard second moment method certain distortions illustrate different failure modes via stylized example use refinement second moment method show achievable distortion values finally refinement technique applied suen correlation inequality prove achievability optimal gaussian exponent index compression sparse superposition codes function gaussian source error exponent second moment method large deviations ntroduction eveloping practical codes lossy compression rates approaching shannon bound long important goal information theory practical compression code requires codebook low storage complexity well encoding decoding low computational complexity sparse superposition codes sparse regression codes sparcs recent class codes introduced barron joseph originally communcation awgn channel subsequently used lossy compression distortion criterion codewords sparc linear combinations columns design matrix storage complexity code proportional size matrix polynomial block length computationally efficient encoder compression sparcs proposed shown achieve rates approaching shannon function gaussian sources work partially supported marie curie career integration grant grant agreement number nsf grant paper presented part ieee international symposium information theory venkataramanan department engineering university cambridge cambridge tatikonda department statistics data science yale university new usa log rate bits jun ramji venkataramanan senior member ieee sekhar tatikonda senior member ieee fig solid line shows previous achievable rate given function shown dashed lines coincides paper study compression performance sparcs distortion criterion optimal encoding show ergodic source variance sparcs optimal encoding achieve given log note optimal ratedistortion function gaussian source variance performance sparcs optimal encoding first studied shown distortionlevel rates greater max log achievable optimal gaussian exponent rate equal strictly larger see fig paper complete result proving sparse regression codes achieve gaussian function distortions also show codes attain optimal exponent gaussian sources rates though encoding practically feasible indeed main motivation sparse regression codes enable encoding decoding characterizing function exponent optimal encoding establishes benchmark compare performance various computationally efficient section columns section columns decoder mapping receiving encoder decoder produces reconstruction since columns sections total number codewords obtain compression rate therefore need section columns enr fig matrix binary vector positions correspond gray columns combine form codeword encoding schemes results paper together show sparcs retain good covering properties gaussian random codebook compact representation terms matrix whose size polynomial block length let specify notation proceeding uppercase letters used denote random variables lowercase letters realizations letters used denote random vectors matrices vectors length source sequence reconstruction sequence kxk denotes normalized version vector kxk denotes gaussian distribution mean variance logarithms base rate measured nats unless otherwise mentioned notation means log log used abbreviate phrase high probability use denote generic positive constants whose exact value needed sparcs optimal encoding sparse regression code defined terms design matrix dimension whose entries block length integers whose values specified terms rate shown fig one think matrix composed sections columns codeword linear combination columns one column section formally codeword expressed vector following property exactly one one forth values set equal constant specified later denote set satisfy property encoder defined mapping given source sequence encoder determines produces codeword closest euclidean distance argmin constructions choose implies log thus log number columns dictionary log polynomial overview approach show rate achieved need show high probability least one enr choices satisfies satisfies call solution denoting number solutions goal show high probability note expressed sum enr indicator random variables ith indicator solution zero otherwise enr analyzing probability challenging indicator random variables dependent codewords dependent share common nonzero terms handle dependence use second moment method second mom technique commonly used prove existence achievability results random graphs random constraint satisfaction problems setting lossy compression second mom used obtain function ldgm codes binary symmetric sources hamming distortion random variable second mom bounds probability event therefore second mom succeeds show shown second mom succeeds defined contrast found second mom fails result clear whether gap due inherent weakness sparse regression codebook limitation second mom proof technique paper demonstrate latter refine second mom prove rates greater achievable inequality follows inequality substituting refinement second mom inspired work finding sharp thresholds random hypergraphs idea follows key ratio expressed denotes total number solutions conditioned event given solution recall solution thus second mom fails ratio goes zero situation expected number solutions much smaller expected number solutions conditioned event solution happens atypical realizations design matrix yield large number solutions total probability matrices small enough significantly affected realizations however conditioning solution increases probability realized design matrix one yields unusually large number solutions low rates conditional probability design matrix atypical large enough make causing second mom key rectifying second mom failure show high probability although apply second mom count good solutions solutions succeeds letting conclude high probability error probability decays exponentially rates smaller channel capacity contrast use refinement second moment method function suen correlation inequality obtain exponent beyond exponent dispersion another quantity interest lossy compression problem fixed probability dispersion specifies fast rate approach function growing block length shown discrete memoryless gaussian sources optimal dispersion equal inverse second derivative exponent given sparcs attain optimal exponent would interesting explore also achieve optimal dispersion gaussian sources distortion rest paper organized follows main results specifying function excessdistortion expoenent sparcs stated section section iii set proof show second mom fails proofs main theorems technical motivate main ideas stylized example section main results proved section proof main technical lemma given section related work mentioned second moment method used analyze function ldgm codes binary symmetric sources hamming distortion idea applying second mom random variable counts good solutions recently used obtain improved thresholds problems random hypergraph random graphs random however key step showing given solution good high probability depends heavily geometry problem considered step requires identifying specific property random object considered sparc design matrix hypergraph boolean formula leads large number solutions atypical realizations object example sparc compression atypical realizations design matrices columns unusually source sequence compressed random hypergraph atypical realizations hypergraphs edge structure allows unusually large number vertices take either color interesting contrast analysis sparc lossy compression sparc awgn channel coding dependence structure sparc codewords makes analysis challenging problems techniques required analyze sparc channel coding different used excess distortion analysis channel coding case authors use modified union bound together novel bounding technique probability pairwise error events lemmas establish similar inspection paradox renewal processes esults probability excess distortion code block length encoder decoder mappings sparc generated described section probability measure respect random source sequence random design matrix sparc definition rate achievable distortion level exists sequence sparcs rate code defined design matrix whose parameter satisfies fixed lbn theorem let drawn ergodic source mean variance let log fix bmin bmin exists sequence rate sparcs defined design matrix lbn determined remark though theorem valid relevant case solution equation log let fix max bmin theorem already guarantees optimal function achieved smaller value required theorem exists sequence rate sparcs parameter achieves exponent log exponent sparc consequently supremum exponents achievable sparcs gaussian sources sources equal optimal one given exponent sequence rate codes given lim sup log defined optimal excessdistortion exponent pair supremum exponents sequences codes rate optimal exponent discrete memoryless sources obtained marton result extended memoryless gaussian sources ihara kubo fact gaussian source distributed distortion criterion optimal exponent rate log lim log log thus decays exponentially comparison exp decays faster exponentially therefore exponent satisfies log log lim inf exp log log lim inf exponent divergence two gaussians distributed respectively next theorem characterizes exponent performance sparcs theorem let drawn ergodic source mean zero variance let log let max bmin bmin defined exists sequence rate sparcs defined design matrix lbn determined whose probability excess distortion distortionlevel bounded follows sufficiently large exp proof theorem know exists sequence rate sparcs exp sufficiently large long parameter satisfies large deviation theorem yields strictly positive universal constants corollary let drawn gaussian source mean zero variance fix rate log since chosen arbitrarily small supremum achievable exponents log optimal fact remark function bmin increasing therefore implies larger values design parameter required achieve exponents closer optimal value smaller values corollary iii nadequacy irect econd first steps proof fix rate greater minimum value specified theorem note since log let number code construction block length pick specified construct design matrix entries drawn codebook consists vectors entries set equal value specified encoding decoding source sequence encoder declares error trivially compressed within distortion using codeword addition extra codeword codebook affects rate negligible way compressed two steps first quantize uniform scalar quantizer support interval input quantizer output conveying scalar quantization index decoder additional log nats allows adjust codebook ance according norm observed psource sequence entries set sparc codeword variance define version note use sparc compress encoder finds argmin decoder receives reconstructs note block length total number bits transmitted encoder log log yielding overall rate logn error analysis overall distortion bounded bounded positive constant overall rate including scalar quantizer logn denoting probability excess distortion random code lim bound second term without loss generality assume source sequence codebook distribution rotationally invariant due design matrix entries enumerate codewords enr define indicator random variables otherwise write fixed dependent see consider codewords corresponding vectors respectively recall vector uniquely defined position value sections overlap positions column sums forming codewords share common terms consequently dependent brevity henceforth denote applying second mom obtained expressing follows enr xui scalar quantization step included simplify analysis fact could use codebook variance satisfy would make forthcoming large deviations analysis quite cumbersome positive constants last inequality holds scalar quantizer let event minimum greater encoder declares error occurs occur overall distortion ergodicity source guarantees max enr last equality holds penr due symmetry code construction implies therefore show need expected number solutions given enr enr since applying lemma obtain bounds enr enr note log versus compute derive general lemma specifying probability randomly chosen codeword within distortion source sequence lemma used parts proof well lemma let vector let random vector independent sufficiently large universal positive constant rate function otherwise proof last equality due rotational invariance distribution joint distribution orthogonal rotation matrix particular matrix rotates vector note using strong version large deviation theorem due bahadur rao rate function given sup log expectation rhs computed using standard calculations obtain log log substituting expression maximizing yields given next consider overlap positions column sums forming codewords share common terms therefore event codewords corresponding share common terms codeword total codewords share exactly common terms obtain enr obtained substituting enr notation means equality appendix also shown log min log log inequality asymptotically tight term may interpreted follows conditioned solution expected number solutions share common terms recall require left side tend therefore need need positive order guarantee however verified solution thus positive log consequently implies second mom fails follows obtained using bayes rule compute second mom ratio therefore equals stylized example describing rectify second mom failure sparc setting present simple example give intuition failure modes second mom proofs next two sections rely discussion consider sequence generic random structures sequence random graphs sparc design matrices denoted suppose realization belongs one two categories category structure solutions category structure solutions case sparc solution codeword within target distortion let probabilities category constant regardless realization note always least solutions examine whether second mom guarantee existence solution problem number solutions expressed sum indicator random variables configuration solution total number configurations sparc context configuration codeword assume configurations symmetric sparc one equal probability solution due symmetry second moment ratio expressed conditional expectation numerator computed examine behavior ratio different values case dominant term numerator denominator get second mom succeeds case dominant term numerator dominant term denominator hence case dominant term numerator dominant term denominator hence enp thus case case second mom fails expected number solutions conditioned solution exponentially larger unconditional expected value however important distinction two cases allows fix failure second mom case case consider conditional distribution number solutions given calculation first term denominator rhs dominates conditional distribution using notation thus conditional probability realization category given slightly smaller unconditional probability however conditioned realization still extremely likely come category solutions therefore conditioning solution change nature typical realization makes possible fix failure second mom case idea define new random variable counts number solutions coming typical realizations category structures second mom applied show strictly positive high probability conditioning solution completely changes distribution dominant term denominator rhs conditional distribution thus conditioned solution typical realization belongs category solutions hand draw unconditional distribution typical realization solutions case second moment method fixed counting solutions realizations category total conditional probability realizations small analog condensation phase found problems random hypergraph coloring phase although solutions may exist even enhanced second mom prove existence fortunately condensation phase sparc compression problem despite failure direct second mom prove lemma conditioning solution significantly alter total number solutions large fraction design matrices analogous case apply second mom new random variable counts solutions coming typical realizations design matrix yields desired result solutions exist rates roofs esults proof theorem code parameters encoding decoding described section build proof section given solution define number solutions share terms total number solutions given solution holds symmetry code construction allows condition generic solution follows note expectations evaluated conditional distribution space design matrices given solution key ingredient proof following lemma shows much smaller particular even lemma let log solution sufficiently large min function bmin defined proof proof lemma given section probability measure lemma conditional distribution space design matrices given solution definition call solution since fixed whether solution determined design matrix lemma guarantees solution solution design matrix number solutions sharing common terms less key proving theorem apply second mom solutions fix enr define indicator random variables otherwise number solutions denoted given venr apply second mom show exg exg second equality obtained writing exg similar lemma exg defined proof due symmetry code construction exg solution follows definitions given solution lemma shows probability least according definition satisfied thus exg lower bounded exg part first observe total number solutions upper bound number solutions therefore given solution expected number solutions expressed codewords share common terms codewords independent thus independent event next note conditioned solution certainty follows definition using conclude combining completes proof lemma using lemma obtain exg min last equality obtained using definition hence probability existence least one good solution goes thus shown quantity tends zero whenever log bmin combining conclude probability goes one chosen arbitrarily close proof theorem complete proof theorem code construction described section parameter chosen satisfy recall definition solution definition follow section count number solutions appropriately defined want upper bound probability event number solutions defined theorem obtained using suen correlation inequality upper bound probability event suen inequality yields sharper upper bound second mom use prove probability decays comparison second mom guarantees polynomial decay begin definitions required suen inequality definition dependency graphs let family random variables defined common probability space dependency graph graph vertex set whose set edges satisfies following property two disjoint subsets edges one vertex families independent fact example suppose family independent random variables function variables subset graph vertex set edge set dependency graph setting fix let indicator random variable defined note one solution set codewords share least one common term ones play role determining whether solution hence graph vertex set enr edge set given codewords share least one common term dependency graph family follows fact observing function columns define codewords share least one common term columns generated independently one another given codeword codewords exactly terms common therefore vertex dependency graph family connected fact suen inequality let bern finite family bernoulli random variables dependency graph write edge define max evk evi evi exg exp min apply suen inequality dependency graph specified compute upper bound penr total number solutions note chosen smaller value used theorem smaller value required prove decay probability via suen inequality also need stronger version lemma lemma let log solution sufficiently large min solution follows given solution lemma shows probability least according definition satisfied thus rhs lower bounded follows solution using expression expected number solutions log constant bmin implies approaches growing second term due symmetry code construction max enr proof proof nearly identical lemma replaced given section terms respectively throughout lemma thus obtain following condition analog enr combining together fact vertices first term evi log min max min log obtain bmin second equality obtained substituting using taylor series bound denominator see result obtained using arguments analogous sec details yields following lower bound sufficiently large compute three terms rhs suen inequality third term lemma growing using conclude probability excess distortion bounded exg exg exg holds symmetry code construction inequality obtained follows number solutions share common terms bounded total number solutions sharing common terms latter quantity expressed sum number solutions sharing exactly common terms conditioned event solution total number solutions share common terms bounded therefore exg exg used fact combining obtain strictly positive constant applying suen inequality using lower bounds obtained obtain exp min log log exp max positive constant recalling logn see exp constant note condition bmin also needed obtain via suen inequality particular condition required provided parameter satisfies max max bmin verified definition bmin strictly increasing therefore maximum rhs bounded max bmin choosing larger value guarantee holds completes proof theorem roof emma begin listing three useful properties function defined recall probability sequence within distortion within distortion sequence fixed strictly decreasing fixed strictly increasing fixed convex attains minimum value log properties straightforward verify definition using elementary calculus let denote restriction set coincides sections indicated remaining entries equal zero example second third sections one entry entries zeros definition given solution define event every size subset solution equation intuition behind choosing according following subset sections design matrix defines sparc rate codeword consisting entries note entries single codeword though codewords dependent due sparc structure probability codeword rate code within distortion source sequence hence expected number codewords rate codebook within distortion strictly decreasing function says smallest expected distortion rate code codeword entries chosen expected number codewords within distortion vanishingly small conditioned idea sections represent distortion less words typical realization design matrix sections contribute roughly equal amounts finding codeword within hand sections sparc represent distortion less remaining sections less work creates proliferation solutions share common sections consequently total number solutions much greater atypical design matrices first step proving lemma show event holds second step showing holds expected number solutions share common terms small compared indeed using write last line follows markov inequality show probability left side small solution showing two terms rhs small first bound lemma log consequently proof last equality holds define function log derivative second therefore strictly concave minimum value attained proves recalling definition implies note function rate codewords chosen optimal variance rate decreasing third argument distortion conclude bound term rhs showing first term small implies sections leave residual distortion least showing second term small implies condition expected number solutions sharing common terms small compared bounding definition event solution union subsets using union bound becomes generic subset say recall sufficiently large denominator bounded expressed log numerator density random variable using cdf bound rhs obtain following upper bound sufficiently large holds sufficiently large obtained using strong version large deviation theorem note linear combination columns hence gaussian random vector entries independent inequality similarly obtained entries independent finally holds overall exponent two exponents equal bound use following lemma decreasing function using sufficiently large bounding codewords share common terms therefore codeword shares exactly common terms set common sections holds sufficiently large obtained follows event norm least vector independent follows rotational invariance distribution inequality obtained using strong version large deviation theorem using obtain sufficiently large overall bound substituting bounds sufficiently large since chosen satisfy lemma solution positive constant given log proof see appendix observe strictly decreasing seen using taylor expansion log write log since log shows strictly positive strictly decreasing lim log substituting exp min taking logarithms dividing sides log obtain log log log log min log log log log log min log log min obtain used bound log min log log log relation right side negative sufficiently large need min log min log arranged choosing large enough since satisfied need log min min log bmin want lower bound solution consider cases separately recall lemma case case terms definition strictly positive write expanding around using taylor theorem obtain since number interval bound obtaining separate lower bounds lower bound using definition second derivative max holds constant order hence maximum attained constant given bmin defined statement theorem satisfies sufficiently large bound becomes log log min bmin log log bmin log bmin log log log therefore min completes proof lemma verified decreasing function hence lower bound note solution ppendix roof emma define function using taylor theorem third argument around point quadratic positive coefficients terms replacing coefficient upper bound solving resulting quadratic yield lower bound since function decreasing coefficient bounded follows finally using lower bounds obtain computed therefore obtain lower bound denoted solving equation thus obtain show bounded obtaining lower upper bounds case case given used fact right hand side equation decreasing therefore sufficient consider order obtain lower bound holds next claim solves equation lies interval indeed observe lhs increasing rhs decreasing since lhs strictly greater rhs solution strictly less hand inequality obtained noting strictly increasing hence taking gives lower bound analogously taking yields upper bound using bounds obtain lhs strictly less rhs therefore solves lies obtain lower bound rhs expand using taylor theorem second argument lies interval using shorthand written acknowledgement thank anonymous referee comments helped improve paper eferences solving quadratic get using get lhs exactly quantity want bound definition second partial derivative respect computed rhs strictly decreasing therefore bound substituting bounds conclude barron joseph least squares superposition codes moderate dictionary size reliable rates capacity ieee trans inf theory vol feb joseph barron fast sparse superposition codes exponentially small error probability ieee trans inf theory vol feb kontoyiannis rad gitzenis sparse superposition codes gaussian vector quantization ieee inf theory workshop venkataramanan joseph tatikonda lossy compression via sparse linear regression performance encoding ieee trans inf thy vol june venkataramanan sarkar tatikonda lossy compression via sparse linear regression computationally efficient encoding decoding ieee trans inf theory vol june alon spencer probabilistic method john wiley sons wainwright maneva martinian lossy source compression using generator matrix codes analysis algorithms ieee trans inf theory vol janson random graphs wiley condensation transition random hypergraph proc annual symp discrete algorithms vilenchik chasing threshold proc ieee annual symposium foundations computer science panagiotou going threshold proc annual acm symposium theory computing ingber kochman dispersion lossy source coding data compression conference march kostina lossy compression finite blocklength regime ieee trans inf theory vol marton error exponent source coding fidelity criterion ieee trans inf theory vol mar ihara kubo error exponent coding memoryless gaussian sources fidelity criterion ieice trans fundamentals vol den hollander large deviations vol amer mathematical society bahadur rao deviations sample mean annals mathematical statistics vol

conference paper assignment problem using order weighted averages assign indivisible goods may jing nicholas renee toby csiro unsw sydney australia lianjingwu ibm watson research center new york usa csiro unsw berlin berlin germany abstract motivated common academic problem allocating papers referees conference reviewing propose novel mechanism solving assignment problem two sided matching problem preferences one side side sides capacity constraints assignment problem fundamental problem computer science economics application many areas including task resource allocation draw inspiration multicriteria decision making voting use order weighted averages owas propose novel flexible class algorithms assignment problem show algorithm finding assignment polynomial time contrast finding egalitarian assignment inspired setting observe interesting connection model classic proportional election problem social choice introduction assigning indivisible items multiple agents fundamental problem many fields including computer science economics operations research algorithms matching assignment used variety application areas including allocating runways airplanes residents hospitals kidneys patients students schools assets individuals divorce jobs machines tasks cloud computing nodes understanding properties underlying algorithms important aspect ensuring participating agents happy allocations attempt misrepresent preferences key area study computational social choice area near many academics hearts problem allocating papers referees peer review results grant journal conference reviewing significant impact careers scientists ensuring papers proposals reviewed referees part ensuring items treated properly participants support outcome processes making sure processes work proposers reviewers important methods improving peer review proposed discussed broadly across sciences number ways one improve quality peer review first ensure reviewers incentivized misreport reviews personal gain along line significant interest recently strategyproof mechanisms peer review unfortunately method discuss paper strategyproof another way ensure reviewers competent provide judgements papers assigned toronto paper matching system designed improve process model third alternative one focus study ensuring reviewers happy papers asked review fundamentally question optimization objectives assignment functions used formally study conference paper assignment problem cpap special resource allocation problem mara propose novel assignment assignment cpap setting market one side preferences side sides possibly infinite upper lower capacities fundamental tension assignment settings tradeoff maximizing social welfare also know utilitarian maximal assignment rawlsian fairness concept maximizing utility worst agent known egalitarian maximal assignment two ideas incompatible optimization objectives diverge computational sense well computing utilitarian assignment additive utilities done polynomial time computing egalitarian assignment perhaps could reason implementers large conference paper assignment software often opt utilitarian assignments supposedly case easychair however also clear egalitarian assignment desirable cpap contributions establish motivation using owa vectors assignment setting define novel notion allocation assignment give algorithm compute maximal assignment polynomial time show owa objective generalizes utilitarian objective show assignments satisfy notion pareto optimality pairwise comparisons objects agents implement algorithm assignments perform experiments real world conference paper assignment data preliminaries use general notation describe setting assignment settings agent provides preference objects reflexive complete transitive preference relation weak order set objects technically unsubstantiated authors contacted easychair understand assignment process told provide information paper assignment easychair implemented information garg may incorrect date none authors worked easychair also access easychair assume complete possible agents may conflicts interest preference particular object assumption often called unacceptable objects literature many cpap settings fixed number equivalence classes agents asked place objects assume number equivalence classes ranks objects given input problem agents tell within rank objects belong agents also provide decreasing utility value main result extended case number equivalence classes fixed formally cpap problem defined set agents set objects reflexive transitive preference relation weak order set objects divided equivalence classes ranks utility vector length assigns decreasing utility let rank object denote value side constraints feasible assignments two practical constraints include model making model general standard mara cpap problems studied computer science upper lower capacities agents objects agent capacity agent possibly equal upper lower bound capacity number objects allocated cnmin cnmax object capacity object possibly equal upper lower bound number agents assigned min cmax respectively define feasible assignment instance given assignment let denote set objects assigned agent let denote set agents assigned object let denote size number elements set vector feasible assignment must obey cnmin cnmax min cmax write set feasible assignments instance individual agent evaluation first formalize individual agent evaluates assigned objects feasible assignment gives rise signature vector agent intuitively signature vector number objects rank assigned formally let assume agents give utilities input however often utilities restricted borda utilities conference paper bidding come fixed budget bidding fake currency course allocation harvard omit arguments clear context indivisible discrete objects lexicographic relation modeled additive utility relation setting agent utilities high enough values formally utility rank lexicographic additive utility relations matter many additional objects rank agent receives one additional object rank preferred define relations referee might consider assignments lexicographic agent lexicographically prefers comes lexicographic order index receives least one paper higher rank lexicographic relation vectors long history assignment literature additive utility agent prefers assignment additive utility objects assigned formally slightly abusing notation alternative formulation using dot product overall assignment evaluation literature several optimization objectives defined assignment implementer may wish consider limit discussion two classical notions additional discussion objectives including imposition various fairness criteria cpap setting found garg mara setting see bouveret utilitarian social welfare maximal assignment often called utilitarian assignment want maximize total social welfare agents assignment utilitarian assignment satisfies arg max arg max egalitarian social welfare maximal assignment often called egalitarian assignment want enforce rawlsian notion fairness making sure worst referee happy possible maximize utility least well agent formally arg max min arg max min discrete mara cpap setting objects divisible problem finding egalitarian assignment finding utilitarian assignment done polynomial time background related work one two sided matching assignment problems studied economics computer science years matching assignment many applications including kidneys exchanges school choice problem often called resource allocation mara problem computer science papers referees formulation problem additional side constraints common economics literature common computer science economics literature problem closely related analogue problem modeling matchings capacities conference paper assignment studied number times years computer science defining refining notions fairness assignment vectors allocation problems build work garg extensively study notion fair paper assignments including leximin assignments within context conference paper assignment garg show setting study finding egalitarian optimal assignment finding leximin optimal assignment three equivalence classes polynomial time computable two also provide approximation algorithm leximin optimal assignments know capacity constraints hard values reviewer must review papers paper must receive exactly reviews resulting version capacitated assignment answer set programming cpap studied amendola encode cpap problem asp show finding solution roughly correspond leximin optimal egalitarian solutions done reasonable time large settings agents cpap also receives considerable attention recommender systems machine learning communities often though work takes approach attempting infer refined utility preference model order distinguish papers fairness efficiency concerns secondary prime example toronto paper matching system designed charlin zemel system attempts increase accuracy matching algorithms papers express preferences reviewers preferences inferred contents papers make use order weighted averages owas often employed decision making owas recently received attention computational social choice voting ranking finding collective set items group voting proportional representation key difference cpap voting using owas comsoc literature cpap select set winners agents share instead agents allocated possibly disjoint set objects assignments formally define owas use defining assignment objectives discuss alternative formulations studied order weighted average owa function defined integer vector numbers let vector numbers let rearrangement say order apply owas setting need define weighted rank signature assignment let defined sorted vector utility referee gets assignment formally sort example included two objects utility one utility one utility would inspiration applying owas comes voting rule known proportional approval voting pav approval voting settings agent approve many candidates wish standard approval voting method approvals agent assign one point candidate cast however lead number pathologies described aziz intuitively seem fair candidate like selected winning set next candidate selected winning set seemingly count less hence pav designed fair voter first approval counts full point second next harmonically decreasing sequence transitioning logic cpap setting motivated find way distribute objects agents increases number agents receive top ranked objects logic pav get candidate winning set count less everyone else candidate winning set desire directly get rank maximal assignment completely ignoring utilities know polynomial result garg however wish modulate using utilities using ranks perhaps use owas use sum agents optimization criteria assignment order cleanly define need place restrictions owa vectors firstly length needs least long maximum agent capacity arg cnmax typically literature owas assumes normalized enforce convention wish study pav setting formally relaxation observe whether owas normalized affect computational results however require owa vector entry ssignment input given assignment setting agent capacities cnmin cnmax object capacities min cmax owa vector cnmax question find feasible assignment arg max formulation owa operator applied vector agent utilities aggregate sum modified utilities give assignment objective hence name observe formulation strictly generalizes utilitarian assignment objective set recover utilitarian assignment one may also wish consider applying owa sorted vector total agent utility allocation one could call version problem indeed formulation problem considered proposed earliest writings owas decision making taking formulation allows one recover utilitarian assignment well egalitarian assignment however formulation generalization egalitarian assignment becomes general think vector kind control knob given implementer market allowing apply transform agent utilities ability may especially useful agents free report normalized utilities ranks via bidding mechanisms many settings utility vector controlled individual agents owa vector control market implementers consider following example example consider setting four agents agents four objects agents let cnmin cnmax objects let min cmax assignment let get following allocations utilitarian owa egalitarian inspecting results example observe set utilitarian maximal assignments assigned set maximal assignments assigned one assigned one set egalitarian maximal assignments agents receives one either along one thus observe following observation set assignments returned three objective functions utilitarian egalitarian owa disjoint instances set assignments set egalitarian assignments disjoint set utilitarian assignments hence interesting direction future work fully characterize assignments discover owa vectors nice properties pareto optimality allocation preferred given agent respect pairwise comparisons allocation result replacing item strictly preferred item note pairwise comparison relation transitive allocation pareto optimal respect pairwise comparisons exists allocation agent weakly prefers least one agent strictly prefers lemma consider agent two allocations equal size least preferred respect pairwise comparison yields least much owa value owa vector matter increasing decreasing proof note viewed transformation item replaced item least preferred hence value item either stays increases either case corresponding owa multiplied value since owa transform bilinear total owa score least much proposition maximal assignment pareto optimal respect pairwise comparison irrespective owa proof assume contradiction maximal assignment pareto optimal respect pairwise comparisons lemma exists another outcome agent weakly prefers least one agent strictly prefers means agent gets least much owa score least one agent gets strictly contradicts fact owa maximal algorithm assignments give algorithm finding assignments using flow networks proof use general formulation problem allowing values upper lower capacities cnmin cnmax vary agent upper lower object capacities min cmax vary object theorem assignment found polynomial time proof reduce problem problem finding minimum cost feasible flow graph upper lower capacities edges polynomial time solvable problem addition polynomial time solvable know flow integral long edge capacities integral even real valued costs figures provide high level view flow network construct cnmin cnmax max min cnmin cnmax cnmin cnmax gadget gadget gadget gadget min cmax cmo min cmax min cmax fig main gadget reduction enforces agent object capacity constraints figure first build tripartite graph two sets nodes one set gadgets per agent agent nodes one agent agent gadgets one illustrated figure agent object nodes one object edge source node agent nodes cost minimum flow capacity cnmin maximum flow capacity cnmax set edges nodes enforces constraint capacity cnmin cnmax also construct edge object node sink edges cost minimum capacity min maximum capacity cmax set edges enforces constraint capacity cmin cmax turn agent gadget depicted figure arbitrary leftmost node rightmost set nodes figure correspond agent nodes fig per agent gadget note costs edges capacities unless otherwise noted object nodes figure respectively agent gadget create tripartite subgraph agent node serving source set object nodes serving sinks create three layers nodes describe turn left right first create set decision nodes labels cnmax intuitively multiplying owa value utility object need keep track values could result arcs nodes set upper capacity minimum capacity cost case cnmax set maximum capacity edges node cnmax enforces value owa vector modify one utility value decision nodes constructed create set nodes denote decision nodes create edge nodes created particular decision node edges maximum capacity cost equal rank object costs negative cost matching agent object weighted rank contributes owa objective finally create one set nodes one denoted nodes connect nodes label corresponding node connect cost maximum capacity connect node corresponding object node main construction cost maximum capacity set nodes edges enforces agent assigned object extract assignment minimum cost feasible flow observing paper allocated agent unit flow passing particular node object node argue correctness algorithm two steps constraints assignment problem enforced minimum cost feasible flow constructed graph gives assignment note since units flow across graph represent assignment explained capacity constraints edges enforce particular constraints imposed definition feasible assignment feasible flow iff flow satisfies constraints observe agent nodes fill flow order owa vector utilities decreasing agent edge costs monotonically increase edges associated edges associated thus agent first unit flow agent use least cost negative edge must associated similarly capacity constraints know one unit flow enters decision node one unit flow leave node means modify one selected must unique agent decision nodes filled order modify value single object know total cost flow across agent gadget equal hence price min cost flow across agents equal thus min cost flow graph assignment generalizations observe two possible generalizations construction allow use constructive proof general instances cpap first proof generalized allow vary agent specifically observe decision nodes agent independent agents means agent class agents could use owa vector ability may useful instance group agents reports extreme utility distribution organizer wishes apply transform utilities second generalization make construction allow agent assigned object ability make sense setting unless sub reviewers could capacitated assignment settings may wish assign agents objects multiple times discrete jobs need done certain number times single agent assigned job multiple times order generalize capacity constraint agent object introduce capacity upper bound encodes number times agent assigned object taking gives original cpap setting order enforce constraint within agent gadget figure add capacity constraint equal edge want lower bound number copies assigned encode lower bound edge well extract assignment minimum cost feasible flow observing paper allocated agent times units flow passing particular node object node argument correctness follows exactly proof theorem corollary assignment found polynomial time even agent unique owa vector object assigned agent number times experiments turn question good assignments practice answer question using real world data three large international conferences ref org focus discussion agents objects implemented algorithm given section using networkx python lemon however still run time giving runtime caused computers crash even memory quite disappointing thought flow argument could used solve problem instances deterred still wanted investigate assignments get owa compare utilitarian egalitarian assignments consequently implemented model mip gurobi ran minute instances settings using cores mip similar one given skowron mara mip bouveret however capacity constraints length owas mip general either encode problem introduce binary variable indicating agent assigned object introduce real valued variable uowa utility agent finally introduce owa matrix notes agent assigned object owa rank mip given max min cmax max min description object capacities agent capacities one object per owa rank objects one rank assignment owa link fcn ranks fill increasing order agent utility must decreasing constraints enforce cardinality constraints agents objects owa rank matrix constraint links agent object assignments positions owa rank matrix line enforces rank matrix fills first position cnmax position agent finally enforces value assignment positions rank matrix must decreasing maximize sum agents owa objective value found utilitarian egalitarian assignments real world datasets object must receive reviews agent must review objects data agent sorts papers equivalence classes gave utility values use pav inspired decreasing harmonic owa vector compute assignment one reasons wanted use assignment allow market designer enforce equitable distribution papers respect ranks hence test statistic number top ranked items average agent expect receive figure shows agent counts cumulative distribution function cdf number top ranked items agents receive looking left side figure see agents receive top ranked papers assignment utilitarian assignment utilitarian assignment agents receive top ranked papers consequently average agents expect get top ranked papers assignment egalitarian assignment utilitarian assignemnt however utilitarian assignment several agents receive entire set top ranked objects egalitarian assignment modulates agents receive top ranked items contrast assignment balance agents receiving top ranked items agents per num top ranked num agents num top ranked cdf num top ranked num top ranked utilitarian egalitarian owa utilitarian egalitarian owa num top ranked fig count agents receiving top ranked papers top cumulative distribution function cdf bottom number agents assigned top ranked objects though agents receive top ranked items egalitarian assignments cdf pdf shows agents receive top ranked items assignment conclusions proposed provided algorithms novel notion assignment assignment using decreasing owa vectors gives central nizer slider move utility maximizing towards rank maximal assignment computationally efficient package important open question future work find axiomatic characterizations good owa vectors additionally owa method methods cpap surveyed treat objects positive utility generally case reviewers conference want review fewer papers consequently would interesting study cpap point view chores called economics literature references pathak roth new york city high school match american economic review ahuja magnanti network flows theory algorithms applications prentice hall amendola dodaro leone ricca application answer set programming conference paper assignment problem proc international conference italian association artificial intelligence aziz brill conitzer elkind freeman walsh justified representation committee voting proc aaai conference aziz gaspers gudmundsson mackenzie mattei walsh computational aspects approval voting proc aamas conference aziz lev mattei rosenschein walsh strategyproof peer selection mechanisms analyses experiments proc aaai conference bouveret chevaleyre lang fair allocation indivisible goods brandt conitzer endriss lang procaccia eds handbook computational social choice chap cambridge university press bouveret characterizing conflicts fair division indivisible goods using scale criteria autonomous agents systems brandt conitzer endriss lang procaccia eds handbook computational social choice cambridge university press budish cantillon assignment problem theory evidence course allocation harvard american economic review charlin zemel toronto paper matching system automated assignment system proc icml workshop peer reviewing publishing models peer charlin zemel boutilier framework optimizing paper matching corr conry koren ramakrishnan recommender systems conference paper assignment problem proc acm conference recommender systems recsys demko hill equitable distribution indivisible objects mathematical social sciences dickerson procaccia sandholm price fairness kidney exchange proc aamas conference elkind ismaili extensions rule proc adt conference elkind faliszewski skowron slinko properties multiwinner voting rules proc aamas conference fishburn lexicographic orders utilities decision rules survey management science garg kavitha kumar mehlhorn mestre assigning papers referees algorithmica golden perny infinite order lorenz dominance fair multiagent optimization proc aamas conference goldsmith lang mattei perny voting rank dependent scoring rules proc aaai conference goldsmith sloan onference paper assignment problem proc aaai conferenceworkshop preference handling artificial intelligence mpref kilgour approval balloting elections handbook approval voting chap springer klaus manlove rossi matching preferences brandt conitzer endriss lang procaccia eds handbook computational social choice chap cambridge university press long wong peng good fair assignment proc ieee international conference data mining icdm manlove algorithmics matching preferences world scientific mattei walsh preflib library preferences http proc adt conference merrifield saari telescope time without tears distributed approach peer review astronomy geophysics price flach computational support academic peer review perspective artificial intelligence cacm rawls theory justice harvard university press roth sotomayor matching study gametheoretic modeling analysis cambridge university press skowron faliszewski lang finding collective set items proportional group recommendation aij yager ordered weighted averaging aggregation operators multicriteria decisionmaking ieee transactions systems man cybernetics

dec estimating monotone probability mass function known flat regions dragi anevski vladimir pastukhov centre mathematical sciences lund university lund sweden abstract propose new estimator discrete monotone probability mass function known flat regions analyse asymptotic properties compare performance grenander estimator monotone rearrangement estimator introduction paper introduce new estimator monotone discrete distribution problem studied particular first study estimation problem also introduced two new estimators problem monotone probability mass function estimation related problem density estimation shape constraints first studied much earlier grenander literature continuous case problem vaste mention results see example discrete case problem recent results discrete continuous case problems one derived particular limit distribution results assumption regions constancy true underlying mass function however knowledge one previously used assumption regions constancy estimation procedure paper use information constructing estimator thus present maximum likelihood estimator mle assumption regions constancy probability mass function derive limit properties new estimator pastuhov paper mainly motivated paper jankowski wellner first study problem estimating discrete monotone distribution introduce estimator suppose monotone decreasing probability mass function support several known flat regions sup number flat regions vector lengths numbers points flat regions true mass function otherwise note strictly decreasing point strictly decreasing whole support otherwise suppose observed random variables probability mass function empirical estimator given also unrestricted maximum likelihood estimator mle argmax gini given vector follows multinomial distributions mult empirical estimator unbiased consistent asymptotically normal see however guaranty order restriction satisfied next discuss two estimators satisfy order restrictions first introduced order restricted mle monotone rearrangement empirical estimator monotone rearrangement empirical estimator defined rear unrestricted mle rear vector vector estimator clearly satisfies order restriction mle order restriction defined argmax fini equivalent isotonic regression unrestricted mle see defined argmin basic estimator unrestricted mle estimator usually called grenander estimator derived using algorithm continuous case problem vector left derivatives leastp concave majorant lcm empirical distribution function estimators introduced studied detail paper jankowski wellner particular jankowski wellner derived consistency estimators analysed asymptotic properties performance estimators distributions different data sets showed converge weakly processes obtained following transform gaussian process space mean zero covariance matrix components periods constancy let rear denotes elements theorem paper construct estimator monotone probability mass function following way argmax fini note vector constitutes lengths flat regions true probability mass function propose following algorithm assume given data set observations random variables vector lengths flat regions true mass function group probabilities required equal flat region single parameters note true values strictly decreasing satisfy following linear constraint next find order restricted mle equivalent isotonic regression weights argmin unrestricted mle defined argmax lemma proof thepequivalence data reduced vector index first element flat region obtained mle finally construct mle letting probabilities flat region equal corresponding values written matrix form matrix elements ones first index flat region length flat region goal investigate estimator compare performance monotone rearrangement estimator defined grenang der estimator defined paper organised follows lemma section prove order restricted mle grouped parameters given isotonic regression unrestricted mle grouped parameters next lemma shows consistency asymptotic normality unrestricted mle grouped parameters lemma show order restricted mle grouped parameters consistent asymptotically normal finally theorem show consistency derive limit distribution new estimator section make comparison previous estimators particular lemma show properly scaled asymptotically smaller risk well hellinger loss compared grenander estimator asymptotically smaller risk compared follows result together result better risk performance respect paper ends small simulation study illustrating small sample behaviour comparison new estimator seems perform better proof characterization estimator asymptotic results section prove statements made algorithm analyse asymptotic properties estimator begin lemma used later section lemma assume sequences random variables taking values metric space endowed borel sigma algebra proof prove statement lemma use portmanteau lemma giving several equivalent characterisations distributional convergence portmanteau lemma follows prove bounded lipschitz functions triangle inequality first term portmanteau lemma next take arbitrary second term bounded using boundness first term right hand side sup sup every since second term right hand side written lipschitz norm smallest number furthermore every since therefore taking limsup left hand side equation obtain lim arbitrary positive number thus goal obtain asymptotic distribution defined true probability mass function satisfies order restrictions let make reparametrisation grouping probabilities required equal flat region single parameters reparametrisation transforms estimation problem becomes argmax index first element flat region lemma solution problem defined given weighted isotonic regression problem argmin unrestricted without order restrictions mle argmax proof result consequence problem maximising product several factors given relations order linear side condition pages pages fact results show product several factors mle order restrictions coincides isotonic regression unrestricted estimates next analyse asymptotic behaviour unrestricted mle lemma unrestricted mle given index first element flat region consistent asymptotically normal matrix wii indicator function proof result lemma case finite support consequently follows directly theorem also see pages next consider case infinite support obviously let introduce notations note sequence processes endowed borel sigma algebra first finite integer sequence vectors converges distribution vector wpii fact follows second show sequence tight metric shown similarly fact lemma enough show two conditions sup lim sup satisfied note bin therefore wjj thus conditions lemma satisfied third since space separable complete prokhorov theorem follows relatively compact means every sequence contains subsequence converges weakly process addition limit processes laws every convergent subsequence converges weakly next show equality laws limit processes convergent subsequences first note since separable space borel equals generated open balls enough show limit laws agree finite intersections open balls since constitute show note open balls written finite support part lemma vectors converge weakly finite implies subsequence converges weakly means law arbitrary fixed subsequence law set note limit law subsequences therefore since continuity set gaussian limit law continuity properties probability measure obtain lim lim lim lim lim law thus shown limit laws convergent subsequences agree open balls therefore also finite intersections open balls since laws agree equal agree borel summarising results previous lemmas obtain final limit result estimator lemma estimator consistent asymptotically normal wpii indicator function proof lemma follows basic estimator consistent theorem follows basic estimator consistent isotonic regression also consistent since consistent since interior point open set furthermore since long equality holds since left hand side inequality goes one shown let clearly applying lemma shows statement lemma theorem estimator consistent asymptotically normal wpii matrix whose elements first index flat region true mass function stands regions length proof lemma follows consistent asymptotically normal estimator given statements theorem follow delta method see example theorem comparison estimators compare estimators consider metric hellinger distance shown grenander estimator smaller risk rearrangement estimator loss next lemma shows new estimator performs better grenander estimator asymptotically expected hellinger distance sense properly normalised lemma metric lim lim hellinger distance lim lim equalities hold true probability mass function strictly monotone proof first theorem continuous mapping theorem second using reduction error property isotonic regression theorem constructed way since every lim sup lim sup lemma also using delta method continuous mapping theorem shown proves sequence asymptotically uniformly integrable see example theorem lim lim sup together proves lim lim sup shows asymptotic uniform integrability sequence third since sequence asymptotically uniformly integrable converges distribution also converges expectation theorem lim furthermore proposed lim pwj obvious finishes proof statement metric prove statement hellinger distance let assume arbitrary sufficient note since weak convergence consistency slutsky theorem continuous mapping theorem follows furthermore asymptotic uniform integrability shown using inequality asymptotic integrability see therefore also convergence expectation lim finally shows hellinger distance estimator converges expectation lim lim note inequality comparison clear equality holds strictly monotone visualisation finite sample performance proposed estimator make small simulation study choose probability mass functions ones chosen figure present results monte carlo simulations samples sample sizes probability mass functions top center bottom stands uniform discrete distribution results shown boxplots hellinger distance metric sample sizes left right fig simulation study clearly illustrates newly proposed estimator better finite sample performance grenander monotone rearrangement estimators distance sense acknowledgements research fully supported research partially supported swedish research council whose support gratefully acknowledged figure boxplots norms hellinger distances estimators empirical estimator white rearrangement estimator grey grenander estimator dark grey estimator shaded references aitchison silvey estimation parameters subject restraints annals mathematical statistics balabdaoui durot koladjo asymptotics discrete convex lse pmf tech barlow bartholomew bremner brunk statistical inference order restrictions john wiley sons bogachev measure theory vol berlin carolan dykstra asymptotic behavior grenander estimator density flat regions canadian journal statistics durot huet koladjo robin estimation convex discrete distribution computational statistics data analysis giguelay estimation discrete probability constraint tech grenander theory mortality measurement skand jankowski wellner estimation discrete monotone distribution electronic journal statistics prakasa estimation unimodal density sankhya series robertson wright dykstra order restricted statistical inference john wiley sons chichester shiryaev probability springer new york silvey statistical inference penguin books baltimore van der vaart asymptotic statistics cambridge university press cambridge

depth prediction sparse depth samples single image feb fangchang sertac consider problem dense depth prediction sparse set depth measurements single rgb image since depth estimation monocular images alone inherently ambiguous unreliable attain higher level robustness accuracy introduce additional sparse depth samples either acquired depth sensor computed via visual simultaneous localization mapping slam algorithms propose use single deep regression network learn directly raw data explore impact number depth samples prediction accuracy experiments show compared using rgb images addition spatially random depth samples reduces prediction error indoor dataset also boosts percentage reliable prediction kitti dataset demonstrate two applications proposed algorithm module slam convert sparse maps dense maps lidars video publicly available ntroduction depth sensing estimation vital importance wide range engineering applications robotics autonomous driving augmented reality mapping however existing depth sensors including lidars depth sensors stereo cameras limitations instance lidars cost per unit yet provide sparse measurements distant objects depth sensors kinect short ranging distance finally stereo cameras require large baseline careful calibration accurate triangulation demands large amount computation usually fails featureless regions limitations always strong interest depth estimation using single camera small ubiquitous consumer electronic products however accuracy reliability methods still far practical despite decade research effort devoted depth prediction including recent improvements deep learning approaches instance depth prediction methods produce average error measured root mean squared error indoor scenarios dataset karaman laboratory information decision systems massachusetts institute technology cambridge usa fcma sertac https https rgb sparse depth ground truth prediction fig develop deep regression model predict dense depth image single rgb image set sparse depth samples method significantly outperforms rgbbased algorithms methods perform even worse outdoors least meters average error kitti datasets address potential fundamental limitations rgbbased depth estimation consider utilization sparse depth measurements along rgb data reconstruct depth full resolution sparse depth measurements readily available many applications instance lowresolution depth sensors lidars provide measurements sparse depth measurements also computed output odometry algorithms work demonstrate effectiveness using sparse depth measurements addition rgb images part input system use single convolutional neural network learn deep regression model depth image prediction experimental results show addition depth samples reduces root mean squared error dataset boosts percentage reliable prediction challenging kitti outdoor dataset general results show addition sparse depth samples drastically improves depth reconstruction performance quantitative results may help inform development typical slam algorithm keeps track hundreds landmarks frame sensors future robotic vehicles consumer devices main contribution paper deep regression model takes sparse set depth samples rgb images input predicts depth image prediction accuracy method significantly outperforms methods including techniques furthermore demonstrate experiments method used module sparse visual odometry slam algorithms create accurate dense point cloud addition show method also used lidars create much denser measurements elated ork depth prediction early works depth estimation using rgb images usually relied features probabilistic graphical models instance saxena estimated absolute scales different image patches inferred depth image using markov random field model approaches also exploited estimate depth query image combining depths images similar photometric content retrieved database recently deep learning successfully applied depth estimation problem eigen suggest convolutional neural network cnn one predicting global coarse scale refining local details eigen fergus incorporate auxiliary prediction tasks architecture liu combined deep cnn continuous conditional random field attained visually sharper transitions local details laina developed deep residual network based resnet achieved higher accuracy unsupervised learning setups also explored disparity image prediction instance godard formulated disparity estimation image reconstruction problem neural networks trained warp left images match right depth reconstruction sparse samples another line related work depth reconstruction sparse samples common ground many approaches area use sparse representations depth signals instance hawe assumed disparity maps sparse wavelet basis reconstructed dense disparity image conjugate method liu combined wavelet contourlet dictionaries accurate reconstruction previous work sparse depth sensing exploited sparsity underlying secondorder derivatives depth images outperformed reconstruction accuracy speed sensor fusion wide range techniques attempted improve depth prediction fusing additional information different sensor modalities instance mancini proposed cnn took rgb images optical flow images input predict distance liao studied use laser scanner mounted mobile ground robot provide additional reference depth signal input obtained higher accuracy using rgb images alone compared approach liao work makes assumption regarding orientation position sensors spatial distribution input depth samples pixel space cadena developed learn three input modalities including rgb depth semantic labels experiments cadena used sparse depth extracted fast corner features part input system produce depth prediction accuracy comparable using rgb alone comparison method predicts depth image learns better representation rgb sparse depth attains significantly higher accuracy iii ethodology section describe architecture convolutional neural network also discuss depth sampling strategy data augmentation techniques loss functions used training cnn architecture found experiments many bottleneck architectures encoder decoder could result good performance chose final structure based sake benchmarking achieved accuracy depth prediction network tailed problem input data different modalities sizes dimensions use two different networks kitti kitti image triple size consequently architecture would require times gpu memory exceeding current hardware capacity final structure illustrated figure feature extraction encoding layers network highlighted blue consist resnet followed convolution layer specifically used kitti used last average pooling layer linear transformation layer original resnet removed second component encoding structure convolution layer kernel size decoding layers highlighted yellow composed upsampling layers followed bilinear upsampling layer use upproj module proposed laina upsampling layer deconvolution larger kernel size also achieve level accuracy empirical comparison different upsampling layers shown section depth sampling section introduce sampling strategy creating input sparse depth image ground truth training input sparse depth sampled randomly ground truth depth image fly particular targeted number depth samples fig cnn architecture kitti datasets respectively cubes feature maps dimensions represented features encoding layers blue consist resnet convolution decoding layers yellow composed upsampling layers upproj followed bilinear upsampling fixed training compute bernoulli probability total number valid depth pixels pixel probability otherwise sampling strategy actual number nonzero depth pixels varies training sample around expectation note sampling strategy different dropout scales output training compensate deactivated neurons purpose sampling strategy increase robustness network different number inputs create training data data augmentation technique worth exploring injection random noise different sampling strategy feature points would affect performance network data augmentation augment training data online manner random transformations including scale color images scaled random number depths divided rotation color depths rotated random degree color jitter brightness contrast saturation color images scaled color normalization rgb normalized mean subtraction division standard deviation flips color depths horizontally flipped chance nearest neighbor interpolation rather common interpolation used scaling rotation avoid creating spurious sparse depth points take center crop augmented image input size network consistent loss function one common default choice loss function regression problems mean squared error sensitive outliers training data since penalizes heavily larger errors experiments found loss function also yields visually undesirable boundaries instead sharp transitions another common choice reversed huber denoted berhu loss function defined otherwise uses parameter computed maximum absolute error pixels batch intuitively berhu acts mean absolute error error falls behaves approximately error exceeds experiments besides aforementioned two loss functions also tested found produced slightly better results depth prediction problem empirical comparison shown section result use default choice throughout paper simplicity performance xperiments implement network using torch models trained kitti odometry datasets using nvidia tesla gpu memory weights resnet encoding layers except first layer different number input channels initialized models pretrained imagenet dataset use small batch size train epochs learning rate starts reduced every epochs small weight decay applied regularization dataset dataset consists rgb depth images collected different indoor scenes microsoft kinect use official split data scenes used training remaining testing particular sake benchmarking small labeled test dataset images used evaluating final performance seen previous work training sample spatially evenly raw video sequence training dataset generating roughly synchronized image pairs depth values projected onto rgb image filter using official toolbox following original frames size first downsampled half producing final size kitti odometry dataset work use odometry dataset includes camera lidar measurements odometry dataset consists sequences among one half used training half evaluation use images training sequences training neural network random subset images test sequences final evaluation use left right rgb cameras unassociated shots velodyne lidar measurements projected onto rgb images bottom crop used since lidar returns measurement upper part images compared even ground truth sparse kitti typically projected measurements image pixels error metrics evaluate method using following metrics rmse root mean squared error rel mean absolute relative error percentage predicted pixels relative error within threshold specifically card max card respectively ground truth prediction card cardinality set higher indicates better prediction esults section present experimental results first evaluate performance proposed method different loss functions network components prediction accuracy section second compare proposed method methods kitti datasets section third section explore impact number sparse depth samples performance finally section section demonstrate two use cases proposed algorithm creating dense maps lidar architecture evaluation section present empirical study impact different loss functions network components depth prediction accuracy results listed table problem rgb loss berhu rgbd encoder conv conv conv conv conv conv chandrop depthwise conv decoder rmse rel upconv upproj upproj upproj upproj table evaluation loss functions upsampling layers first convolution layer rgbd average sparse depth input samples comparison loss functions listed row comparison upsampling layers row comparison first convolution layers bottom rows loss functions compare loss functions use network architecture upsampling layers simple deconvolution kernel denoted berhu loss functions listed first three rows table comparison shown table berhu significantly outperform addition produces slightly better results berhu therefore use default choice loss function upsampling layers perform empirical evaluation different upsampling layers including deconvolution kernels different sizes well upconv upproj modules proposed laina results listed row table make several observations firstly deconvolution kernel outperforms component kernel every single metric secondly since upconv receptive field meaning output neuron computed neighborhood input neurons comparable performance thirdly even larger receptive field upproj module outperforms others choose use upproj default choice first convolution layer since rgbd input data comes different sensing modalities input channels depth vastly different distributions support perform simple analysis first convolution layer explore three different options first option regular spatial convolution conv second option depthwise separable convolution denoted depthwise consists spatial convolution performed independently input channel followed pointwise convolution across different channels window size third choice channel dropout denoted chandrop input channel preserved probability zeroed probability bottom rows compare results options networks trained using rgbd input average sparse input samples depthwise conv yield similar results significantly outperform chandrop layer since difference small sake comparison consistency use convolution layer experiments comparison section compare existing methods dataset compare approaches well fusion approach utilizes additional laser scanner mounted ground robot quantitative results listed table problem samples rgb rgbd method rmse rel roy eigen laina liao table comparison dataset values originally reported authors respective paper first observation row row network architecture achieve slightly better result albeit higher rel replacing berhu loss function proposed simple secondly comparing problem group rgb row problem group row draw conclusion extremely small set sparse depth samples without color information already produces significantly better predictions using rgb thirdly comparing problem group proble group rgbd row row number samples clear color information help improve prediction accuracy words proposed method able learn suitable representation rgb images sparse depth images finally compare bottom row proposed method even using samples outperforms laser measurements samples spatially uniform thus provides information line measurement examples predictions different inputs displayed figure kitti dataset kitti dataset challenging depth prediction since maximum distance meters opposed meters dataset greater performance boost obtained using approach although training test data across different methods scenes similar sense come sensor setup car data collected driving report values work table iii results first rgb group demonstrate rgbbased depth prediction methods fail outdoor scenarios rmse close meters note fig predictions top bottom rgb images prediction prediction rgb rgbd prediction sparse depth rgb ground truth depth problem samples method rgb mancini eigen rgbd liao rmse rel table iii comparison kitti dataset values reported use sparsely labeled depth image projected lidar instead dense disparity maps computed stereo cameras words much smaller training dataset compared additional depth samples bring rmse meters half rgb approach boosts performance also compares favorably fusion techniques including time demands fewer samples number depth samples section explore relation prediction accuracy number available depth samples train network different input size optimal rmse color sparse input depth rgbd sparse depth rgb rgbd sparse depth rgb number depth samples rel number depth samples ground truth fig example prediction kitti top bottom rgb sparse depth rgbd dense prediction ground truth depth projected lidar performance compare performance three kinds input data including rgb rgbd performance depth prediction independent input sample size thus plotted horizontal line benchmarking rmse rgbd sparse depth rgb rel number depth samples rgbd sparse depth rgb number depth samples rgbd sparse depth rgb number depth samples rgbd sparse depth rgb number depth samples fig impact number depth sample prediction accuracy dataset left column lower better right column higher better dataset figure rgbd outperforms rgb depth samples performance gap quickly increases number samples set samples rmse rgbd decreases around half rgb rel sees larger rgbd sparse depth rgb number depth samples rgbd sparse depth rgb number depth samples fig impact number depth sample prediction accuracy kitti dataset left column lower better right column higher better improvement reduced two thirds one hand rgbd approach consistently outperforms indicates learned model indeed able extract information sparse samples alone also colors hand performance gap rgbd shrinks sample size increases approaches perform equally well sample size goes accounts less image pixels still small number compared image size observation indicates information extracted sparse sample set dominates prediction sample size sufficiently large case color cue becomes almost irrelevant performance gain kitti dataset almost identical shown figure samples rmse rgbd decreases meters half meters percentage improvement dataset similarly rel reduced percentage improvement datasets accuracy saturates number depth samples increases additionally prediction blurry boundaries even many depth samples see figure believe phenomena attributed fact fine details lost bottleneck network architectures remains study additional skip connections encoders decoders help improve performance application dense map visual odometry features section demonstrate use case proposed method sparse visual slam visual inertial odometry vio algorithms slam vio usually sparse methods represent environment sparse landmarks although sparse algorithms robust efficient output map form sparse point clouds useful applications motion planning rgb sparse landmarks fig application lidar creating denser point cloud raw measurements top bottom rgb raw depth predicted depth distant cars almost invisible raw depth easily recognizable predicted depth application lidar ground truth map prediction fig application sparse slam visual inertial odometry vio create dense point clouds sparse landmarks rgb sparse landmarks ground truth point cloud prediction point cloud created stitching rgbd predictions frame present another demonstration method superresolution lidar measurements lidars low vertical angular resolution thus generate vertically sparse point cloud use measurements sparse depth image rgb images input network average rel compared using rgb example shown figure cars much recognizable prediction raw scans conclusion demonstrate effectiveness proposed methods implement simple visual odometry algorithm data one test scenes dataset simplicity absolute scale derived ground truth depth image first frame landmarks produced onto rgb image space create sparse depth image use rgb sparse depth images input prediction pixels within trusted region define convex hull pixel space formed input sparse depth samples preserved since well constrained thus reliable dense point clouds reconstructed reliable predictions stitched together using trajectory estimation vio introduced new depth prediction method predicting dense depth images rgb images sparse depth images well suited sensor fusion sparse slam demonstrated method significantly outperforms depth prediction using rgb images existing fusion techniques method used module sparse slam visual inertial odometry algorithms well superresolution lidar measurements believe new method opens important avenue research rgbd learning general perception problems might benefit substantially sparse depth samples results displayed figure prediction map resembles closely ground truth map much denser sparse point cloud major difference prediction ground truth prediction map points white wall feature extracted tracked result pixels corresponding white walls fall outside trusted region thus removed work supported part office naval research onr onr yip program also gratefully acknowledge support nvidia corporation donation used research acknowledgment eferences liu shen lin deep convolutional neural fields depth estimation single image proceedings ieee conference computer vision pattern recognition eigen fergus predicting depth surface normals semantic labels common convolutional architecture proceedings ieee international conference computer vision laina rupprecht deeper depth prediction fully convolutional residual networks vision fourth international conference ieee silberman hoiem indoor segmentation support inference rgbd images computer saxena sun learning scene structure single still image ieee transactions pattern analysis machine intelligence vol geiger lenz urtasun ready autonomous driving kitti vision benchmark suite conference computer vision pattern recognition cvpr montiel tardos orbslam versatile accurate monocular slam system ieee transactions robotics vol saxena chung learning depth single monocular images advances neural information processing systems karsch liu kang depth extraction video using sampling european conference computer vision springer konrad wang ishwar image conversion learning depth examples computer vision pattern recognition workshops cvprw ieee computer society conference ieee karsch liu kang depthtransfer depth extraction video using sampling pattern analysis machine intelligence ieee transactions liu salzmann depth estimation single image proceedings ieee conference computer vision pattern recognition eigen puhrsch fergus depth map prediction single image using deep network advances neural information processing systems zhang deep residual learning image recognition proceedings ieee conference computer vision pattern recognition kuznietsov leibe semisupervised deep learning monocular depth map prediction arxiv preprint zhou brown unsupervised learning depth video arxiv preprint garg carneiro reid unsupervised cnn single view depth estimation geometry rescue european conference computer vision springer godard mac aodha brostow unsupervised monocular depth estimation consistency arxiv preprint hawe kleinsteuber diepold dense disparity maps sparse disparity measurements computer vision iccv ieee international conference ieee liu chan nguyen depth reconstruction sparse samples representation algorithm sampling ieee transactions image processing vol carlone sparse sensing resourceconstrained depth reconstruction intelligent robots systems iros international conference ieee sparse depth sensing robots arxiv preprint mancini costante fast robust monocular depth estimation obstacle detection fully convolutional networks intelligent robots systems iros international conference ieee liao huang parse geometry line monocular depth estimation partial laser observation robotics automation icra ieee international conference ieee cadena dick reid joint estimators robotics scene robotics science systems srivastava hinton dropout simple way prevent neural networks journal machine learning research vol owen robust hybrid lasso ridge regression contemporary mathematics vol collobert kavukcuoglu farabet environment machine learning biglearn nips workshop russakovsky deng imagenet large scale visual recognition challenge international journal computer vision vol roy todorovic monocular depth estimation using neural regression forest proceedings ieee conference computer vision pattern recognition

entropy rate estimation markov chains large state space feb yanjun jiantao tsachy yihong tiancheng february abstract estimating entropy based data one prototypical problems distribution property testing estimation estimating shannon entropy distribution elements independent samples showed sample complexity sublinear showed consistent estimation shannon entropy possible sample size far exceeds logs paper consider problem estimating entropy rate stationary reversible markov chain states sample path observations show long markov chain mixes slowly relaxation time consistent estimation achievable log long markov chain slight dependency relaxation time least consistent estimation impossible log assumptions optimal estimation accuracy shown log parison empirical entropy rate requires least samples consistent even markov chain memoryless addition synthetic experiments also apply estimators achieve optimal sample complexity estimate entropy rate english language penn treebank google one billion words corpora provides natural benchmark language modeling relates directly widely used perplexity measure introduction consider stationary stochastic process takes values finite alphabet size shannon entropy rate simply entropy rate process defined lim yanjun han jiantao jiao lee tsachy weissman department electrical engineering stanford university email jiantao yjhan czlee tsachy yihong department statistics data science yale university email tiancheng department electronic engineering tsinghua university email shannon entropy entropy random vector joint probability mass function since entropy random variable depends distribution also refer entropy discrete distribution defined shannon entropy rate fundamental limit expected logarithmic loss predicting next symbol given past symbols also fundamental limit data compressing stationary stochastic processes terms average number bits required represent symbol estimating entropy rate stochastic process fundamental problem information theory statistics machine learning diverse example exists extensive literature entropy rate estimation known data compression theory normalized codelength universal code consistent estimator entropy rate number samples approaches infinity observation inspired large variety entropy rate estimators see however work asymptotic regime attention analysis recent date almost data little work performance entropy rate estimator dependent alphabet size large making asymptotically large datasets infeasible stochastic process memory understanding regime increasingly important modern machine learning applications example substantial recent advances probabilistic language models used applications machine translation search query completion entropy rate say english language represents fundamental limit efficacy language model measured perplexity interest language model researchers obtain accurate estimate entropy rate sheds light much room left improvement however since alphabet size size entire english lexicon google one billion words corpus includes two million unique unrealistic assume asymptotics especially dealing combinations words bigrams trigrams etc therefore significant practical importance investigate optimal entropy rate estimator limited sample size context analysis samples paninski first showed shannon entropy consistently estimated samples alphabet size approaches infinity seminal work showed estimating entropy rate source logs samples necessary sufficient consistency entropy estimators proposed refined based linear programming shown achieve minimax estimation rate another estimator proposed authors shown achieve minimax rate restrictive regime lnss sln using idea best polynomial approximation independent work log obtained estimators achieve minimax error log entropy estimation intuition logs sample complexity independent case interpreted follows opposed estimating entire distribution exceeds estimated vocabulary english language partly different forms word count different words language models partly edge cases tokenization automatic splitting text words parameters requires samples estimating scalar functional entropy done logarithmic factor reduction samples markov chains characterized transition matrix consisting free parameters reasonable expect log sample complexity indeed show correct provided mixing time slow estimating entropy rate markov chain falls general area property testing estimation dependent data prior work provided analysis estimation entropy rate markov chains showed necessary assume certain assumptions mixing time otherwise entropy rate impossible estimate progress related questions estimating mixing time sample path estimating transition matrix current paper makes contribution growing field particular main results paper highlighted follows provide tight analysis sample complexity empirical entropy rate markov chains mixing time large refines results shows mixing slow sample complexity empirical entropy depend mixing time obtain characterization optimal sample complexity estimating entropy rate stationary reversible markov chain terms sample size state space size mixing time partially resolve one open questions raised particular show mixing neither fast slow sample complexity depend mixing time regime performance optimal estimator samples essentially empirical entropy rate log samples opposed lower bound estimating mixing time obtained applying cam method two markov chains produce statistically indistinguishable minimax lower bound current paper much involved addition series reductions means simulation relies constructing two stationary reversible markov chains random transition matrices marginal distributions sample paths statistically indistinguishable construct estimators efficiently computable achieve minimax sample complexity key step connect entropy rate estimation problem shannon entropy estimation large alphabets samples analysis uses alternative probabilistic descriptions markov chains billingsley concentration inequalities markov chains compare empirical performance various estimators entropy rate variety synthetic data sets demonstrate superior performances informationtheoretically optimal estimators compared empirical entropy rate apply optimal estimators estimate entropy rate penn treebank ptb google one billion words datasets show even estimates using may exist language models achieve better perplexity current rest paper organized follows setting preliminary definitions section present summary main results section analyze empirical entropy rate prove achievability theorems empirical entropy rate entropy rate estimator section lower bound sample complexity empirical entropy rate proven section minimax lower bound proven section section provides empirical results performance various entropy rate estimators synthetic data section applies estimators estimate entropy rate penn treebank ptb google one billion words datasets auxiliary lemmas used throughout paper collected section proofs lemmas presented section preliminaries denote multi multinomial distribution number trials event probability vector consider markov chain finite state space transition kernel denote entries tij tij let denote ith row conditional law given throughout paper focus markov chains since markov chain converted one extending state space say markov chain stationary distribution denoted satisfies tij say markov chain reversible satisfies detailed balance equations tij tji markov chain reversible left spectrum transition matrix contains real eigenvalues denote define spectral gap reversible markov chain absolute spectral gap defined max clearly follows reversible markov chain relaxation time reversible markov chain defined relaxation time reversible markov chain approximately captures mixing time roughly speaking smallest marginal distribution close markov chain stationary distribution refer survey fact always markov chain memory intuitively speaking shorter relaxation time faster markov chain mixes shorter memory sooner evolutions markov chain different initial states begin look similar consider following observation model observe sample path stationary finitestate markov chain since limit exists stationary process assume markov chain starts rather order simplify formulae entropy rate stationary markov chain always without additional assumptions mixing time irreducibility aperiodicity furthermore markov chains shannon entropy rate reduces tij tij stationary distribution markov chain denote set discrete distributions alphabet size probability simplex set markov chain transition matrices state space size let rev set transition matrices stationary reversible markov chains state space size define following class stationary markov reversible chains whose relaxation time rev rev goal characterize sample complexity entropy rate estimation function estimation accuracy throughout paper sometimes slightly abuse notation using rev rev also denote spaces probability measures corresponding stochastic processes generated according parameter sets given sample path let denote empirical distribution states subsequence containing elements following occurrence state since entropy rate markov chain written natural idea use estimate appropriate shannon entropy estimator estimate empirical entropy rate defined computes shannon entropy empirical distribution argument fact also interpreted maximum likelihood estimate lemma entropy rate estimator proposed paper differs replace estimator minimax estimator shannon entropy samples entropy rate estimator defined minimax shannon entropy estimator designed data found main property needs satisfy concentration property presented lemma main results motivated results independent case one might expect bias dominates close empirical entropy total error including bias variance depend mixing properties mixing slow intuition supported following theorem theorem suppose sample path stationary reversible markov chain spectral gap exists constant independent entropy rate estimator conditions exists constant independent empirical entropy rate satisfies remark theorem shows number samples large mixing slow suffices take lns estimator achieve vanishing error empirical entropy rate theorem improves analysis empirical entropy rate sense unlike error term dominating term depend mixing time emphasize constraint large restrictive theory entropy estimation data shows bias dominates optimal estimator empirical entropy theorem characterizing regime bias dominates next result shows bias empirical entropy rate unless even data independent theorem suppose empirical entropy rate defined denotes true entropy rate let mutually independent uniformly distributed following corollary immediate corollary exists universal constant absolute value bias bounded away zero even markov chain memoryless next theorem presents minimax lower bound entropy rate estimation quantifies limit estimation scheme beat asymptotic results section interpreted parameterizing subject conditions theorem theorem lim inf inf sup rev universal constants theorem following corollary follows theorem presents critical scaling determines whether consistent estimation entropy rate possible corollary exists estimator estimates entropy rate uniformly vanishing error markov chains rev conclude section summarize result terms sample complexity estimating entropy rate within bits classified according relaxation time case sample complexity lnss narrow regime sample complexity lns matching lower bound known sample complexity lns sample complexity lns matching upper bound known case chain mixes slowly likely variance dominate upper bound analysis proof theorem performance terms shannon entropy estimation collected following lemma lemma suppose one observes samples exists entropy estimator exp universal constants shannon entropy defined moreover empirical entropy satisfies exp consequently proof part pertaining concentration follows part pertaining empirical entropy follows proposition eqn alternative probabilistic description markov chains one key step analysis identify event ensures accuracy estimator refer event good event end adopt following view generation markov chain process viewed generated following fashion consider independent collection random variables win pwin tij imagine variables win set following array wsn first sampled first variable ith row array sampled result assigned definition first variable jth row sampled unless case second variable sampled case result sampling definition next variable sampled first one row yet sampled process thus continues follows definition joint distribution sampled model number elements among equal due independence assumptions taj collecting model assume last variable sampled row wini clear subsequence sample path defined simply wini analysis next define two events ensure proposed entropy rate estimator empirical entropy rate accurate respectively definition good event estimation let universal constants take every define event max every define event wim lemma finally define good event intersection events gopt analogously define good event gemp empirical entropy rate similar fashion replaced wim following lemma shows good events defined definition indeed occur high probability lemma gopt gemp definition occur probability least present main upper bound implies theorem corollary theorem suppose comes stationary reversible markov chain spectral gap probability least value entropy rate estimator satisfies constants definition introduced take similarly empirical entropy rate probability least value proof write wini write next bound two terms separately condition good event gopt definition occurs note function increasing function thus whenever let note decreasing let max max wini wim key observation fixed wim taking intersection gopt note effectively taking union value instead conditioning fact conditioned wim longer therefore event gopt last step follows fact event gopt max combining using lemma completes proof proof follows entirely analogously gopt replaced gemp impossibility results lower bound empirical entropy rate first prove theorem quantifies performance limit empirical entropy rate lemma section shows min denotes set markov chain transition matrices state space size since know specify true distribution product distribution suffices lower bound min empirical distribution counts marginal distribution shown choosing uniform distribution min used fact uniform distribution elements entropy maximizes entropy among distribution supported elements minimax lower bound use sequence reductions prove lower bound markov chains specifically introduce two auxiliary models namely independent multinomial independent poisson model show sample complexity markov chain model lower bounded independent multinomial model lemma lower bounded independent poisson model lemma finally theorem follows lower bound independent poisson model theorem precise use notation pmc pim pip denote probability measure corresponding three models respectively reduction markov chain independent multinomial definition independent multinomial model given stationary reversible markov chain transition matrix tij rev stationary distribution absolute spectral gap fix integer independent multinomial model statistician observes following arrays independent random variables wsms constant within ith row random variables wimi number observations ith row max equivalently observations summarized following sufficient statistic matrix cij row independently distributed multi hence name independent multinomial model following lemma relates independent multinomial model markov chain model lemma exists estimator markov chain model parameter sup rev pmc exists another estimator independent multinomial model parameter sup rev pim constant definition reduction independent multinomial independent poisson introduce independent poisson model parametrized symmetric matrix integer parameter definition independent poisson model given symmetric matrix rij rij parameter independent poisson model observe matrix cij independent entries distributed cij poi rij symmetric matrix define transition matrix normalizing rows tij rij rij thanks symmetry transition matrix reversible markov chain stationary distribution indeed detailed balance equation satisfied rij rij rji rji used fact rij rji upon observing poisson matrix functional estimated entropy rate normalized transition matrix simplified follows rij rij rij rij rij rij rij rij given define following collection symmetric matrices rij normalized transition matrix defined mini parameters ensure exist sufficiently many observations simulate independent multinomial model made precise next lemma lemma exists estimator independent multinomial model parameter sup rev pim exists another estimator independent poisson model parameter sup provided pip constant definition minimax lower bounds independent poisson model task reduced lower bounding sample complexity independent poisson model general strategy method fuzzy hypotheses extension lecam methods following version adapted theorem see also lemma lemma let random variable distributed according let pair probability measures necessarily supported let arbitrary estimator functional based observation suppose exist inf sup marginal distributions induced prior total variation distance distributions apply method independent poisson model parameter symmetric matrix function estimated observation sufficient statistic cij cji cii goal construct two symmetric random matrices whose distributions serve priors sufficiently concentrated near desired parameter space properly chosen parameters entropy rates different values induced marginal laws statistically inseparable end need following results proof proposition lemma let let absolute constants exist random variables supported lemma lemma let random variables taking values poi poi poi poi denotes poisson mixture respect distribution positive random variable ready define priors independent poisson model simplicity assume cardinality state space introduce new state definition prior construction suppose set constant lemma recall random variables introduced lemma use construction akin studied define symmetric random matrices uij uij copies copies respectively let let laws respectively parameters chosen later set independent poisson model lemma construction pair priors achieves following three goals statistical indistinguishablility note distributions first row column identical hence sufficient statistics cij cji cii denote marginal distribution prior following lemma shows distributions sufficient statistic indistinguishable lemma functional value separation two priors corresponding entropy rates independent poisson model differ constant factor nsln explain intuition view rij rij rij similarly rij show close common mean furthermore also concentrate common mean thus view lemma precise statement summarized following lemma lemma assume lns exist universal constants concentration parameter space although random matrices may take values outside desired space show mass concentrated set appropriately chosen parameters following lemma core argument lower bound makes statement precise lemma assume exist universal constants fitting lemma lemma lemma main lemma following minimax lower bound holds independent poisson model theorem exist universal constants lim inf inf sup pip proof theorem choice nsln lemma combination lemma lemma gives similarly lemma theorem follows lemma directly proof theorem combine previous results prove theorem firstly theorem shows inf sup ensures inf sup rev moreover since larger results smaller set parameters models may always assume pip pim choice assumption thus lemma implies finally application lemma gives inf sup rev pmc completing proof theorem experiments entropy rate estimator proposed paper achieves minimax rates viewed conditional approach words apply shannon entropy estimator observations corresponding state average estimates using empirical frequency states generally estimator shannon entropy data conditional approach follows idea defined list several choices empirical entropy estimator simply evaluates shannon entropy empirical distribution input sequence shown achieve minimax rates shannon entropy estimation also achieve optimal sample complexity estimating entropy rate theorem corollary jvhw estimator based best polynomial approximation proved minimax independent work based similar ideas estimator based linear programming proved achieve lnss phase transition shannon entropy profile maximum likelihood estimator pml proved achieve lnss phase transition however exist efficient algorithm even approximately compute pml provably ganrantees another estimator entropy rate estimator lie category conditional approaches estimator estimates entropy compression well known universal lossless compression scheme codelength per symbol would approach shannon entropy rate length sample path grows infinity specifically following random matching length defined lni max shown stationary ergodic markov chains lni lim use alphabet size vary sample size demonstrate performance varies sample size increases compare performance estimators measuring root mean square error rmse following four different scenarios via monte carlo simulations uniform eigenvalue transition matrix uniformly distributed except largest one transition matrix generated using method use spectral gap zipf transition probability tij geometric transition probability tij memoryless transition matrix consists identical rows four cases jvhw estimator outperforms empirical entropy rate results included due considerable longer running time example try estimate entropy rate single trajectory markov chain empirical entropy jvhw estimator evaluated less seconds evaluation estimator conditional method terminate main reason slowness methods context markov chains context needs call original entropy estimator times total experiment needs solve linear programming entropy rate estimation error various estimators entropy rate estimation error various estimators rmse rmse empirical jvhw conditional empirical jvhw conditional number samples uniform number samples zipf entropy rate estimation error various estimators rmse rmse entropy rate estimation error various estimators empirical jvhw conditional empirical jvhw conditional number samples number samples geometric memoryless figure comparison performances empirical entropy rate jvhw estimator different parameter configurations use matlab implementation https use matlab implementation http use cores server cpu frequency application fundamental limits language modeling section apply entropy rate estimators estimate fundamental limits language modeling lot recent interest progress developing probabilistic models natural languages applications machine translation search query completion mostly using recurrent neural networks language model specifies joint probability distribution sequence words common use markov assumption train models using sequences words known sometimes latin prefixes unigrams bigrams etc values measure efficacy model researchers commonly use metric called perplexity normalized inverse probability model test set perplexityq logarithm perplexity also known rate seen logarithmic loss function log perplexityq log particular language stationary ergodic stochastic process entropy rate drawn language true distribution lim inf log lim inf log perplexity equality section logarithms respect base entropy measured bits entropy rate english language therefore significant interest language model researchers since tight lower bound perplexity quantity indicates close given language model optimum several researchers presented estimates bits per character language models trained words estimates directly relevant present task one earliest papers topic claude shannon gave estimate bits per word latter figure comprehensively beaten recent models example achieved perplexity corresponding rate bits per word produce estimate entropy rate english used two linguistic corpora penn treebank ptb google one billion words benchmark results based corpora particularly relevant widespread use training models used conditional approach proposed paper jvhw estimator describe section ptb corpus contains million words unique corpus contains million words million unique obviously english language markov process however since stationary stochastic processes necessarily markov entropy rate limit lim language modeling literature typically known use avoid conflict size dataset estimated cond entropy ptb best known model ptb best known model memory length figure estimates conditional entropy based linguistic corpora estimate entropy rate using estimates conditional entropy successively increasing successively longer equivalently augment state space use estimator relaxing markov assumption kth order markov assumption worth noting corpora comprise individual sentences disconnected corpora effect markov order length sentence results shown figure estimated conditional entropy provides refined analysis intrinsic uncertainty language prediction context length using jvhw estimator corpus estimate bits per word current models trained corpus rate bits per word indicates language models still least bits per word away fundamental limit note since decreasing similarly much smaller ptb corpus estimate entropy rate bits per word compared models achieve rate bits per word least bits away fundamental limit since number words english language alphabet size huge view log result showed theory natural question whether corpus vast corpus enough allow reliable estimates conditional entropy quick answer question theory far focused analysis demonstrated natural language data much nicer sample complexity accurate estimation much lower minimax theory predicts specifically computed conditional entropy estimates figure time restricting sample subset corpus plot resulting estimate function sample size shown figures sentences corpus randomized order subset corpus taken randomly chosen interpret results first note number distinct unigrams words corpus two million recall case samples necessary even worst case dataset million words adequate provide reliable estimate entropy million indeed plot unigrams jvhw estimator figure supports case entropy estimates figure estimates conditional entropy versus sample size unigrams dotted lines estimate using entire corpus final estimate note axes table points entropy estimates within bit final estimate sample size corpus sample sizes greater words within bits entropy estimate using entire corpus takes corpus reach estimate within bits true value note also empirical entropy rate converges value within two decimal places also shown figure dotted lines indicate final entropy estimate estimator using entire corpus words results similar experiments bigrams trigrams shown figure table since state space bigrams trigrams much larger convergence naturally slower nonetheless appears fast enough entropy estimate within order bits true value observations believe estimates based corpus enough samples produce reasonably reliable entropy estimates one measure approximate variance entropy estimates also ran bootstraps memory length bootstrap size size original dataset sampling replacement corpus bootstraps range estimates highest less lowest memory length never exceeded bit standard deviation estimates error ranges implied bootstraps figure estimates conditional entropy versus sample size bigrams trigrams dotted lines estimate using entire corpus final estimate table bootstrap estimates error range estimate ptb dev range estimate dev range small show legibly figure ptb corpus range never exceeded bit details bootstrap estimates given table auxiliary lemmas lemma arbitrary sequence define empirical distribution consecutive pairs let marginal distribution empirical frequency state denote empirical conditional distribution whenever let given shannon entropy defined min given transition matrix following lemma gives tail bounds poisson binomial random variables lemma exercise poi following lemma hoeffding inequality lemma let independent random variables takes value almost surely let exp proofs main lemmas proof lemma lemma concentration empirical distribution reversible markov chains lemma consider reversible stationary markov chain spectral gap every every constant event happens probability max proof lemma recall following bernstein inequality reversible chains theorem stationary reversible markov chain spectral gap exp split proof two parts invoking setting exp exp invoking setting exp exp ready prove lemma consider gopt upper bound gemp follows steps union bound suffices upper bound probability complement event definition good event gopt definition first part definition probability bad events eic upper bounded eic lemma since assumed second part definition applying lemma overall probability bad events hic upper bounded hic second step follows fact increasing proof lemma simulate markov chain sample path transition matrix tij stationary distribution independent multinomial model described section define estimator follows output zero event happen events defined definition otherwise set wij wij note valid definition since implies result pim pim pim pim follows lemma pim suffices upper bound pim crucial observation joint distribution wij identical two models thus pim pmc pim pmc definition estimator satisfies pmc combination previous inequalities gives pim desired proof lemma simulate independent multinomial model independent poisson model conp ditioning row sum conditioned cij random vector cis follows multinomial distribution multi ris transition matrix obtained normalizing particular rij furthermore conditionally independent thus apply estimator designed independent multinomial model parameter fulfills guarantee need guarantee max probability least constant definition rij rij note poi rij due assumption assumption max max poi exp follows lemma follows proof completes proof lemma dependence diagram random variables follows stationary distribution defined obtained normalizing matrix recall denotes joint distribution sufficient statistic prior goal show note dependent however key observation concentration distribution close fixed distribution state space thus approximately independent clarity denote triangle inequality total variation distance upper bound first term note forms markov chain hence convexity total variation distance epc epr start showing row sums concentrate let rij follows hoeffding inequality lemma uij exp provided lnns next consider entrywise sum write rij uii note follows hoeffding inequality lemma provided exp henceforth set hence probability tending one conditioning event therefore view similarly also remains show note products poisson mixtures triangle inequality total variation distance poi uij poi upper bound individual terms total variation distance poisson mixtures note random variables uij uij match moments order supported follows lemma poi uij set poi used fact establishing desired lemma proof lemma let log constant lemma recall log view rij rij uii uij last step follows symmetry matrix first term note log thus conditioned put probability tending one second term definition uij supported supported lemma hence follows hoeffding inequality uij uii exp exp thus uij snln snln provided put using fact snln third term condition event absolute constant put finally combining well probability tending one absolute constant likewise probability tending one view lemma completes proof proof lemma consider random matrix rij distributed according prior case entirely analogous first lower bound high probability recall definition since sln probability tending one furthermore consequently min desired next deal spectral gap recall normalized version given let diag diag rij rij rri furthermore reversiblity symmetric matrix since similarity transform share spectrum let recall matrix view crucially choice symmetric positive semidefinite matrix thus note also symmetric positive semidefinite matrix let weyl inequality stands spectral norm largest singular values far everything hasqbeen determinimistic next show high probability rhs note wigner matrix furthermore uij takes values absolute constant follows standard tail estimate spectral norm wigner ensemble see corollary exist universal constants combining absolute spectral gap satisfies union bound shown chosen lemma proof lemma representation follows definition conditional entropy remains show let denote transition matrix corresponding empirical conditional distribution transition matrix pij pij pij pij last step pkq pqii stands divergence probability vectors follows fact nonnegativity divergence references jayadev acharya hirakendu das 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arxiv nov predicting rna secondary structures arbitrary pseudoknots maximizing number stacking pairs samuel abstract paper investigates computational problem predicting rna secondary structures general belief allowing pseudoknots makes problem hard existing algorithms heuristic algorithms performance guarantee handle limited types pseudoknots paper initiate study predicting rna secondary structures maximum number stacking pairs allowing arbitrary pseudoknots obtain two approximation algorithms approximation ratios planar general secondary structures respectively rna sequence bases approximation algorithm planar secondary structures runs time general case runs linear time furthermore prove allowing pseudoknots makes maximize number stacking pairs planar secondary structure result contrast recent results psuedoknots based optimizing general complicated energy functions introduction ribonucleic acids rnas molecules responsible regulating many genetic metabolic activities cells rna considered sequence nucleotides also known bases four basic nucleotides namely adenine cytosine guanine uracil rna folds structure forming pairs bases paired bases tend stabilize rna negative free energy yet base pairing occur arbitrarily particular form stable pairs known base pairs base pairings less stable often ignored example folded rna shown figure note figure schematic practice rnas molecules department computer science yale university new department computer science northwestern university evanston kao research supported part nsf grant department computer science university hong kong hong kong twlam smyiu research supported part hong kong rgc grant department computer science national university singapore science drive singapore ksung stacking pair hairpin loop internal loop bulge loop figure example folded rna structure related function rna yet existing experimental techniques determining structures rnas often costly time consuming see secondary structure rna set base pairings formed structure determine structure given rna sequence useful determine corresponding secondary structure result important design efficient algorithms predict secondary structure computers computational viewpoint challenge rna secondary structure prediction problem arises special structures called pseudoknots defined follows let rna sequence pseudoknot composed two interleaving base pairs see figure examples assume secondary structure rna contains pseudoknots secondary structure decomposed types loops stacking pairs hairpins bulges internal loops multiple loops see tompa lecture notes waterman book stacking pair loop formed two pairs consecutive bases see figure example definition stacking pair contains unpaired bases kinds loops contain one unpaired bases since unpaired bases destabilizing positive free energy stacking pairs type loops negative free energy stabilize secondary structure also natural assume free energies loops independent optimal secondary structure computed using dynamic programming time however pseudoknots known exist rnas predicting secondary structures pseudoknots nussinov studied case energy function minimized number base pairs maximized obtained algorithm predicting secondary structures based special energy functions lyngso pedersen proven determining optimal secondary figure examples pseudoknots structure possibly pseudoknots akutsu shown determine optimal planar secondary structure secondary structure planar graph formed base pairings backbone connections adjacent bases planar see section detailed definition rivas eddy uemura akutsu also proposed algorithms handle limited types pseudoknots note exact types pseudoknots implicit algorithms difficult determine although might desirable better classification pseudoknots better algorithms handle wider class pseudoknots paper approaches problem different general direction initiate study predicting rna secondary structures allow arbitrary pseudoknots maximizing number stacking pairs simple energy function meaningful stacking pairs loops stabilize secondary structures obtain two approximation algorithms ratios planar general secondary structures respectively planar approximation algorithm makes use geometric observation allows visualize planarity stacking pairs rectangular grid interestingly observation hold aim maximize number base pairs algorithm runs time second approximation algorithm complicated based combination multiple greedy strategies straightforward analysis lead approximation ratio make use amortization different steps obtain desired ratio algorithm runs time complement two algorithms also prove allowing pseudoknots makes find planar secondary structure largest number stacking pairs proof makes use reduction problem called tripartite matching result indicates hardness rna secondary structure prediction problem may inherent pseudoknot structures may necessarily due complication energy functions contrast results discussed earlier rest paper organized four sections section discusses basic properties sections present approximation algorithms planar general secondary structures respectively section details result section concludes paper open problems preliminaries let rna sequence bases secondary structure set pairs sip sjp sir sjr two pairs share base denote consecutive stacking pairs definition given secondary structure define undirected graph bases nodes edge base pair definition secondary structure planar planar graph definition secondary structure said contain interleaving block contains three stacking pairs lemma secondary structure contains interleaving block proof suppose contains interleaving block without loss generality assume contains stacking pairs figure shows subgraph corresponding stacking pairs since subgraph contains homeomorphic copy see figure figure interleaving block approximation algorithm planar secondary structures present algorithm given rna sequence constructs planar secondary structure approximate one maximum number stacking pairs ratio least approximation algorithm based subtle observation lemma secondary structure planar subgraph contains stacking pairs embedded grid useful property property enables consider secondary structure without pseudoknots order achieve approximation ratio definition given secondary structure define stacking pair embedding grid follows represent bases consecutive grid points horizontal grid line connected directly horizontal grid edge stacking pair connected respectively sequence grid edges two sequences must either figure shows stacking pair embedding figure given secondary structure figure note form stacking pair base pair connected stacking pair embedding similarly connected embedding figure example stacking pair embedding definition stacking pair embedding said planar drawn way lines cross overlap grid embedding shown figure planar lemma let secondary structure rna sequence let stacking pair embedding planar must planar proof planar stacking pair embedding claim contains interleaving block let horizontal grid line contains bases since planar stacking pair embedding assume two stacking pairs intersect see figure figure stacking pair embedding stacking pair underneath two pairs flip one pairs shown figure must least one stacking pair underneath two pairs checking possible cases cases shown figures shown redrawn without crossing overlapping lines contains interleaving block figures lemma lemma relate two secondary structures maximum number stacking pairs without pseudoknots following lemma lemma given rna sequence let maximum number stacking pairs formed planar secondary structure let maximum number stacking pairs formed without pseudoknots proof let planar secondary structure stacking pairs since planar lemma stacking pair embedding planar let stacking pair embedding lines cross grid let horizontal grid line contains bases let number stacking pairs drawn respectively without loss generality assume construct another planar secondary structure deleting stacking pairs drawn obviously planar secondary structure without pseudoknots since based lemma present dynamic programming algorithm axsp computes maximium number stacking pairs formed rna sequence without pseudoknots algorithm axsp define maximum number stacking pairs without pseudoknots formed form pair let maximum number stacking pairs without pseudoknots formed obviously gives maximum number stacking pairs formed without pseudoknots basis form pair recurrence max max form pair form pair lemma given rna sequence length algorithm axsp computes maximum number stacking pairs formed without pseudoknots time space proof entries filled fill entry check values fill entry time suffices total time complexity filling entries storing entries requires space although algorithm axsp presented computes number stacking pairs easily modified compute secondary structure thus following theorem theorem algorithm axsp algorithm problem constructing secondary structure maximizes number stacking pairs rna sequence approximation algorithm general secondary structures present algorithm greedysp given rna sequence constructs secondary structure necessarily planar least maximum possible number stacking pairs approximation algorithm uses greedy approach figure shows algorithm greedysp let input rna sequence initially unmarked let set base pairs output algorithm initially greedysp repeatedly find leftmost consecutive stacking pairs find small possible formed unmarked bases add mark bases downto repeatedly find consecutive stacking pairs formed unmarked bases add mark bases repeatedly find leftmost stacking pair formed unmarked bases add mark bases figure algorithm following analyze approximation ratio algorithm algorithm greedysp generate sequence denoted sph fact spj spk stacking pairs spj share base spk spj define two intervals indexes respectively order compare number stacking pairs formed optimal case following definition definition let optimal secondary structure maximum number stacking pairs let set stacking pairs spj computed greedysp let least one indexes note may disjoint lemma xij xjj proof prove lemma contradiction suppose exists stacking pair xij xjj definition none indexes contradicts step algorithm greedysp definition xij let xij xik xjk let xjj xik xjk xij let number stacking pairs represented spj let numbers indexes intervals respectively lemma let number stacking pairs computed algorithm greedysp maximum number stacking pairs formed proof definition xik xjk fact thus xik xjk lemma lemma spj computed greedysp proof three cases follows case spj computed greedysp step note spj leftmost consecutive stacking pairs smallest possible definition claim prove claim contradiction assume integer consecutive stacking pairs furthermore none bases marked spj chosen otherwise suppose one base says marked algorithm chooses stacking pair adjacent belong belong instead therefore leftmost consecutive stacking pairs formed unmarked bases spj chosen spj leftmost consecutive stacking pairs contradicts selection criteria spj claim follows case spj computed greedysp step let let spj definition claim least show contradiction assume thus integer exist consecutive stacking pairs similarly case show none bases marked spj chosen thus greedysp select consecutive stacking pairs instead chosen consecutive stacking pairs reaching contradiction similarly show case spj computed greedysp step spj leftmost stacking pair chosen let spj approach case show claim verify consider possible cases two consecutive stacking pairs possible case integers belong spj leftmost stacking pair formed unmarked bases contradicting selection criteria spj theorem let rna sequence let maximum number stacking pairs formed secondary structure let number stacking pairs output greedysp proof lemmas result follows remark setting greedysp already achieve approximation ratio following theorem gives time space complexity algorithm theorem given rna sequence length constant algorithm greedysp implemented time space proof recall bases rna sequence chosen alphabet constant constant number different patterns consecutive stacking pairs must consider different strings formed four characters locations occurrences possible strings rna sequence recorded array linked lists indexed pattern string using time preprocessing linked lists fixed entries linked lists total entries linked lists possible values fix constant locate consecutive stacking pairs scan rna sequence left right substring consecutive characters look array see whether form consecutive stacking pairs simple bookkeeping keep track bases used already entry linked lists scanned whole procedure takes time since constant repeat whole procedure different values total time complexity still time section show find planar secondary structure largest number stacking pairs consider following decision problem given rna sequence integer wish determine whether largest possible number stacking pairs planar secondary structure denoted least show decision problem reducing tripartite matching problem defined follows given three node sets cardinality edge set size tripartite matching problem determine whether contains perfect matching set edges touches every node exactly remainder section organized follows section shows construct polynomial time rna sequence integer given instance tripartite matching problem depends section shows contains perfect matching section part showing contain perfect matching combining three sections conclude maximize number stacking pairs planar rna secondary structures construction rna sequence consider instance tripartite matching problem construct rna sequence integer follows let furthermore let edge xpj yqj zrj recall rna sequence contains characters chosen alphabet denote positive integer sequence furthermore means sequence one let max define following four rna sequences every positive integer sequence sequence sequence agc sequence fragments note sequences composed two substrings form separated character two substrings called fragment similarly two substrings form separated two substrings form separated character also called fragments node encoding node three node sets associated unique sequence let hxi hyi hzi denote sequences respectively intuitively hxi encoding node similarly hyi hzi nodes respectively furthermore define hxi hyi hzi node set associated two sequences ghxn hxn let hxn hxn node similarly node sets associated sequences respectively edge encoding edge define four delimiter sequences namely assume xpj yqj zrj encoded sequence defined zrj yqj xpj let special sequence defined following discussion referred region finally define sequence let let note characters constructed time sections show contains perfect matching correctness section shows perfect matching construct planar secondary structure containing least stacking pairs therefore first establish several basic steps constructing stacking pairs form stacking pairs together form stacking pairs together form stacking pairs together form stacking pairs lemma perfect matching proof let ejn perfect matching without loss generality assume define obtain planar secondary structure least stacking pairs consider regions one one three cases case consider region goal show stacking pairs formed within note edges thus obtain total stacking pairs case details follows assume xpj yqj zrj stacking pairs formed stacking pairs formed hxi hxi hyi hyi hzi hzi hxpj hyqj hzrj form stacking pairs total number stacking pairs formed within case consider edges ejn goal show corresponding region accounts stacking pairs thus obtain total stacking pairs case details follows unlike case region sjk may bases paired stacking pairs formed wjk sjk stacking pairs formed vjk sjk vjk sjk stacking pairs paired hxi sjk hxi sjk pjk hyi sjk hyi sjk qjk hzi sjk hzi sjk rjk stacking pairs paired hxi sjk hxi pjk hyi sjk hyi qjk hzi hzi rjk total number stacking pairs charged sjk case consider form stacking pairs stacking pairs number stacking pairs combining three cases number stacking pairs formed exactly notice two stacking pairs formed cross thus correctness part section shows perfect matching first give framework proof section basic definitions concepts presented section proof part given section framework proof let opt secondary structure maximum number stacking pairs let opt number stacking pairs opt opt section establish upper bound opt recall consider base pairs pairs define conjugate substring follows conjugates every substring conjugate example conjugate conjugate form stacking pair two adjacent bases must paired another two adjacent bases concentrate possible patterns adjacent bases two adjacent characters referred construction ten different types form stacking pair conjugate actually form stacking pair said paired since conjugates exist stacking pair involves need consider table shows numbers occurrences total occurrences substrings substring total number occurrences table number occurrences different let denote number occurrences use notation types similarly following fact gives straightforward upper bound opt fact opt min min note opt may pair let number paired opt use notaion types fact strengthened follows fact opt min upper bound given fact forms basis proof showing opt following sections consider possible structure opt possible case show lower bounds values sufficiently large opt shown less particular one cases must make use fact perfect matching order prove lower bound give basic definitions concepts section lower bounds proof given section definitions concepts section give definitions concepts useful deriving lower bounds values first classify region either open closed respect opt extending definitions fragments conjugates introduce conjugate fragments delimiter fragments finally present property delimiter fragments open regions open closed regions respect opt region said open region paired outside otherwise closed region lemma closed region opt proof closed region paired opt thus fact opt recall sequence composed respectively consists two substrings form substrings called fragment furthermore resp consists two substrings form respectively subtrings also called fragment conjugate fragments delimiter fragments consider fragment another fragment called conjugate fragment conjugate note fragment certian resp appears respectively vice versa construction fragment delimiter sequence unique conjugate fragment located respectively however fragment sequence says hxi every instance hxi contains one conjugate fragment hxi fragment said paired conjugate fragment opt opt includes pairs bases fragment called delimiter fragment note delimiter fragment form following lemma shows property delimiter fragments open regions lemma open region delimiter fragments either must pair conjugate fragments opt proof prove statement contradiction suppose one fragment one fragment paired conjugate fragments let particular stacking pairs respectively since open region identify stacking pair within outside respectively note three stacking pairs form interleaving block lemma opt planar reaching contradiction proof part lemma suffices assume open region give proof part let consider following lemma lemma let number delimiter fragments paired conjugate fragments proof construction must next left end delimiter fragment form exist paired leftmost must paired ggc pattern thus must one delimiter fragments paired conjugate fragments based observation classify delimiter fragments two groups delimiter fragments whose left end paired delimiter fragments whose left end paired delimiter fragment group since left paired leftmost must paired opt remaining either find paired opt paired fragment thus paired therefore delimiter fragment group introduces either two unpaired one unpaired one unpaired hence total number unpaired due delimiter fragments group delimiter fragment group consider similar argument show delimiter fragment group introduces either one unpaired one unpaired hence total number unpaired due delimiter fragments group total state lemma shows lower bounds values terms number open regions opt lemma let number open regions opt open region max open region perfect matching either proof statement within closed region paired opt closed regions paired opt thus statement lemma identify fragments open regions paired conjugate fragments lemma thus max statement similar argument proof statement within closed regions paired opt open regions one must let sjn remaining open regions recall ejn corresponding edges open regions since edges form perfect matching node says adjacent edges thus within sjn hxk hxk therefore least two fragments hxk paired conjugate fragments let one fragments note form since paired conjugate fragment one following three cases occurs opt case paired case paired case paired case paired fragment paired substring fragment fragment paired summary either based lemma prove part case analysis following lemma lemma prefect matching opt proof recall closed region opt suppose open region show opt three cases case lemma fact conclude opt case lemma max fact opt smaller case lemma either fact opt max lemma opt conclude prefect matching opt equivalently opt prefect matching conclusions paper studied problem predicting rna secondary structures allow arbitrary pseudoknots simple free energy function minimized number stacking pairs maximized proved problem secondary structure required planar conjecture problem also general case also given two approximation algorithms problem approximation ratios planar general secondary structures respectively would interest improve approximation ratios another direction study problem using energy function minimized number base pairs maximized known problem solved cubic time secondary structure however computational complexity problem still open secondary structure required planar conjecture problem becomes additional condition would like point observation enabled visualize planarity stacking pairs rectangular grid hold case maximizing base pairs references akutsu dynamic programming algorithms rna secondary structure prediction pseudoknots discrete applied mathematics garey johnson computers intractability guide theory freeman new york zuker pedersen internal loops rna secondary structure prediction proceedings annual international conference computational molecular biology pages lyon france pedersen rna pseudoknot prediction energy based models journal computational biology zuker pedersen fast evaluation internal loops rna secondary structure prediction bioinformatics meidanis setubal introduction computational molecular biology international thomson publishing new york nussinov pieczenik griggs kleitman algorithms loop matchings siam journal applied mathematics rivas eddy dynamic programming algorithm rna structure prediction including pseudoknots journal molecular biology tompa lecture notes biological sequence analysis technical report department computer science engineering university washington seattle uemura hasegawa kobayashi yokomori tree adjoining grammars rna structure prediction theoretical computer science waterman introduction computational biology maps sequences genomes chapman hall new york zuker use dynamic algorithms rna secondary structure prediction waterman editor mathematical methods dna sequences pages crc press boca raton zuker sankoff rna secondary structures prediction bulletin mathematical biology

estimating linear quadratic forms via indirect observations apr anatoli juditsky arkadi nemirovski abstract paper develop approach originating statistical estimation via convex focus estimating linear quadratic form unknown signal known belong given convex compact set via noisy indirect observations signal classical theoretical results subject deal precisely stated statistical models aim designing statistical inferences quantifying performance closed analytic form contrast traditional highly instructive descriptive framework approach promote qualified operational estimation routines risks available closed form yielded efficient computation know advance favorable circumstances risk resulting estimate whether high low provably circumstances compensation lack explanatory power approach applicable much wider family observation schemes closed form descriptive analysis possible discuss applications approach classical problems estimating linear forms parameters distribution quadratic forms partameters gaussian discrete distributions performance constructed estimates illustrated computation experiments compare risks constructed estimates numerical lower bounds corresponding minimax risks randomly sampled estimation problems introduction paper considered paper dealing hypothesis testing simple families families distributions specified terms upper bounds momentgenerating functions follows work simple families distributions focus estimation linear quadratic forms unknown signal partly parameterizing distribution question give impression approach results let consider subgaussian case one given random observation drawn distribution parameters affinely parameterized signal goal given observation stemming unknown signal known belong given convex compact set recover value given linear form estimate build affine function observation coefficients function upper bound estimate stem optimal solution explicit convex optimization ljk grenoble alpes avenue centrale domaine universitaire france georgia institute technology atlanta georgia usa nemirovs first author supported labex pgmo grant research second author supported nsf grants time given estimate defined width interval yielded estimate problem thus specified computationally efficient fashion moreover mild structural assumptions affine mapping resulting estimate provably nearoptimal minimax sense see section details latter statement extension fundamental result donoho affine recovery linear form signal gaussian observation scheme paper contributes long line research estimating linear see references therein quadratic among others functionals parameters probability distributions via observations drawn distributions majority cited papers objective provide closed analytical form lower risk bounds problems hand upper risk bounds proposed estimates good cases matching lower bounds paradigm referred descriptive relies upon analytical risk analysis estimate design possesses strong explanation power however imposes severe restrictions structure statistical model restrictions making estimation problem amenable complete analytical treatment exists another operational line research initiated donoho spirit operational approach perfectly well illustrated main result stating recovering linear form unknown signal known belong given convex compact set via indirect gaussian observation worstcase risk affine estimate yielded optimal solution explicit convex optimization problem within factor minimax optimal risk subsequent operational literature similar spirit recommended estimate risk given efficient computation typically stem solutions explicit convex optimization problems addition good situations know advance resulting risk whether large small nearly minimax optimal explanation power operational results almost nonexisting compensation scope operational results usually much wider one analytical results example cited result donoho imposes restrictions except convexity compactness contrast known analytical results problem subject severe structural restrictions terms outlined descriptive operational dichotomy paper operational instance problem estimating linear functional signal affinely parameterising parameters distribution started allow quite general affine mapping general enough signal set restrictions convexity compactness technically approach use paper combines machinery developed techniques risk affine estimate developed hand approach also viewed extension theoretical results cramer tests supplied conjunction techniques exploits attractive opinion feature line research potential applicability wide variety observation schemes convex signal sets rest paper organized follows section following describe families distributions working present estimate construction study general properties section section discuss applications estimating linear forms subgaussian distributions section apply proposed construction estimating quadratic forms parameters gaussian discrete distributions illustrate performance proposed approach describe results preliminary numerical experiments compare bounds risk estimates supplied machinery numerically computed lower bounds minimax risk streamline presentation proofs collected appendix handle case estimates quadratic observation treat affine functions quadratic lifting actual observation notation follows stand spaces real vectors real symmetric matrices respectively spaces equipped standard inner products relation means symmetric matrices size positive semidefinite positive definite denote int use matlab notation means vertical concatenation matrices width means horizontal concatenation matrices height particular reals column vector entries probability distributions product distribution direct product corresponding probability spaces denote given positive integer denote family parameters probability distributions family borel probability distributions exp use shorthand notation express fact probability distribution random vector belongs family simple families probability distributions let int closed convex set symmetric origin closed convex set continuous function convex concave following refer satisfying restrictions regular data regular data define family borel probability distributions exp say distributions satisfying simple given regular data refer simple family distributions associated data standard examples simple families supplied good observation schemes defined include families gaussian poisson discrete distributions instructive examples algorithmic calculus simple families reader referred present three examples simple families use sequel distributions let closed convex subset set let case contains distributions parameters particular contains gaussian distributions quadratically lifted gaussian observations let nonempty convex compact subset set gives rise family distributions quadratic liftings random vectors goal build regular data associated simple family distributions contains end select one spectral norm restrictions smaller better observe hence lose nothing assuming required regular data given following proposition described situation let let set det form regular data every holds besides function coercive convex argument whenever proof see appendix quadratically lifted discrete observations consider random variable taking values standard basic orths identify probability distribution variable point probabilistic simplex prob let drawn independently across let point regular data associated simple family distributions contains distributions quadratic lifts random vectors proposition let zij zij let set positive semidefinite matrices denote zij exp hij set exp words simple family contains distributions random variables proof see appendix estimating linear forms situation goal consider situation follows given euclidean spaces along regular data nonempty set contained convex compact set affine mapping nothing convenient way thinking discrete random variable taking values set vector constant specifying linear form tolerance let family borel probability distributions given random observation associated unknown signal known belong association meaning want recover quantity given call estimate borel function pairs satisfying holds infimum estimate clearly refer estimate data min prob clear context shorten setting section build computationally efficient fashion affine estimate along estimate construction let set nonempty convex set let sup sup convex functions recall continuous compact subset functions give rise convex functions given inf inf convex optimization problem opt min approach presumably good estimate risk given optimal nearly solution latter problem corresponding result follows denotes inner product vectors belonging euclidean space space always clear context proposition situation section let satisfy relation inf max inf inf max inf convex furthermore feasible solution system functions convex constraints variables induces estimate relation thus risk bound clearly holds true candidate solution problem result properly selecting make upper bound estimate arbitrarily close opt equal opt optimization problem solvable proof see appendix estimation repeated observations assume situation described section access observations sampled independently probability distribution allowed build estimate based observations rather single observation immediately reduce new situation previous one simply redefining data specifically given see section positive integer let replace efk replace immediately seen updated data satisfy requirements imposed data section furthermore whenever borel probability distribution satisfy distribution sample drawn linked relation ehfi efk applying new data construction section arrive repeated observations version proposition note resulting convex symmetric permutations components implying lose nothing restricting collections equal components convenient denote common value components observations proposition becomes statements follows use assumptions notation previous section proposition situation described section let satisfy relation let positive integer given functions inf inf inf inf convex real valued furthermore let feasible solution system convex constraints variables setting get estimate via independent observations meaning whenever borel probability distribution associated sense one relation clearly holds true candidate solution convex optimization problem opt min result properly selecting make upper bound estimate arbitrarily close opt equal opt optimization problem solvable otherwise explicitly stated deal observations get back case suffices set application estimating linear form parameters distributions situation apply construction form section situation observation subgaussian parameters affinely parameterized signal goal recover linear function specifically consider situation described section data follows family distributions rnx nonempty convex compact set matrix affinely depending symmetric matrix affine function section goal recover value given linear function unknown signal via observation drawn independently across distribution associated means parameters refer gaussian case special case described problem distribution associated signal exactly case question takes place left hand sides constraints inf inf thus system reads max max arrive following version proposition proposition situation described given let feasible solution convex optimization problem opt min max max let set affine estimate taken data listed beginning section immediately seen optimization problem solvable provided ker optimal solution problem taken along yields affine estimate data listed beginning section opt consistency easily answer natural question proposed estimation scheme consistent meaning every allows achieve arbitrarily small provided large enough specifically denote proposition immediately seen sufficient condition consistency existence equivalently orthogonal intersection kernel linear span indeed assumption every fixed clearly implying opt opt given condition question necessary consistency well since condition violated properly selected making low risk recovery impossible already case zero noise observations observation stemming signal identically equal direct product case simplifications possible direct product case addition assumed beginning section convex compact sets rnu rnv depends solely immediately seen direct product case problem reads opt min max note gaussian case depending condition general necessary consistency since nontrivial information thus principle extracted covariance matrix estimated observations max assuming ker problem solvable optimal solution gives rise affine estimate opt addition assumption direct product case assume sake simplicity whenever case reads opt min max whence taking account clearly convex concave convex compact set theorem get also opt max opt min consider problem recovering observation independently sampled unknown known belong known let minimax recovery inf inf taken borel functions rkd invoking proposition immediately seen whenever one opt standard normal distribution since family parameters distributions contains gaussian distributions induced arrive following conclusion proposition described situation minimax optimal riskopt inf recovering parameters random observations within moderate factor upper bound opt taken data affine estimate yielded optimal solution namely opt riskopt factor worth mentioning general setting good observation schemes described numerical illustration section consider problem estimating linear form signal known belong given convex compact subset via indirect observations affected relative specifically observation given matrices words situation small signal results low observation noise linear form recovered observation entities reals degree smoothness noise intensity parameters estimation problem intend process parameters generated follows selected random normalized max consider case deficient observations nonzero singular values set condition number parameter orthonormal systems first left respectively right singular vectors drawn random rotationally invariant distributions positive semidefinite matrices orthogonal projectors randomly selected subspaces dimension experiments deal case note possesses point whence whenever result subgaussian distributions matrix parameter thought also matrix parameter one goals present experiment compare risk affine estimate model performance envelope model fact small signals result observations ignored present figure results experiment given set parameters generate random estimation problems collections problem compute two affine estimates yielded optimal solution first problem described left boxplot group second aforementioned direct product envelope problem mapping right boxplot note noise amplification replaced effect risk times level observation noise significant variability risk across experiments seemingly phenomena due deficient observation model combined random interplay directions coordinate axes along directions becomes thin orientation kernel opt affine estimate constructed following rules section satisfies bound opt riskopt riskopt corresponding minimax risk figure empirical distribution affine estimation estimation problems group distribution risks problem left problem right quadratic lifting estimating quadratic forms section apply approach section situation given sample distribution depending unknown signal goal estimate quadratic functional signal consider two situations gaussian case gaussian distribution parameters affinely depending discrete case discrete distribution corresponding probabilistic vector given stochastic matrix estimation strategy apply techniques developed section quadratic liftings actual observations gaussian case resulting estimates affine functions first focus implementing program gaussian case estimating quadratic forms gaussian case section focus problem follows given nonempty bounded set nonempty convex compact set affine mapping maps onto convex compact subset affine mapping given matrix functional interest known symmetric matrix vector respectively tolerance observe sample gaussian distribution depending unknown signal known belong goal estimate observation candidate estimate borel function rkd defined smallest construction course actions follows specify convex compact subset matrix real section set select set bzb canonic basis vector adding entities function defined conclude proposition form regular data exp inner product defined observe bzb affine mapping maps linear functional result steps get disposal entities participating setup described section immediately seen entities meet requirements imposed setup bottom line estimation problem stated beginning section reduces problem considered section result applying resulting data proposition legitimate since clearly satisfies arrive result follows proposition described situation let set max inf bzb max inf bzb functions convex furthermore whenever form feasible solution system convex constraints variables setting get estimate functional interest via independent observations exceeding particular setting obtain estimate exceeding proof see section remark situation described beginning section let set given assume interested recovering functional interest points reducing domain interest hopefully reduce recovery assuming point convex compact set straightforwardly verified case conclusion proposition remains valid set replaced set replaced modification enlarges feasible set thus reduces attainable risk bound discussion estimating quadratic forms observations applied literally construction section thus restricting estimates affine quadratic liftings alternative basic approach let consider estimates affine full quadratic lifting thus extending family candidate estimates affine affine vice versa unless note alternative covered approach need replace original components setup section extensions diag diag set easily seen modification reduce risk resulting estimates price increase design dimension thus computational complexity optimization problems yielding estimates illustrate difference two approaches consider situation revisited section interested recover energy signal observation unknown diagonal matrix diagonal entries range priori information known assume section given reliability tolerance assumptions one easily verify case estimate estimate yielded proposed approach absolute constant factors optimal namely let look case observe two independent copies observation naive estimate estimate obtained applying basic approach absolute constant factors better case risks still contrast intelligent estimate risk whenever much smaller easily seen outlined alternative implementation approach also results estimate correct consistency present simple sufficient condition estimator suggested proposition consistent sense section specifically assume singleton allows satisfy allows assume first columns matrix linearly independent consistency estimation procedure given following simple statement proposition described situation assumptions given consider estimate gbk goes given proof see section numerical illustration direct observations problem first illustration deliberately selected extremely simple given direct noisy observation unknown signal known belong given set want recover energy interested quadratic estimate small possible given design parameter note situation dimension observation equal dimension signal underlying observation details setup follows spherical layer given result main ingredient constructions section convex compact subset containing matrices see specified matrix known diagonal diagonal entries satisfying known terms setup section case diag functional interest given processing problem easily seen situation question construction section boils following lose nothing restricting estimates form properly selected scalars supplied convex optimization problem variables min max max max max quantity specifically feasible solution augmented yields estimate exceeding energy estimation problem known belong given range well studied literature available results investigate analytically interplay dimension signal range noise intensity parameters offer provably optimal absolute constant factors estimates example consider case assume sake definiteness otherwise already trivial identically zero estimate near optimal high dimensional regime well known case optimal absolute constant factor achieved absolute constant factor estimate easily seen circumstances similar risk bound holds true estimate yielded optimal solution nice property proposed approach automatically takes care parameters results estimates seemingly performance witnessed numerical results present numerical results experiments reporting compute different sets parameters experiments attainable proposed estimators gaussian case optimal values problem along suboptimality ratios risks lower bounds best possible circumstances compute lower bounds use following construction consider problem estimating given observation section optimal riskopt problem defined infimum estimates let select somehow let two distributions observations follows distribution random vector independent uniformly distributed sphere immediately seen test decide hypotheses via observation total risk defined sum two hypotheses probabilities reject hypothesis true quantity lower bound optimal riskopt words denoting density min riskopt densities spherically symmetric whence denoting univariate density energy observation min min conclude min riskopt closest inspection convolution two univariate densities representable explicit formulas implying given check numerically whether premise indeed takes place whenever case quantity lower bound riskopt experiments used simple search strategy described aimed crude maximizing bound used resulting lower bounds riskopt compute suboptimality ratios figures present typical simulation results illustrating dependence risks problem dimension figure ratio figure parameter figure different curves plot correspond different values parameter varying parameters fixed believe quite moderate values optimality ratios presented figures results typical much larger series experiments conducted attest rather good performance proposed apparatus figure estimation risks functions problem dimension different curves parameters left plot estimation risks right plot suboptimality ratios numerical illustration indirect observations problem estimation problem address section follows observations given matrix observations signal known belong given compact set observation noise positive semidefinite matrix known belong given convex compact set reader surprised singular numerical spectrum optimality ratios lower bounding scheme restricted identify actual optimality ratios among candidate values figure estimation risks functions ratio different curves parameters left plot estimation risks right plot suboptimality ratios figure estimation risks functions different curves parameters left plot estimation risks right plot suboptimality ratios goal estimate energy signal given single observation experiment data specified follows assume discretization smooth function continuous argument use role ellipsoid selected natural version ball make matrix form randomly selected orthogonal matrices diagonal entries diagonal matrix form condition number design parameter set allowed values covariance matrices set diagonal matrices diagonal entries varying noise intensity design parameter processing problem estimating problem clearly covered setups considered section terms setups specify identity mapping onto mapping becomes mapping set convex compact subset set containing matrices form becomes set zdiag suggested proposition linear lifted observation estimates stem optimal solution convex optimization problem opt min given applied resulting estimate estimate opt problem saddle point problem beyond immediate scope standard convex programming software toolboxes primarily aimed solving convex minimization problems however applying conic duality one easily eliminate inner maxima arrive reformulation solved numerically cvx processed experiments numerical results quantify performance proposed approach present along upper risk bounds simple lower bounds best achievable circumstances origin lower bounds follows let let standard normal quantile max latter due origin implies test decides hypotheses via observation risk immediate consequence quantity lower bound whatever estimate try maximize resulting lower risk bound thus arriving lower bound lwbnd max closest inspection latter problem convex one prevent building suboptimal solution note experiments even fixed design parameters still deal families estimation problems differing sensing matrices orientation system right singular vectors respect axes random matrices varies essentially simulation simulation affects significantly attainable estimation risks display figure typical results experiments see theoretical upper bounds estimates varying significantly parameters experiment time stay within moderate factor lower risk bounds figure empirical distribution random estimation problems upper risk bound opt corresponding suboptimality ratios estimation quadratic functionals discrete distribution section consider situation follows given sensing matrix stochastic columns belonging probabilistic simplex nonempty closed subset along observation drawn independently across discrete distribution unknown probabilistic vector signal known belong always assume treat discrete distribution set distribution vertices possible values basic orths goal recover observation value given quadratic form construction observe uut vector observation allows rewrite homogeneous quadratic form goal construct estimate specifically estimate form quadratic lifting observation parameters estimate end set uut uut specify convex compact subset intersection symmetric matrix simplex see cone positive semidefinite matrices put thus proposition defined form regular data setting holds exp auu exp frobenius inner product observe axat affine mapping setting get linear functional ensure uut uut relation obvious proposition combines proposition yield following result proposition situation question given let let max axat max axat inf max inf axat max inf axat inf max inf axat max inf axat real valued convex every candidate solution convex functions optimization problem opt min induces estimate functional interest via observation exceeding numerical illustration illustrate construction consider following problem observe independent across realizations discrete random variable taking values distribution linearly parameterized signal probability distribution discrete square urs known coefficients given two sets consider events objective quantify deviation events probability distribution independence specifically estimate via observations quantity fij urs urs urs quadratic function experiments report estimation carried via straightforward implementation construction presented earlier section setup follows use sensing matrix corresponding observations generated according matrix control parameter selected random normalizing columns matrix independent entries drawn uniform distribution set xrs xrs simplest convex outer approximation set uut use present figure results experiments taking values things equal smaller larger condition number cond sensing matrix thus larger upper bound risk estimate optimal value note variation fij exactly maximal risk worthy note simple compared much involved results bounds proposition laplace functional distribution result fairly good approximations risk estimate boxplots empirical distributions estimation error right plot figure identify discrete square allows treat probability distribution vector figure estimation independence upper risk bound value opt linear estimate function condition number cond data risk linear estimation function along boxplots empirical error distributions simulations cond references bickel ritov estimating integrated squared density derivatives sharp best order convergence estimates indian journal statistics series pages vitesses maximales des erreurs tests optimaux zeitschrift wahrscheinlichkeitstheorie und verwandte gebiete sur minimax son application aux tests probab math approximation dans les espaces estimation zeitschrift wahrscheinlichkeitstheorie und verwandte gebiete model selection via testing alternative penalized maximum likelihood estimators annales institut henri poincare probability statistics volume pages elsevier massart estimation integral functionals density annals statistics pages butucea comte adaptive estimation linear functionals convolution model applications bernoulli butucea meziani quadratic functional estimation inverse problems statistical methodology cao nemirovski xie guigues juditsky change detection via affine quadratic detectors electronic journal statistics donoho statistical estimation optimal recovery annals statistics donoho liu geometrizing rates convergence annals statistics pages donoho liu geometrizing rates convergence iii annals statistics pages donoho liu macgibbon minimax risk hyperrectangles implications annals statistics pages donoho nussbaum minimax quadratic estimation quadratic functional journal complexity efromovich low optimal adaptive estimation quadratic functional annals statistics efromovich low adaptive estimates linear functionals probability theory related fields fan estimation quadratic functionals annals statistics pages gayraud tribouley wavelet methods estimate integrated quadratic functional adaptivity asymptotic law statistics probability letters goldenshluger juditsky nemirovski hypothesis testing convex optimization electronic journal statistics grant boyd cvx users guide release http hasminskii ibragimov estimation problems stochastic differential equations stochastic differential systems filtering control pages springer exponential inequalities constants order two stochastic inequalities applications pages springer huang fan nonparametric estimation quadratic regression functionals bernoulli ibragimov khas minskii estimation linear functionals gaussian noise theory probability applications ibragimov nemirovskii khas minskii problems nonparametric estimation gaussian white noise theory probability applications juditsky nemirovski nonparametric estimation convex programming annals statistics juditsky nemirovski hypothesis testing via affine detectors electronic journal statistics sharp adaptive estimation quadratic functionals probability theory related fields klemela tsybakov sharp adaptive estimation linear functionals annals statistics pages laurent estimation integral functionals density derivatives bernoulli laurent adaptive estimation quadratic functional density model selection esaim probability statistics laurent massart adaptive estimation quadratic functional model selection annals statistics pages lepski new ideas nonparametric estimation arxiv preprint lepski willer estimation convolution structure density model part oracle inequalities arxiv preprint lepski spokoiny optimal pointwise adaptive methods nonparametric estimation annals statistics pages levit conditional estimation linear functionals problemy peredachi informatsii proofs use notation uut proof proposition proposition nothing proposition make paper reproduce proof start proving item proposition exp exp det det observe implies det det det need following lemma let symmetric positive definite matrix let let closed convex subset let also det det spectral frobenius norm matrix addition continuous function convex concave fact affine proof fixed used fact implies kabkf kakkbkf similar computation yields noting besides setting det int equipping frobenius inner product properly selected conclude denoting eigenvalues noting max get therefore eigenvalues satisfy whence noting max see conclude substituting get furthermore matrix satisfies kdk consequently combines relation yield arrive remains prove continuous component claim completely evident convexity function see indeed case note det concave interior semidefinite cone function convex nondecreasing convex domain function obtained convex substitution variables mapping combining origin see conclude exp complete proof need verify claim regular data boils checking continuous let verify continuity recalling indeed continuous verification question reduces checking continuous continuity concavity evident need prove whenever function convex schur complement lemma implying convex since due convex epigraph claimed item proposition proved remains verify item proposition stating coercive let let prove looking expression immediately seen terms expression except terms coming remain bounded grows need verify observe sequence bounded due implying khi denoting last basic orth taking satisfy positive due account matrices observe hti khi kri khi result zpi khi eet khi zeet kri khi kzkf concluding quantity tends due khi proof proposition continuity obvious let verify relations let fix let let denote set permutations let symmetry argument clearly number permutations particular pair met among pairs comparing total number left right hand sides latter equality get card combines equality imply card card let identity permutation due exp exp card inequality exp equally distributed exp definition exp since exp distribution random variable clearly exp exp hij latter relation combines imply proof proposition let first verify identities function continuous compact hence theorem inf inf inf inf required know continuous function convex provided function let let subgradient taken used therefore bounded set addition set nonempty since contains neighbourhood origin thus convex verification fact convex function completely similar given feasible solution let select somehow taking account find definition implying collection feasible solution need following statement lemma given let feasible solution system convex constraints variables estimate proof let satisfy premise lemma let satisfy thus definition due conclude similarly whence definition due invoking lemma get satisfying since selected arbitrarily close indeed estimate proof proposition premise proposition let fix denoting distribution invoking see defined relation takes place applying proposition conclude remains note construction holds particular part proposition immediate given clearly satisfy proof proposition columns matrix see linearly independent find matrix let define relation matrix matrix obtained replacing entry zero let fix setting invoking proposition need prove case one lim sup end note current situation simplify det inf max inf max hence inf max inf max det inf max det nonzero entry matrix due structure see conclude nonzero element recall whenever one whence implying quantity zero provided consequently becomes inf max det appropriately selected independent real det recall bounded consequently given find large enough ensure combines imply follows

apr stochastic variance reduction nonconvex optimization sashank reddi sjakkamr carnegie mellon university ahmed hefny ahefny carnegie mellon university suvrit sra suvrit massachusetts institute technology bapoczos carnegie mellon university alex smola alex carnegie mellon university original circulated date february abstract study nonconvex problems analyze stochastic variance reduced gradient svrg methods svrg related methods recently surged prominence convex optimization given edge stochastic gradient descent sgd theoretical analysis almost exclusively assumes convexity contrast prove rates convergence stationary points svrg nonconvex optimization show provably faster sgd gradient descent also analyze subclass nonconvex problems svrg attains linear convergence global optimum extend analysis variants svrg showing theoretical linear speedup due parallel settings introduction study nonconvex problems form min neither individual necessarily convex lipschitz smooth lipschitz continuous gradients use denote functions form optimize functions incremental oracle ifo framework agarwal bottou defined definition ifo takes index point returns pair algorithm sgd gradientdescent svrg msvrg nonconvex convex gradient dominated min log min log fixed step size table table comparing ifo complexity different algorithms discussed paper complexity measured terms number oracle calls required achieve solution see definition fixed step size mean step size algorithm fixed dependent alternatively total number iterations complexity gradient dominated functions refers number ifo calls required obtain solution dominated function see section definition sgd aware specific results gradient dominated functions also initial point optimal solution assumed constant clean comparison results marked red contributions paper ifo based complexity analysis introduced study lower bounds problems algorithms use ifos favored applications require small amount information iteration two fundamental models machine learning profit ifo algorithms empirical risk minimization typically uses convex models deep learning uses nonconvex ones prototypical ifo algorithm stochastic gradient descent sgd witnessed tremendous progress recent years variety accelerated parallel faster converging versions known among particular importance variance reduced stochastic methods schmidt johnson zhang defazio delivered exciting progress linear convergence rates strongly convex functions opposed sublinear rates ordinary sgd robbins monro nemirovski similar benefits methods also seen smooth convex functions svrg algorithm johnson zhang particularly attractive low storage requirement comparison algorithms schmidt defazio despite meteoric rise methods analysis general nonconvex problems largely missing johnson zhang remark convergence svrg locally strongly convex provide compelling experimental results fig johnson zhang however problems encountered practice typically even locally convex let alone strongly convex current analysis svrg extend nonconvex functions relies heavily convexity controlling variance given dominance stochastic gradient methods optimizing deep neural nets large nonconvex models theoretical investigation faster nonconvex stochastic methods much needed convex methods known enjoy faster convergence rate gradientdescent much weaker dependence without compromising rate like sgd however clear benefits carry beyond convex problems prompting central question paper nonconvex functions one achieve convergence rates faster sgd gradientdescent using ifo rate depend number iterations performed algorithm perhaps surprisingly provide affirmative answer question showing careful selection parameters svrg leads faster convergence sgd gradientdescent use incremental gradient stochastic gradient interchangeably though interested problems knowledge first work improve convergence rates sgd gradientdescent nonconvex optimization main contributions summarize main contributions also list key results table analyze nonconvex stochastic variance reduced gradient svrg prove faster rates convergence gradientdescent ordinary sgd show svrg faster gradientdescent factor see table provide new theoretical insights interplay iteration complexity convergence nonconvex svrg see corollary interesting nonconvex subclass called gradient dominated functions polyak nesterov polyak propose variant svrg attains global linear rate convergence improve upon many prior results subclass functions see section best knowledge first work shows stochastic method linear convergence gradient dominated functions analyze nonconvex svrg show provably benefits specifically show theoretical linear speedups parallel settings large sizes using size show nonconvex svrg faster factor theorem aware prior work stochastic methods shows linear speedup parallel settings nonconvex optimization analysis yields byproduct direct convergence analysis svrg smooth convex functions section examine variant svrg called msvrg faster rates gradientdescent sgd related work convex bertsekas surveys several incremental gradient methods convex problems key reference stochastic convex optimization min nemirovski faster rates convergence attained problems methods see defazio johnson zhang schmidt zhang defazio asynchronous frameworks developed reddi agarwal bottou lan zhou study convex problems shalevshwartz prove linear convergence stochastic dual coordinate ascent individual nonconvex strongly convex study general nonconvex case moreover even special setting results improve upon high condition number regime nonconvex sgd dates least seminal work robbins monro since developed several directions poljak tsypkin ljung bottou kushner clark nonsmooth setting sra considers proximal splitting methods analyzes asymptotic convergence nonvanishing gradient errors hong studies distributed nonconvex incremental admm algorithm works however prove expected convergence stationary points often lack analysis rates first nonasymptotic convergence rate analysis sgd ghadimi lan show sgd ensures iterations similar rate parallel distributed sgd shown recently lian gradientdescent known ensure iterations nesterov chap first analysis nonconvex svrg seems due shamir considers special problem computing leading eigenvectors pca see also follow work shamir finally note another interesting example stochastic optimization locally functions hazan wherein actually convergence function value shown background problem setup say constant lkx throughout assume functions lkx assumption common analysis methods lipschitz constant assumed independent function called convex quantity called condition number whenever convex say convex convex also recall class gradient dominated functions polyak nesterov polyak function called dominated global minimizer note function need convex also easy show convex function dominated analyze convergence rates classes functions following nesterov ghadimi lan use judge iterate approximately stationary contrast sgd convex one uses convergence criterion unfortunately criteria used nonconvex functions due hardness problem quantities comparable general see ghadimi lan typically assumed similar magnitude throughout analysis assume constant report dependence results analysis need following definition definition point called stochastic iterative algorithm said achieve iterations expectation stochasticity algorithm introduce one definition useful analysis sgd methods bounding variance definition say gradient nonconvex sgd convergence rate stochastic gradient descent sgd one simplest algorithms solving algorithm lists pseudocode using uniformly randomly chosen replacement index sgd algorithm sgd input sequence uniformly randomly pick end uses unbiased estimate gradient iteration appropriate conditions ghadimi lan establish convergence rate sgd stationary point results include following theorem theorem suppose gradient let optimal solution iterates algorithm satisfy min completeness present proof appendix note choice step size requires knowing total number iterations advance practical approach use bound ifo calls made algorithm follows corollary theorem corollary suppose function gradient ifo complexity algorithm obtain solution seen theorem sgd convergence rate rate improvable general even function convex nemirovski yudin barrier due variance introduced stochasticity gradients clear better rates obtained sgd even convex nonconvex svrg turn focus variance reduced methods use svrg johnson zhang algorithm recently shown effective reducing variance convex problems result gained considerable interest machine learning optimization communities seek understand benefits nonconvex optimization reference algorithm presents svrg pseudocode observe algorithm operates epochs end epoch full gradient calculated point requiring calls ifo within inner loop svrg performs stochastic updates total number ifo calls epoch thus algorithm reduces classic gradientdescent algorithm suppose chosen typically used practice total ifo calls per epoch enable fair comparison sgd assume total number inner iterations across epochs algorithm also note simple important implementation detail written algorithm requires storing iterates storage avoided keeping running average respect probability distribution algorithm attains linear convergence strongly convex johnson zhang nonstrongly convex functions rates faster sgd shown using indirect perturbation xiao zhang first state intermediate result iterates nonconvex svrg ease exposition define parameters defined shortly first main result following theorem provides convergence rate algorithm theorem let let define quantity mint let let multiple output algorithm optimal solution algorithm svrg input epoch length step sizes discrete probability distribution xsmp uniformly randomly pick end forp end output iterate chosen uniformly random furthermore also show nonconvex svrg exhibits expected descent objective every epoch condition multiple solely convenience removed slight modification theorem statement note value depend obtain explicit dependence simplify using specific choices formalized theorem suppose let multiple exists universal constants following theorem optimal solution problem output algorithm rewriting result terms ifo calls get following general corollary nonconvex svrg corollary suppose ifo complexity algorithm parameters theorem achieving solution ifo calls corollary shows interplay step size ifo complexity observe number ifo calls minimized corollary gives rise following key results paper corollary suppose let multiple exists universal constants theorem following optimal solution problem output algorithm corollary ifo complexity algorithm parameters corollary obtain solution note rate results opposed slower rate sgd theorem comprehensive comparison rates refer section algorithm input epoch length step sizes discrete probability distribution svrg end output gradient dominated functions ending discussion convergence nonconvex svrg prove linear convergence rate class dominated functions ease exposition assume property analogous high condition number regime strongly convex functions typical machine learning note gradient dominated functions nonconvex theorem suppose dominated iterates algorithm satisfy constants used corollary fact dominated functions prove stronger result global linear convergence theorem dominated iterates algorithm satisfy corollary optimal solution immediate consequence following corollary dominated ifo complexity algorithm parameters theorem compute solution log note gradientdescent also achieve linear convergence rate gradient dominated functions polyak however gradientdescent requires log ifo calls obtain solution opposed log svrg similar gains seen svrg strongly convex functions johnson zhang also notice assume anything except smoothness individual functions results particular following corollary also immediate consequence corollary convex functions possibly nonconvex number ifo calls made algorithm parameters theorem compute solution log recall denotes condition number convex function corollary follows corollary upon noting convex function dominated theorem generalizes linear convergence result johnson zhang since allows nonconvex observe corollary also applies strongly convex though case refined result proved johnson zhang finally note result also improves recent result sdca setting corollary condition number reasonably large case typically arises machine learning precisely empirical loss minimization show sdca requires log iterations possibly nonconvex sum strongly convex comparison show algorithm requires log iterations improvement sdca convex case previous section showed nonconvex svrg converges stationary point rate natural question whether rate improved assume convexity provide affirmative answer convex functions yields direct analysis based strongly convex perturbations svrg state results terms stationarity gap ease comparison analysis also provides rates respect optimality gap see proof theorem appendix theorem convex algorithm optimal output algorithm state corollaries theorem explicitly show dependence convergence rates corollary theorem following bound optimal output algorithm result uses step size depends convex case also use step sizes independent following corollary states associated result corollary theorem following bound optimal output algorithm rewrite corollaries terms ifo complexity get following corollaries corollary convex ifo complexity algorithm parameters corollary compute solution corollary convex ifo complexity algorithm parameters corollary compute solution results follow corollary corollary noting total ifo calls made algorithm instructive quantitatively compare corollary step size independent convergence rate svrg dependence order corollary dependence reduced either carefully selecting step size diminishes corollary using good initial point obtained say running iterations sgd emphasize convergence rate convex case improved significantly slightly modifying algorithm either adding appropriate strongly convex perturbation xiao zhang using choice changes epoch zhu yuan however clear strategies provide theoretical gains general nonconvex case nonconvex svrg section study version algorithm popular strategy especially multicore distributed settings greatly helps one exploit parallelism reduce communication costs pseudocode nonconvex svrg algorithm provided supplement due lack space key difference svrg algorithm lies lines use replace line sampling replacement size lines replaced following updates reduces algorithm typically used reduce variance stochastic gradient increase parallelism lemma section appendix shows reduction variance stochastic gradients size using lemma one derive equivalents lemma theorem theorem however sake brevity directly state following main result svrg theorem let denote following quantity min suppose multiple version algorithm size exists universal constants following optimal important result sgd batch size sgd obtains rate dekel obtainable simple modification theorem specifically sgd dependence batch size contrast theorem shows svrg much better dependence batch size hence compared sgd svrg allows efficient formally terms ifo queries following result corollary ifo complexity version algorithm parameters theorem size obtain solution corollary shows interesting property svrg first note ifo calls required calculating gradient size hence svrg gain ifo complexity using however gradients calculated parallel leads theoretical linear speedup multicore distributed settings contrast sgd yield efficient strategy requires ifo calls achieving solution thus performance sgd degrades comparison convergence rates section give comprehensive comparison results obtained paper particular compare key aspects convergence rates sgd gradientdescent svrg comparison based ifo complexity achieve solution dependence number ifo calls svrg gradientdescent depend explicitly contrast number oracle calls sgd independent theorem however comes expense worse dependence number ifo calls gradientdescent proportional svrg dependence reduces convex corollary nonconvex corollary problems whether difference dependence due nonconvexity artifact analysis interesting open problem dependence dependence alternatively follows convergence rates algorithms sgd seen depend regardless convexity nonconvexity contrast convex nonconvex settings svrg gradientdescent converge furthermore gradient dominated functions svrg gradientdescent global linear convergence speedup convergence sgd especially significant medium high accuracy solutions required small assumptions used analysis important understand assumptions used deriving convergence rates algorithms assume lipschitz continuous gradients however sgd requires two additional subtle important assumptions gradients advance knowledge since step sizes depend hand svrg gradientdescent require assumptions thus flexible step size learning rates valuable compare step sizes used algorithms step sizes sgd shrink number iterations undesirable property hand step sizes svrg gradientdescent independent hence algorithms executed fixed step size however svrg uses step sizes depend see corollary corollary step size independent used svrg convex albeit cost worse dependence corollary gradientdescent issue step size independent dependence initial point svrg sensitive initial point comparison sgd seen comparing corollary svrg theorem sgd hence important use good initial point svrg similarly good beneficial svrg moreover provides parallelism also good theoretical guarantees see theorem contrast performance gain sgd pronounced see section best two worlds seen previous section svrg combines benefits gradientdescent sgd show benefits svrg made pronounced appropriate step size additional assumptions case ifo complexity svrg lower sgd gradientdescent variant svrg msvrg chooses step size based total number iterations alternatively discussion assume theorem let gradients let max universal constant corollary let multiple output algorithm satisfies min universal constant universal constant corollary optimal solution corollary gradients ifo complexity algorithm parameters theorem achieve solution min sgd svrg grad sgd svrg grad grad sgd svrg grad sgd svrg grad sgd svrg training loss sgd svrg training loss training loss test error grad figure neural network results mnist datasets top row represents results dataset bottom left middle figures represent results mnist dataset bottom right figure represents result almost identical reasoning applied convex get bounds specified table hence omit details directly state following result corollary suppose convex gradients ifo complexity algorithm step size max achieve solution min msvrg convergence rate faster sgd svrg though benefit without cost msvrg contrast svrg uses additional assumption gradients furthermore step size fixed since depends number iterations often difficult practice compute step size msvrg theorem typical try multiple step sizes choose one best results experiments present empirical results section experiments study problem multiclass classification using neural networks typical nonconvex problem encountered machine learning experimental setup train neural networks one hidden layer nodes softmax output nodes use training use datasets experiments datasets standard neural networks literature regularization mnist features datasets normalized interval datasets come predefined split training test datasets compare sgd algorithm training neural networks nonconvex svrg step size learning rate critical sgd set learning rate sgd using popular schedule chosen sgd gives best http https performance training loss experiments also use results fixed step size sgd svrg use fixed step size suggested analysis step size chosen svrg gives best performance training loss initialization initialization critical training neural networks use normalized initialization glorot bengio parameters chosen uniformly number input output layers neural network respectively svrg use iterations sgd minst iterations sgd running algorithm initialization standard variance reduced schemes even convex problems johnson zhang schmidt noted earlier section svrg sensitive sgd initial point initialization typically helpful use size experiments sgd common training neural networks note training especially beneficial svrg shown analysis section along lines theoretical analysis provided theorem use epoch size experiments results report objective function training loss test error classification error test set convergence criterion throughout analysis datasets algorithms compare criteria number effective passes data ifo calls divided includes cost calculating full gradient end epoch svrg due sgd initialization svrg svrg plots start value mnist figure shows results experiment seen svrg lower compared sgd suggesting faster convergence stationary point furthermore training loss also lower compared sgd datasets notably test error lower svrg indicating better generalization notice substantial difference test error mnist see section appendix overall results network one hidden layer promising interesting study svrg deep neural networks future discussion paper examined scheme nonconvex optimization showed employing stochastic methods one perform better sgd gradientdescent context nonconvex optimization function gradient dominated proposed variant svrg linear convergence global minimum analysis shows svrg number interesting properties include convergence fixed step size descent property every epoch property need hold sgd also showed svrg contrast sgd enjoys efficient attaining speedups linear size minibatches parallel settings analysis also reveals initial point use important svrg concluding paper would like discuss implications work caveats one exercise caution interpreting results paper theoretical results based stationarity gap general necessarily translate optimality gap low training loss test error one criticism schemes nonconvex optimization general wisdom variance stochastic gradients sgd actually help escape local minimum saddle points fact add additional noise stochastic gradient order escape saddle points however one reap benefit schemes even scenarios example one envision algorithm uses sgd exploration tool obtain good initial point uses algorithm exploitation tool quickly converge good local minimum either case believe variance reduction used important tool alongside tools like momentum adaptive learning rates faster better nonconvex optimization references agarwal alekh bottou leon lower bound optimization finite sums bertsekas dimitri incremental gradient subgradient proximal methods convex optimization survey sra nowozin wright optimization machine learning mit press bottou stochastic gradient learning neural networks proceedings defazio aaron bach francis simon saga fast incremental gradient method support convex composite objectives nips defazio aaron caetano domke justin finito faster permutable incremental gradient method big data problems dekel ofer ran shamir ohad xiao lin optimal distributed online prediction using journal machine learning research january issn rong huang furong jin chi yuan yang escaping saddle points online stochastic gradient tensor decomposition proceedings conference learning theory colt ghadimi saeed lan guanghui stochastic methods nonconvex stochastic programming siam journal optimization doi glorot xavier bengio yoshua understanding difficulty training deep feedforward neural networks proceedings international conference artificial intelligence statistics hazan elad levy kfir shai beyond convexity stochastic optimization advances neural information processing systems hong mingyi distributed asynchronous incremental algorithm nonconvex optimization admm based approach arxiv preprint johnson rie zhang tong accelerating stochastic gradient descent using predictive variance reduction nips jakub peter gradient descent methods jakub liu jie peter martin gradient descent proximal setting kushner harold joseph clark dean stochastic approximation methods constrained unconstrained systems volume springer science business media lan guanghui zhou optimal randomized incremental gradient method zhang tong chen yuqiang smola alexander efficient training stochastic optimization proceedings acm sigkdd international conference knowledge discovery data mining kdd acm lian xiangru huang yijun yuncheng liu asynchronous parallel stochastic gradient nonconvex optimization nips ljung lennart analysis recursive stochastic algorithms automatic control ieee transactions nemirovski juditsky lan shapiro robust stochastic approximation approach stochastic programming siam journal optimization nemirovski arkadi yudin problem complexity method efficiency optimization john wiley sons nesterov yurii introductory lectures convex optimization basic course springer nesterov yurii polyak boris cubic regularization newton method global performance mathematical programming poljak tsypkin pseudogradient adaptation training algorithms automation remote control polyak gradient methods minimisation functionals ussr computational mathematics mathematical physics january reddi sashank hefny ahmed sra suvrit poczos barnabas smola alex variance reduction stochastic gradient descent asynchronous variants nips robbins monro stochastic approximation method annals mathematical statistics schmidt mark roux nicolas bach francis minimizing finite sums stochastic average gradient shai sdca without duality corr shai zhang tong stochastic dual coordinate ascent methods regularized loss journal machine learning research shamir ohad stochastic pca svd algorithm exponential convergence rate shamir ohad fast stochastic algorithms svd pca convergence properties convexity sra suvrit scalable nonconvex inexact proximal splitting nips xiao lin zhang tong proximal stochastic gradient method progressive variance reduction siam journal optimization zhu zeyuan allen yuan yang univr universal variance reduction framework proximal stochastic gradient method corr appendix nonconvex sgd convergence rate proof theorem theorem suppose gradient let optimal solution iterates algorithm satisfy min proof include proof completeness please refer ghadimi lan general result iterates algorithm satisfy following bound first inequality follows lipschitz continuity second inequality follows update algorithm since eit unbiasedness stochastic gradient last step uses assumption gradient boundedness rearranging equation obtain summing equation using constant obtain min first step holds minimum less average second third steps obtained equation fact respectively final inequality follows upon using setting inequality get desired result nonconvex svrg section provide proofs results nonconvex svrg first start useful lemmas proceed towards main results lemma suppose let chosen equation iterate algorithm satisfy bound proof since using svrg update algorithm unbiasedness right hand side upper bounded kvt consider lyapunov function bounding require following second equality follows unbiasedness update svrg last inequality follows simple application young inequality plugging equation equation obtain following bound kvt kxt bound quantity use lemma bound upon substituting equation see second inequality follows definition thus concluding proof proof theorem theorem let let define quantity mint let let multiple output algorithm optimal solution proof since using lemma telescoping sum obtain inequality turn implies since used since sum epochs obtain inequality used fact using inequality definition algorithm obtain desired result proof theorem theorem suppose let multiple exists universal constants following theorem optimal solution problem output algorithm proof analysis require upper bound observe obtained using relation fact using specified values inequality follows since using bound get wherein second inequality follows upon noting increasing euler number lower bound min constant independent first inequality holds since decreases second inequality holds since upper bounded constant independent follows equation follows equation choosing independent appropriately one ensure universal constant example choosing substituting lower bound equation obtain desired result proof corollary corollary suppose ifo complexity algorithm parameters theorem achieving solution ifo calls proof result follows theorem fact suppose however ifo calls invested calculating average gradient end epoch words computation average gradient requires ifo calls every iterations algorithm using relationship get case hand total number ifo calls made algorithm epoch since hence oracle calculating average gradient per epoch lower order leading ifo calls proof theorem theorem suppose dominated iterates algorithm satisfy constants used corollary proof corollary shows iterates algorithm satisfy substituting specified value inequality second inequality follows dominance function proof theorem theorem dominated iterates algorithm satisfy corollary optimal solution proof proof mimics theorem following condition iterates algorithm however dominated combined equation concludes proof convex svrg convergence rate proof theorem theorem convex algorithm optimal output algorithm proof consider following sequence inequalities kxt second inequality uses unbiasedness svrg update convexity third inequality follows lemma defining lyapunov function kxsm summing inequality get algorithm svrg input epoch length step sizes discrete probability distribution size xsmp choose uniformly random replacement size end forp end output iterate chosen uniformly random due fact equality uses fact summing epochs telescoping obtain inequality also uses definition given alg inequality use lemma yields easy see obtain convergence rates reasoning leads direct analysis svrg convex functions minibatch nonconvex svrg proof theorem proofs essentially follow along lines lemma theorem theorem added complexity first prove intermediate results proceeding proof theorem lemma suppose parameters chosen iterates version algorithm algorithm size satisfy bound proof using essentially argument proof lemma equation use lemma order bound inequality substituting equat tion see second inequality follows definition thus concluding proof intermediate key result following theorem provides convergence rate svrg theorem let denote following quantity min suppose let output version algorithm size optimal solution proof since using lemma telescoping sum obtain inequality turn implies used since since sum epochs using fact get desired result present proof theorem using results theorem let denote following quantity min suppose multiple version algorithm size exists universal constants following optimal proof theorem first observe using specified values obtain inequality follows since analysis require following bound wherein first equality holds due relation inequality follows upon noting increasing lower bound min constant independent first inequality holds since decreases second one holds since upper bounded constant independent due equation due equation fact choosing appropriately small constant independent one ensure universal constant example choosing substituting lower bound theorem get desired result msvrg convergence rate proof theorem theorem let gradients let max universal constant corollary let multiple output algorithm satisfies min universal constant universal constant corollary optimal solution proof first observe step size chosen max suppose obtain convergence rate corollary lets consider case case following bound first inequality follows lemma second inequality follows gradient property fact random variable rest proof along exactly lines theorem provides convergence rate similar theorem specifically using step size get thing remains proved size choice max minimum two bounds hold consider case case following max constant corollary inequality holds since rearranging inequality case left hand side inequality precisely bound obtained using step size see equation similarly inequality holds direction using two observations desired result key lemmatta lemma intermediate iterates computed algorithm following proof proof simply follows proof lemma present result bound variance svrg lemma let computed version algorithm algorithm size kxt proof ease exposition use following notation use definition get first inequality follows lemma fact inequality get first inequality follows fact indices drawn uniformly randomly independently noting random variable last inequality follows fit experiments figure shows remaining plots mnist datasets seen plots significant difference test error svrg sgd datasets sgd svrg grad sgd svrg test error sgd svrg test error grad grad figure neural network results mnist leftmost result mnist remaining two plots lemmas need lemma results convex case lemma johnson zhang let convex continuous gradient proof consider arbitrary observe also continuous note since alternatively since defines bregman divergence follows min min rewriting terms obtain required result lemma bounds variance svrg convex case please refer johnson zhang details lemma johnson zhang suppose convex updates algorithm following inequality proof proof follows upon observing following first inequality follows young inequality second one third one lemma lemma random variables kzr

may automating embedded analysis capabilities managing software complexity multiphysics simulation part application partial differential equations roger pawlowski eric phipps andrew steven owen christopher siefert matthew staten sandia national april keywords generic programming templating operator overloading automatic differentiation partial differential equations finite element analysis optimization uncertainty quantification abstract generic programming approach presented previous paper separates development effort programming physical model computing additional quantities derivatives needed embedded analysis algorithms paper describe implementation details using generic programming approach simulation analysis partial differential equations pdes detail several hurdles encountered software infrastructure developed overcome end demonstration present shape optimization uncertainty quantification results pde application introduction computational science potential provide much numerical solutions set equations set analysis opportunities beyond simulation include parameter studies stability analysis optimization uncertainty quantification capabilities demand application code required single simulation typically form extra derivative information addition computational design analysis often entail modification governing equations refinement model hierarchy fidelities previous paper described generic programming tbgp approach paper provides conceptual framework upon paper builds thus prerequisite work described showed assembly templatebased automatic differentiation technology work together deliver flexible assembly engine corresponding author sandia national laboratories numerical analysis applications department box albuquerque new mexico usa tel fax agsalin sandia national laboratories laboratory managed operated sandia corporation wholly owned subsidiary lockheed martin corporation department energy national nuclear security administration contract model equations rapidly composed basic building blocks residual needs explicitly programmed approach based templating scalar operations within simulation instantiation template code various data types effect code transformations needed embedded analysis operator overloading often application operator overloading manner assumed introduce significant runtime overhead simulation however demonstrated careful implementation overloaded operators using techniques expression templates completely eliminate overhead results single templated code base must developed tested combined appropriate seeding extracting speciallydesigned overloaded data types see section definitions terms allows manner additional quantities generated additional software development time paper extend description approach simulation analysis partial differential equations pdes discussed previous paper number projects implemented embedded analysis capabilities leverage domain specific language specifically finite elements fenics sundance projects demonstrated capability respect derivative evlauation pdes provide additional challenges regards data structures scalability large systems paper deal specifically galerkin finite element approach though approach follow directly assemblies analogy assemblies section discuss approach begins ends relates global local data structures section present many details approach finite element assembly particular seed compute extract phases section addresses advanced issues dealt codes use approach specifically includes infrastructure exposing model parameters needed continuation bifurcation optimization uncertainty quantification approaches dealing templated code stack approaches dealing code templated finally section demonstrate whole process example pde application sliding electromagnetic contact problem show results shape optimization embedded uncertainty quantification critical main message paper fact infrastructure computing extra quantities needed analysis capabilities implemented independently work implementing pde model infrastructure includes seed extract phases approach also includes solver libraries implemented trilinos framework linear nonlinear transient optimization solvers place application codes new pdes readily generated born analytic derivatives embedded analysis capabilities novel interface exposed computational scientists allowing templated data types passed equation assembly tremendous potential interface exploited derivatives operation counting polynomial propagation expect developers find innovative ways exploit interface beyond currently imagine note transforming legacy implementation use templates manner involve significant effort thus would consider approach appropriate new development efforts however transformations necessary type specifications function variable declarations approach finite element codes first paper series explained generic programming approach included illustrative demonstration applied ode problem section present basic details approach used context pde applications implementation details restricted discretization strategies assembly kernels finite element fem control volume finite element cvfem methods details approach would need adapted discretizations finite difference methods integral equations methods extending approach odes pdes gives rise many issues core design principle still evaluation equations separated three phases seed compute extract seed extract phases need specialized template type extra information data types derivative information must initialized retrieved compute phase equations implemented written fully generic fashion also issues regard data structures sparse matrices parallelism use discretization libraries potential dependency libraries property data issues addressed following sections assembly primary issue arises using generic programming tbgp approach pdes sparsity derivative dependencies automatic differentiation approach computing jacobian matrix using sacado package requires relevant variables sacado forward automatic differentiation data type includes dense array partial derivatives respect independent variables problem problem sizes easily extend millions beyond yet nonzero entries per row stay bounded feasible adopt approach second issue requirement ability run codes parallel architectures adding layers within infrastructure would also challenging two issues circumvented invoking generic programming local level fem methods single element entire pde assembly phase performed summing contributions individual elements within element typically bad assumption local jacobian often referred element stiffness matrix dense jacobian matrices performed element level array partial derivatives sized number degrees freedom element dense contributions row matrix subsequently scattered global sparse matrix structure similarly quantities computed generic programming approach also calculated element element summed global data choice implementing generic programming local level also nullifies second issue distributed memory parallelism typical distributed memory implementation information neighboring elements often called ghost overlap halo data templating infrastructure loop falls messagepassing layer implementation communication performed within templated code note tools including sacado compute residual along jacobian allowing quantities computed simultaneously general evaluation nth derivative also involves simultaneous evaluation derivatives order well table embedded analysis algorithms require variety quantities computed pde assembly table shows list linear algebra quantities computed well required inputs table solution vector time derivative one vectors chosen analysis algorithm one system parameters one random variables residual vector discretized pde system stochastic expansion residual vector evaluation residual steady jacobian transient jacobian directional derivative sensitivity stochastic galerkin residual stochastic galerkin jacobian input vector input output vector output matrix note local approach solution problems use sparse derivative arrays compression techniques see overview approaches references relevant literature automatic differentiation applied directly global level furthermore message passing libraries distributed memory parallelism augmented support communication derivative quantities however due extra level indirection introduced use sparse derivative arrays significantly degrade performance moreover compression techniques require first computing derivative sparsity pattern solving optimization problem compress sparse derivative nearly dense one practice approximate solutions optimization problem attained however solution problem fact known priori precisely equivalent local approach assuming element derivative dense thus found local approach significantly simpler global one particularly pde discretization software tools support templated data types developed intrepid package trilinos data structures purpose pde assembly engine fill linear algebra objects primarily vectors sparse matrices data structures used solvers analysis algorithms instance newton based solver need residual vector jacobian matrix algorithm need directional derivative explicit time integration algorithm need forcing vector polynomial chaos propagation creates vector vectors polynomial coefficients sensitivity solve computes multiple vectors single multivector dense column matrix derivatives respect handful design parameters input pde assembly also vectors solution vector vector design parameters coefficient vectors polynomial expansions parameters critical note tbgp machinery applied linear algebra structures used solvers analysis algorithms none expression templating infrastructure comes play level tbgp applied locally single element assembly process used fill linear algebra data structures example implementations vectors matrices objects epetra tpetra libraries convenient support parallelism compatibility solvers trilinos linear solvers analysis algorithms however none subsequent implementation tbgp code dependent choice inside pde assembly finite element codes natural storage layout computations discretized pde equations operate arrays mdarrays data accesses local indexing local nodes local quadrature points local equation number current mdarray domain model specified implemented shards package trilinos tbgp computations occur locally within element assembly element contributions linear algebra objects done local blocks elements called worksets workset homogeneous set elements share local bookkeeping material information computations within element workset independent ability loop workset amortizes overhead function calls gives flexibility obtain speedups vectorization cache utilization threading based parallelism restricting workset elements homogeneous avoid excessive conditional tests indirect addressing within workset loops number elements workset chosen based number criteria including runtime performance optimization memory limitations dimensions mdarrays include number local nodes number quadrature points number local equations unknowns neq number spatial dimensions instance nodal basis function mdarray dimensions gradient solution vector evaluated quadrature points dimensioned neq mdarrays templated scalar data type called scalart code examples depending specific scalar type instantiated hold value also hold information derivatives case jacobian evaluations sensitivities polynomial chaos coefficients point hope reader understanding generic programming approach previous paper paradigm current section motivated application tbgp approach local element level defined distinction global linear algebra objects matrices vectors span mesh typically double data type mdarrays data structures quantities templated scalar type foundation main concept paper presented following section template based element assembly generic programming approach requires seed phase scalar data types initialized appropriately described section different data structures need exist solution phase assembly phase notably gather routine needed pull global information solution vector nonlinear solver local element data structures solution values local nodes element design perform gather seed operations routine pull global data local data storage copy local storage also seed scalar data types needed seeding dependent scalar data type gather operation must template specialized code example jacobian evaluation partial derivative array associated solution vector seeded identity matrix inverse also true end pde assembly contributions global residual vector depending scalar type information jacobian polynomial chaos expansion evaluation types contained data structure well quantities need extracted data structures well scattered back local global data storage containers combine scatter extract operations single step require code jacobian example derivative array associated residual entries rows element stiffness matrix compute phase operates solely local mdarray data structures data templated scalar type phase written entirely generic template type attempted capture concept schematically figure phase must take global data depending template type seed local arrays appropriately compute phase broken five distinct evaluations cartoon blue boxes performs element level finite element calculations specific pdes written generic template type uniqueness gather coordinates box addressed later section phase takes results assembly loads data appropriate global quantities dictated specific evaluation type execution phases initiated application code handled phalanx package traversing evaluation kernels directed acyclic graph details three phases typical finite element assembly described subsequent sections finite element assembly process tbgp framing finite element assembly process terms generic programming concept best explained example apply galerkin finite element method generic scalar multidimensional conservation equation see example unknown solved degree freedom time derivative flux source term functions time position exact form flux important comment flux strongly convecting additonal terms supg may required damp oscillations simplify analysis ignore terms valid systems convection dominant low reynolds number flows heat conduction solid equation put variational form integration parts ignoring boundary contributions sake simplicity yields residual equations figure schematic template based generic programming model pdes gather seed evaluator takes global data copies local storage seeding embedded data types template specialized codes scatter extract phase inverse compute phase written generic template type operating local data domain problem solved finite element basis functions unknown time derivative spatial derivative computed using unknown coefficients discretization coordinate direction number basis functions integrations performed using numerical quadrature total number elements domain number quadrature points element integration order determinant jacobian transformation physical space element reference space quadrature weights finite element assembly algorithm defined process redefined terms operations assembly algorithm loops elements domain sums partial contributions form residual equations complete set residual equations constitute global residual reformulating terms workset concept assembly process evaluating residual defined skt number worksets partial residual associated finite element contributions elements workset gather operation maps global solution vector local solution vector workset mentioned software implementation gather routine also performs seeding scalar types skt scatter operation maps local element residual elements workset global residual contribution skt noted software implementation extraction process occurs scatter element residual contributions come elements workset function local workset solution vector number elements workset important point note code written used evaluating residual bulk code reused evaluation types jacobians parameter sensitivities stochastic residuals etc accomplished merely writing additional specialization gather scatter skt operations code residual evaluation written generic template argument scalar type reused evaluation type following sections show examples explain assembly steps seed gather phase template specialization first phase approach gather operation pulling quantities global vector seed phase initializing template type desired embedded operation block code example adaptation working code phalanx evaluator called gathersolution operation occurs within evaluatefields method particular described previous paper trilinos package phalanx used build governing equations separate pieces computation broken phalanx evaluator objects field evaluated called local solution vector local nodes element depends global solution vector vector data layout figure gathersolution class specialized residual evaluation type routine simply copies values one data structure another use bookkeeping function connectivitymap figure code gathersolution class specialized jacobian evaluation type shown addition gather operation load value void gathersolution evaluationtype evaluatefields note mdarray dimension numberofelements numberoflocalnodes data type double int elem numberofelements int node numberoflocalnodes elem node connectivitymap elem node figure seed gather code residual evaluation connectivitymap function degree freedom connectivity map gets global element number local node number seed phase trivial copy value local data structure also seed phase initialize partial derivatives identity matrix independent variables defined initializing partial derivative array sacado automatic differentiation data type two nested loops local nodes dxi used set otherwise number output quantities produced finite element assembly increases defined rows table number template specialized implementations gathersolution object need written syntax dependent implementation sacado package trilinos concept seeding automatic differentiation calculations general compute phase generic template compute phase computes local contributions pde application operates data exists local data structures code written entirely generic evaluation template type evalt one must write code needed evaluate residual equation using scalart data type corresponding evaluation type evalt instead raw double data type shown section overloaded data type together specializations seed extract phases enable code compute manner quantities outputs table section give two examples evaluatefields method phalanx evaluator class first shown figure calculates source term heat equation source term equation parameters model solution field code presupposes field computed another evaluator mdarray elements quadrature points factors example scalar values vary domain discuss later section expose parameters design analysis note code templated generic evalt evaluation type one implementation needed general pde codes common efficient use discretization libraries perform common operations finite element code includes basis function calculations calculating transformation reference physical elements supplying quadrature schemes void gathersolution evaluationtype evaluatefields note mdarray dimension numberofelements numberoflocalnodes data type sacado allocated space numberoflocalnodes partial derivatives int elem numberofelements int node numberoflocalnodes elem node connectivitymap elem node loop nodes seed int numberoflocalnodes node elem node else elem node figure seed gather code jacobian evaluation gather operation residual calculation seed phase involves local automatic differentiation data type method local accesses value local accessing ith partial derivative example assumes one equation one unknown per local node extent operations occur within loop must support templating tbgp approach work trilinos intrepid discretization library serves roles written templated generic scalart data type figure shows example evaluator method final assembly residual equation heat balance code uses integrate method intrepid finite element library accumulate summations quadrature points four terms integrals variational formulation pdes terms matched galerkin basis functions adapted equation using common notation heat transfer basis functions three terms right hand side correspond diffusive source accumulation terms source term equation function solution function position simple constant dependencies must defined source term evaluator however dependencies need described heat equation residual piece code accurately propagate derivatives seeded gather phase accumulated source term evaluator also note possible write code compute phase instance one would like jacobian fill efficiency leave terms preconditioner could simply write function heatequationresidual evaluationtype void sourceterm evalt evaluatefields note evaluator depends properties alpha beta scalar quantity compute scalar source int elem numberofelements int numberofquadpoints source elem alpha beta elem elem figure example evaluation kernel compute phase note code templated generic evalt evaluation type code propagate auxiliary information contained evalt data type embedded capabilities template specialization needed void heatequationresidual evalt evaluatefields note evaluator depends several precomputed fields flux source tdot time derivative wbf basis function quadrature transformation weights wgradbf gradient basis functions weights result tresidual field element contribution heat equation residual typedef intrepid fst fst scalart tresidual flux wgradbf fst scalart tresidual source wbf fst scalart tresidual tdot wbf figure evaluator final assembly heat equations terms correspond order variable tresidual accumulated step variables mdarrays depending template parameter evalt corresponding data type scalart block code used accumulating residual jacobian output quantities listed table void scatterresidual evaluationtype evaluatefields note mdarray dimension numberofelements numberoflocalnodes data type double int elem numberofelements int node numberoflocalnodes connectivitymap elem node elem node figure extract scatter code residual evaluation connectivitymap function also called degree freedom map gets global element number local node number extract phase trivial evaluation type copy value generic template type evalt replaced template specialized evaluation type jacobian extract scatter phase template specialization section closely mimics section transpose operations local element contributions finite element residual scattered global data structure time additional information stored templated data types extracted scattered global linear algebra objects example evaluator object called scatterresidual operation occurs within evaluatefields method particular culmination compute steps resulted computation field local represents local element contribution global residual vector may also contain additional information scalart data type figure scatterresidual class specialized residual evaluation type routine simply copies values element data structure global data structure use bookkeeping function connectivitymap used section figure code scatterresidual class specialized jacobian evaluation type shown local dense stiffness matrix extracted sacado automatic dfi differentiation data type two nested loops local nodes used extract load global sparse matrix set template specialized implementations scatterresidual object need written match gathersolution class two shown implementations specific interface global data structures used encapsulated connectivitymap addsparsematrixentry methods central point paper concept generic programming implementations seed gather extract scatter sections written agnostic physics solved work correctly programming two phases evaluation types trivial development effort completely orthogonal work adding terms pdes code set implementations new evaluation type pdes assembled compute phase note examples shown trivial design assembly engine void scatterresidual evaluationtype evaluatefields note mdarray dimension numberofelements numberoflocalnodes data type sacado allocated space numberoflocalnodes partial derivatives int elem numberofelements int node numberoflocalnodes int elem node int numberoflocalnodes int elem double val elem node addsparsematrixentry row col val figure extract scatter code jacobian evaluation extract phase involves local automatic differentiation data type method data type accesses ith partial derivative fictitious method called addsparsematrixentry int row int col int value shows jacobian information scattered global sparse storage general allows complex multiphsyics problems particular unknowns bound basis mixed basis problems demonstrated phalanx assembly engine design evaluator graph completely controlled user thus allowing algorithm local workset implemented note evaluation routines loops explicitly written evaluator general ideal since likely repeated across multiple evaluators introduce additional points error loops could eliminated using utility functions expression templates future area research furthermore optimizing ordering nested loops corresponding data layouts field data embedded scalar types also areas future research finally note design phalanx package great care given designing library users little template experiience could easily add new physics feel extremely successful exists distinct physics applications using tbgp packages trilinos drawback however initial setup tbgp process requires programmer strong background templates contrast codes sundance fenics automate entire assembly process users lowering barrier adoption feel extra work setting tbgp machinery worth effort quickly extended embedded analysis support new types stochastic galerkin methods extensions tbgp finite element code design basic implementation details generic programming approach finite element code described previous section implemented approach application codes run across many issues implemented solutions important described following sections infrastructure exposing one main selling points generic programming approach ability perform design analysis involving system parameters continuation sensitivity analysis optimization require model parameters manipulated analysis algorithms parameters model specific commonly include value boundary condition dimensionless group reynolds number model parameter arrhenius rate coefficient shape parameter radius cylindrical part section briefly describe infrastructure exposing parameters infrastructure dealing parameters try meet following design requirements simple interface model developers expose new parameters integration templatebase approach derivatives respect parameters captured seamless exposure parameters design algorithms optimization approach successful codes use parameterlibrary class sacado package trilinos utility stores available parameters string name value handle multiple data types needed approach developer register parameters parameter library identified strings simply calling register method construction phase expose parameters example labels alpha beta constructor sourceterm evaluator simply needs add lines alpha beta assuming parameterlibrary object scope end problem construction parameter library queried list registered parameters include two addition registered elsewhere analysis algorithms manipulate values parameters parameterlibrary choice using push pull paradigm value changed parameter library immediately pushed evaluator eventually used model pull parameter values parameter library needed chosen push approach since choice performance penalty exposing numerous parameters potential design variables parameters pushed one location parameters used multiple evaluators must root evaluator registered evaluators must dependency one evaluator class registers parameter must inherit abstract parameteraccessor class single method called scalart getvalue std name parameter gets registered parameterlibrary needs send pointer parameteraccessor class parameter library push new values parameter manipulated scalart sourceterm evalt std name alpha return alpha elseif beta return beta figure example implementation getvalue method provides hook analysis algorithms manipulate design parameters analysis algorithm example handled argument registration call example getvalue method simply implemented shown figure assuming parameters alpha beta member data generic template type scalart sensitivities residual equation respect parameters calculated automatic differentiation evaluated tangent evaluation type like jacobian evaluation type associated scalar data type sacado type however length derivative array number parameters seed extract phases require different specializations shape optimization second scalar type quantities pde assembly might nonzero partial derivatives respect independent variable whether parameter part solution vector must templated data type way derivatives propagated using object overloading approach constants hardwired realtype data type avoid expense propagating partial derivatives know zero bulk calculations coordinates nodes finite element mesh fixed quantities solely function coordinates basis function gradients mapping element reference element set realtype however began shape optimization coordinates node could nonzero derivatives respect shape parameter tangent sensitivity evaluation simply make quantities dependent coordinates scalart type would trigger excessive amount computations particularly jacobian calculations chain rule would propagating zeroes large part finite element assembly solution create second generic data type meshscalart quantities derivatives respect coordinates dependency solution vector traits class defined previous paper extended include meshscalart well scalart follows struct usertraits public phx scalar types typedef double realtype typedef sacado fadtype evaluation types default scalar type struct residual typedef realtype scalart typedef realtype meshscalart struct jacobian typedef fadtype scalart typedef realtype meshscalart struct tangent typedef fadtype scalart typedef fadtype meshscalart future decide moving mesh problem coordinate vector heat equation depend current displacement field calculated elasticity equation jacobian evaluation could switched traits class typedef fadtype meshscalart automatically code would calculate accurate jacobian fully coupled moving mesh formulation complicated pick correct data type quantities approach sacado implementation data types useful feature code compile attempt assign derivative data type real type casting away derivative information must done explicitly done accident illustrated following code fragment annotated comments realtype fadtype derivatives set zero report error template infrastructure generic programming places number additional requirements code base building manipulating objects intrusive additionally compile times excessive template types added infrastructure address issues extensible infrastructure infrastructure assembly process must designed extensibility addition new evaluation types scalar types minimally invasive code support requirement template manager class developed automate construction manipulation templated class given list template types template manager instantiates particular class template type using user supplied factory class instantiated objects stored std inside manager stored vector class instantiated must inherit base class objects instantiated template manager provides functionality similar std return iterators base class objects allows random access based template type using templated accessor methods returning object either base derived class list types template manager must build fixed compile time use template metaprogramming techniques implemented boost mpl library sacado types defined traits class object discussed detail note template manager described simplified version tuple manipulation tools supplied boost fusion library future plan transition code using fusion library compile time efficiency class templated evaluation type compiler must build object code template types result extremely long compile times even minor code changes reason explicit template instantiation highly recommended classes templated evaluation type experience compilers support explicit instantiation therefore inclusion model explicit template instantiation supported objects see chapter details downside system class implements explicit instantiation declaration definition must split separate header files third file must also added code base incorporating code situations code may provide based implementations analysis evaluations example straightforward differentiate fortran codes source transformation tools adifor provide analytic evaluations first higher derivatives however clearly resulting derivative code use procedure sacado operator overloading library similarly libraries derivatives either case mechanism necessary translate derivative evaluation governed sacado one provided library providing translation relatively straightforward procedure using template specialization techniques already discussed briefly phalanx evaluator written wraps code phalanx evaluation hierarchy evaluator specialized evaluation type library provides mechanism evaluating specialization extracts requisite information corresponding scalar type derivative values copies whatever data structure specified library evaluating quantities situations layout data given scalar type matches layout required library case copy necessary sacado provides forward data type layout matches layout required adifor case pointer derivative values needs extracted however always case situations copy necessary approach work evaluation types library provides mechanism evaluating however clearly situations arise library provides mechanism certain evaluation types case specializations must written way generate required information example library provide derivatives approximated scheme jacobian evaluation example jacobian specialization evaluator library would make several calls library perturbation input data library combine derivatives inputs dependent fields evaluator using chain rule place derivative arrays corresponding outputs evaluated fields evaluator similarly polynomial chaos expansions code computed spectral projection mesh morphing importing coordinate derivatives shape optimization capability demonstrated section requires sensitivities residual equation respect shape parameters implementation uses external library moving mesh coordinates function shape parameters mesh morphing active research area paper prepared detailing six different approaches variety applications briefly desired capability application code able manipulate shape parameters length curvature part solid model mesh morphing utility provide mesh conforms geometry avoid changes data structures discontinuities objective function calculation desirable mesh topology connectivity stay fixed algorithm must find balance maintaining good mesh quality preserving grading original mesh anisotropy mesh designed capture boundary layer large shape changes require remeshing beyond scope work would need accommodated remeshing restarting optimization run variety mesh morphing algorithms developed investigated one end spectrum smoothing approach surface nodes moved accommodate new shape parameters resulting mesh smoothed elements regain acceptable quality end spectrum femwarp algorithm finite element projection used warp mesh requiring global linear solve determine new node locations paper used weighted residual method new node coordinates based boundary nodes neighborhood moved chose always morph mesh original meshed configuration chosen configuration even intermediate mesh already computed nearby shape parameters new mesh uniquely defined shape parameters since mesh morphing algorithm local calculation element operates across entire mesh one time derivatives calculated within phalanx evaluator using approach using methods described section approach calculating sensitivities residual vector respect shape parameters use chain rule coordinate vector outside section templated code mesh sensitivities finite difference algorithm around mesh morphing algorithm fed residual calculation global data part typical assembly gathercoordinates evaluator takes coordinate vector mesh database gathers local mdarray data structure shape sensitivities coordinate vector sacado data type gather operation imports values coordinates also seeds derivative components vectors shown schematically figure gathercoordinates box shown template specialized version shape optimization addition generic implementation evaluation types rest calculation proceeds directional derivative direction computed implementation extracting described section works case set model parameters demonstration sliding electromagnetic contact generic programming approach demonstrated prototype pde application sliding electromagnetic contact problem geometry problem shown figure nominal design geometry simply extruded third dimension slider light blue situated two conductors green yellow given shapes contact pads thin red blue regions potential difference device prescribed electrical current flows system general direction dashed red line electrical current generates magnetic field turn propels slider forward addition current generates heat figure front view geometry sliding electromagnetic contact application imposed gradient potential causes electric current joule heating magnetic field propels slider currently modeled design problem investigated find shape slider given volume minimizes maximum temperature achieved inside governing equations objective function demonstration simplify system decoupling magnetics solve model slider velocity given model reduced two coupled pdes first potential equation electric potential second heat balance accounts conduction convection joule heating source term depends current electrical conductivity varies function local temperature field based simplified version knoepfel model take different values slider conductor pads dependency results coupled pair equations choosing frame reference stays slider impose fixed convective velocity beams slider dirichlet boundary conditions potential temperature shown figure others natural boundary conditions since equations geometry symmetric mid plane vertical line figure solve half geometry impose along axis pde model implemented using phalanx evaluators specifying dependencies evaluators evaluation tree automatically constructed full graph problem shown figure ode example previous paper gather scatter functions need written template specialization seed extract phases intermediate compute quantities written generic evaluation type example interpret graph oval marked computes temperature field quadrature points using basis functions temperature field nodes implementing equation subsequently used compute another evaluator implements equation objective function minimized design problem simply design parameters modify shape slider shape fixed match rectangular pads end volume fixed rectangular box shape allowed vary parabolically optimization problem matched parabolic profiles top bottom slider free vary single maximum deflection parameter symmetry problem defines parabola optimization problem two parabolas allowed vary independently third bulge slider plane figure adjusted volume constraint met optimization optimization problem solved using optimization algorithm dakota framework dakota built part trilinos using build system adaptors trikota package goal minimize objective function equation function shape parameters addition problem constrained discretized pdes figure dependency graph phalanx evaluators build equation set shown box represents separate class quantities bottom must computed derivatives automatically propagated chain rule using sacado automatic differentiation data types coordinate vector temperature potential subscript indicating node quadrature point data satisfied represents finite element residuals equation set specified equations solution vector combined vector discretized potential temperature fields shape parameters appear explicitly objective function even governing equations instead effect geometry problem appear discretization written vector coordinates nodes mesh addition objective function algorithm depends reduced gradient objective function respect parameters formula term expanded terms computed different way starting end computed finite differences around mesh morphing algorithm described section sensitivity residual vector respect shape parameters directional tive direction computed using automatic differentiation using infrastructure described section sacado automatic derivative data type coordinate vector seeded derivative vectors result jacobian matrix computed automatic differentiation sacado data types allocated derivative arrays length number independent variables hexahedral element trilinear basis functions two degrees freedom per node local element solution vector seeded action inverse jacobian performed preconditioned iterative linear solver using belos ifpack packages trilinos gradient objective function computed hand since max operator half corresponding temperature unknown max operator respect changes parameter general issue however problems encountered application since location maximum move significantly itrations finally term identically zero parameters appear explicitly objective function first optimization problem run find parabolic deflection top bottom slider minimized maximum temperature addition gradientbased optimization algorithm described continuation run performed using loca package trilinos results shown figure continuation run shows smooth response surface wide range deflections positive negative optimization iteration rapidly converges minimum optimum occurs small positive value deflection parameter corresponding shape slightly arched upwards rather near nominal shape rectangular box figure results continuation run loca minimization run using dakota shape optimization problem optimization run also performed top bottom parabolas freed vary independently bulge slider adjusted conserve volume mesh figure shows initial configuration rather arched bottom surface nearly flat upper surfaces moderate bulge optimal shape shown figure case optimal shape found close rectangular box temperature contouring two figures share color map shows noticeable reduction temperature optimal shape figure initial mesh configuration temperature profiles optimization problem deflections top bottom surface slider varied independently deflection bulge plane chosen constrain volume rectangular brick results addition optimization runs embedded uncertainty quantification performed model scalart data type stochastic galerkin residual evaluation hold polynomial coefficients expansion quantities spectral basis nesting stochastic galerkin automatic differentiation data types jacobian stochastic galerkin expansions also calculated seed extract phases computation well subsequent nonlinear solve stochastic fem system required significant development however generic programming approach work completely orthogonal implementation pdes additional coding needed perform embedded application needed residuals governing equations demonstration chose electrical conductivity pad uncertain variable pad region thin rectangular region edge slider fixed shape run chosen uniform distribution within nominal value figure final configuration color map optimization problem maximum temperature significantly decreased variables legendre polynomials computation run polynomial basis newton iteration performed nonlinear system discretized domain fem space stochastic dimension spectral basis resulting probability distribution maximum temperature unknown computed figure shows mean temperature profile distribution figure show variation temperature respect parameter note reduced range color bar results show variation electrical conductivity pad region large effect temperature middle slider large dependence total current pad region strongly controlled convective cooling beam due moving frame reference conclusions paper related experience using generic programming tbgp approach pdes finite element code used approach local element level dependencies jacobian evaluations small dense combining gather phase finite element calculation seed phase tbgp approach scatter extract infrastructure tbgp well contained infrastructure place transformational analysis capabilities optimization embedded immediately available new pde modes implemented figure results embedded uncertainty quantification using stokhos electrical conductivity thin pad region given distribution figure temperature profile mean solution shown also presented implementation details approach includes infrastructure dealing parameters dealing templated code stack dealing code demonstrated approach example sliding electromagnetic contact problem pair coupled nonlinear equations performed optimization algorithms embedded gradients also embedded stochastic finite element calculations paper appear special issue along trilinos capabilities would like mention explicitly trilinos packages underlined used calculations paper centered use phalanx assembly engine sacado automatic differentiation stokhos embedded linear algebra used epetra data structures ifpack preconditioners belos iterative solver linear algebra accessed stratimikos linear algebra strategy layer using thyra abstraction layer stk mesh package used parallel mesh database stk packages together ioss exodus seacas used partitioning mesh intrepid package used finite element discretization operating multidimensional arrays shards package utility packages teuchos used parameter list specification memory management piro package managed solver analysis algorithms makes heavy use epetraext model evaluator abstraction piro turn calls nox nonlinear solver loca library continuation algorithms trikota interface dakota optimization algorithms stokhos presenting stochastic galerkin system single nonlinear problem results also relied several products outside trilinos including cubit mesh generator associated mesh morphing software dakota framework paraview visualization package figure variation temperature field shown embedded uncertainty quantification calculation previous figure netcdf mesh library acknowledgements work funded department energy nnsa advanced scientific computing office science advanced 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faster algorithms svp cvp norm divesh aggarwal priyanka mukhopadhyay jan january abstract blomer naewe modified randomized sieving algorithm ajtai kumar sivakumar solve shortest vector problem svp algorithm starts randomly chosen vectors lattice employs sieving procedure iteratively obtain shorter vectors lattice running time sieving procedure quadratic study problem special important case norm give new sieving procedure runs time linear thereby significantly improving running time algorithm svp norm also extend algorithm obtain significantly faster algorithms approximate versions shortest vector problem closest vector problem cvp norm also show heuristic sieving algorithms nguyen vidick wang also analyzed norm main technical contribution part calculate expected volume intersection unit ball centred origin another ball different radius centred uniformly random point boundary unit ball might independent interest introduction lattice set integer combinations linearly independent vectors call rank lattice dimension lattice matrix called basis write lattice generated lattice said work consider lattices unless otherwise stated two important computational problems lattices shortest vector problem svp closest vector problem cvp given basis lattice svp asks compute vector minimal length cvp asks compute lattice vector centre quantum technologies school computing national university singapore singapore dcsdiva centre quantum technologies national university singapore singapore minimum distance target vector typically defined terms norm kxkp max popular well studied euclidean norm corresponds starting seminal work algorithms solving problems either exactly approximately studied intensely algorithms found applications factoring polynomials rationals integer programming cryptanalysis checking solvability radicals solving problems recently many powerful cryptographic primitives constructed whose security based hardness related lattice problems ducas proposed signature scheme based modulesis problem security cryptosystem authors choose parameters assumption svp norm appropriate dimension infeasible due lack sufficient work complexity analysis svp norm choose parameters based best known algorithms svp norm variants algorithm rationale svp norm likely harder norm results paper show assumption ducas correct perhaps generous particular show time space complexity version least significantly larger best known algorithms svp norm closest vector problem norm particularly important since equivalent integer programming problem focus work study complexity closest vector problem shortest vector problem norm given importance problems complexity quite well studied prior work algorithms euclidean norm fastest known algorithms solving problems run time rank lattice constant first algorithm solve svp time exponential dimension lattice given ajtai kumar sivakumar devised method based randomized sieving whereby exponentially many randomly generated lattice vectors iteratively combined create shorter shorter vectors eventually resulting shortest vector lattice sequence works given improvements sieving technique thereby improving constant exponent current fastest provable algorithm exact svp runs time fastest algorithm gives constant approximation runs time heuristic algorithms run time cvp considered harder problem svp since simple dimension preserving reduction svp cvp based technique due kannan ajtai kumar sivakumar gave sieving based algorithm gives approximation cvp time later exact exponential time algorithms cvp discovered current fastest algorithm cvp runs time due algorithms norms generalized aks algorithm give exact algorithms svp run time fact gave exact algorithm general problem called subspace avoidance problem sap particular showed several lattice problems particular approximate svp approximate cvp easily reducible approximate sap thus give approximation algorithms cvp norms run time special case eisenbrand gave log algorithm cvp hardness results first hardness result cvp norms svp norm given van emde boas subsequent results shown hardness approximating cvp factor log log norms also hardness svp similar approximating factor obtained plausible stronger complexity assumptions recently showed almost cvp norm solved time strong exponential time hypothesis similar hardness result also obtained svp norm contribution provable algorithms modify sieving algorithm svp approximate cvp norm results substantial improvement prior results describing idea give brief description sieving procedure algorithm starts randomly generating set vector pairs vector difference pair lattice vector length runs sieving procedure usually polynomial number times ith iteration algorithm maintains updates list centre pairs initialized empty set second vector centre pair usually referred centre vector algorithm checks whether distance say exists centre pair vector pair replaced otherwise deleted added results lattice vectors shorter beginning sieving iteration number lattice vectors end sieving iterations thus continuing manner eventually obtain shortest vector crucial step algorithm find vector close problem called nearest neighbor search nns problem well studied especially context heuristic algorithms svp trivial bound running time aforementioned heuristic algorithms spent considerable effort trying improve bound reasonable heuristic assumptions since require heuristic assumptions improved algorithms nns used improve provable algorithms svp make simple powerful observation special case norm partition ambient space easy see partition contain one centre thus find centre distance given vector need find partition belongs check whether partition contains centre easily done checking interval belongs drastically improves running time sieving procedure svp algorithm idea also used obtain significantly faster approximation algorithms svp cvp must noted prior provable algorithms using aks sieve lacked explicit value constant exponent space time complexity used quadratic sieve modified sieving procedure linear size input list thus yields better compared prior algorithms order get best possible running time optimize several steps specialized case norm analysis algorithms see theorems explicit running times detailed description emphasise results nearly best possible using techniques notice large enough constant obtain running time space close svp put things context best algorithm constant approx svp norm runs time space algorithm crucially uses fact best known upper bound kissing number lattice number shortest vectors lattice norm however norm kissing number would analyze algorithm norm without improvement would obtain space complexity time complexity heuristic algorithms sieving step algorithm length lattice vectors reduce constant factor seems like continue reduce length lattice vectors get vectors length length shortest vector obtain shortest vector sieving procedure however risk vectors output sieving procedure copies zero vector reason aks algorithm needs start much vectors order provably argue obtain shortest vector nguyen vidick observed view perhaps pessimistic practice randomness initial set vectors ensure basic sieving procedure output shortest vector lattices main ingredient analyze space time complexity algorithm compute expected number centres necessary point distance one centres note heuristic setting unlike aks algorithm stores lattice vectors instead vector pairs number roughly reciprocal fraction ball radius centred origin covered ball radius centred uniformly random point maximum length lattice vector sieving iterations work show heuristic algorithm also analyzed norm similar assumptions main technical contribution order analyze time space complexity algorithm compute expected fraction ball radius centered origin covered ball radius centered uniformly random point order improve running time sieve modified sieve introduced wang first partition lattice sets vectors larger norm within set carry sieving procedure similar analyzed norm obtain algorithms significantly faster provable algorithms particular sieve algorithm runs time would like mention result contradict near lower bound svp obtained strong exponential time hypothesis reason lattice obtained reduction lattice dimension significantly larger rank lattice organization paper section give basic definitions results used paper section introduce sieving procedure apply provably solve exact svp section describe approximate algorithms svp cvp using sieving technique results used analysis given section reader look referenced section talk heuristic sieving algorithms svp preliminaries notations write natural logarithm log logarithm base dimension may vary specified use bold lower case letters vectors bold upper case letters matrices may drop dimension superscript whenever clear context sometimes represent matrix vector column vectors ispan vector ith denoted given vector representation size respect maximum binary lengths numerators denominators coefficients set vectors norm let ksk ksi denotes volume geometric body cardinality set norm definition norm norm vector defined kvkp max fact kxkp nkxkp kxkp kxkp definition ball ball set points within fixed distance radius defined metric fixed point centre precisely define closed ball centered radius xkp boundary set xkp may drop first argument ball centered origin drop arguments unit ball centered origin let xkp drop first argument spherical shell corona centered origin fact algorithm dyer frieze kannan selects almost uniformly point convex body polynomial time membership oracle given sake simplicity ignore implementation detail assume able uniformly select point polynomial time definition lattice discrete additive subgroup lattice basis algorithmic purposes assume call rank dimension lattice said though results generalized arbitrary lattices rest paper consider full rank lattices definition lattice basis define fundamental parallelepiped kykp nkbkp easily seen triangle inequality exists unique vector denoted mod computed polynomial time given definition ith successive minimum defined smallest real number contains linearly independent vectors length inf dim span thus first successive minimum lattice length shortest vector lattice min kvkp consider following lattice problems problems defined arbitrary approximation factor usually specified subscript constant function parameter lattice usually rank exact versions problems drop subscript definition shortest vector problem svpc given lattice find vector kvkp ckukp definition closest vector problem cvpc given lattice rank target vector find tkp ckw tkp lemma lll algorithm used solve polynomial time proof let lattice length shortest vector shown lll algorithm used obtain estimate length shortest vector satisfying using fact get hence result follows following result shows order solve sufficient consider case done appropriately scaling lattice lemma lemma norms algorithm lattices solves time algorithm solves lattices time volume estimates infinity norm section prove results volume intersection balls used analysis later reader may skip section look referenced lemma let lattice proof note integers region contains one lattice values region intersects result follows following lemma derive expected volume intersection assuming centre uniformly distributed lemma let hence proof hyperrectangle therefore volume product edges let event since due symmetry thus since let variable denoting length hyperrectangle direction ith eri lim lim eri lim max min thus let consider eri note expression eri eri eri eri eri eri eri lim eri lim lim similarly eri thus lim theorem follows next deduce similar result except consider volume intersection big ball radius unit ball big ball centred uniformly distributed point corona lemma let lim hence lim proof proof similar lemma use similar notations max min lim lim eri lim eri thus eri eri eri eri lim eri lim lim lim also bound expression lim eri lim thus conclude eri similarly eri lemma follows hence following result gives bound size intersection two balls given radius norm lemma let let let proof easy see intersection two balls norm hyperrectangles also hyperrectangle length side hyperrectangle result follows faster algorithm svp section present algorithm svp uses framework aks algorithm uses different sieving procedure yields faster running time using lemma obtain estimate thus try polynomially many different values one rest section assume know guess length shortest vector correct upto factor aks algorithm initially samples uniformly lot perturbation vectors perturbation vector maintains vector close lattice thus initially set many pairs desired situation polynomial number sieving iterations left set vector pairs finally take differences lattice vectors corresponding remaining vector pairs output one smallest norm shown overwhelming probability shortest vector lattice one main usually expensive step algorithm sieving procedure given list vector pairs iteration outputs list vector pairs sieving iteration number vector pairs usually exponential identified centre pairs second element centre pair referred centre map remaining vector pair associated centre pair certain operations like subtraction vectors get pair vector difference yielding lattice vector norm less start iteration say vector pairs identify number centre pairs output consists vector pairs original aks algorithm variants running time sieving procedure dominant part total running time algorithm roughly quadratic number sampled vectors reduce running time norm use different sieving approach give brief description sieving procedure algorithm details found algorithm two algorithm algorithm partition interval intervals length intervals note thejlast may smaller rest ball thus partitioned regions two vectors region distance greater norm list pairs maintained first entry pair array second one initialized emptyset storing centre pair think tuple index call intuition following want associate vector pair centre pair note condition satisfied since partitioned intervals length given map linear time indicates interval belong access say constant time list exists implying add output list else add vector pair centre pair finally return lemma let number centre pairs algorithm always algorithm exact algorithm svp input basis lattice iii output shortest vector sample using algorithm end maxi kbi sieve using algorithm end compute vector smallest norm return algorithm sample input basis lattice output pair mod return algorithm faster sieve norm input set triplet output set max else find integer end else end end end return satisfies log proof partitioned range intervals length intervals thus number index set cardinality hence theorem follows claim following two invariants maintained algorithm proof first invariant maintained beginning sieving iterations algorithm due choice step algorithm since centre pair belonged thus step sieving procedure algorithm second invariant maintained step algorithm hence kbi maxi kbi claim invariant also maintained iteration sieving procedure consider pair let let associated centre pair algorithm hence kec claim follows variable step algorithm following lemma bound length remaining lattice vectors sieving iterations lemma end iterations algorithm length lattice vectors proof let value iterations thus iterations hence iterations using lemma assuming get upper bound number vectors length log lemma along invariants imply beginning step algorithm short lattice vectors vectors norm bounded want start sufficient number vector pairs end zero vectors end sieving iterations work following conceptual modification proposed regev let define bijection maps else analysis algorithm assume perturbation vector chosen algorithm replace probability remains unchanged probability call procedure tossing vector assume replacement perturbation vectors happens step first time effect algorithm particular step algorithm identified centre pair apply probability beginning step algorithm apply pairs distribution remains unchanged procedure mod somewhat detailed explanation found following result lemma theorem modification outlined change output distribution actual procedure note since conceptual modification intended ease analysis concerned actual running time modified procedure even fact need shortest vector begin mapping matter following lemma help estimate number vector pairs sample beginning algorithm lemma lemma let denote probability random point contained points chosen uniformly random least probability larger points property lemma log thus probability least iterations least pairs sieving lemma probability least algorithm outputs shortest vector respect norm proof vector pairs sampled step algorithm consider already seen least pairs probability least remove vector pairs sieve iterations step algorithm pairs process lemma contained within ball radius lattice vectors exists least one lattice vector perturbation appears twice beginning step probability remains probability becomes either thus taking difference step probability least find shortest vector theorem let let given full rank lattice randomized algorithm svp success probability least space complexity running time cspace max ctime max cspace log log log proof start pairs stated lemma space complexity cspace max kiteration sieving algorithm takes time initialize index vector pair takes time calculate time taken process vector pair thus total time taken per iteration algorithm poly iterations time complexity computation pairwise differences overall time complexity ctime max cspace improvement using birthday paradox get better running time space complexity use birthday paradox decrease number sampled vectors get least two vector pairs corresponding lattice vector sieving iterations ensure vectors independent identically distributed step algorithm incorporate following modification assume start sampled pairs initial sampling sieving iterations fix pairs used centre pairs following way let kyi maintain lists pairs list similar already described ith list partition range intervals length first calculate check list potentially belong say map already described add empty else subtract vectors step algorithm using analysis similar get following improvement running time theorem let let given full rank lattice randomized algorithm svp success probability least space complexity ctime running time space max max cspace log log log particular algorithm runs time sponding space requirement faster approximation algorithms algorithm approximate svp notice algorithm end sieving procedure obtains lattice vectors length long ensure one vectors obtained end sieving procedure obtain shortest vector consider new algorithm identical algorithm except step replaced following find vector show start sufficiently many vectors must obtain vector lemma probability least algorithm outputs vector length respect norm proof vector pairs sampled step algorithm consider already seen least pairs remove vector pairs sieve iterations step algorithm pairs process probability hence replaced either thus probability vector zero vector thus obtain following theorem let given full rank lattice randomized success probability least algorithm approximates space time complexity log log particular algorithm runs time algorithm approximate cvp given lattice target vector let denote distance closest vector section assume know value within factor get rid assumption using babai algorithm guess value within factor run algorithm polynomially many values within factor previous one define following lattice let lattice vector closest sample vector pairs using algorithm basis next run number iterations sieving algorithm get number vector pairs details found algorithm note algorithm vector obtained restricting first respect computational basis lemma seen iterations kbi thus sieving iterations set consists vector pairs corresponding lattice vector order ensure sieving algorithm return vectors choose parameters follows every vector either lattice vector need argue must least vectors sieving iterations use tossing argument section let lattice vector closest let let lemma probability random perturbation vector bounded log thus long max algorithm approximate algorithm cvp input basis lattice target vector iii approximation factor max small constant vii output closest vector maxi kbi span sample using algorithm end maxi kbi sieve using algorithm end compute min end let vector min return else return end least pairs sieving iterations thus using argument section obtain following theorem let let max given full rank lattice randomized algorithm approximates cvp success probability least space time complexity log particular algorithm runs time log heuristic algorithm svp nguyen vidick introduced heuristic variant aks sieving algorithm used solve svp basic framework similar aks except work perturbation vectors start set uniformly sampled lattice vectors norm iteratively fed sieving procedure algorithm provided list lattice vectors norm say return list lattice vectors norm iteration sieve number vectors identified centres vector within distance centre subtract centre add resultant output list iterations continue till list vectors currently consideration empty size decrease either due elimination zero vectors steps algorithm due removal centres algorithm linear number iterations expect left list short vectors output one minimum norm order shortest vector proper approximation good probability ensure end list indicating end sieving iterations soon say number iterations make following assumption distribution vectors stage algorithm heuristic stage algorithm vectors uniformly distributed sieving iteration get zero vector collision vector centre vector assumption following estimate expected number collisions lemma let vectors randomly chosen replacement set cardinality expected number different vectors picked expected number vectors lost collisions number negligible since expected number lattice points inside ball radius effect collisions remain negligible till shown sufficient take gives collisions expected become significant already good estimate even collisions imply good proportion lattice vectors previous iteration algorithm svp algorithm norm using lattice sieve input basis lattice sieve factor iii number output short vector sampling using klein algorithm end remove zero vectors latticesieve using algorithm remove zero vectors end compute return algorithm lattice sieve input subset sieve factor output subset else else end end end return thus good probability expect get shortest vector constant approximation step algorithm would like make comments initial sampling lattice vectors step algorithm due assumption heuristic ensure lattice points uniformly distributed spherical shell corona stage use klein randomized variant babai nearest plane algorithm intuitively ensure sampled points biased towards single direction gentry gave detailed analysis klein algorithm proved following theorem let basis lattice exists randomized polynomial time algorithm whose output distribution statistical distance restriction gaussian centered variance density proportional exp using fact exp exp exp exp assuming kbkp conclude algorithm used uniformly sample lattice points norm step algorithm analyze complexity algorithm crucial part assess number centres done following lemma lemma let points picked independently random uniform distribution uniformly distributed points cardinality least following high probability proof assuming heuristic holds every iteration sieve expected fraction covered balls radius centered randomly chosen points lemma log log thus expected fraction corona covered balls least expected number uncovered points less since number integer probability least theorem expected space complexity running time algorithm respectively defined proof let expected number centers iteration poly lemma thus time lattice sieve invoked steps algorithm expect size provided satisfies heuristic decrease aproximately use lll algorithm lemma obtain estimate approxima tion factor start vectors norm iteration lattice sieve norm vectors decrease factor start vectors linear number iterations expect left short vectors since running time lattice sieve quadratic expected running time algorithm heuristic sieving algorithm svp order improve running time mostly dictated number centres wang introduced sieving procedure improves upon sieve large first level identify set centres associate vectors within distance within radius big ball another set vectors call centre subtract vectors add resultant output list analysed sieve algorithm norm also found similar improvement running time analyze complexity algorithm sieving procedure algorithm need count number centres first level given lemma count number centres lemma let points picked independently random uniform distribution uniformly distributed points cardinality least following high probability proof proof similar lemma cover smaller balls let apply lemma conclude lim algorithm heuristic sieve input subset sieve factors output subset else else end else end end end return lim hence fraction covered using similar arguments lemma lemma estimate number centres second level following lemma bound number centres within radius big ball centred point say lemma let points picked independently random uniform distribution uniformly distributed points cardinality least following high probability finally analyze complexity algorithm theorem space complexity algorithm using sieve algorithm poly defined lemma respectively also time complexity optimal value attained yielding time complexity space complexity proof expected number centres iteration algorithm poly use lll algorithm lemma obtain approximation thus initially sample poly vectors norm assuming heuristic holds iteration sieve norm vectors decrease factor also expected size decreases 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table marks direct product finite groups feb brendan masterson pfeiffer abstract present method computing table marks direct product finite groups contrast character table direct product two finite groups table marks simply kronecker product tables marks two groups based decomposition inclusion order subgroup lattice direct product relation product three smaller partial orders describe table marks direct product essentially matrix product three class incidence matrices matrices turn described sparse block diagonal matrix application use variant matrix product construct ghost ring mark homomorphism rational double burnside algebra symmetric group introduction table marks finite group first introduced william burnside book theory groups finite order table characterizes actions transitive bijection conjugacy classes subgroups thus table marks provides complete classification permutation representations finite group equivalence burnside ring grothendieck ring category finite table marks arises matrix mark homomorphism free number conjugacy classes subgroups like character table table marks important invariant group classical theorem dress solvable prime ideal spectrum connected nontrivial idempotents property easily derived table marks table marks finite group determined counting inclusions conjugacy classes subgroups subgroup lattice needs known cost complete knowledge subgroups increases drastically order rather number prime factors order approach limited small groups alternative methods computation table marks developed avoid excessive computations subgroup lattice includes method computing table marks tables marks maximal subgroups method computing table marks cyclic extension table marks purpose article develop tools computation table marks direct product finite groups obvious idea relate subgroup lattice subgroup lattice compute table marks using relationship many properties derived properties little effort conjugacy classes date february mathematics subject classification key words phrases burnside ring table marks subgroup lattice double burnside ring ghost ring mark homomorphism brendan masterson pfeiffer elements example simply pairs conjugacy classes character table simply kronecker product character tables however relationship table marks tables marks much intricate flavour complexity expected already given classical result known goursat lemma lemma according subgroups direct product finite groups correspond isomorphisms sections article presents first general systematic study subgroup lattice direct product finite groups beyond goursat lemma special cases subgroup lattices considered far schmidt zacher view goursat lemma seems appropriate first develop theory sections finite groups section finite group pair subgroups normal subgroup study sections first defining partial order set sections componentwise inclusion subgroups canonical homomorphism decomposes product three maps epimorhism isomorphism monomorphism show induces decomposition partial order product three partial orders denote reasons become clear section thus decomposition partial order compatible conjugation action set sections description subgroups terms sections allows transfer decomposition partial orders sections set subgroups show section subgroups exist unique intermediary subgroups partial orders set subgroups defined terms corresponding relations sections gives decomposition partial order subgroups three partial orders compatible conjugation action section show one main results yields corresponding decomposition table marks matrix product three class incidence matrices individually class incidence matrices block diagonal structure significantly easier compute subgroup lattice rest paper arranged follows section collect useful known results section study sections finite group discuss properties lattice sections partially ordered componentwise show decomposition partial order relation product three partial orders leads corresponding decomposition class incidence matrix sections matrix product section concludes brief discussion interesting variant partial order sections class incidence matrix section considers isomorphisms sections particular group subgroups determine structure set isomorphisms aut section study subgroups pairs isomorphisms one section one allows determine structure set subgroups aut also derive decomposition subgroup inclusion order relation product three partial orders corresponding table marks direct product finite groups decomposition partial orders sections section section develop methods computing individual class incidence matrices partial orders subgroups use matrices compute table marks essentially product finally section present application theory double burnside ring finite group defined grothendieck ring transitive addition defined disjoint union multiplication tensor product double burnside ring currently centre much research important invariant group see study particular case use partial orders construct explicit ghost ring mark homomorphism sense boltje danz acknowledgement much work article based first author phd thesis see research supported college science national university ireland galway preliminaries notation denote symmetric group degree alternating group degree cyclic group order simply use various forms composition paper group homomorphisms act right composed accordingly product defined group relation product relations relation sets section product subgroups defined mop lop rop denotes opposite subgroups relations following classical result describes subgroups direct product isomorphisms section quotients section finite group pair subgroups lemma goursat lemma let groups bijective correspondence subgroups direct product isomorphisms form section proof let let projection onto binary relation writing easy see partition cosets normal subgroup similarly sets cosets normal subgroup relation thus difunctional establishes bijection section quotients fact group homomorphism conversely isomorphism sections yields relation fact subgroup subgroup corresponds isomorphism write call sections goursat sections isomorphism type goursat type finally called graph conversely goursat isomorphism next lemma illustrated fig derived lemma see brendan masterson pfeiffer figure butterfly lemma lemma butterfly lemma let sections set canonical map isomorphism refer section butterfly meet let finite groups product subgroups defined fact subgroup thanks obtain goursat isomorphism composing goursat isomorphisms follows suppose graph isomorphism graph sections let subgroups isomorphisms butterfly lemma let isomorphism obtained restricting defined moreover let defined graph subgroup although necessarily graph subgroup lemma notation graph composite isomorphism use subgroup product goursat isomorphism proof theorem bisets biset products action direct product set sometimes conveniently described two groups acting set one left one right table marks direct product finite groups definition let groups left right actions commute suitable conditions bisets composed follows definition let groups biset tensor product set action given tensor product bisets used section describe certain sets subgroups also provides multiplication double burnside ring group subject section action pairs also need deal one group acting two sets following parametrization orbits group acting set pairs lemma let finite group acting finite sets suppose set pairs pairs thus represented pairs represent orbits fixed represent orbits stabilizer set proof note disjoint union intersections whence corresponding disjoint union orbit spaces lemma map bijection hence every bijection class incidence matrices let finite partially ordered set poset incidence matrix axy axy else incidence matrix lower triangular order rows columns extends partial order suppose equivalence relation partitions classes transversal say partial order compatible equivalence relation classes number axy brendan masterson pfeiffer depend choice representatives axy axy case define class incidence matrix partial order matrix axy whose rows columns labelled chosen transversal matrix multiplication relates matrices following way lemma define row summing matrix rxy column picking matrix cxy entries rxy cxy else else identity matrix iii proof rxy cyz rxz matrices equal axy represents class iii follows remark examples compatible posets provided group actions suppose finite group acts poset way called partial order compatible partition since write equivalence given remark generally square matrix rows columns indexed set equivalence relation choosing transversal equivalence classes yields product say matrix compatible equivalence product depend choice transversal equivalence induced action group matrix axy compatible axy matrices subject proposition theorem burnside ring table marks burnside ring finite group grothendieck ring category finite free abelian group basis consisting isomorphism classes transitive disjoint union addition cartesian product multiplication multiplication transitive described mackey formula lemma rational burnside algebra isomorphic direct sum copies one conjugacy class subgroups products basis elements determined formula table marks direct product finite groups mark subgroup number fixed points obviously whenever conjugate subgroups map assigns vector transversal conjugacy classes subgroups context ring componentwise addition multiplication called ghost ring latter product componentwise multiplication thus homomorphism rings called mark homomorphism table marks rows mark vectors transitive isomorphism regarding linear map table marks matrix relative natural basis standard basis subgroups table marks provides compact description subgroup lattice fact diagonal matrix entries class incidence matrix group acting lattice subgroups conjugation example let conjugacy classes subgroups sections let finite group denote set subgroups set conjugacy classes subgroups section pair subgroups call top group bottom group section refer quotient group quotient section isomorphism type section isomorphism type quotient size section size quotient denote set sections group acts set pairs conjugation sections classify orbits action describe automorphisms induced stabilizer section quotient partial order induces partial order pairs section show partial order fact lattice decomposed product three smaller partial order relations section determine class incidence matrix show decomposition partial order corresponds decomposition class incidence matrix matrix product three class incidence matrices section use smaller partial orders define new partial order consistent notion size section brendan masterson pfeiffer conjugacy classes sections finite group naturally acts sections componentwise conjugation via write conjugacy class section denote set conjugacy classes sections conjugacy classes sections parametrized different ways terms simpler actions follows proposition let let sep sep let ske ske proof note set pairs stabilizer result follows lemma follows similar way write finite group isomorphic subquotient denote set sections isomorphism type classes naturally three partitions used sequel section automizers automizer subgroup quotient group section automizer isomorphic subgroup aut induced conjugation action analogously define automizer section section whose quotient isomorphic subgroup automorphisms induced conjugation definition let set using natural homomorphism let define section normalizer inverse image section centralizer section automizer section moreover denote autg subgroup aut automorphisms induced conjugation see fig table marks direct product finite groups aut autg inn figure section automorphisms following properties groups obvious lemma let section set induce identity automorphism iii inn autg aut sections lattice subgroup inclusion induces partial order set sections inherits lattice property subgroup lattice follows definition poset partial order defined componentwise sections subgroups write join subgroup lattice hhaiib normal closure proposition poset lattice componentwise meet join given sections proof clearly normal subgroup section unique greatest lower bound sections also easy see least section iip theorem let sections finite group largest section top group smallest section bottom group iii map isomorphism section quotients brendan masterson pfeiffer figure proof canonical homomorphism given according homomorphism theorem decomposed surjective bijective injective part uniquely determined see fig motivated result define following three partial orders definition let write sections top groups sections bottom groups iii map isomorphism reformulate theorem terms three relations corollary partial order product three relations let denote incidence matrix partial order stronger property also holds proof theorem exists unique intermediate sections remark note correspondence theorem bijective correspondence subgroups sections similarly bijective correspondence normal subgroups hence factor groups sections class incidence matrices denote class incidence matrix note set sections also respect partial orders definition respective class incidence matrices theorem notation table marks direct product finite groups proof set lemma iii corollary lemma gives classes incidence matrices direct sum smaller class incidence matrices following results show theorem denote class incidence matrix sep proof let proposition classes containing section top group represented sections runs transversal sep order count suffices note example let theorem denote class incidence matrix ske proof similar proof theorem example let lemma class incidence matrix brendan masterson pfeiffer proof implies example let class incidence matrix product matrix class incidence matrices examples according theorem sections lattice revisited partial order compatible section size implies turns effectively replacing partial order opposite one obtains new partial order compatible section size proposition define relation sections proof relation clearly reflexive antisymmetric compatible action hence remains shown relation transitive let sections order show need clear intersecting sides gives desired example let denote three subgroups order klein sections unique infimum poset lattice proposition let sections finite group uniquely determined sections iii proof definition implies second isomorphism theorem normal subgroup subgroup isomorphic table marks direct product finite groups figure hence desired properties see fig corollary partial order product three relations moreover example let contrast class incidence matrix example matrix lower triangular rows columns sorted section size moreover sections remark whenever canonical isomorphism let goursat isomorphism subgroup suppose canonical isomorphism determines unique restriction goursat isomorphism similarly section unique goursat isomorphism butterfly meet sections group satisfies lemma product subgroups goursat isomorphisms composition restriction butterfly meet morphisms let finite group isomorphism section group set forms aut section describe set identification brendan masterson pfeiffer certain subgroups section induces partial order section compute class incidence matrix partial order classes induces isomorphism automorphism groups aut aut define automizer isomorphism quotient automizer section corresponding subgroup aut definition given denote let aut image autg aut automizer maps coset automorphism corresponding conjugation moreover denote group outer automorphisms induced via noting inn group acts via conjugation map induced denote set section denote set domain action aut set decomposes regular aut one section action aut commutes induces aut set action used classify follows proposition let aut disjoint union transitive aut one sections let transversal right cosets aut note abuse notation set full morphisms although general proof let let identified ginvariant subset consisting pairs domain lemma disjoint union aut one sections table marks direct product finite groups let stabilizer section normalizer automizer transforms action subgroup aut aut acts regularly set correspond cosets aut inn aut regarded thus section set isomorphic example let aut makes two orbits form comparing morphisms goursat lemma lemma corresponds subgroup call graph partial order subgroups induces natural partial order follows graphs define partial order closely related order proposition let homomorphism defined proof let graph let assume first clearly implies moreover unique element first component hence isomorphism whence conversely clearly whence generally finite groups suppose sections canonical homomorphism isomorphisms composition obviously homomorphism see fig figure brendan masterson pfeiffer case previous lemma says idu incomparable however following connections partial orders lemma let induces order preserving bijection sections subgroups order preserving bijection sections normal subgroups proof immediate consequence remark correspondence theorem partial order morphism classes partial order compatible sense section conjugation action hence yields class incidence matrix matrix submatrix class incidence matrix subgroup lattice corresponding classes subgroups occur graphs proposition suppose graphs proof result follows show larger let hence aut matrix compatible sense section action fact relates class incidence matrix follows proposition row summing column picking matrices corresponding proof proposition union classes aut set form set contains conjugate exactly one proposition example let consists three classes one two permuted table marks direct product finite groups subgroups direct product let finite groups section describe subgroups conjugacy classes subgroups direct product terms properties groups goursat lemma subgroups correspond isomorphisms sections isomorphism arises composition two suitable finite group motivates study subgroups pairs pairs morphisms let finite group call denote goursat type set given morphisms mgi composition yields isomorphism whose graph hence map defined fact tensor product aut opposite aut proposition proof exist mgi moreover mgi aut convenient express order terms lemma let comparing subgroups let finite groups describe analyze partial order subgroups terms pairs morphisms proposition let mgi mgi morphisms let corresponding subgroups sections homomorphism defined figure brendan masterson pfeiffer proof write see fig corollary notation proposition sections partial orders sections introduced definition give rise relations subgroups follows definition let subgroups suppose write sections top groups sections bottom groups iii canonical homomorphisms isomorphisms three relations obviously partial orders moreover decompose partial order subgroups analogy corollary theorem let define map whenever map whenever isomorphisms corresponding graphs unique subgroups proof denote canonical homomorphism proof theorem product epimorphism ker isomorphism ker monomorphism corollary follows ker ker thus restricts isomorphism induces isomorphism ker ker following diagram commutes proposition ker corollary partial order product three relations moreover denotes incidence matrix relation stronger property also holds proof like corollary follows uniqueness intermediate subgroups theorem table marks direct product finite groups ker ker figure lemma let mgi set order preserving bijective correspondence subgroups set order preserving bijective correspondence quotients iii set order preserving bijective correspondence set order preserving bijective correspondence proof follows lemma correspondences induced together proposition theorem classes subgroups conjugacy classes described aut pairs classes theorem let disjoint union sets one pair section classes qgi let transversal cosets aut proof classes aut direct product proposition direct product disjoint union aut invariant direct products one choice sections qgi let aut note first image class form show classes correspond brendan masterson pfeiffer cosets aut let aut assume proposition case lie coset example let therefore theorem exists exactly one conjugacy class subgroups pair classes isomorphic sections transversal conjugacy classes subgroups labelled pairs sections follows subgroup row column goursat isomorphism form normalizer subgroup described quotient two described quotient automizers two theorem let let mgi proof suppose let agi agi aut automizer let one hand consists elements induce automorphisms agi aut hand lemma restriction isomorphism agi preimage hence subgroup product consists elements follows desired immediate consequence determine normalizer index subgroup terms corollary let ngi proof lemma notation preceding proof thus table marks direct product finite groups definition moreover table marks position assemble table marks collection smaller class incidence matrices theorem let finite groups table marks diagonal matrix entries running transversal conjugacy classes subgroups proof proof similar theorem combination corollary remainder section determine block diagonal structure matrices class incidence matrix block diagonal matrix one block pair conjugacy classes subgroups theorem denote class incidence matrix acting subposet consisting subgroups bottom groups proof let identify lemma yields partition conjugacy classes subgroups indexed sgi stabilizer let order count conjugates subgroup bottom groups suffices note finally definition incidences subgroups different giving block diagonal structure example let block sum matrices table rows columns labelled conjugacy classes subgroups within row label subgroup form brevity column labels identical omitted brendan masterson pfeiffer class incidence matrix block diagonal matrix one block group isomorphism definition finite group finite let square matrix rows columns labelled action permutes rows columns kronecker product matrices compatible kronecker product define row summing column picking matrices constructed lemma respect consider class incidence matrices mgi section proposition matrices hence compatible action rows columns theorem class incidence matrix mgi proof let subgroup goursat type select lemma subgroups correspond pairs sections section set canonical isomorphism proposition unique number conjugates subgroup thus equal number pairs conjugate theorem set pairs uif morphisms mapping conjugate number given entry class incidence matrix proposition aut mgi isomorphic aut hence transversal right cosets aut also used represent right cosets number appears matrix example let block sum following matrices acts trivially matrices simply kronecker squares matrices example column labels table marks direct product finite groups identical row labels omitted example continuing example illustrating effect class incidence matrix block diagonal matrix one block pair conjugacy classes subgroups theorem denote class incidence matrix acting sub poset consisting subgroups proof similar proof theorem example let block sum matrices table rows columns labelled conjugacy classes subgroups similar example within row label subgroup form brevity column labels identical omitted brendan masterson pfeiffer example combining matrices examples according theorem yields table marks rows columns sorted section size example double burnside algebra application ideas previous sections construct mark homomorphism rational double burnside algebra double burnside ring let finite groups grothendieck group category finite denoted identified abelian group identified burnside group hence transitive bisets runs transversal conjugacy classes subgroups form map given multiplication transitive bisets described following proposition let let transversal cosets multiplication particular ring double burnside ring rational double burnside algebra known semisimple cyclic proposition little known structure general table marks direct product finite groups mark homomorphism double burnside ring ordinary burnside ring table marks matrix mark isomorphism rational burnside algebra ghost algebra open question whether exist equivalent constructions ghost algebras mark homomorphims double burnside ring boltje danz investigated role table marks direct product context use decomposition table marks theorem idea transposing part section order build satisfying ghost algebra group purpose first set labelling natural basis follows set let conjugacy class representatives example rational burnside algebra consisting elements multiplication defined theorem table marks matrix product diagonal matrix entries three class incidence matrices purpose modify product set diag diag diagonal matrices resulting matrix matrix mij obviously invertible hence unique elements mij forming new brendan masterson pfeiffer theorem let linear map defined defined injective homomorphism algebras proof claim validated explicit calculation whose details omit general strategy follows let matrix right regular representation computed help mackey formula proposition let equivalence relation corresponding kernel map sends conjugacy class subgroup conjugacy class section partitions turns transposed matrices cti compatible equivalence sense section hence choosing transversal using corresponding row summing column picking matrices map defined independent choice transversal fact lemma showing homomorphism injectivity follows dimension count might worth pointing equivalence hence notion compatibility map depend basis used matrices right regular representation case natural basis also yields compatible matrices corresponding map injective base change table marks gives matrices compatible changing basis matrix product yields compatible matrices injective homomorphism like matrices added benefit normalized extremely sparse exposing representation theoretic properties algebra following corollary let denote jacobsen radical rational burnside algebra basis map regarded mark homomorphism double burnside ring assigns square matrix rational marks example table marks direct product finite groups image identity matrix case provides small example construction involves hoc measures expect many finite groups mark homomorphism rational double burnside algebra constructed similar way subject future research references robert boltje susanne danz ghost ring double burnside ring application fusion systems adv math ghost algebra double burnside algebra characteristic zero pure appl algebra serge bouc biset functors finite groups lecture notes mathematics vol springerverlag berlin serge bouc radu stancu jacques simple biset functors double burnside ring pure appl algebra william burnside theory groups finite order cambridge university press cambridge andreas dress characterisation solvable groups math goursat sur les substitutions orthogonales les divisions espace ann sci norm sup bertram huppert endliche gruppen die grundlehren der mathematischen wissenschaften band york joachim lambek goursat theorem zassenhaus lemma canad math klaus lux herbert pahlings representations groups computational approach cambridge university press cambridge brendan masterson table marks direct product finite groups thesis national university ireland galway liam naughton pfeiffer computing table marks cyclic extension math comp pfeiffer subgroups compute table marks finite group experiment math ragnarsson radu stancu saturated fusion systems idempotents double burnside ring geom topol roland schmidt direkter produkte von gruppen arch math basel giovanni zacher lattice subgroups cartesian square simple group rend sem mat univ padova brendan masterson pfeiffer department design engineering mathematics middlesex university london boroughs london united kingdom address school mathematics statistics applied mathematics national university ireland galway university road galway ireland address

apr merging joint distributions via causal model classes low dimension dominik janzing max planck institute intelligent systems germany april abstract denote sets random variables two different data sources may contain samples respectively argue causal inference help inferring properties unobserved joint distributions properties may conditional independences integrative causal inference also quantitative statements dependences generally define learning scenario input subset variables label statistical property subset sets jointly observed variables define training points unobserved sets possible test points solve learning task infer intermediate step causal model observations entails properties unobserved sets accordingly define dimension class causal models derive generalization bounds predictions causal inference becomes modest better accessible empirical tests usual rather trying find causal hypothesis true problematic term unclear define interventions causal hypothesis useful whenever correctly predicts statistical properties unobserved joint distributions within pragmatic application causal inference popular heuristic approaches become justified retrospect instance allowed infer dags partial correlations instead conditional independences dags used predict partial correlations hypothesize pragmatic view causality may even cover usual meaning terms interventions sketch predicting impact interventions sometimes also phrased task type introduction difficulty inferring causal relations purely observational data lies fact observations drawn joint distribution supposed imply statements system behaves interventions pearl spirtes specificly one may interested new joint distribution obtained setting subset variables specific values induces different joint distribution task causal inference phrased way actually lies outside typical domain statistics thus requires assumptions link statistics causaliy render task feasible certain limitations instance one infer causal directed acyclic graph dag markov equivalence class observed conditional statistical independences spirtes pearl moreover also distinguish dags markov equivalence class certain model assumptions linear models noise kano shimizu additive noise hoyer made relevance causal information without reference interventions goal causal inference need necessarily consist predicting impact interventions instead causal information could help transferring knowledge accross data sets different distributions underlying idea modularity assumption peters according conditional distribiutions causal bayesian network may change others remain fixed among many tasks causal information could help particularly emphasize integrative causal inference tsamardinos work closest present paper tsamardinos use causal inference combine knowledge different data sets idea reads follows given data sets containing observations different overlapping sets variables causal inference algorithms applied independently afterwards joint causal model constructed entails independences subsets variables joint observations available slightly abusing terminology refer sets variables observed together unobserved sets variables keep mind although observed jointly usually observed individually part observed set explain idea explicitly sketch example tsamardinos combines knowledge two data sets contains variables one observes conditional unconditional independences data set contains variables one observes independence one constructs set maximal ancestral graphs mags set consistent observed pattern independences result mag implies given subset variables although never observed together perspective inference procedure thus reads statistical properties observed subsets causal models consistent statistical properties unobserved subsets contrast tsamardinos term statistical properties need necessarily refer conditional independences one hand meanwhile broad variety new approaches infer causal directions statistical properties conditional independences kano shimizu sun hoyer mags define class graphical causal models closed marginalization conditioning subsets variables richardson spirtes zhang daniusis janzing mooij peters mooij hand causal model inferred observations may entail statistical properties conditional independences subject model assumptions inference procedures rely regardless kind statistical properties meant scheme describes sense causal model tested within usual scenario way causal model entails statements empirically tested without referring interventional scenario consequently drop ambitious demand finding true causal model replace modest goal finding causal models properly predict unseen joint distributions reinterpreting causal inference way also becomes directly accessible statistical learning theory assume found causal model consistent statistical properties large number observed subsets hope also correctly predicts properties unobserved subsets provided causal model taken sufficiently small class avoid overfitting radical empirical point view developed even rather asking whether statistical property like statistical independence true ask whether test hand rejects accepts hence replace term statistical properties scheme test results point view may also justify several common pragmatic solutions following issues linear causal models relations perspective justifies apply multivariate gaussian causal models data sets clearly assume hypothetical causal graph inferred conditional independence pattern obtained via partial correlation tests correct multivariate gaussians done common causal inference software tetrad even one knows graph represents partial correlations correctly conditional independences may predict well partial correlations unseen variable sets way linear causal model helpful goal predict linear statistics good news particularly general conditional independence tests remain difficult issue see instance zhang recent proposal tuning confidence levels also another heuristic solution difficult question causal inference justified inferring causal dags based causal markov condition causal faithfulness spirtes relies setting confidence levels accepting conditional dependence practice one usually adjust level enough independences accepted enough rejected sample size hand otherwise inference impossible problematic however perspective common justification causal faithfulness one rejects causal hypotheses accidental conditional independences occur measure zero meek becomes questionable set confidence level high enough one wants get independences argue follows instead assume given arbitrary confidence level threshold conditional independence tests assume found dag asking whether two variables fact statistically independent make sense empirical sample unless sample thought part infinite sample ridiculous finite world detailed discussion causal conclusions several causal inference algorithms may repeatedly change increasing sample size see kelly sufficiently small model class consistent outcomes conditional independence tests large number subsets justified assume correctly predict outcomes test unobserved variable sets methodological justification causal faithfulness learning scenarios dags used predict choice variables xjk whether xjk without faithfulness dag entail independence never entail dependence rather stating unfaithful distributions unlikely need faithfulness simply obtain definite prediction first place paper structured follows section explains causal models sometimes entail strong statements regarding composition data sets motivates use causal inference intermediate step actual task predict properties unobserved joint distributions section formalizes scenario standard prediction task input subset ordered tuple variables want test statistical property output statistical property subset tuple way observed variable set defines training point inferring causal model unobserved variable sets test instances accordingly classes causal models define function classes described section whose richness measured via dimension starightforward application learning theory section derives error bounds predicted statistical properties discusses used guidance constructing causal hypotheses classes hypotheses section argue use causal models linked usual interpretation causality terms interventions raises philosophical questions whether empirical content causality reduces providing rules merge probability distributions causal models particularly helpful obvious inferring properties unobserved joimnt distributions observed ones take detour via causal models visualized one could also define class statistical models class joint distributions without nay causal interpretation sufficiently small yield definite predictions desired properties example however suggests causal models typically entail particularly strong predictions regarding properties joint distribution among reasons causal models subsets variables sometimes imply simple joint causal model make point consider following toy example example merging two pairs chain assume given variables observed extension heavily underdetermined assume additional causal information causes causes see figure left sense pairs causally sufficient words neither common cause information result bivariate causal inference algorithm able exclude confunding given instance additive noise model kano shimizu hoyer figure simplest example causal information allows glue two distributions two unique joint distribution confounder unlikely would typically destroy independence additive noise term entire causal structure infer entire causal structure causal chain following reasons first show causally sufficient set variables common cause would common cause pair common causes assumption one checks easily dag arrows leaves pairs unconfounded checking dags arrows path end causal chain figure middle option resulting joint distribution implies therefore note presentation example neglected subtle issue several different notions means causes causally sufficient way used purely graphical criterion asking whether variable directed paths alternative option defining influences causally sufficient way would demand condition called interventional sufficiency peters condition testable interventions without referring larger background dag embedded condition however weaker graphical one sufficient argument one could add link chain still observe detailed example peters therefore stick graphical criterion causal sufficiency justify fact generic parameter values coincides interventional sufficiency would actually reasonable criterion causal marginal problem probabilistic marginal problem given marginal distributions psk sets variables problem existence uniqueness joint distribution consistent marginals usually referred marginal problem vorob kellerer call probabilistic marginal problem motivated terminology informally inroduce causal marginal problem follows given distributions psk together causal models unique joint distribution causal model consistent marginal model definition informal specified notion causal model neither specify marginalization causal models dags marginalization requires general graphical model class mags richardson spirtes already mentioned marginalization sructural equations require structural equations dependent noise terms rubenstein without formalizing claim example suggests causal marginal problem may unique solution even probabilistic marginal problem janzing procedure constructing joint distribution example described following special case scheme statistical properties observed subsets causal model observed subsets joint causal model statistical properties unobserved subsets whether joint causal model inferred first inferring marginal causal models whether directly inferred statistical properties marginal disributions irrelevant discussion example detour marginal causal models particularly simple formal setting usually refer given set variables xjk whose subsets considered whenever cause confusion carefully distinguish set vector xjk also use term joint distribution although order variables certainly matters statistical properties statistical properties crucial concept work one hand used infer causal structure hand causal structure used predict definition statistical property statistical property range given function denotes joint distribution variables consideration output space often consider binary properties respectively slightly abusing terminology term statistical property sometimes refer value output function hopefully cause confusion may defined fixed size general moreover consider properties depend ordering variables depend invariant permutations variables clear context refer tuples part order matters partly ordered tuples given impression variety statistical properties conclude section list examples start example binary property refer ordering example statistical independence jointly independent otherwise following binary property allows permutations variables example conditional independence otherwise emphasize causal models used predict conditional independences also statistical properties also mention linear additive noise models kano shimizu example existence linear additive noise models matrix entries aij lower triangular permutation basis vectors aij jointly independent noise variables additive linear model exists set lower triangularity means dag entries aij whenever arrow entire order variables matters linear structural equation whenever noise variables linear additive noise models allow unique identification causal dag kano shimizu one assumes true generating process linear holds orderings variables compatible true dag way statistical propery directly linked causal structure subject strong assumption course following simple binary property also play role later example sign correlations whether pair random variables positively negatively correlated defines simple binary property scenario variables correlated cov cov finally mention statistical property binary example covariances correlations variables let set positive matrices define denotes joint covariance matrix one also get property focusing term one may define map cov alternatively one prefers correlations define corr statistical causal models idea paper causal models used predict statistical properties priori models need causal one use bayesian networks instance encode conditional statistical independences without interpreting arrows formalizing causal influence formalism introduced section matter whether one interprets models causal example however suggested model classes come causal semantics particularly intuitive regarding statistical properties predict introduce notion models definition models statistical property given set variables statistical property model class joint distributions coincide regarding output accordingly property predicted model given function runs allowed input partly ordered tuples formally partly ordered tuples equivalence classes equivalence corresponds irrelevant reorderings tuple avoid cumbersome formalism refer allowed inputs later model instance dag property formalizes conditional independences hold respective markov equivalence class understand terminology note receives distribution input output tells respective property distribution whether independence holds contrast receives set nodes variables dag inputs tells property entailed goal find model coincide majority observed tuples variables prominent example reads example dag model conditional independences let dag nodes set conditional independences example let function defined markov condition implies otherwise note mean markov condition implies dependence says imply independence however think causal dag common assumption causal faithfulness spirtes states dependences allowed markov condition occur reality adopting assumption therefore interpret function predicts dependence independence instead making prediction otherwise also mention particularly simple class dags appear interesting example later example dags consisting single colliderfree path let set dags consist single colliderfree path directions arrows variable two arrowheads colliderfree paths important property dependence two nodes screened variable lies two nodes whenever lies one assumes addition joint distribution gaussian partial correlation given vanishes one show correlation coefficient two nodes given product pairwise correlations along path corr corr follows easily induction corr corr corr three variables therefore dag together correlations adjacent nodes predicts pairwise correlations therefore specify model ordering nodes correlations adjacent nodes following example shows dag entail also properties sophisticated conditional independences correlations example dags linear additive noise let dag nodes linear additive noise property example let function defined following two conditions hold causally sufficient subset two different common ancestor ordering consistent ancestor example predicts graphical structure whether joint distribution subset variables admits linear additive noise model idea following assuming entire joint distribution variables generated linear additive noise model kano shimizu also admits linear additive noise model provided hold marginalizations linear additive noise models remain linear additive noise models whenever one marginalize common hence conditions clearly sufficient generic parameter values underlying linear model two conditions also necessary linear models render causal directions uniquely identifiable also admit detection hidden common causes hoyer testing properties data far introduced statistical properties mathematical properties distributions applications however want predict outcome test empirical data task longer predict whether set variables really conditionally independent want predict whether statistical test hand accepts independence whether test appropriate respective mathematical property relevant generalization bounds derived later one infers dags instance partial correlations uses dags infer partial correlations matter relations actually prohibit replace conditional independences partial correlations reader may get confused remarks seems requirement tests supposed good test mathematical property difficult question one say however test entirely unrelated property guidance outcomes test causal hypothesis predict fact partial correlations despite limitations approximate conditional independence provide justification expecting vanishing partial correlations many cases causal dag first specify information provided data set definition data set data set matrix observations denotes sample size number variables dataset contains tuple values specifying variables ykj samples refer check whether variables consideration fact satisfy property predicted model need statistical test case binary properties estimator case properties let say given test estimator property formally defined follows definition statistical test estimator test respective estimator properties statistical property range map data set involves observed instances partly ordered tuple defines allowed input thought indicate outcome test estimated value respectively phrasing task standard prediction problem learning problem reads given data sets variables find model data sets less note class additive noise models hoyer closed marginalization demanding data sets however importantly would like choose also hold future data set problem constructing causal model becomes standard learning problem training well test examples data sets note also phrased causal inference problem standard learning problem task classify two variables cause effect getting large number pairs training examples however data sets refer observations different subsets variables actually assumed follow joint distribution union variables occuring data sets phrased problem standard prediction scenario whose inputs subsets variables introduce usual notion empirical error training data accordingly definition empirical error let statistical property statistical test collection data sets referring variable tuples empirical training error model defined sdj finding model training error small guarantee however error also small future test data chosen rich class models small training error may result overfitting fortunately phrased learning problem way richness class causal models quantified standard concepts statistical learning theory discussed following section capacity classes causal models formally phrased problem prediction problem task predict outcome test applied unobserved variable set assume given class models defining statistical properties supposed predict outcomes binary properties given binary statistical property straightforwardly apply notion vcdimension vapnik classes define definition dimension model class binary properties let set variables binary property let class models defines map dimension largest number allowed inputs restriction runs possible binary functions since model classes thought given causal hypotheses following class important example although later restrict class get stronger generalization bounds lemma dimension conditional independences entailed dags let set dags nodes every define example dimension satisfies proof number dags labeled nodes easily upper bounded number orderings times number choices draw edge yields using stirling formula obtain thus since dimension class larger binary logarithm number elements contains easily follows note number possible conditional independence tests form already grows faster dimension namely third power therefore class dags sufficiently restrictive since able explain possible patterns conditional dependences even one conditions one variable nevertheless set dags may large number data sets hand therefore mention following restrictive class given polytrees dags whose skeleton tree hence contain undirected cycles lemma dimension cond independences entailed polytrees let set polyntrees nodes every define example dimension satisfies proof according cayley formula number trees nodes reads aigner ziegler number markov equivalence classes polytrees bounded radhakrishnan bound follows taking logarithm later use following result lemma dimension sign correlations along path consider set dags consist single colliderfree path example sign pairwise correlations determined permutation aligns graph sign correlations adjacent pairs thus parameterize model vector denotes signs adjacent nodes full model class obtained runs entire group permutations combinations let property indicating sign correlation two variables example dimension proof defining sign corr obtain sign corr due therefore signs computed since possible assignments values thus induces functions thus dimension statistical properties also want obtain quantitative statements strength dependences therefore consider also correlation example property lemma correlations along path let model class whose elements colliderfree paths correlations adjacent nodes specified see example already explained specification determines uniquely pairwise correlations thus define model induced property corrm term right hand side denotes correlation determined model introduced example dimension proof assume simplicity correlations specify absolute value correlation adjacent nodes define parameters log specify sign correlations define binary values corrm otherwise convenient introduce parameters cumulative versions adjacent log correlations likewise introduce binaries mod indicate whether number negative correlations along chain beginning odd even way correlations two nodes computed corrm technical reasons define corr formally function ordered pairs variables although actually symmetric interested dimension family functions defined corrm defined dimension set classifiers otherwise estimate dimension compose classifiers whose dimension easier estimate first define family classifiers given otherwise likewise define otherwise dimensions given linear functions space possible vapnik section example define set classifiers classify according sign correlations otherwise otherwise likewise set since components dimension dimension equivalent log log therefore denotes intersection concept classes van der wart wellner given likewise union concept classes given opposed unions intersections equivalent log log hence obtain hence finite union intersection concept classes set theoertic union dimension therefore dimension van der wart wellner generalization bounds binary properties seen scenario causal models like dags define classifiers sense standard learning scenarios use usual bounds like theorem vapnik guarantee generalization future data sets end need assume data sets sampled distribution data sets assumption discussed end section theorem generalization bound let statistical test statistical binary property model class dimension defining property given data sets sampled distribution sdj probability thus suffices increase number data sets slightly faster dimension illustrate apply theorem recall class polytrees lemma interesting property polytrees every pair nodes already rendered conditional independent one appropriate intermediate node always one undirected path connecting moreover two nodes close together dag realistic chance randomly chosen satisfies therefore consider following scenario draw triples uniformly random check whether search polytree consistent observed dependences number independence tests number nodes figure red curce shows number tests required bound grows number variables blue one shows number possible tests grows predict conditional independences unobserved triples via since number points training set increase slightly faster dimension see lemma know small fraction possible independence tests grows third power already sufficient predict conditional independences red curve figure provides rough estimate needs grow want ensure term blue curve shows number possible tests grows significantly exceeds required ones variables fraction possible tests needed predict also remaining ones hold high probability conditional independences used causal inference already since decades recently became popular use properties distributions infer causal dags particular several methods proposed distinguish cause effect bivariate distributions kano shimizu hoyer zhang daniusis peters mooij tempting multivariate causal inference finding dags consistent bivariate causal direction test motivates following example lemma bivariate directionality test dags let class dags nodes directed path pairs nodes define property iff directed path iff directed path proof dimension maximal number pairs variables causal directions oriented possible ways take pairs undirected graph defined connecting pair contains cycle however causal directions possible would directed cycle result used infer causal directions pairs observed together apply bivariate causality test randomly chosen ordered pairs needs grow slightly faster search dag consistent last fraction outcomes infer outcome bivariate causality tests remarkable generalization bound holds regardless bivariate causality tested whether one understands statistical features used infer causal direction solely fact causal hypothesis class low dimension matches majority bivariate tests ensures generalizes well future tests properties bounds subsection referred binary statistical properties consider also properties note dimension class functions defined dimension set binary functions see section vapnik combining vapnik obtain theorem bound statistical properties let class properties dimension given data sets sampled distribution sdj probability least bound easily applied prediction correlations via paths due lemma since correlations set interpretation setting learning theory practical applications scenario usually somehow different one choose observed unobserved subsets randomly instead observed sets defined available data sets one may object considerations therefore inapplicable formal argument objection however may reasons believe observed variable sets hand substantially different unobserved ones whose properties supposed predicted apart fact observed based belief one may still use generalization bounds guidance richness class causal hypotheses allowed obtain good generalization properties predicting impact interventions merging distributions argued causal hypotheses provide strong guidance merge probability distributions thus become empirically testable without resorting interventions one may wonder whether view causality completely disconnected interventions argue sense estimating impact intervention also phrased problem inferring properties unobserved joint distributions assume want test whether causal hypothesis true would check distribution changes randomized interventions let formally introduce variable pearl attain possible values indicating value set value idle intervention made whether influences equivalent demand causal relation unconfounded usually intended notation test condition intervention made conditions refer unobserved distribution inferring whether true thus amounts inferring unobserved distribution plus additional background knowledge regarding statistical causal relation based knowledge action made fact desired intervention applications question action considered intervention target variable hand instance complex interactions one assumes based purely observational data maybe earlier past reduced problem predicting impact interventions entirely problem merging joint distributions conclusions described different scenarios causal models used infer statistical properties joint distributions variables never observed together causal models taken class sufficiently low dimension justified generalization bounds statistical learning theory opens new pragmatic perspective causality essential empirical content causal model may consist prediction regarding merge distributions overlapping data sets pragmatic use causal concepts may helpful domains interventional definition causality raises difficult questions one claims age person causally influences income assumed mooij unclear means intervene variable age moreover argued pragmatic view causal models related usual concept causality terms interventions even possible view causality could also relevant foundational questions physics language causal models plays increasing role recently leifer spekkens chaves ried wood spekkens janzing references aigner ziegler proofs book springer berlin chaves majenz gross implications quantum causal structures nat commun daniusis janzing mooij zscheischler steudel zhang inferring deterministic causal relations proceedings annual conference uncertainty artificial intelligence uai pages auai press hoyer shimizu 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faster computation via rle keita kuboi yuta fujishige shunsuke inenaga hideo bannai masayuki takeda department informatics kyushu university japan inenaga bannai takeda mar abstract constrained lcs problem asks one find longest common subsequence two input strings constraints problem variant constrained lcs problem solution must include given constraint string substring given two strings respective lengths constraint string length min best known algorithm problem proposed deorowicz inf process runs time work present solution problem denote sizes encodings respectively since always hold algorithm always fast deorowicz algorithm faster input strings compressible via rle introduction longest common subsequence lcs one basic measures similarity strings vast amount literature concerning efficient computation lcs two strings lengths respectively longest string subsequence well known time space dynamic programming algorithm compute lcs two strings lcs applications bioinformatics file comparisons pattern recognition etc recently several variants problem try find longest common subsequence satisfy constraints considered tsai proposed constrained lcs clcs problem given strings respective lengths constraint string length problem find longest string contains subsequence also common subsequence tsai gave time solution improved chin time variants constrained lcs problem called considered chen chao problem considers input three strings problem find longest string includes excludes subsequence seq substring str common subsequence clcs equivalent problem best solution problems shown table table time complexities best known solutions various constrained lcs problems problem solution solution using rle min work order speed lcs computation one direction research received much attention apply compression namely encoding rle strings bunke csirik one first consider scenario proposed time algorithm sizes rle input strings lengths respectively notice since rle computed linear time algorithm always asymptotically faster standard time dynamic programming algorithm especially strings compressible rle furthermore ahsan proposed algorithm runs log log log log time total number pairs runs character two rle strings algorithm much faster strings compressible rle constrained lcs problems rle based solutions problem proposed time algorithm proposed ann later liu proposed faster min time algorithm paper present first rle based solution problem runs time since rle computed linear time proposed algorithm always asymptotically faster best known solution problem deorowicz runs time common criticism rle based solutions claim although theoretically interesting since strings real world compressible rle applicability limited useful extreme artificial cases believe entirely true cases rle natural encoding data example music melody expressed string pitches duration furthermore data mining community exist popular preprocessing schemes analyzing various types time series data convert time series strings fairly small alphabet approximation original data various analyses conducted sax symbolic aggregate approximation clipped bit representation conversions likely produce strings compressible rle fact shown effective indicating rle based solutions may wider range application commonly perceived preliminaries let finite set characters set strings string let length let ith character let denote substring especially denotes prefix denotes suffix string subsequence obtained removing zero characters two string string longest common subsequence lcs longest string subsequence let lpref denote length lcs let lsuf denote length lcs lcs problem compute length lcs given two strings well known solution dynamic programming computes time table call table size stores values lpref table lsuf computed similarly two strings constraint string string longest string includes substring also subsequence problem compute length given three strings example abacab babcaba abcab bacab lcss abb encoding rle string kind compressed representation maximal run character represented pair character length run let rle denote rle string size rle number runs denoted definition always less equal next section consider problem strings constraint string let assume min min since case solution also assume case problem becomes normal lcs problem algorithm section first introduce slightly modified version deorowicz algorithm problem propose algorithm based dynamic programming approach uses rle deorowicz algorithm first define notion minimal string definition strings interval minimal subsequence subsequence deorowicz algorithm based lemma used implicitly lemma implicit exist minimal respectively xcy lcs lcs proof definition substring therefore exist possibly empty strings xcy also since common subsequence exist monotonically increasing sequences since subsequence exist minimal cintervals respectively satisfy let lcs lcs since must common subsequence common subsequence however since otherwise would string longer contains substring common subsequence contradicting thus implying also lcs also lcs proving lemma algorithm consists following two steps whose correctness follows lemma step compute minimal step pairs minimal minimal compute length lcs corresponding prefixes lpref corresponding suffixes lsuf largest sum lcs lengths plus lpref lsuf length steps executed following running times step respectively minimal enumerated time step precompute time two dynamic programming tables respectively contain values lpref lsuf using tables value lpref lsuf computed constant time possible pairs minimal step done time total since problem solved time note original presentation deorowicz algorithm intervals subsequence computed instead minimal defined definition although number considered intervals changes influence asymptotic complexities case however see lemma section essential difference rle case since number minimal bounded number bounded algorithm via rle subsection propose efficient algorithm based deorowicz algorithm explained subsection extended strings expressed rle two main cases consider consists one type character contains least two different characters case theorem let strings let compute length time step execute following procedure enumerate minimal let first find right minimal starting smallest position subsequence next starting position search backwards find left minimal ending largest position subsequence process repeated find smallest position subsequence search backwards find largest position subsequence easy see intervals obtained repeating procedure reaching end minimal since interval found distinct exist another minimal found procedure done strings takes time lemma shows procedure implemented efficiently using rle lemma let strings number minimal enumerated time proof easy see backward search procedure described minimal unique run last character first run corresponds last character run therefore number minimal compute rle rle time remains show search procedure described compute minimal implemented time algorithm described shown algorithm forward search scan rle find right minimal greedily matching runs rle rle maintain character exponent rest first run crest rle suffix yet matched comparing run rle crest characters different know entire run match thus consider next run suppose characters rest entire run matched consider next run also rest updated accordingly constant time simple arithmetic furthermore since fact skip next run rest entire run crest matched consider next run also since skip rest consider next run thus spend constant time run scanned forward search holds backward search finish proof show total number times run scanned procedure bounded number minimal intersects given run since minimal contained thus minimal intersect run must cross either left boundary run right boundary run minimal cross left boundary run must strings occurs subsequence occurs subsequence minimal corresponds union left minimal ending left boundary run right minimal starting left boundary run thus unique similar arguments also hold minimal cross right boundary since choices claim holds thus proving lemma deorowicz algorithm two tables computed step took time algorithm use compressed representation table proposed bunke csirik instead normal table note bunke csirik actually solved edit distance problem cost insertion deletion substitution easily translates lcs lpref pref pref denotes edit distance costs definition let strings length respectively rle rle compressed table cdp table compressed representation table holds values table algorithm computing minimal input strings output minimal rle rle index run respectively rest number rest searching characters number minimal rest true forward search else rest rest else rest else rest rest break rest rest else rest rest backward search else rest rest else rest else rest rest rest rest return figure example compressed lpref table strings bbbaaaa aaaabbbaa figure illustrates values stored cdp table strings bbbaaaa aaaabbbaa note although figure depicts sparsely filled table size values actually stored two completely filled tables one size holding values another size holding values total space results adapted use lemma theorem let strings compressed table computed time space lemma lemma let let strings integer lpref lpref lemma lemma let let strings integers lpref max lpref lpref lemmas easily obtain following lemma lemma let strings entry table retrieved time using compressed table lemma compute time two cdp tables respectively hold values lpref lpref lsuf lsuf taking space lemma obtain lpref lsuf time actually make lemma work also need able convert indexes cdp constant time values largest easy preparing arrays time space ready show running time algorithm case compute rle rle rle time step lemma number minimal respectively computed time preprocessing step build cdp tables holding values lpref lsuf computed time space lemma tables obtain values lpref lsuf constant time lemma since pairs minimal minimal total time step computing lpref lsuf pairs since assume total time thus theorem holds case next consider case consists one run theorem let strings let compute length time step compute minimal lemma note difference lemma case lemma number minimal respectively enumerated time respectively proof let let number times occurs number minimal minimal enumerated time checking positions applies lemma see number pairs minimal afford consider pairs step overcome problem follows let set minimal consider partition equivalence classes induced following equivalence relation words equivalence class start run end run noticing minimal completely contained another assume lemma let partition set minimal induced equivalence relation proof let let satisfy since intervals equivalent either must hold thus equivalently consider set minimal partition based analogous equivalence relation let minimal smallest largest start positions since definition said observation show following lemma lemma let positive integer lpref lsuf lpref lsuf proof since lemma lpref lpref lsuf lsuf lemma see need compute lpref lsuf pairs let gmin minimal respectively smallest starting position need consider combination gmin combination therefore combinations minimal need consider combination gmin number combinations clearly example consider rle rle rle minimal minimal also gmin figure shows lengths lcs prefixes suffixes combination minimal gray part values referred values denoted inside parentheses stored cdp table computed time lemma figure shows sum lcs prefixes suffixes corresponding gray part due lemma values along diagonal equal thus combinations minimal need consider six combinations ready show running time algorithm case compute rle rle rle time respectively minimal assigned one equivalence classes total time preprocessing cdp table case done time lemma reduce number combinations minimal consider finally lemma lcs lengths combination computed using cdp table therefore total running time proving theorem theorems following theorem holds proposed algorithm shown algorithm written appendix theorem let strings let compute length time although showed compute length note algorithm modified obtain rle time provided rle figure example depicting lcss corresponding prefixes left suffixes right combinations strings rle rle rle values denoted inside parentheses stored cdp table computed time figure sum lengths lcss corresponding prefixes suffixes shown figure values along diagonal equal value equal value upper right precomputed simply storing minimal respectively maximizes lpref lsuf lemmas simulate standard table obtaining lcss cdp table obtain rle lcss time finally rle obtained combining three rle strings two lcss rle middle appropriately merging boundary runs necessary conclusion work proposed new algorithm solve problem using rle representation compute length strings time space using algorithm result better deorowicz time space use rle want know length also retrieve time references shegufta bakht ahsan syeda persia aziz sohel rahman longest common subsequence problem strings jcp ann yang tseng hor fast algorithms computing constrained lcs encoded strings theor comput anthony bagnall chotirat ann ratanamahatana eamonn keogh stefano lonardi gareth janacek bit level representation time series data mining shape based similarity data mining knowledge discovery horst bunke csirik improved algorithm computing edit distance coded strings inf process chen chao generalized constrained longest common subsequence problems comb francis chin alfredo santis anna lisa ferrara kim simple algorithm constrained sequence problems inf process sebastian deorowicz algorithm string constrained lcs problem inf process paul heckel technique isolating differences files communications acm james hunt malcolm douglas mcilroy algorithm differential file comparison technical report compt sci techn dmitry korkin lev goldfarb multiple genome rearrangement general approach via evolutionary genome graph bioinformatics suppl jessica lin eamonn keogh wei stefano lonardi experiencing sax novel symbolic representation time series data mining knowledge discovery jia jie liu wang chiu constrained longest common subsequences strings comput helman stern merav shmueli sigal berman discriminating segment longest common subsequence mdslcs algorithm dynamic hand gesture classification pattern recognition letters tsai constrained longest common subsequence problem inf process robert wagner michael fischer correction problem acm congmao wang dabing zhang novel compression tool efficient storage genome resequencing data nucleic acids research lei wang xiaodong wang yingjie daxin zhu dynamic programming solution generalized lcs problem inf process appendix show proposed algorithm algorithm proposed time algorithm input strings output length minimal minimal number minimal respectively gmin minimum element respectively number sets respectively make compressed tables compute minimal use algorithm lmax lsum lpref lsuf lmax lsum lmax lsum else gmin gmin gmin lmax gmin gmin lsum lpref lsuf min min lmax lsum lmax lsum lsum lpref sgmin lsuf fgmin lmax lsum lmax lsum return lmax

jun eine charakterisierung der moduln helmut mathematisches institut der theresienstr germany zoeschinger abstract let noetherian local ring injective hull homr matlis dual canonical monomorphism surjective known called reflexive help bass numbers homr respect show reflexive spec follows every exists monomorphism epimorphism already reflexive key words modules bass numbers associated prime ideals torsion modules cotorsion modules mathematics subject classification der rang eines moduls stets sei noethersch und lokal jeden extir spec die von siehe und weil wir sie folgenden nur brauchen schreiben wir statt kurz ist ein mit ist dimk offenbar der rang von ist beliebig wird annm einem modul dem und folgt insbesondere ist mit ass allgemeiner gilt mit einem untermodul von spec ist satz ist ein und die kanonische einbettung gilt jedes primideal ass kok beweis sei ein und ein mit rang rang dann ist rang endlich zum beweis man einen freien untermodul von mit torsion aus rang rang auch rang rang folgt mit ist also auch ein zerfallender monomorphismus insbesondere rang rang daraus folgt die endlichkeit von mit und ist auch flach nach theorem einen freien zwischenmodul gibt mit rein klar ist rang dimk andererseits rang dimk dimk dimk aus rang rang mit die behauptung folgt sei ein und ein beliebiger dann gilt rang rang kok ist torsion der beweis folgt unmittelbar aus der gleichung rang rang rang kok denn bei sind nach dem ersten schritt alle kardinalzahlen endlich also rang kok und ist klar sind jetzt und beliebig gibt jedem primideal nach lemma ein kommutatives diagramm kok kok mit exakten zeilen dem also auch ein isomorphismus ist wobei das dem lokalen ring sei genau dann ist jetzt wenn und denselben rang haben nach dem zweiten schritt kok torsion ist kok torsion ist ass kok folgerung genau dann ist reflexiv wenn gilt alle primideale folgerung gibt einen monomorphismus oder einen epimorphismus ist bereits reflexiv beweis bei der ersten folgerung ist kok mit der rechten seite bei der zweiten gilt jeden monomorphismus ist alle bei also wie ist aber ein epimorphismus wird ein monomorphismus nach eben reflexiv ist also auch bemerkung einigen sich sofort entscheiden wann ein primideal die bedingung satz alle ass gilt denn beide seiten sind null ist mit gilt genau dann wenn ist denn die von liefert ass kok ass und bekanntlich ist ass koass spec ist ist mit unendlich gibt kein primideal mit denn auch ist nicht reflexiv also nach beispiel rang rang starke torsionsmoduln ist ein ist die klasse aller mit rang rang nach dem untermoduln faktormoduln und gruppenerweiterungen abgeschlossen ein torsionsfreier genau dann wenn kok reflexiv ist siehe die genauere beschreibung ein torsionsmodul genau dann wenn torsion ist und haben wir keine beschreibung nur wenn sogar stark torsion ass ist siehe wir die struktur von angeben satz ist ein kein sind einen ist stark torsion ist artinsch und reflexiv ist ass koass ist nach voraussetzung beweis wegen koass sinne von stark kotorsion nach dem dortigen satz der exakten folge das erste glied und das dritte endlich erzeugt ist den divisiblen anteil ist dann torsionsfrei als faktormodul von also endlich erzeugt und reflexiv sogar also selbst artinsch und reflexiv der exakten folge ist dann das erste glied artinsch und das dritte also artinsch ein folgt artinsch also auch wie behauptet gibt nach voraussetzung ein mit jedes ass gilt dann ass oder ass oder also ass wie verlangt bemerkung ist ein und nur torsion wir wenigstens zeigen stets ist artinsch ist ein nagataring ist sogar besitzt koass eine endliche finale teilmenge ist gut sinne von ist bereits stark torsion beispiel ist ein injektiv und rang rang ist bereits reflexiv beweis ist abgeschlossen also direkter summand und ist dann torsionsfrei also reflexiv sei jetzt gleich injektiv und torsion nach voraussetzung ist torsion flach also gut sogar stark torsion ist und mit die behauptung folgt beispiel ist ein diskreter bewertungsring gilt jeden rang rang mit reflexiv beweis nur ist zeigen und weil jeder modul gut ist ist sogar stark torsion also nach artinsch und reflexiv damit ist also mit reflexiv und sei wieder beliebig und reflexiv dann folgt aus der bijektion annm jeden untermodul von die menge ein minimales element besitzt ein sogenanntes komplement von allein aus dieser komplementeigenschaft folgt sehr viel die struktur von satz ist ein mit und ein komplementierter gilt wobei endlich erzeugt und von der form ist mit artinsch beweis sei der radikalvolle untermodul von weil dann jeder zwischenmodul mit radikalvoll ein komplement besitzt folgt bereits ist koatomar ein komplement von ist dann ebenfalls koatomar also von der form mit endlich erzeugt ein weil auch komplementiert ist lemma besitzt wie eben keine teilbaren faktormoduln ist also kotorsion nach theorem ist nun minimax ein komplement von also sogar stark kotorsion radikalvoll nach satz schon damit ist auch und bleibt zerlegen auch ist minimax besitzt also einen endlich erzeugten untermodul artinsch ist und aus mit folgt auch also selbst artinsch ist weil teilbar torsionsfrei und minimax ist also isomorph ist als reiner artinscher untermodul von sogar direkter summand und wie bemerkung seien und wie satz falls dim ist bekanntlich als nicht minimax also und daher selbst artinsch literatur local cohomology algebraic introduction geometric applications cambridge univ press commutative ring theory cambridge univ press structure couniform complemented modules pure appl algebra gelfandringe und koabgeschlossene untermoduln bayer akad wiss starke kotorsionsmoduln arch math

flying insect classification inexpensive sensors yanping chen department computer science engineering university california riverside adena department entomology university california riverside gustavo batista university paulo usp gbatista agenor isca technologies president eamonn keogh department computer science engineering university california riverside eamonn abstract ability use inexpensive noninvasive sensors accurately classify flying insects would significant implications entomological research allow development many useful applications vector control medical agricultural entomology given last sixty years seen many research efforts task date however none research lasting impact work explain lack progress attribute stagnation problem several factors including use acoustic sensing devices overreliance single feature wingbeat frequency attempts learn complex models relatively little data contrast show optical sensors produce vastly superior data exploit additional features intrinsic extrinsic insect flight behavior bayesian classification approach allows efficiently learn classification models robust overfitting demonstrate findings large scale experiments dwarf previous works combined measured number insects number species considered keywords automate insect classification insect flight sound insect wingbeat bayesian classifier flight activity circadian rhythm introduction idea automatically classifying insects using incidental sound flight opposed deliberate insect sounds produced stridulation hao dates back dawn computers commercially available audio recording equipment three researchers cornell university medical college kahn celestin offenhauser used equipment donated oliver buckley president bell telephone laboratories record analyze mosquito sounds kahn authors later wrote authors considered opinion intensive application apparatus make possible precise rapid simple observation natural phenomena related sounds mosquitoes lead effective control mosquitoes diseases kahn offenhauser retrospect given importance insects human affairs seems astonishing progress problem made intervening decades even earlier paper reed makes similar suggestion however authors determined wingbeat frequencies manually aided stroboscope sporadic efforts flying insect classification audio features sawedal hall schaefer bent unwin ellington moore especially last decade moore miller repasky however little real progress seems made lack progress mean suggest pioneering research efforts fruitful however would like automatic classification become simple inexpensive ubiquitous current mechanical traps sticky traps interception traps capinera advantages offered digital device higher accuracy low cost monitoring ability ability collect additional information time feel lack progress pursuit attributed three related factors efforts collect data used acoustic microphones reed belton mankin raman sound attenuates according inverse squared law example insect flies three times away microphone sound intensity informally loudness drops one ninth attempt mitigate using sensitive microphone invariably results extreme sensitivity wind noise ambient noise environment moreover difficulty collecting data devices seems led researchers obtain data unnatural conditions example nocturnal insects forced fly tapping prodding bright halogen lights insects recorded confined spaces extreme temperatures belton moore miller cases insects tethered string confine within range microphone reed hard imagine insect handling could result data would generalize insects natural conditions unsurprisingly difficultly obtaining data noted meant many researchers attempted build classification models limited data instances moore less however known building commercially available rotator bottle trap made allow researchers measure time arrival granularity hours however shall show section additional feature circadian rhythm flight activity measure time arrival granularity exploit improve classification accuracy classification models data better halevy banko brill shotton compounding poor quality data issue sparse data issue fact many researchers attempted learn complicated classification models especially neural networks moore moore miller however neural networks many including interconnection pattern different layers neurons learning process updating weights interconnections activation function converts neuron weighted input output activation etc learning say classification problem millions training data difficult zhan attempting learn insect classification problem mere twenty examples recipe overfitting figure difficult overstate optimistic results neural network experiments unless rigorous protocols followed prechelt work demonstrate largely solved problems show use optical sensors record sound insect flight meters away complete invariance wind noise ambient sounds demonstrate sensors allowed record order millions labeled training instances far data previous efforts combined thus allow avoid overfitting plagued previous research efforts introduce principled method incorporate additional information classification model additional information quotidian yet still produce significant gains accuracy finally demonstrate enormous amounts data collected allow take advantage unreasonable effectiveness data halevy produce simple accurate robust classifiers summary believe flying insect classification moved beyond dubious claims created research lab ready deployment sensors formal framework define complexity classification model dimension vapnik chervonenkis informally think complicated complex model one requires many parameters set learned software present work provide researchers worldwide robust tools accelerate research background related work vast majority attempts classify insects flight sounds explicitly implicitly used wingbeat frequency reed sotavalta sawedal hall schaefer bent unwin ellington moore moore however approach limited applications insects discriminated different frequencies consider figure shows histogram created measuring wingbeat frequencies three sexed species insects culex stigmatosoma female aedes aegypti female culex tarsalis male defer details data collected later paper aegypti stigmatosoma tarsalis aegypti stigmatosoma tarsalis figure histograms wingbeat frequencies three species insects stigmatosoma aegypti tarsalis histogram derived based wingbeat sound snippets gaussian curves fit wingbeat frequency histograms visually obvious asked separate stigmatosoma tarsalis wingbeat frequency could produce accurate classification two species different frequencies minimal overlap see compute optimal bayes error rate fukunaga strict lower bound actual error rate obtained classifier considers feature bayes error rate half overlapping area curves divided total area two curves tiny overlap wingbeat frequency distributions two species bayes error rate correspondingly small use raw histograms use derived gaussians however task separate stigmatosoma aegypti wingbeat frequency well frequencies two species overlap greatly case bayes error rate much larger use raw histograms use derived gaussians problem get worse consider species increasing overlap among wingbeat frequencies phenomenon understood version pigeonhole principle grimaldi concrete example assume limit attention mosquitoes species mosquitoes described worldwide assume range mosquito wingbeat frequency species takes integer wingbeat frequency least species must share wingbeat frequency another species possible frequency values actual overlap rate even higher typical wingbeat frequency species distribution peaking value rather single integer value shown figure given unsurprising doubt utility wingbeat sounds classify insects however show analysis pessimistic insect flight sounds allow much higher classification rates suggests information flight sound signal wingbeat frequency analogy humans problem distinguishing middle piano middle saxophone even though fundamental frequency bayes error rate classify three species figure using wingbeat frequency however shall see section titled flying insect classification using additional features signal obtain error rate believe experiments first explicit demonstration actionable information signal beyond wingbeat frequency augment wingbeat sounds additional features help improve classification performance example many species may different flight activity circadian rhythms shall see section titled additional feature circadian rhythm flight activity simply incorporating information significantly improve performance classification ability allow incorporation auxiliary features one reasons argue bayesian classifier ideal task section flying insect classification gracefully incorporate evidence multiple sources multiple formats one bayesian classifier incorporate prior probability seeing particular insect cases may able improve accuracy classification adjusting prior probabilities intervention example may use attractants repellants may construct sensors mechanical barriers limit entry large insects fans discourage weak flyers etc leave considerations future work materials methods insect colony rearing six species insects studied work tarsalis stigmatosoma aegypti culex quinquefasciatus musca domestica drosophila simulans adult insects reared laboratory colonies derived wild individuals collected various locations tarsalis colony derived wild individuals collected eastern municipal water district demonstration constructed treatment wetland san jacinto quinquefasciatus colony derived wild individuals collected southern california georghiou wirth stigmatosoma colony derived wild individuals collected university california riverside aquatic research facility riverside aegypti colony started eggs thailand van dam walton musca domestica colony derived wild individuals collected san jacinto drosophila simulans colony derived wild individuals caught riverside larvae tarsalis quinquefasciatus stigmatosoma aegypti reared enamel pans standard laboratory conditions light dark cycle hour periods fed libitum mixture ground rodent chow brewer yeast musca domestica larvae kept standard laboratory conditions light dark cycle reared mixture water bran meal alfalfa yeast powdered milk drosophila simulans larvae fed libitum mixture rotting fruit mosquito pupae collected cups solo cup chicago placed experimental chambers alternatively adults aspirated experimental chambers within week emergence adult mosquitoes allowed feed libitum sucrose water mixture food replaced weekly cotton towels moistened twice week placed top experimental chambers cup tap water solo cup chicago kept chamber times maintain higher level humidity within cage musca domestica adults fed libitum mixture sugar dried milk free access water drosophila simulans adults fed libitum mixture rotting fruit experimental chambers consisted kritter keepers lee aquarium pet products san marcos modified include sensor apparatus well sleeve bug dorm sleeve bioquip rancho dominguez attached piece pvc piping allow access insects two different sizes experimental chambers used larger smaller lids experimental chambers modified piece mesh cloth affixed inside order prevent escape insects shown figure experimental chambers maintained light dark cycle duration experiment experimental chamber contained individuals species order capture many flying sounds possible limiting possibility capturing one sound time insect handling portal phototransistor array lid circuit board laser source laser beam power supply recording device figure one cages used gather data project logical version setup components annotated instruments record flying sounds used sensor described batista capture insect flying sounds logic design sensor consists phototransistor array connected electronic board laser line pointing phototransistor array insect flies across laser beam wings partially occlude light causing small light fluctuations light fluctuations captured phototransistor array changes current signal filtered amplified custom designed electronic board physical version sensor shown figure output electronic board feeds digital sound recorder zoom handy recorder recorded audio data format file hours long new file starts recording immediately file recorded hours data continuous length file limited device firmware rather disk space standard lossy format optimized human perception speech music however flying insects produce sounds well within range human hearing careful comparisons lossless recordings suggest lose exploitable indeed detectable information sensor data processing downloaded sound files twice week used detection algorithm automatically extract brief insect flight sounds raw recording data detection algorithm used sliding window slide raw data data point used decide whether audio segment contains insect flying sound important note classifier used stage solving relatively simple task differentiating discuss sophisticated classifier attempts differentiate species sex next section used problem nearest neighbor classifier based frequency spectrum ground truth data used ten flying sounds extracted early experiments training data insect sounds ten segments raw recording background noise training data sounds number training data limited ten training data would slow algorithm fewer data would represent variability observed note training data background sounds different minute minute frequency spectrum background sound little variance within short time interval change greatly unpredictably long run variability called concept drift machine learning community tsymbal widmer kubat may due effects temperature change electronics slow decline battery output power etc fortunately given high ratio audio high variation sounds cause significant problem figure shows example audio clip containing flying insect generated sensor see signal insects flying across laser well distinguished background signal amplitude much higher range frequency quite different background sound length sliding window detection algorithm set average length flying sound detected insect sound saved onesecond long wav format audio file centering insect flying signal padding zeros elsewhere makes flying sounds length simplifies future archiving processing data note converted audio format wav stage simply publicly release data community confirm extend results vast majority signal processing community uses matlab matlab provides native functions working wav files obvious choice archiving format figure shows saved audio insect sound shown figure flying sounds detected raw recordings may contaminated background noise noise american domestic electricity bleeds recording due inadequate filtering power transformers obtain cleaner signal applied spectral subtraction technique boll ephraim malah detected flying sound reduce noise flying insect classification section showed simple nearest neighbor classifier detect sound insects pass sound snippet inspection discuss algorithms actually classify snippets species cases sex level host classification algorithms literature decision trees neural networks nearest neighbor etc bayes classifier optimal minimizing probability misclassification devroye assumption independence features bayes classifier simple probabilistic classifier predicts class membership probabilities based bayes theorem addition excellent classification performance bayesian classifier several properties make extremely useful practice particularly suitable task hand bayes classifier undemanding cpu memory requirements devices deployed field large quantities typically small devices limited resources limited memory cpu power battery life bayesian classifier constructed offline lab requires time space resources linear number features bayes classifier easy implement unlike neural networks moore miller bayes classifier many parameters must carefully tuned addition model fast build requires small amount training data estimate distribution parameters necessary accurate classification means variances gaussian distributions unlike classification methods essentially black box bayesian classifier allows graceful introduction user knowledge example external training data set knowledge given particular location deployed insect sensor expect twice likely encounter tarsalis aegypti tell algorithm algorithm use information improve accuracy means cases augment classifier information gleaned text journal papers simply experiences field technicians section tentative additional feature geographic distribution give concrete example bayesian classifier simplifies task flagging anomalies classifiers must make classification decision even object classified vastly different anything observed training phase contrast slightly modify bayesian classifier produce unknown classification one two classifications per day could ignored spate could investigated case indicative infestation completely unexpected invasive species multiple features used classification need consider possibility missing values happens features observed example discuss use feature however dead clock battery could deny feature even rest system working perfectly missing values problem learner may cause serious difficulties however bayesian classifier trivially handle problem simply dynamically ignoring feature question classification time considerations listed argue bayesian classifier best problem hand note decision use bayesian classifier informed advantages also informed extensive empirical comparison accuracy achievable methods given situations accuracy trumps considerations omit exhaustive results brevity figure show comparison neural network classifier frequently used technique literature moore miller considered frequency spectrum wingbeat snippets three species discussed figure training data randomly sampled pool objects test data completely disjoint set objects tested random resamplings neural network used single hidden layer size ten seemed approximately default parameters literature mean performance bayesian classifier mean performance neural network worst performance bayesian classifier worst performance neural network number items training set figure comparison mean worst performance bayesian versus neural networks classifiers datasets ranging size five fifty results show neural network classifier eventually converges performance bayesian classifier significantly worse smaller datasets moreover dataset size range examined occasionally produce pathologically poor results worse default rate note concern performance small datasets apparently conflict claim sensors produce massive datasets cases dealing new insect species may necessary bootstrap modeling species using handful annotated examples find unannotated examples archives process known learning chen intuition behind bayesian classification find mostly likely class given data observed probability observed data belongs class computed using bayes rule prior probability class estimated frequencies database probability observing data class probability occurrence observed data probability usually unknown since depend class usually understood normalization factor thus numerator considered classification probability proportional numerator called posterior probability bayesian classifier assigns data class highest posterior probability argmax set classes stigmatosoma aegypti gambiae bayesian classifier represented using graph called bayesian network bayesian network uses single feature classification shown figure direction arrow graph encodes fact probability insect member class depends value feature observed etc figure bayesian network uses single feature classification classifier based single feature posterior probability observed data belongs class calculated probability observing feature class insect classification primary data observed flight sounds illustrated figure flying sound signal amplitude section center audio represented sequence signal sampled instance total number samples signal sequence contains lot acoustic information features extracted mosquito flying across laser sensor captured flying sound background noise flying signal extracted centered paddings elsewhere make sound long iii wingbeat frequency amplitude spectrum flying sound harmonics figure example audio clip containing flying sound generated sensor sound produced female stigmatosoma insect sound highlighted insect sound cleaned saved long audio clip centering insect signal padding elsewhere iii frequency spectrum insect sound obtained using dft obvious feature extract sound snippet wingbeat frequency compute wingbeat frequency first transform audio signal frequency spectrum using discrete fourier transform dft bracewell bracewell shown figure frequency spectrum sound figure peak fundamental frequency harmonics integer multiples fundamental frequency highest peak represents frequency interest insect wingbeat frequency wingbeat frequency distribution univariate density function easily estimated using histogram figure shows wingbeat frequency histogram plot three species insects single sex observe histogram species well modeled gaussian distribution hence fit gaussian distribution estimated means variances using frequency data fitted gaussian distributions shown figure note hinted introduction bayesian classifier use idealized gaussian distribution could use raw histograms estimate probabilities instead however using gaussian distributions computationally cheaper classification time helps guard overfitting distributions calculate probability observing flying sound class given wingbeat frequency example suppose class insect shown figure unknown measured wingbeat frequency suppose previously measured mean standard deviation female stigmatosoma wingbeat frequency respectively calculate probability observing wingbeat frequency female stigmatosoma insect using gaussian distribution function calculate probabilities classes similar way predict unknown insect likely class using equation example prior probability equal class unknown insect times likely female stigmatosoma female aegypti second likely class thus case correctly classified female stigmatosoma note wingbeat frequency scalar learning density functions feature typically dimensions easy either fitted using distribution models gaussian distribution approximated using histogram constructed small amount training data however feature need multivariate density function estimation methods usually idea distribution model fit distributions features building histogram feature requires prohibitively large size training dataset size training dataset grows exponentially increase dimensionality literature offered multivariate density function estimation methods highdimensional variables window method rosenblatt parzen knn approach mack rosenblatt knn approach simple leads approximation optimal bayes classifier hence use work estimate density functions features knn approach require training phase learn density function directly uses training data estimate probability observing unknown data class specifically given observed data knn approach first searches training data find top nearest neighbors computes probability observing class fraction top nearest neighbors labeled class parameter specifying number nearest neighbors number neighbors labeled class among top nearest neighbors equation calculate probability plug bayesian classifier example imagine use entire spectrum feature insect sound figure given unknown insect first transform flight sound spectrum representation search entire training data find top nearest neighbors suppose set among eight nearest neighbors three belong female stigmatosoma one belongs female aegypti four belong male tarsalis calculate conditional probability using equation stigmatosoma conditional probabilities multiplied class prior probability calculate posterior probability observed insect predicted class highest posterior probability able estimate probability features format including feature distance returned opaque similarity function thus generalize bayesian classifier subsume advantages nearest neighbor classifier table outlines bayesian classification algorithm using nearest neighbor distance feature algorithm begins lines estimating prior probability class done counting number occurrences class training data set estimates conditional probability unknown data using knn approach outlined specifically given unknown insect sound algorithm first searches entire training data find top nearest neighbors using distance measure lines counts class number neighbors belong class calculates probability using equation prior probability probability known class algorithm calculates posterior probability class lines predicts unknown data belong class highest posterior probability line table bayesian classification algorithm using feature notation number nearest neighbors knn approach disfunc distance function calculate distance two data set classes train training dataset tci number training data belong class tci prior probability end unknown data disfunc trainj distance training data end sort ascend distance ascending order find nearest neighbors kci number data labeled class kci calculate conditional probability knn approach calculate posterior probability end normalize posterior probability end argmax assign unknown data class end algorithm outlined table requires two inputs including parameter goal choose value minimizes probability estimation error one way use validation kohavi idea keep part training data apart validation data evaluate different values based estimation accuracy validation data value achieves best estimation accuracy chosen used classification leaves question distance measure use decide distance two insect sounds find good distance measure flying sounds turned crowdsourcing organizing contest july november chen asked participants create best distance measure insect sounds fifteen teams worldwide participated contest received eighty submissions team allowed multiple submissions evaluated submission final score one submission scored team result contest suggested best distance measure simple algorithm computes euclidean distance frequency spectrums insect sounds building crowdsourcing efforts found truncated frequency spectrums exclude data outside range possible wingbeat frequencies table could improve accuracy note crowdsourcing participants somewhat similar less explicit ideas distance measure explained table given two flying sounds first transform sound frequency spectrums using dft lines spectrums truncated include corresponding frequency range lines frequency range thus chosen according entomological frequencies unlikely result insect activity probably reflect noise sensor compute euclidean distance two truncated spectrums line return distance two flying sounds table distance measure two insect flight sounds notation two sound sequences dis distance two sounds function dis disfunc dft dft frequency frequency dis insect classification algorithm obtained plugging distance measure explained table bayesian classification framework outlined table demonstrate effectiveness algorithm considered data used generate plot figure data randomly sampled dataset sounds generated sensor sampled total flying sounds sounds species prior probability class using insect classification algorithm set eight selected based validation result achieved error rate using compared algorithm optimal result possible using wingbeat frequency commonly used approach previous research efforts optimal bayes classify insects using wingbeat frequency lower bound algorithm uses feature means using truncated frequency spectrum able reduce error rate almost third best knowledge first explicit demonstration exploitable information flight sounds beyond wingbeat frequency important note claim distance measure used work optimal may better measures could discovered additional research many large insects members odonata lepidoptera wingbeat frequencies significantly slower choice truncation level reflects special interest culicidae revisiting crowdsourcing etc moreover possible may better distance measures confining attention culicidae tipulidae etc however better distance measure found simply plug distance measure bayesian classification framework get better classification performance additional feature circadian rhythm flight activity addition insect flight sounds features used reduce error rate features cheap obtain simple noting yet improvement significant long noted different insects often different circadian flight activity patterns taylor thus time flying sound intercepted used help classify insects example house flies musca domestica active daytime night whereas tarsalis active dawn dusk unknown insect sound captured noon probable produced house fly tarsalis based information given additional feature must consider incorporate classification algorithm one advantages using bayesian classifier offers principled way gracefully combine multiple features thus solution straightforward simplicity temporarily assume two features conditionally independent independent given class revisit reasonableness assumption later independence feature assumption bayesian classifier called bayesian classifier illustrated figure two arrows graph encode fact probability unknown data class depends features whereas lack arrows means two features independent figure bayesian network uses two independent features classification observed object include two values posterior probability belonging class calculated independence assumption probability proportional probability observing pair class concreteness feature algorithm insect sound time sound produced previous section shown calculate probability using knn estimation method prior probability incorporate additional feature need calculate probability note scalar discussed section learning distributions feature easily done constructing histogram histogram species simply insect flight activity circadian rhythm literature many attempts quantify patterns various insects however due difficulty obtaining data attempts made counting activity observed time period window otherwise taylor jones rowland lindsay without distinguishing number observations period resulting patterns course granularity contrast using sensors able collect order hundreds thousands observations per species count exact number observations sub second granularity producing believe densest circadian rhythms ever recorded figure shows flight activity circadian rhythms stigmatosoma female tarsalis male agypti female circadian rhythms learned based observations collected one month results consistent report mian taylor jones much finer temporal resolution minutes note although three species active dawn dusk aegypti females significantly active daylight hours dusk dawn stigmatosoma tarsalis aegypti figure flight activity circadian rhythms stigmatosoma female tarsalis male aegypti female learned based observations generated sensor collected one month insects discussed work constructed circadian rhythms based many hundreds thousands individual observations however obvious sensors become broadly used community always expect finegrained data example approximately mosquito species worldwide unlikely high quality circadian rhythms collected coming years however absence gold standard circadian rhythms hinder using useful additional feature instead may consider using approximate rhythms one idea use circadian rhythms closely related insect species taxonomically data example suppose circadian rhythm stigmatosoma use rhythm tarsalis approximation latter available cases circadian rhythms insects construct approximate rhythms simply based text descriptions entomological literature frequently encountered descriptions periods insects active include diurnal nocturnal crepuscular etc offline build dictionary templates based descriptions process converting text probabilities course subjective however show lead improvements using information simple first attempt work quantifying different levels activities numbers representing low medium high example insect described diurnal active dawn dusk use three degrees quantify activities highest degree dawn dusk second daytime low activity night resulting template shown figure note circadian rhythm probability distribution tells likely capture certain insect species flights certain time thus template normalized area template sums one figure show approximate circadian rhythm stigmatosoma constructed way spend two minutes searching web academic paper describes stigmatosoma flight activity discovering according mian stigmatosoma active dawn dusk crepuscular dawn dusk diurnal crepuscular iii nocturne diurnal active dawn dusk figure examples approximation templates insects flight activity circadian rhythm markings shown templates normalized area curve one smallest value epsilon greater zero worst case glean knowledge insect circadian rhythm species simply use constant line approximation constant line encodes information activity hours insect enables incorporation familiar insects circadian rhythms classifier improve lowest level flight activity represent low zero bayesian classifier never want assign zero probabilities event unless sure logically impossible occur practice technique called laplacian correction typically used prevent probability estimate exactly zero performance pathological case circadian rhythms approximated using constant line classifier degenerates bayesian classifier use additional feature given almost always incorporate circadian rhythm information classifier given observation calculate probability observing activity species simply looking flight activity circadian rhythms species example suppose insect sound detected probability observing activity tarsalis male three times probability observing aegypti female according circadian rhythms shown figure concreteness insect classification algorithm uses two features outlined table similar algorithm outlined table uses single feature five modifications made table insect classification algorithm using two features algorithm similar one outlined table three modifications needed listed modifications highlighted line unknown data line calculate posterior probability line normalize posterior probability line line argmax assign unknown data class demonstrate benefit incorporating additional feature classification revisit toy example figure feature incorporated accurate flight activity circadian rhythms learned using sensor data achieve classification accuracy recall classification accuracy using paragraph right table simply incorporating feature reduce classification error rate test effect using proxies learned flight activity circadian rhythm imagine flight activity circadian rhythm stigmatosoma female must use one approximate rhythms discussed results shown table see even constant line approximation classification accuracy slightly better using feature although algorithm knowledge circadian rhythm stigmatosoma females knowledge two species circadian rhythms approximation created based text description achieve accuracy better using constant line hoped approximate rhythm carries useful information insects activity pattern even though coarse granularity even better classification accuracy achieved using circadian rhythm tarsalis males approximation seen figure circadian rhythm tarsalis males quite close stigmatosoma females table classification performance using different approximations stigmatosoma female flight activity circadian rhythm flight activity circadian rhythm constant line description using learned rhythm approximations used approximation based using approximation insect rhythm sensor data classification accuracy note classification accuracy approximation worse using accurate circadian rhythm accuracy surprising accurate estimated distribution accurate classification reveals great utility sensor allows inexpensive collection massive amounts training data used learn accurate distributions tentative additional feature geographic distribution addition also use additional feature reduce classification error rate also simply location sensor deployed must preempt possible confusion behalf reader one application sensors estimating relative abundance various species insects particular location however suggesting use estimates relative abundance species insects location accurately appears chicken egg however contradiction classifier attempting optimize accuracy individual decisions particular insect observed knowing even approximately expected prevalence species improve accuracy carries useful information classification insects rarely evenly distributed spatial granularity consider example tarsalis relatively rare east mississippi river reisen whereas aedes albopictus asian tiger mosquito become established states area novak insect captured state east mississippi river probable albopictus tarsalis finer spatial granularity may leverage knowledge since trap next dairy farm animal manure source five times likely see sylvicola fenestralis window gnat anopheles bayesian classifier uses three features illustrated figure assume three features independent figure bayesian network uses three independent features classification based figure probability observed object belonging class calculated probability observing insect class location probability reflects geographic distribution insects classification need true absolute densities insect prevalence need ratio densities different species observation location ratio calculate ratio posterior probability species predict observation belong species highest posterior probability case need actual posterior probability values always calculate posterior probability ratio based constraint sum posterior probabilities different classes one shown lines table obtain ratio given location simple glean information text relevant journal papers simply experiences local field technicians example suppose deploy insect sensor location expect twice likely encounter tarsalis stigmatosoma ratio tarsalis stigmatosoma case glean knowledge local insect population temporarily augment sensor various insect traps physically capture insects cdc trap mosquitoes yellow sticky cards traps sharpshooters etc use manually counted number observations species estimate ratios demonstrate utility incorporating feature simple simulation experiment note features real data simulated assuming two species insects stigmatosoma female aegypti female geographically distributed shown figure assumed sensors deployed three different locations distance centers close one centers model location distributions gaussian density bumps simplicity however necessary assumption use density distribution distribution stigmatosoma insect sensor locations distribution aegypti figure assumptions geographic distributions insect species sensor locations simulation demonstrate effectiveness using feature classification simulate data captured three sensors project ten thousand insect exemplars species onto map according geographic distribution assumption sample insects within capture range sensors assumed square region centering sensor shown figure sampled insect assumed fly across sensor data captured experiment sampled stigmatosoma females aegypti females location location location using frequency spectrum classify sampled insects achieved error rate however incorporating reduced error rate impressive information yet reduced error rate half general framework adding features previous two sections shown extend insect flight sound classifier incorporate two additional features however may dozens additional features could help improve classification performance potential features domain application specific give representative examples long known species insects preferred height fly example njie noted anopheline mosquitoes much likely observed culicine flying height two meters approximate height required enter eave house imagine using feature placing two sensors known distance apart observing lag sensor observations obtain approximation speed insect flying approximation insect may fly angle light beam feature may help discriminate speedy members genus culicoides flying relatively sluggish members family culicidae max bidlingmayer section generalize classifier framework easily extendable incorporate arbitrarily many specialized features compare figure figure figure show bayesian networks increasing number features see adding feature classifier represented adding node bayesian network bayesian network uses independent features classification shown figure figure general bayesian network uses features classification positive integer independent features posterior probability observation belongs class calculated probability observing class note posterior probability calculated incrementally number features increases used features classify objects later discovered useful features would like add new features classifier objects entire classification scratch instead keep posterior probability obtained previous classification based old features update posterior probability multiplying corresponding probability new features objects using new posterior probabilities example suppose first used classify observation posterior probability belonging aegypti belonging tarsalis later find useful would like incorporate new feature suppose probability observing aegypti location intercepted probability observing tarsalis location case update posterior probability belonging agypti belonging tarsalis case belong agypti advantage incremental calculation incorporating new feature fast simple multiplication required calculate posterior probability discussions thus far assumed features independent given class practice features seldom independent given class however shown domingos pazzani even independence assumption hold bayesian classifier may still optimal minimizing misclassification error empirical evidence recent years also showed bayesian classifier works quite well even domains clear feature dependences exist work prove three features used conditionally independent however shall show independence assumption features used classifier reasonable bayesian classifier work well revisiting independent assumption recall naive bayesian classifier optimal assumption features independent domingos pazzani majority experiments work consider two features frequency spectrum order test two features conditionally independent check satisfy constraint concretely task constraint given certain insect species equivalently properties frequency spectrum given species must timestamp spectrum scalar value mass length could use standard test test see properties observed two different time windows distribution however spectrums vectors complicates issue greatly thus see constraint satisfied following experiment indirectly forcefully tests constraint sampled insect sounds captured dawn sounds captured dusk sounds generated aegypti females classified sounds captured different time periods using frequency spectrum hypothesis distribution frequency spectrum species dawn dusk impossible distinguish sounds captured two different periods thus classification error rate would around experiment sounds sampled pool objects averaged ten samplings replacement average classification error rate suggests perceptible difference frequency spectrum insect sounds captured dawn dusk note experiment conducted insects observed constant temperature humidity insectary may generalize insects observed field however experiment increases confidence conditional independence assumption two features least reasonable nevertheless clear general framework possible users may wish use features clearly violate assumption example sensor augmented obtain insect mass generally useful feature clear basic principles allometric scaling frequency spectrum feature would independent deakin good news shown figure bayesian network generalized encode dependencies among features cases clear dependence features consider adding arrow dependent features represent dependence example suppose dependence features add arrow shown red arrow figure direction arrow represents causality example insect larger mass causes slower wingbeat drawback augmented bayesian classifier keogh pazzani training data required learn classification model feature dependences distribution parameters need estimated covariance matrix required instead standard deviation figure bayesian network uses features classification feature conditionally dependent case study sexing mosquitoes sexing mosquitoes required entomological applications example sterile insect technique method eliminates large populations breeding insects releasing sterile males wild separate male mosquitoes females released papathanos conducted experiment see well possible distinguish female male mosquitoes single species using proposed classifier experiment would like distinguish male aegypti mosquitoes females feature used experiment frequency spectrum use obvious difference flight activity circadian rhythms males females belong species recent paper offers evidence minor measurable differences related species anopheles gambiae rund however ignore possibility simplicity data used randomly sampled pool exemplars varied number exemplars sex averaged runs time using random sampling replacement average classification performance using cross validation shown figure average accuracy spectrum wingbeat number sex training data figure classification accuracy sex discrimination agypti mosquitoes different numbers training data using proposed classifier classifier see classifier quite accurate sex separation training data sex achieved classification accuracy using truncated frequency spectrum classifier used separate mosquitoes make eight misclassifications note amount training data increases classification accuracy increases additional confirmation claim data improves classification halevy compared classifier classifier using wingbeat frequency shown figure classifier consistently outperforms wingbeat frequency classifier across entire range number training data classification accuracy using wingbeat classifier training data sex recall accuracy using proposed classifier using frequency spectrum instead wingbeat frequency reduced error rate important recall comparison data basic classifier identical thus improvement attributed additional information available frequency spectrum beyond wingbeat frequency offers additional evidence claim wingbeat frequency insufficient accurate classification experiment assume cost female misclassification misclassifying female male cost male misclassification misclassifying male female confusion matrix classifying mosquitoes equal size sex cost assumption one experiment shown table table confusion matrix sex discrimination aegypti mosquitoes decision threshold female cost assumption confusion matrix sexing mosquitoes decision threshold female predicted class balanced cost actual female class male female male predicted class asymmetric cost female male actual female class male however cases misclassification costs asymmetric example sterile insect technique applied mosquito control failing release occasional male mosquito mistakenly thought female matter much contrast releasing female wild serious mistake females pose threat human health cases deal asymmetric misclassification costs change decision boundary classifier lower number misclassifications principled manner course free lunch reduction number misclassifications accompanied increase number misclassifications previous experiment equal misclassification costs unknown insect predicted belong class higher posterior probability equivalent saying threshold predict unknown insect female posterior probability belonging class females larger unknown insect predicted female equivalently replace line table code table setting threshold table decision making policy sex separation experiment threshold female else male end change threshold minimize total cost costs different misclassifications different sterile insect technique goal reduce number female misclassifications achieved lowering threshold required predict exemplar female example set threshold probability unknown exemplar belonging female less value predicted female changing threshold may result lower overall accuracy males misclassified females reduces number females misclassified male examining experiment summarized table predict setting threshold reduce female misclassification rate male misclassification rate rising chose threshold value gives approximately one thousand chance releasing female however domain specific threshold value used practitioner simply needs state preference one two intuitive equivalent ways threshold gives one value chance misclassifying female male problem misclassifying male female value times worse type mistake threshold elkan applied threshold data used produce confusion matrix shown table obtained confusion matrix shown table see insects experiment males zero females misclassified numbers close agreement theory experiment insect classification increasing number species discussing invariably asked accurate answer depends insects classified example classifier used distinguish stigmatosoma female tarsalis male achieve near perfect accuracy two classes radically different wingbeat sounds whereas used separate stigmatosoma female aegypti female classification accuracy much lower given two species quite similar sounds hinted figure therefore single absolute value classification accuracy give reader good intuition performance system instead section rather reporting classifier accuracy fixed set insects applied classifier datasets incrementally increasing number species therefore increasing classification difficulty began classifying two species insects step added one species single sex sexually dimorphic species used classifier classify increased number species considered total ten classes insects different sexes species counting different classes exemplars class classifier used frequency spectrum classification classification accuracy measured step relevant class added shown table note classification accuracy step accuracy classifying species come step example classification accuracy last step accuracy classifying ten classes insects table classification accuracy increasing number classes step species added classification classification step species added quinquefasciatus accuracy accuracy aegypti musca domestica stigmatosoma aegypti tarsalis stigmatosoma tarsalis drosophila simulans see classifier achieves accuracy classifying five species insects significantly higher default rate accuracy even number classes considered increases ten classification accuracy never lower significantly higher default rate note ten classes easy separate even human inspection among ten species eight mosquitoes six genus utility automatic insect classification reader may already appreciate utility automatic insect classification however completeness give examples technology may used electrical discharge insect control systems edics bug zappers insect traps attract electrocute mosquitoes popular consumers presumably gratified hearing characteristic buzz sound produced insect electrocuted commercial devices sold mosquito deterrents studies shown little insects killed mosquitoes frick tallamy surprising since attractant typically ultraviolet light augmenting traps chemical attractants helps still allows needless electrocution beneficial insects isca technologies owned author experimenting building smart trap classifies insects approach trap selectively killing target insects blowing insects away compressed air noted sterile insect technique used reduce populations certain target insects notably screwworm flies cochliomyia hominovorax mediterranean fruit fly ceratitis capitata basic idea release sterile males wild mate wild females males sterile females lay eggs either unfertilized produce smaller proportion fertilized eggs leading population declines eventual eradication certain areas benedict robinson note important release females sexing mosquitoes notoriously difficult researchers university kentucky experimenting sensors create insectaries male hatchlings escape idea use modified edics high powered laser selectively turns allow males pass kills females much research insect behavior regard color odor done human observers count insects move dual choice olfactometer landing strips etc example cooperband notes virgin female wasps individually released downwind color landed recorded human observer several problems human time becomes bottleneck research human error possibility host seeking insects presence human nearby may affect outcome experiment unless costly isolation used envision sensor used accelerate research making significantly cheaper conduct types experiments moreover unique abilities system allow researchers conduct experiments currently impossible example recent paper rund attempted see differences daily flight activity patterns anopheles gambiae mosquitoes authors placed individual sexed mosquitoes small glass tubes record behavior however possible small size glass tubes fact insects isolation affected result moreover even act physically sexing mosquitoes may affect due metabolic stress etc contrast using sensors allow unsexed pupae hatch adults fly cages order magnitude larger volumes way automatically noninvasively sex produce daily flight activity plots conclusion future work work introduced framework allows inexpensive scalable classification flying insects shown experimentally accuracies achievable system good enough allow development commercial products useful tool entomological research encourage adoption extension ideas making code data sensor schematics freely available ucr computational entomology page chen moreover within limits budget continue practice giving complete system shown figure research entomologist requests one acknowledgements would like thank vodafone americas foundation bill melinda gates foundation paulo research foundation fapesp funding research many faculties department entomology ucr offered advice expertise references banko brill mitigating problem exploring effect training corpus size classifier performance natural language processing proceedings first international conference human language technology research association computational linguistics batista keogh rowton sigkdd demo sensors software allow computational entomology emerging application data mining proceedings acm sigkdd international conference knowledge discovery data mining belton costello flight sounds females mosquitoes western canada entomologia experimentalis applicata benedict robinson first releases transgenic mosquitoes argument sterile insect technique trends parasitology accessed march bidlingmayer day evans effect wind velocity suction trap catches florida mosquitoes journal american mosquito control association boll suppression acoustic noise speech using spectral subtraction acoustics speech signal processing ieee transactions bracewell bracewell fourier transform applications new york vol capinera encyclopedia entomology springer epsky morrill mankin traps capturing insects encyclopedia entomology springer netherlands chen supporting materials https chen keogh batista time series learning single example proceedings acm sigkdd international conference knowledge discovery data mining chen keogh batista http ucr insect classification contest cooperband hartness lelito cosse landing surface color preferences spathius agrili hymenoptera braconidae parasitoid emerald ash borer agrilus planipennis coleoptera buprestidae journal insect behavior deakin formulae insect wingbeat frequency journal insect devroye probabilistic theory pattern recognition springer vol domingos pazzani optimality simple bayesian classifier loss machine learning elkan foundations learning international joint conference artificial intelligence vol lawrence erlbaum associates ephraim malah speech enhancement using square error spectral amplitude estimator acoustics speech signal processing ieee transactions frick tallamy density diversity insects killed suburban electric insect traps entomological news fukunaga introduction statistical pattern recognition access online via elsevier grimaldi discrete combinatoral mathematics applied introduction addisonwesley longman publishing halevy norvig pereira unreasonable effectiveness data ieee intelligent systems hao campana keogh monitoring mining animal sounds visual space journal insect behavior kahn celestin offenhauser recording sounds produced certain mosquitoes science kahn offenhauser identification certain west african mosquitos sound amer trop ivied keogh pazzani learning augmented bayesian classifiers comparison approaches proceedings seventh international workshop artificial intelligence statistics kohavi study bootstrap accuracy estimation model selection ijcai vol zhou shen yao automated mosquito diptera culicidae wingbeat waveform neural network intelligence applications innovations mack rosenblatt multivariate neighbor density estimates journal multivariate analysis mankin machan jones field testing prototype acoustic device detection mediterranean fruit flies flying trap proc int symp fruit flies economic importance mermelstein distance measures speech recognition psychological instrumental pattern recognition artificial intelligence mian mulla axelrod chaney dhillon studies bioecological aspects adult mosquitoes prado basin southern california journal american mosquito control association moore artificial neural network trained identify mosquitoes flight journal insect behavior moore miller automated identification optically sensed aphid homoptera aphidae wingbeat waveforms ann entomol soc moore miller tabashnik gage automated identification flying insects analysis wingbeat frequencies journal economic entomology njie dilger lindsay kirby importance eaves house entry anopheline culicine mosquitoes med entomol novak asian tiger mosquito aedes albopictus wing beats vol papathanos bossin benedict catteruccia malcolm alphey crisanti sex separation strategies past experience new approaches malar suppl parzen estimation probability density function annals mathematical statistics prechelt quantitative study neural network learning algorithm evaluation practices proceedings int conference artificial neural networks raman gerhardt wilkerson detecting insect flight sounds field implications acoustical counting asabe reed williams chadwick frequency character separating species races geographic varieties drosophila genetics reisen western encephalitis mosquito culex tarsalis wing beats vol repasky shaw scheppele melton carsten spangler optical detection honeybees use modulation scattered laser light locating explosives land mines appl rosenblatt remarks nonparametric estimates density function annals mathematical statistics rowland lindsay circadian flight activity aedes aegypti parasitized filarial nematode brugia entomology rund ssc lee bush duffield differences daily flight activity circadian clock anopheles gambiae mosquitoes journal insect physiology sawedal hall flight tone taxonomic character chironomidae diptera entomol scand suppl schaefer bent remote sensing system active detection automatic determination insect flight trajectories iradit bull entomol res shotton sharp kipman fitzgibbon finocchio blake cook moore human pose recognition parts single depth images communications acm sotavalta frequency insects contributions problem insect flight acta entomol fenn taylor geographical range circadian rhythm nature taylor jones mdr circadian rhythm flight activity mosquito aedes aegypti effects journal experimental biology tsymbal problem concept drift definitions related work computer science department trinity college dublin unwin ellington optical tachometer measurement frequency freeflying insects journal experimental biology vapnik chervonenkis uniform convergence relative frequencies events probabilities theory probability applications widmer kubat learning presence concept drift hidden contexts machine learning zhan hou zhou neural network email approach acm sigops oper syst rev issn

polylogarithmic approximation algorithms akanksha daniel pranabendu saket meirav zehavik abstract jul let family graphs canonical vertex deletion problem corresponding defined follows given undirected graph weight function find minimum weight subset belongs known weighted vertex deletion problem paper devise recursive scheme obtain logo algorithms problems building upon classic technique finding balanced separators graph roughly speaking scheme applies problems optimum solution together set form balanced separator input graph paper obtain first logo approximation algorithms following vertex deletion problems give approximation algorithm weighted chordal vertex deletion wcvd vertex deletion problem family chordal graphs way algorithm also obtain constant factor approximation algorithm multicut chordal graphs give approximation algorithm weighted distance hereditary vertex deletion wdhvd also known weighted vertex deletion vertex deletion problem family distance hereditary graphs equivalently family graphs rankwidth methods also allow obtain clean fashion algorithm weighted vertex deletion problem minor closed family excluding least one planar graph unweighted version problem constant factor approximation algorithms known fomin focs weighted version considered log log log algorithm follows bansal soda believe recursive scheme applied obtain logo algorithms many problems well research leading results received funding european research council european unions seventh framework programme erc grant agreement university bergen bergen norway university bergen bergen norway daniello institute mathematical sciences chennai india pranabendu university bergen bergen norway institute mathematical sciences hbni chennai india saket university bergen bergen norway introduction let family undirected graphs natural optimization problem follows weighted vertex deletion input undirected graph weight function question find minimum weight subset belongs weighted vertex deletion problem captures wide class node vertex deletion problems studied example family independent sets forests bipartite graphs planar graphs chordal graphs corresponding vertex deletion problem corresponds weighted vertex cover weighted feedback vertex set weighted vertex bipartization also called weighted odd cycle transversal weighted planar vertex deletion weighted chordal vertex deletion respectively classic theorem lewis yannakakis decision version weighted vertex deletion whether exists set weight removing results graph property every hereditary characterizing graph properties corresponding vertex deletion problems approximated within bounded factor polynomial time long standing open problem approximation algorithms spite long history research still far complete characterization constant factor approximation algorithms weighted vertex cover known since lund yannakakis observed vertex deletion problem hereditary property finite number minimal forbidden induced subgraphs approximated within constant ratio conjectured every nontrivial hereditary property infinite forbidden set corresponding vertex deletion problem approximated within constant ratio however later shown weighted feedback vertex set finite forbidden set admits constant factor approximation thus disproving conjecture hand result yannakakis shows wide range graph properties approximating minimum number vertices delete order obtain connected graph property within factor refer precise list graph properties result applies worth mentioning list includes class acyclic graphs class outerplanar graphs paper explore approximability weighted vertex deletion several different families design logo approximation algorithms problems precisely results follows let finite set graphs includes planar graph let family graphs every graph contain graph minor vertex deletion problem corresponding known weighted planar deletion wpf wpf problem generic problem selecting different sets forbidden minors one obtain various fundamental problems weighted vertex cover weighted feedback vertex set weighted treewidth first result randomized deterministic approximation algorithm wpf finite contains planar graph remark different approximation algorithm class problems graph property simply family graphs called exists infinite number graphs well infinite number graphs graph property called hereditary implies every induced subgraph also slightly better approximation ratio log log log follows recent work bansal reichman umboh see also discussion following theorem therefore first result interpreted clean gentle introduction methods give approximation algorithm weighted chordal vertex deletion wcvd vertex deletion problem corresponding family chordal graphs way algorithm also obtain constant factor approximation algorithm weighted multicut chordal graphs give approximation algorithm weighted distance hereditary vertex deletion wdhvd also known weighted vertex deletion problem vertex deletion problem corresponding family distance hereditary graphs equivalently graphs rankwidth algorithms follow recursive scheme find well structured balanced separators graph exploiting properties family following first describe methodology design approximation algorithms give brief overview consisting known results contributions problem study methods multicommodity theorems classical technique designing approximation algorithms pioneered leighton rao seminal paper approach viewed using balanced vertex edge graph obtain approximation algorithm typical application optimum solution forms balanced separator graph thus idea find minimum cost balanced separator graph add solution recursively solve problem connected components leads logo approximation algorithm problem question recursive scheme strengthening approach exploits structural properties family optimum solution need balanced separator graph indeed balanced separator graph could much larger rather along possibly large subset vertices forms balanced separator graph exploit presence balanced separator graph compute approximate solution consider family weighted vertex deletion amenable approach let instance problem let approximate solution compute approximation algorithm following steps find set balanced separator costly next compute balanced separator using known factor log approximation algorithm deterministic log algorithm weighted vertex separators add solution set recursively solve problem connected component let solutions returned recursive calls add solution finally add back graph consider instance observe partitioned belongs set call instances special case weighted vertex deletion apply approximation algorithm exploits structural properties special case compute solution consider problem finding structure one way enumerate candidates pick one balanced vertex separator least cost balanced vertex separator set vertices every connected component contains half vertices separator plays role however number candidates graph could many enumerate polynomial time example case weighted chordal vertex deletion set clique graph number maximal cliques graph vertices could many hence enumerate test every candidate structure polynomial time however exploit certain structural properties family reduce number candidates graph problems tidy graph removing short obstructions forbid graph belonging family one obtain upper bound number candidate structures example recall graph chordal induced cycles length known graph without induced cycle length maximal cliques observe greedily compute set vertices intersects induced cycles length graph therefore cost factor approximation ratio ensure graph polynomially many maximal cliques hence one enumerate maximal cliques remaining graph test next consider task solving instance special case problem apply recursive scheme advantage much structured graph careful modification solution instance eventually reduce instances weighted multicut example weighted chordal vertex deletion obtain instances weighted multicut chordal graph follow approach three problems study paper believe recursive scheme applied obtain logo algorithms weighted vertex edge deletion corresponding several graph families weighted planar deletion let finite set graphs containing planar graph formally weighted planar deletion defined follows weighted planar deletion wpf input undirected graph weight function question find minimum weight subset contain graph minor wpf problem generic problem encompasses several known problems explain versatility problem require definitions graph called minor graph obtain sequence vertex deletions edge deletions edge contractions family graphs called minor closed implies every minor also given graph family forbidminor denote family graphs contain graph forbidminor minor celebrated graph minor theorem robertson seymour every minor closed family characterized finite family forbidden minors forbidminor finite size indeed size forbidminor depends family finite collection graphs may define weighted deletion problem observe even though definition weighted deletion consider finite sized problem actually encompasses deletion every minor closed family graphs let set finite undirected graphs let family finite subsets thus every element finite set graphs throughout paper assume explicitly given paper show contains least one planar graph possible obtain logo approximation algorithm wpf case contains planar graph considerably restricted general case already encompasses number instances wpf example complete graph two vertices weighted vertex cover problem cycle three vertices weighted feedback vertex set problem another fundamental problem also special case wpf mfd weighted vertex deletion weighted task delete minimum weight vertex subset obtain graph treewidth since graph treewidth excludes grid minor set forbidden minors treewidth graphs contains planar graph vertex deletion plays important role generic efficient polynomial time approximation schemes based bidimensionality theory among examples planar deletion problems found literature approximation parameterized algorithms cases correspond removing vertices obtain outerplanar graph graph diamond graph graph pathwidth respectively apart case weighted vertex cover weighted feedback vertex set much progress wpf work fiorini joret pietropaoli gave constant factor approximation algorithm case wpf diamond graph graph two vertices three parallel edges fomin considered planar deletion unweighted version wpf full generality designed randomized deterministic approximation algorithm later fomin gave randomized constant factor approximation algorithm planar deletion algorithm wpf extends result weighted setting cost increasing approximation factor logo theorem every set wpf admits randomized deterministic factor approximation algorithm mention theorem subsumed recent related result bansal reichman umboh studied edge deletion version vertex deletion problem name bounded treewidth interdiction problem gave bicriteria approximation algorithm particular graph integer gave polynomial time algorithm finds subset edges log log log opt treewidth log additional effort algorithm made work weighted vertex deletion problem well setting fixed constant immediately implies factor log log log approximation algorithm wpf statement theorem subsumed proof gives simple clean introduction methods weighted chordal vertex deletion formally weighted chordal vertex deletion problem defined follows weighted chordal vertex deletion wcvd input undirected graph weight function question find minimum weight subset chordal graph class chordal graphs natural class graphs extensively studied viewpoints graph theory algorithm design many important problems general graphs independent set graph coloring solvable polynomial time restricted class chordal graphs recall graph chordal induced cycle length thus chordal vertex deletion cvd viewed natural variant classic feedback vertex set one run algorithm first remove solution output algorithm obtain graph treewidth log one find optimal solution using standard dynamic programming fvs indeed objective fvs eliminate cycles cvd problem asks eliminate induced cycles length despite apparent similarity objectives two problems design approximation algorithms wcvd challenging particular chordal graphs clique chordal graph rely sparsity output approach must deviate employed approximation algorithms fvs said chordal graphs still retain properties resemble trees properties utilized algorithm prior work two approximation algorithms cvd known first one jansen pilipczuk deterministic log opt log approximation algorithm second one agrawal deterministic opt approximation algorithm second result implies cvd admits log approximation paper obtain first logo algorithm wcvd theorem cvd admits deterministic approximation algorithm approximation algorithm follows general scheme also requires incorporate several new ideas particular implement third step scheme need design different log approximation algorithm special case wcvd input graph partitioned two sets clique chordal graph approximation algorithm based recursion involved recursive call carefully manipulates fractional solution special form moreover ensure current problem instance divided two subinstances independent simpler origin introduce multicut constraints addition constraints keep track complexity subinstances measured via cardinality maximum independent set graph multicut constraints result instance weighted multicut ensure chordal graph formally weighted multicut problem defined follows weighted multicut input undirected graph weight function set pairs vertices question find minimum weight subset pair path weighted multicut chordal graphs approximation algorithm previously known remark weighted multicut trees hence also chordal graphs design first algorithm main algorithm employs black box theorem weighted multicut admits approximation algorithm chordal graphs algorithm inspired work garg vazirani yannakakis weighted multicut trees carefully exploit characterization class chordal graphs class graphs admit clique forests believe result independent interest algorithm garg vazirani yannakakis classic algorithm recent algorithm golovin nagarajan singh uses total modularity obtain different algorithm multicut trees opt log output greedy solution input graph otherwise opt log hence call opt approximation algorithm weighted distance hereditary vertex deletion start formally defining weighted distance hereditary vertex deletion problem weighted distance hereditary vertex deletion wdhvd input undirected graph weight function question find minimum weight subset distance hereditary graph graph distance hereditary graph also called completely separable graph distances vertices every connected induced subgraph graph distance hereditary graphs named first studied hworka however equivalent family graphs earlier studied olaru sachs shown perfect later discovered graphs precisely graphs rankwidth rankwidth graph parameter introduced oum seymour approximate yet another graph parameter called cliquewidth notion cliquewidth defined courcelle olariu measure input graph similar notion treewidth measures input graph one main motivations several problems become tractable family cliques complete graphs assumption algorithmic properties extend graphs however computing cliquewidth corresponding cliquewidth decomposition seems computationally intractable motivated notion rankwidth graph parameter approximates cliquewidth well also algorithmically tractable information cliquewidth rankwidth refer surveys oum algorithms vertex deletion applied subroutines solve many graph problems believe algorithms weighted vertex deletion useful respect particular vertex deletion considered designing efficient approximation kernelization fixed parameter tractable algorithms wpf unweighted counterpart planar deletion along similar lines believe unweighted counterpart useful designing efficient approximation kernelization fixed parameter tractable algorithms weighted vertex deletion characterized finite family forbidden vertex minors recently kim kwon designed log approximation algorithm distance hereditary vertex deletion dhvd result implies dhvd admits log approximation algorithm paper take first step towards obtaining good approximation algorithm designing logo approximation algorithm wdhvd theorem wdhvd admits approximation algorithm note several steps approximation algorithm generalized approximation algorithm thus believe approach yield logo approximation algorithm leave interesting open problem future preliminaries positive integer use shorthand given function subset let denote function restricted domain graphs given graph let denote respectively paper consider undirected graphs let denote number vertices graph clear context open neighborhood simply neighborhood vertex defined closed neighborhood defined degree defined extend definition set vertices follows given neighborhood vertexs subset induced subgraph graph moreover define induced subgraph omit subscripts graph clear context graphs denote graph vertex set edge set independent set set vertices edge pair vertices set independence number denoted defined cardinality largest independent set clique set vertices edge every pair vertices set path subgraph vertices called endpoints path remaining vertices called internal vertices also say path connects cycle subgraph path additional edge graph connected path every pair vertices otherwise disconnected connected graph without cycles tree collection trees forest maximal connected subgraph called connected component given function subset denote moreover say subset balanced separator connected component holds refer reader details standard graph theoretic notations terminologies explicitly defined forest decompositions forest decomposition graph pair forest function satisfies following edge node iii collection nodes subtree call bag sake clarity presentation sometimes use interchangeably refer vertices nodes tree decomposition forest decomposition tree graph denote minimum possible tree decompositions maximum size bag minus one tree decomposition minors given graph edge graph denotes graph obtained contracting edge vertices deleted new vertex added adjacent neighbors previously except graph obtained sequence edge contractions said contraction graph minor contraction subgraph say graph free minor given family graphs say graph free minor well known minor graph planar free clique vertices complete bipartite graph sides bipartition vertices chordal graphs let graph cycle least four vertices say chord cycle chordless contains least four vertices chords graph chordal graph chordless cycle induced subgraph clique forest forest decomposition every bag maximal clique following lemma shows class chordal graphs exactly class graphs clique forest lemma graph chordal graph clique forest moreover clique forest chordal graph constructed polynomial time given subset say intersects chordless cycle observe intersects every chordless cycle chordal graph approximation algorithm wpf section prove theorem assume weight vertex positive else insert solution state result useful algorithm proposition let finite set graphs contains planar graph graph excludes graph minor satisfies let constant returned proposition approximation algorithm wpf comprises two components first component handles special case vertex set input graph partitioned two sets free graph note edges vertices vertices show special instances polynomial time compute size optimum solution set realizing second component recursive algorithm solves general instances problem gradually disintegrate general instance becomes instance special type resolve polynomial time precisely guess sized subgraph find small separator using approximation algorithm together breaks input graph two graphs significantly smaller origin first removes solves two resulting subinstances calling recursively inserts back graph uses solutions obtained recursive calls construct instance special case solved first component constant sized graph free graph first handle special case input graph consists graph size free graph refer algorithm precisely along input graph weight function also given graph vertices free graph disjoint note may contain edges vertices vertices show instances may solved optimally polynomial time start following easy observation observation let graph free graph treewidth lemma let graph treewidth weight function vertices let finite family graphs one compute minimum weight vertex set free time number vertices constant depends proof follows fact finding set expressible formula whose length depends family theorem compute optimal sized set time apply lemma graph family obtain minimum weight set free general graphs proceed handle general instances developing approximation algorithm wpf thus proving correctness theorem exact value constant determined later recursion define call algorithm form instance wpf induced subgraph denote goal recursive call aim prove following lemma returns solution least opt moreover returns subset realizes solution opt recursive call size graph becomes smaller thus prove lemma true current call assume approximation factor bounded opt call size graph strictly smaller log termination polynomial time test whether minor furthermore size check minor free special instance minor free constant sized graph optimally resolve instance polynomial time using algorithm since output optimal sized solution base cases thus ensure base case induction lemma holds recursive call analysis recursive call let denote hypothetical set realizes optimal solution opt current instance let forest decomposition width whose existence guaranteed proposition using standard arguments forests following observation observation exists node balanced separator observation know exists node balanced separator together fact treewidth results following observation observation exist subset size subset weight opt balanced separator gives polynomial time algorithm stated following lemma lemma deterministic randomized algorithm finds size subset weight log opt log opt fixed constant balanced separator size time proof note enumerate every either run randomized log approximation algorithm feige deterministic log approximation algorithm leighton rao find balanced separator fixed constants observation set log opt thus desired output pair one vertex subset size call algorithm lemma obtain pair since balanced separator partitionsthe set connected components two sets holds remark use different algorithms finding balanced separator lemma based whether looking randomized algorithm deterministic algorithm next define two inputs general case wpf let denote optimal solutions respectively observe since holds opt solve subinstances recursively calling algorithm inductive hypothesis thus obtain two sets free graphs proceed defining input special case wpf observe free graphs edges vertices vertices constant size therefore resolve instance calling algorithm thus graph opt obtain set since optimal solution special subinstances opt observe obstruction either completely contained completely contained contains least one vertex observation along free graph implies fact thus sufficient show free graph log opt discussion log opt log log opt recall opt thus log opt log opt opt log opt log opt log log opt sufficient ensure log log done fixing log log use log approximation algorithm feige finding balance separator lemma analysis similar deterministic case obtain randomized approximation algorithm wpf overall conclude ensure weighted chordal vertex deletion general graphs section prove theorem clearly assume weight vertex positive else insert solution roughly speaking approximation algorithm consists two components first component handles special case input graph consists clique chordal graph also assume input graph short chordless cycle component comprised recursive algorithm based method divide conquer algorithm keeps track fractional solution special form carefully manipulated recursive call used analyze approximation ratio particular ensure assign high values assigns vertices clique well vertices cliques divide problem instance two instances find maximal clique chordal graph breaks two simpler chordal graphs clique remains intact recursive call maximal clique also part resulting instances thus ensure simplified problem measure complexity instances examining maximum size independent set graphs since input graph short chordless cycle maximum depth recursion tree bounded log moreover guarantee obtain instances independent incorporate multicut constraints ensuring sufficient budget satisfy ensure multicut constraints associated chordal graphs allows utilize algorithm design section second component recursive algorithm solves general instances problem initially easily handles short chordless cycles gradually disintegrates general instance becomes instance special form solved using first component precisely given problem instance algorithm divides finding maximal clique using exhaustive search relies guarantee short chordless cycle small separator using approximation algorithm together break input graph two graphs significantly smaller origin first removes solves two resulting subinstances calling recursively inserts back graph uses solutions obtained recursive calls construct instance special case solved first component graphs subsection handle special case input graph consists clique chordal graph precisely along input graph weight function also given clique chordal graph disjoint also assume chordless cycle vertices note may contain edges vertices vertices call special case special case objective prove following result lemma special case wcvd admits log approximation algorithm assume else input instance solve let fixed constant determined rest subsection design log approximation algorithm special case wcvd recursion approximation algorithm recursive algorithm call algorithm define call form induced assumption simplifies calculations ahead subgraph induced subgraph argument discussed remark continue use refer size input graph rather current graph arguments execution algorithm progresses keep track two arguments size maximum independent set current graph denoted fractional solution due special structure computation simple observation measure computed polynomial time proof maximum independent set consists one vertex independent set well known computation size maximum independent set chordal graph performed polynomial time thus compute polynomial time next iterate every vertex graph polynomial time since chordal compute graph overall return max necessity tracking stems fact recursive algorithm based method ensure divide current instance two instances obtain two simpler instances need argue aspect instances indeed simplified although aspect size instance since two instances share many common vertices show size maximum independent set fractional solution function every chordless cycle optimal fractional solution minimizes weight holds clearly solution instance wcvd least large weight optimal fractional solution although initially compute optimal fractional solution initialization phase described execution algorithm manipulate solution may longer optimal prior call exception first call ensure satisfies following invariants invariant holds depth current recursive call recursion invariant holds goal depth recursion tree bounded log fixed constant correctness claim proved explain perform recursive call recursive call exception first call aim prove following lemma log recursive call depth returns solution least opt log moreover returns subset realizes solution initialization phase see order prove lemma sufficient prove lemma initialization initially graphs simply set input graphs weight function simply set input weight function moreover compute optimal fractional solution xinit using ellipsoid method recall following claim holds depth first call defined observation solution instance wcvd lower bounded xinit thus prove lemma sufficient return solution least opt log would like proceed calling algorithm recursively purpose first need ensure satisfies invariants end use following notation let log denote set vertices assigns high values moreover given clique let denote function assigns vertex max vertex adjust desired form phase later recursive calls rely two following lemmata lemma define log log proof definition holds log thus log thus safe update ensure obtain solution new instance add solution set realizing lemma given clique function valid fractional solution proof prove valid fractional solution let chordless cycle need show since clique contain two vertices thus since valid fractional solution holds definition fact implies min min log log last inequality relies assumption proof second part claim note next possible call recursively fractional solution context invariant observe indeed holds similarly lemma also clear thus lemma true return solution least opt log desired words prove lemma sufficient next focus proof lemma proof lemma done induction consider recursive call assume solutions returned additional recursive calls performs comply demands associated graphs lemma termination becomes chordal graph return solution set realizes clearly thus satisfy demands lemma fact thus also ensure execution algorithm terminates lemma chordal graph figure subinstances created recursive call proof suppose way contradiction chordal graph contains chordless cycle since induced subgraph assumed exclude chordless cycle vertices note traverse direction insert every second vertex set excluding last vertex case odd obtain independent set thus contradiction thus since ensure recursive calls associated graph whose independence number independence number current graph following observation observation maximum depth recursion tree bounded log fixed constant recursive call since chordal graph admits clique forest lemma particular contains maximal cliques one find set maximal cliques polynomial time standard arguments trees deduce maximal clique remove obtain two necessarily connected graphs clique found polynomial time let observe last inequality holds else lemma execution already terminated proceed replacing sake clarity denote lemmata prove lemma sufficient return solution least opt log log log along log log log log set realizes moreover holds log log note observation setting log log log log log log therefore log log log log particular prove lemma sufficient return solution least opt log log log figure illustration bad cycle next define two subinstances see figure solve subinstances recursive call discussion calls valid satisfy invariants thus obtain two solutions two sets realize solutions inductive hypothesis following observations observation intersects chordless cycle lies entirely either observation given log moreover since also following observation observation say cycle bad chordless cycle belongs entirely neither see figure next show intersect bad cycles bad cycles pair vertices let denote set simple path whose internal vertices belong contain vertex vertex symmetrically let denote set path whose internal vertices belong contain vertex vertex note first examine relation bad cycles pairs vertices lemma bad cycle exist pair vertices path path proof let bad cycle definition bad cycle must contain least one vertex least one vertex since cliques contain two vertices two vertices contains two vertices resp two vertices neighbors moreover since set contains vertices common must contain least one vertex least one vertex overall conclude subpath contains belongs subpath contains belongs light lemma intersect bad cycles examine fractional solution handles pairs vertices lemma pair vertices exists path proof suppose way contradiction lemma incorrect thus exist pair vertices path path since valid fractional solution deduce contain chordless cycle consider shortest subpath vertex vertex shortest subpath vertex vertex since neither contains edge one endpoints belongs endpoint belongs furthermore since vertices common must belong contain internal vertices belong adjacent internal vertices overall since cliques deduce contains chordless cycle see let vertex closest neighbor observe exists neighbors moreover assume without loss generality neighbor apart let vertex closest subpath neighbor subpath together induce chordless cycle else let vertex closest neighbor subpath subpath together induce chordless cycle since induced subgraph reached contradiction given let denote fractional solution assigns vertex value assigned times moreover let observe chordal graphs every pair perform following operation initialize next consider every pair path insert pair remark vertices pair necessarily distinct definition symmetric one following lemma translates lemma algorithm lemma pair vertices one compute polynomial time index path proof let pair vertices trivially obtained required index otherwise proceed follows index perform following procedure pair use dijkstra algorithm weights compute minimum weight path graph given case every pair minimum weight least found desired index moreover lemma since holds least one index maximum weight among minimum weights associated pairs least value least indices arbitrarily decide fix point need rely approximate solutions weighted multicut chordal graphs context employ algorithm given theorem section fractional solution function every pair path pti holds optimal fractional solution minimizes weight let fopt denote weight optimal fractional solution first employing algorithm given lemma next construct two instances second instance weighted multicut first instance sets defined follows initialize every pair insert pair definition symmetric one lemma since holds deduce valid solutions respectively thus calling algorithm given theorem instance obtain solution first instance along set realizes also obtain solution second instance along set realizes fixed constant observation lemma obtained set following observation observation intersects chordless cycle holds recall prove lemma need show log furthermore log log together lemma implies enough show log recall thus observation since log log log observation deduce log log log log log remains show log log log log log equivalent log log recall log observation thus log log sufficient show log however term log lower bounded words sufficient fix log general graphs subsection handle general instances developing approximation algorithm wcvd thus proving correctness theorem exact value constant max determined algorithm based recursion execution often encounter instances form special case wcvd dealt using algorithm section recall constant fixed ensure approximation ratio bounded log recursion define call algorithm form instance wcvd induced subgraph denote ensure initialization phase graph never contains chordless cycles vertices call invariant invariant particular guarantee ensures graph always contains small number maximal cliques lemma number maximal cliques graph chordless cycles four vertices bounded enumerated polynomial time using polynomial delay algorithm goal recursive call aim prove following lemma returns solution least opt moreover returns subset realizes solution log opt recursive call size graph becomes smaller thus prove lemma true current call assume approximation factor bounded opt call size graph strictly smaller log initialization initially set however need ensure invariant satisfied purpose update follows first let denote set chordless cycles vertices clearly computed polynomial time holds construct instance weighted set universe family weight function since chordless cycle must intersected clear optimal solution weighted set instance opt using standard algorithm weighted set suitable fixed constant obtain set intersects cycles whose weight opt set remove vertices invariant satisfied implies recursively call algorithm outputted solution add lemma true obtain solution opt opt opt allows conclude correctness theorem remark execution algorithm update removing vertices thus always safe assume invariant satisfied termination observe due lemma test polynomial time whether consists clique chordal graph examine maximal clique check whether removal obtain chordal graph becomes graph consists chordal graph clique solve instance calling algorithm since log thus ensure base case induction lemma holds recursive call analysis recursive call let denote hypothetical set realizes optimal solution opt current instance moreover let clique forest whose existence guaranteed lemma using standard arguments forests following observation observation exist maximal clique subset weight opt balanced separator following lemma translates observation algorithm lemma algorithm finds maximal clique subset weight log opt fixed constant balanced separator proof examine every maximal clique lemma need consider maximal cliques cliques enumerated polynomial time clique run log approximation algorithm leighton rao find balanced separator fixed constant let denote set minimum weight among sets maximal clique observation log opt thus desired output pair one examined maximal cliques call algorithm lemma obtain pair since balanced separator partitionsthe set connected components two sets holds remark used log approximation algorithm leighton rao lemma find balanced separator instead log approximation algorithm feige algorithm feige randomized next define two inputs general case wcvd let denote optimal solutions respectively observe since holds opt solve subinstances recursively calling algorithm inductive hypothesis thus obtain two sets chordal graphs proceed defining input special case wcvd observe since chordal graphs clique indeed instance special case wcvd solve instance calling algorithm thus obtain chordal graphs log opt set since optimal solution subinstances opt observe since clique edge vertex vertex chordless cycle entirely belongs either observation along fact chordal graphs implies chordal graphs thus sufficient show opt discussion log opt log log opt recall opt thus log opt opt log opt opt log log opt opt sufficient ensure done fixing log overall conclude ensure log weighted multicut chordal graphs section prove theorem let denote recall weighted multicut fractional solution function every pair path holds optimal fractional solution minimizes weight let fopt denote weight optimal fractional solution theorem follows next result whose proof focus section lemma given instance weighted multicut chordal graphs one find polynomial time solution least opt fopt along set realizes preprocessing using ellipsoid method may next assume optimal fractional solution hand say nice exists let denote set vertices assigns high values follows lemma define function otherwise smallest value form fractional solution least fractional solution consider path proof show let sufficient show thus show since fractional solution holds thus since conclude second part claim follows observation preprocessing step also relies following standard accordingly update lemma lemma define proof definition holds thus ensure obtain thus update solution new instance add solution set realizing overall may next focus proof following lemma lemma let instance weighted multicut chordal graphs nice fractional solution one find polynomial time solution least opt along set realizes algorithm since chordal graph first construct polynomial time clique forest lemma without loss generality may assume tree else connected graph handle connected components separately arbitrarily root node arbitrarily choose vertex use dijkstra algorithm compute polynomial time vertex value min set paths define bins bin contains every vertex exists let bin minimizes output consists set contains every vertex approximation factor given let exists start following claim lemma exists proof observe exists exactly one exists denote suppose choose uniformly random consider vertex since probability exists equal probability probability equal expected weight thus exists proof approximation factor follows next claim lemma exists proof let smallest index consider vertex since since nice holds exists thus holds choice therefore implies feasibility need prove pair path consider path suppose way contradiction holds let closest node satisfies since clique tree path node uniquely defined let vbi vertex sake clarity let denote subpath vbi vbi let smallest value satisfies note thus well defined let denote largest index first suppose holds thus obtain statement implies contradiction suppose note minimality get words let des denote set consisting descendants since clique tree thus path realizes contains vertex since exists path realizes deduce exists path realizes contains vertex let denote subpath let denote path starts traverses note therefore since get symmetric analysis house gem domino cycle least vertices figure obstruction set distance hereditary graphs subpath vbi shows exists path overall get exists path since reach contradiction assumption fractional solution vertex deletion section prove theorem start preliminaries preliminaries graph distance hereditary every connected induced subgraph number vertices shortest path number vertices shortest path another characterization distance hereditary graphs graph containing induced isomorphic house gem domino induced cycle vertices refer figure refer house gem domino induced cycle least vertices vertices small biclique graph vertex bipartition note need independent sets biclique clearly assume weight vertex positive else insert solution approximation algorithm wdhvd comprises two components first component handles special case input graph consists biclique distance hereditary also assume input graph small show input restricted special instances wdhvd admits approximation algorithm second component recursive algorithm solves general instances problem initially easily handles small gradually disintegrates general instance becomes instance special form solved polynomial time precisely given problem instance algorithm divides finding maximal biclique using exhaustive search relies guarantee small small separator using approximation algorithm together break input graph two graphs significantly smaller origin distance hereditary graph subsection handle special case input graph consists biclique distance hereditary graph precisely along input graph weight function also given biclique distance hereditary graph disjoint also assume vertices means every chordless cycle strictly vertices note may contain edges vertices vertices call special case biclique distance hereditary special case objective prove following result lemma biclique distance hereditary special case wdhvd admits factor approximation algorithm assume else input instance solve let fixed constant determined later rest subsection design log approximation algorithm biclique distance hereditary special case wdhvd recursion approximation algorithm recursive algorithm call algorithm define call form induced subgraph induced subgraph argument discussed remark continue use refer size input graph rather current graph arguments execution algorithm progresses keep track two arguments number vertices current distance hereditary graph assigned value denote fractional solution observation measure computed polynomial time fractional solution function every chordless cycle least vertices holds optimal fractional solution minimizes weight clearly solution instance wdhvd least large weight optimal fractional solution although initially compute optimal fractional solution initialization phase described execution algorithm manipulate solution may longer optimal prior call exception first call ensure satisfies following invariants invariant holds log invariant holds note invariant used simpler one used section since enough purpose section goal depth recursion tree bounded log depth initial call correctness claim proved explain perform recursive call recursive call aim prove following lemma recursive call depth returns solution least opt log log moreover returns subset realizes solution initialization phase see order prove lemma sufficient prove lemma initialization initially graphs simply set input graphs weight function simply set input weight function moreover compute optimal fractional solution xinit using ellipsoid method recall following claim holds assumption simplifies calculations ahead observation solution instance wdhvd lower bounded xinit moreover holds therefore prove lemma sufficient return solution least opt log log along subset realizes solution part necessity stronger claim given lemma become clear end initialization phase would like proceed calling algorithm recursively purpose first need ensure satisfies invariants end use following notation let log denote set vertices assigns high values note assume moreover given biclique let denote function assigns vertex vertex adjust desired form phase later recursive calls rely following lemmata lemma define log log log log since proof definition holds thus log log induced subgraph also holds log thus safe update ensure obtain solution new instance add solution set realizing lemma let chordless cycle least vertices biclique vertex partitions chordless cycle least vertices intersects vertices furthermore one following three types single vertex edge induced path vertices proof observe chordless cycle vertices may contain two vertices would imply chord chordless cycle already satisfies required conditions output first consider case contains exactly two vertices edge two vertices say either suppose consider vertex let longer two path segments note must length least observe contains different distances depending included induced subgraph easy see contains induced path however small obstructions removed graph chordless cycle least vertices furthermore induced path consider case contains exactly three vertices observe contain two vertices one vertex vice versa satisfy required conditions therefore contains exactly three vertices form induced path length independent set size obtain chordless cycle least vertices consider cases following claim claim let biclique vertex partition induced proof let induced path length either consider path vertex contains size contradiction fact small obstructions next let contain vertices note case vertices either since otherwise would chordless cycle least vertices let assume vertices lie case symmetric let sequence vertices obtained traverse starting arbitrary vertex claim form induced path vertices consists least two connected components without loss generality may assume different components observe possible edges vertices may two edges hence conclude either distance least let assume distance case symmetric paths containing respectively notice graph contains since graph free small obstruction denoted must chordless cycle least vertices furthermore obstruction contain vertices otherwise would chord hence contains strictly fewer vertices moreover recursive application lemma obtain required consequence lemma whenever biclique may safely ignore intersects vertices leads following lemma lemma given biclique function valid fractional solution proof prove valid fractional solution let chordless cycle vertices need show assumption contain vertices thus since valid fractional solution holds definition fact implies last inequality relies assumption proof second part claim note call recursively fractional solution lemma lemma true return solution least opt log log log log desired words prove lemma sufficient next focus proof lemma proof lemma done induction consider recursive call assume solutions returned additional recursive calls performs complies conclusion lemma associated graphs termination becomes distance hereditary graph return solution set realizes clearly thus satisfy demands lemma fact thus also ensure execution algorithm terminates log lemma log distance hereditary graph proof suppose distance hereditary graph contains obstruction since valid fractional solution holds satisfies invariant therefore holds log two observations imply log furthermore least log vertices assigned value log therefore log must distance hereditary graph fact recursive calls made onto graphs distance hereditary subgraph contains number vertices current distance hereditary subgraph observe following observation maximum depth recursion tree bounded log fixed constant recursive call since distance hereditary graph decomposition binary tree bijection leaves furthermore means edge tree deleting obtain partition leaves partition induces cut graph set edges crossing cut forms biclique vertex partition graph standard arguments trees deduce edge defines partition remove biclique edges obtain two necessarily connected graphs note bicliques vertex disjoint proceed replacing fractional solution sake clarity denote let adjust current instance relying lemma satisfies invariant manner adjusted initialization phase particular remove let denote resulting instance graphs observe return solution least opt log log along set realizes analysis argue enough purposes next define two subinstances solve subinstances recursive call thus obtain two solutions sizes two sets realize solutions inductive hypothesis following observations observation intersects chordless cycle least vertices lies entirely either observation given log log moreover since also following observation observation log log coefficient log replaced smaller coefficient log say cycle bad chordless cycle four vertices belongs entirely neither next show intersect bad cycles bad cycles let recall current state graph partitioned biclique distance hereditary graph furthermore biclique vertex bipartition deleting edges gives balanced partition lemma may ignore chordless cycle intersects either two bicliques three vertices allows update fractional feasible solution recursively solve instances remove returned solution consider remaining graph obstructions left graph longer contains small obstructions clear remaining obstruction chordless cycle least vertices bad cycle first examine relation bad cycles pairs vertices lemma bad cycle exists must also bad cycle union two internally vertex disjoint connect pair vertices proof let bad cycle let recall input graph partitioned biclique distance hereditary graph hence furthermore preserved means either present hence bad cycle contradiction hence well finally contains vertices implies well applying lemma obtain bad cycle either single vertex edge induced path length three since apply lemma obtain bad cycle either single vertex edge induced path length three hence pair internally disjoint paths whose endpoints furthermore one paths denoted contained denoted contained lemma lemma implies safe ignore bad cycles satisfy conclusion lemma proceed enumerate helpful properties bad cycles satisfy lemma call path segments bad cycle lemma suppose path segments bad cycle solution respectively induced path endpoints also bad cycle proof observe paths endpoints distance hereditary graph therefore induced path length furthermore vertex adjacent vertex hence also bad cycle lemma allows reduce problem computing solution intersects computing solution instance weighted multicut formally let bad cycle path segments feasible fractional solution assigns total value least vertices assigns every vertex least one assigned total value least suppose assigns total value fractional solution solution weighted multicut problem defined pairs vertices separated whose description given given let denote fractional solution assigns vertex value assigned times pair vertices call important pair bad cycle path segments connects let solution respectively obtained recursively important pair let denote set simple path whose internal vertices belong contain edge one endpoints belongs endpoint belongs symmetrically let denote set path whose internal vertices belong contain edge one endpoints belongs endpoint belongs lemma important pair vertices polynomial time compute index path proof let important pair vertices start arguing index exists assuming contradiction suppose exists recall bad cycle bad cycle paths segments connects implies contradicting feasible solution therefore index always exists index use dijkstra algorithm compute minimum weight weights given case minimum path graph weight least found desired index moreover know least one index minimum weight least minimum weight least induces arbitrarily decide fix say important pair separated index assigned lemma assigns every important pair perform following operation check pair separated initialize pair neighbors add pair set similarly defined point need rely approximate solutions weighted multicut problem given theorem theorem theorem given instance weighted multicut one find polynomial time solution least opt log fopt fixed constant along set realizes fractional solution function every pair path holds optimal fractional solution minimizes weight let fopt denote weight optimal fractional solution employing algorithm given lemma next construct two instances weighted multicut first instance important pair second instance important pair lemma valid solutions respectively thus calling algorithm given theorem instance obtain solution first instance along set realizes log also obtain solution second instance along set realizes log observation lemma obtained set following observation observation intersects chordless cycle holds start showing recall log thus observation since log log log log log observation deduce log log log log log log remains show equivalent log observe log fixed constant indeed initially holds recursive call number vertices assigned value decreases factor previous value execution terminates value drops log thus sufficient choose log term constant lower bounded sufficient fix log note log therefore together lemma implies log proves lemma general graphs section handle general instances developing approximation algorithm wdhvd thus proving correctness theorem recursive algorithm define call algorithm form instance wdhvd induced subgraph denote ensure initialization phase graph never contains vertices call invariant invariant particular guarantee ensures graph always contains small number maximal bicliques stated following lemma lemma lemma let graph vertices vertices contains maximal bicliques enumerated polynomial time goal recursive call aim prove following lemma returns solution least opt moreover returns subset realizes solution constant determined later recursive call size graph becomes smaller thus prove lemma true current call assume approximation factor bounded opt call size graph strictly smaller log initialization given input first need ensure invariant satisfied purpose update follows first let denote set vertices clearly computed polynomial time holds construct instance weighted set universe family setsof size weight function since must intersected therefore optimal solution weighted set instance opt using standard algorithm weighted set suitable fixed constant obtain set intersects whose weight opt set remove vertices obtain graph invariant satisfied call outputted solution add note execution algorithm update removing vertices thus always safe assume invariant satisfied lemma obtain solution weight opt opt opt combined allows conclude correctness theorem termination observe due lemma test polynomial time current graph special kind partitioned biclique distance hereditary graph examine maximal biclique check whether removal obtain distance hereditary graph becomes graph consists biclique distance hereditary graph solve instance calling algorithm observe returns solution value opt also opt recursive call similar case wcvd instead computing balanced separators maximal clique additional vertices find balanced separator comprises biclique additional small number vertices existence separator guaranteed lemma lemma follows graph size contains maximal bicliques enumerated polynomial time use weighted variant lemma lemma proof lemma remains exactly lemma lemma lemma let connected graph vertices containing size weight function polynomial time find balanced vertex separator following conditions satisfied biclique empty set log opt fixed constant opt weight optimum solution wdhvd note used log approximation algorithm leighton rao lemma find balanced separator instead log approximation algorithm feige algorithm feige randomized let also remark biclique bipartition vertices crucially required later arguments next apply lemma obtain pair since balanced separator partitionsthe set connected components two sets holds define two inputs general case wdhvd let denote optimal solutions respectively observe since holds opt solve two recursively calling algorithm inductive hypothesis obtain two sets distance hereditary graphs empty set easy see feasible solution instance let bound total weight subset log opt recall opt log opt opt opt interesting case biclique first remove graph note bound also holds subset vertices observe graph partitioned biclique distance hereditary graph along weight function thus instance biclique distance hereditary graph spacial case wdhvd furthermore note retained fractional feasible solution initial input upperbounds value fractional feasible solution instance apply algorithm outputs solution opt observe obstruction either completely contained completely contained contains least one vertex observation distance hereditary graph implies along fact thus sufficient distance hereditary graph show log opt discussion returns solution value opt constant log opt opt recall opt thus log opt opt opt opt log opt overall conclude ensure log done fixing opt sufficient ensure log conclusion paper designed logo algorithms weighted planar deletion weighted chordal vertex deletion weighted distance hereditary vertex deletion weighted vertex deletion algorithms first ones problems whose approximation factors bounded logo along way also obtained approximation algorithm weighted multicut chordal graphs algorithms based recursive scheme believe scope applicability approach wide would like conclude paper following concrete open problems weighted planar deletion admit approximation algorithm furthermore studying families necessarily contain planar graph another direction research weighted chordal vertex deletion admit approximation algorithm weighted vertex deletion admit logo approximation algorithm graph classes weighted multicut admits approximation references agrawal lokshtanov misra saurabh zehavi feedback vertex set inspired kernel chordal vertex deletion proceedings symposium discrete algorithms soda bafna berman fujito algorithm undirected feedback vertex set problem siam journal discrete 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streaming data hdd gpus sustained peak performance lucas beyer paolo bientinesi aachen institute advanced study computational engineering science financial support deutsche forschungsgemeinschaft german research foundation grant gsc gratefully acknowledged streaming data hdd gpus sustained peak performance lucas beyer paolo bientinesi rwth aachen university aachen institute advanced study computational engineering science germany beyer pauldj abstract context association studies gwas one solve long sequences generalized problems task two limiting factors execution time range days data management sets order terabytes present algorithm obviates issues pipelining computation thanks sophisticated transfer strategy stream data hard disk main memory gpus achieve sustained peak performance respect cpu implementation algorithm shows speedup moreover approach lends multiple gpus attains almost perfect scalability using gpus observe speedups aforementioned implementation widespread biology library keywords gwas generalized computational biology computation multiple gpus data transfer multibuffering streaming big data gwas importance current implementations nutshell goal association study gwas find association genetic variants specific trait disease since tremendous amount genetic variants computation involved gwas takes long time ranging days weeks even months paper look currently fastest algorithm available show possible speed exploiting computational power offered modern graphics accelerators solution gwas boils sequence generalized least squares gls problems involving huge amounts data order terabytes challenge lies sustaining gpu performance avoiding idle time due data transfers hard disk hdd main memory solution cugwas combines three ideas computation pipelined gpu cpu transfers executed asynchronously data streamed hdd main memory gpus means buffering strategy combined mechanisms allow cugwas attain almost perfect scalability respect number gpus compared another widespread gwas library code respectively times faster first section paper introduce reader gwas computations involved therein give overview upon build cugwas whose key techniques explain section time section provide closing remarks section biological introduction gwas segments dna contain information protein synthesis called genes encode traits features physical appearance organism eye hair well internal features organism blood type resistances diseases hereditary information species consists genes dna called genome visualized book containing instructions body following analogy letters book called nucleotides determining order referred sequencing genome even though genome sequence every individual different within one species humans stays single nucleotide dna differs two individuals species difference called polymorphism snp pronounced snip two variants snp referred alleles association studies compare dna two groups individuals individuals case group trait example specific disease individuals control group trait snps individuals groups compared one variant snp frequent case group control group said snp associated trait disease contrast methods linking traits snps inheritance studies genetic association studies gwas consider whole genome importance gwas gathered insightful statistics published gwas since first gwas started appear amount yearly published studies constantly increased reaching studies trend summarized left panel fig showing median year studies along first second quartiles one observe gwa studies started relatively small since amount analyzed snps growing tremendously besides number snps parameter relevant implementation algorithm sample size total number individuals case control group seen fig grown first past four years median sample size seems settled around individuals apparent contrast snp count growth sample size negligible data well discussions biologists confirm need algorithms software compute gwas even snps faster currently possible mathematics gwas gwas expressed variance component model whose solution formulated xit xit millions variables side known sequence equations used compute relations variations phenotype phenotype observed value certain trait individual example studied trait hair color phenotype individual would one blonde brown black red median snp count median sample size fig median first second quartile sample size studies year variations genotype equation responsible one snp meaning number equations corresponds number snps considered study figure captures dimensions objects involved one equation height xit xit fig dimensions single instance matrices vector corresponds number samples thus row corresponds piece individual genetic makeup information one snp entry corresponds individual models relations amongst individuals two individuals family finally important feature matrices partitioned contains fixed covariates age sex thus stays xri single column vector containing genotypes snp considered individuals example body height trait entries would heights individuals even though computed every single snp right part designmatrix xri changes stay amount data computation involved analyze storage size requirements data involved gwas typical values range one entry varies according analysis section consider size study june snp database dbsnp lists known snps humans consider numbers assuming data stored double precision floating point therefore size respectively fit main memory gpu memory output reaches coming close limit current systems main memory big fit gpu memory weighting big fit memory system foreseeable future streamed disk field bioinformatics probabel library frequently used association studies sun fire server intel xeon cpu ghz authors report runtime almost hours problem estimate runtime roughly two days compared current demand million reasonable amount snps population size individuals clearly much smaller present median fig authors state runtime grows linearly fact tripling sample size increased runtime factor coupling fact median sample size individuals computation time bound reach weeks even months prior work algorithm presently fastest available algorithm solving since work builds upon algorithm describe salient features algorithmic features exploits symmetry positive definiteness matrix decomposing cholesky factorization llt since depend decomposition computed preprocessing step reused every instance substituting llt rearranging obtain effectively replacing inversion multiplication solution triangular linear system trsv may may optimal storage type discussion biologists analysis operations necessary order find whether float precise enough case sizes halved consider authors called linear model mmscore option solves exact problem tackle second piece knowledge exploited structure stays constant varies plugging moving constant parts loop leads algorithm takes advantage structure sequence gls shown listing acronyms correspond blas calls listing solution sequence gls stl potrf trsm trsv gemv syrk xri trsv xri sbl dot xri sbr syrk xri dot xri posv llt xri sbli xbri implementation features two implementation features allow attain efficiency first packing multiple vectors xri matrix xrb slow routine solve triangular linear system trsv line transformed fast trsm listing algorithm deal fit main memory limitation overcome turning algorithm one case using doublebuffering technique cpu busy computing block primary buffer next block already loaded secondary buffer asynchronous full algorithm shown listing algorithm attains efficiency listing full algorithm stl aio potrf trsm trsv gemv syrk read blockcount aio read aio wait xrb trsm xrb xri sbl gemm xri sbr syrk xri gemv xri llt sbli xbri posv aio wait aio write aio wait blockcount increasing performance using gpus efficiency algorithm satisfactory computations sped even leveraging multiple gpus help profiler determined confirming intuition trsm line listing bottleneck since cublas provides implementation routines trsm best candidate executed gpus section introduce cugwas algorithm single gpu extend arbitrary number gpus trsm executed gpu algorithm transfer necessary data since size around matrix sent preprocessing step kept gpu throughout entire computation unfortunately whole matrix weights several way per buffer limit modern gpu holds true result trsm needs sent back main memory thus choice send fashion block xrb weighting profiled implementation algorithm displays pattern fig typical applications gpu green cpu gray need wait data transfer orange furthermore cpu idle gpu busy fig profiled timings implementation first objective make use cpu gpu computes trsm regrettably operations following trsm lines listing call dependent result thus executed parallel way break dependency delay one block pipeline fashion relative block delayed executed cpu gpu executes trsm block thanks pipeline broken dependency introduced parallelism completely removing gray part fig streaming data hdd gpu second problem aforementioned implementation time wasted due data transfers modern gpus capable overlapping data transfers computation properly exploited feature allows eliminate overhead thus attain sustained peak performance gpu major obstacle data already main memory quick analysis shows targeting two layers one layer disk main memory transfers another layer main memory gpu transfers two buffers layer sufficient anymore idea two buffers gpu three buffers cpu buffering illustrated two perspectives tasks executed buffers involved former presented fig refer reader thorough description discuss technique terms buffers gpu gpu trsm gpu trsm send cpu hdd recv send cpu comp recv read cpu read write cpu gpu transfer gpu computation data dependencies hdd cpu transfer cpu computation asynchronous dispatch fig algorithm sizes unrelated runtime scenario size blocks xrb used gpu computation equal cpu using multiple gpus case anymore cpu loads one large block distributes portions gpus gpu buffers used way cpu buffers simple algorithm one buffer used computation data transferred buffer cpu level ram three buffers necessary sake simplicity avoid explanation initial final iterations start iteration reference fig assume blocks already reside gpu buffers cpu buffer respectively block buffer contains solution trsm block point algorithm proceeds dispatching read block disk buffer computation trsm gpu buffer receiving result buffer buffer first two operations dispatched executed asynchronously memory system gpu last one executed synchronously results needed immediately following step soon synchronous transfer completes transfer next block cpu buffer gpu buffer dispatched executed cpu previous block buffer cpu see fig trsm trsm gpu gpus hdd data results hdd data results retrieve previous result gpu block data disk computation send next block ram gpu execute cpu trsm gpus hdd data results gpus write results disk hdd data results switch buffers levels next iteration fig algorithm seen buffer perspective soon cpu done computing results written disk fig finally transfers done buffers rotated pointer index rotations copies according fig loop continues using multiple gpus technique achieves sustained peak performance one gpu since boards many gpus becoming common computing explain algorithm adapted take advantage available parallelism idea increase size xrb blocks factor big number available gpus split trsm among gpus long solving trsm gpu takes longer loading large enough block xrb hdd cpu parallelization strategy holds number gpus since systems loading data hdd order magnitude faster computation trsm algorithm scales gpus available listing shows final version conditions first last pair iterations provided parentheses right results order show speedups obtained single gpu compare hybrid algorithm presented listing using one gpu determine scalability cugwas compare runtimes leveraging gpus timings time initialize gpu preprocessing lines listing order seconds measured gpu usually takes fully initialize preprocessing takes seconds depends omission irrelevant computations run hours listing cugwas black bullet placeholder gpus potrf llt cublas send trsm trsv gemv stl syrk gpubs trsm wait blockcount send wait trsm async blockcount aio read gpu ngpus recv gpu gpubs gpubs aio wait gpu ngpus send async gpu gpubs gpubs xri sbl gemm xri sbli sbr syrk xri xbri gemv xri posv aio wait aio write results experiments performed quadro cluster rwth aachen university cluster equipped two nvidia quadro gpus two intel xeon cpus per node gpus powered fermi chips ram theoretical computational power gflops total cluster gpu peak tflops cpus six cores amount total gflops supported ram cost combined gpus estimated combined cpus cost around figure shows runtime along cugwas using one gpu thanks strategy leverage gpu peak performance achieve speedup implementation cublas trsm implementation attains gpu peak performance gflops peak performance cpu system amounts gflops whole computation performed gpu trsm rate largest speedup possible would achieve computation pipelined executed cpu perfect overlap gpu means performance cugwas perfectly line theoretical peak addition figure indicates algorithm linear runtime allows cope arbitrary red vertical line figure marks largest value two blocks fit gpu memory without presented multibuffering technique would possible compute gwas snps cugwas allows solution gwas given amount snps scalability multiple gpus experiment multiple gpus used tesla cluster universitat jaume spain since equipped nvidia tesla contains four fermi chips model quadro system combined gpu compute power tflops ram host cpu intel xeon delivering approximately gflops order evaluate scalability cugwas solved gwas tesla cluster varying number gpus seen fig scalability algorithm respect number gpus almost ideal doubling amount gpus reduces runtime factor runtime respect snp count scalability respect gpu count runtime runtime snp count cugwas number gpus cugwas ideal scalability fig runtime cugwas algorithm using compared using using varying amount gpus conclusion future work presented strategy makes possible sustain peak performance gpu data big gpu memory also main memory addition shown well strategy lends exploit arbitrary number gpus described developers probabel solution problem size described section gwfgls algorithm took hours contrast cugwas solved problem even accounting seconds initialization moore law doubling runtime probabel timings difference dramatic believe contribution cugwas important step towards making gwas practical software code implementing strategy explained paper freely available http http acknowledgements financial support deutsche forschungsgemeinschaft german research association grant gsc gratefully acknowledged authors thank diego providing center computing communication rwth aachen resources enrique granting access tesla system well yurii aulchenko intorducing 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