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coarsening graded local cohomology jan fred rohrer abstract criteria graded local cohomology commute coarsening functors proven example given graded local cohomology commute coarsening let commutative group let commutative ring category grmodg abelian fulfils grothendieck axiom projective generator hence enough projectives injectives hom bifunctor homr maps two onto ggraded homr homgrmodg bifunctor left exact arguments turns exact functor projective injective plugged first second argument respectively thus general nonsense get ext bifunctors bifunctor extir every extr extir right derived cohomological functors homr homr respectively every furthermore projective system grmodg right filtering ordered set composition yields every functor limj extir follows general nonsense limj extir right derived mological functor limj homr particular given graded ideal consider projective system morphisms canonical ones construction yields local cohomology functors hai right derived cohomological functor functor evokes natural question get first apply graded local cohomology forget graduation mathematics subject classification primary secondary key words phrases coarsening graded local cohomology author supported swiss national science foundation fred rohrer resulting module first forget graduation given module apply ungraded local cohomology main reason nontrivial functor forgets graduation necessarily preserve injectivity objects see counterexample case noetherian question extensively discussed positively answered chapter study question case arbitrary present criteria positive answer importance stems mainly toric geometry arbitrary groups degrees finite type ubiquitous see since seems also useful enlightening consider general situation namely replace functor forgets graduation coarsening functors remind reader let epimorphism groups define ring analogously get functor grmodg grmodh called functor functor faithful conservative exact right adjoint thus commutes inductive limits finite projective limits question whether graded local cohomology commutes coarsening whether diagram categories grmodg grmodg grmodh grmodh quasicommutes first approach uses definition graded local cohomology hence comes analogous question graded ext functors known facts analogous question graded hom functors see together techniques allow quickly get first two criteria second approach uses relation local cohomology cohomology result third criterion start defining canonical morphisms functors fact loc cit supposed ungraded ring underlying noetherian special situation equivalent noetherian graded ring every increasing sequence graded ideals stationary coarsening graded local cohomology monomorphism groups homgrmodg homgrmodh inducing monomorphism homr homr varying get monomorphism bifunctors homr homr projective system grmodg right filtering ordered set induces monomorphism functors lim lim homr lim homr one may note monomorphisms necessarily isomorphisms easy example phenomenon obtained considering infinite trivially coincides canonical injection fact follows isomorphism finite exactness yields exact extir extir extir extir implies effaceability extir extir extir effaceable preserves projectivity thus extir extir extir right derived cohomological functors homr homr homr respectively unique morphisms extir extir extir extir called small homgrmodg commutes direct sums fred rohrer define every morphism bifunctors extir extir furthermore isomorphism thus isomorphism every projective system grmodg right filtering ordered set exactness yields exact lim extir lim extir first universal hence unique morphism hif lim extir lim extir let graded ideal consider projective system grmodg canonical morphism hai hai identity morphism related graded local cohomology graded higher ideal transformations dai extir right derived cohomological functor homr plays important role relation local cohomology sheaf cohomology projective schemes case toric schemes case general groups degree setting get morphism hia dai dai show local cohomology commutes coarsening higher ideal transformation proposition morphism hia dai dai isomorphism morphism hai hai coarsening graded local cohomology proof analogously get exact sequence functors idgrmodg morphism dai isomorphism five lemma yields claim proposition ker finite hai hai isomorphism proof suffices show homr homr epimorphism let let homr finite subset finiteness ker implies finite thus homr hypothesis fulfilled finite type projection onto modulo torsion subgroup sense local cohomology care torsion group degrees proposition projective resolution components finite type every hai hai isomorphism proof suffices show isomorphism every every projective resolution components finite type suffices show isomorphism every finite type every situation finite type hence homr homr homr homr fred rohrer isomorphisms follows homr homr isomorphism since coarsening functors conservative implies isomorphism hypothesis fulfilled coherent finite type thus particular noetherian third criterion makes use iti property ring said iti respect graded submodule injective injective property strictly weaker graded noetherianness lies heart many basic properties local cohomology thus natural hypothesis details examples iti refer reader proposition finite set homogeneous generators iti respect hai every hai hai isomorphism proof choose counting write cohomology respect exact obtained taking homology cocomplex since cocomplex obtained taking direct sums modules fractions homogeneous denominators commutes coarsening unique morphism gai hai idg suffices show gai isomorphism iti respect hai every suffices show effaceable shown analogously ungraded case schenzel characterised ideals local cohomology respect isomorphic cohomology respect one sees proof also applies every power projective resolution small components since projective small finite type yields improvement loc cit one makes use iti property noetherianness coarsening graded local cohomology generating system proof together graded analogue schenzel result yields coarsening criterion end note example showing graded local cohomology necessarily commute coarsening example let field let denote algebra additive monoid positive rational numbers furnished canonical let denote canonical basis space consider coarsening respect zero morphism graded ideal idempotent countable type finite type thus small hence exists canonical monomorphism homr homr epimorphism idempotency equals canonical morphism exact sequence idgrmodq together snake lemma shows canonical morphism monomorphism epimorm phism acknowledgement grateful markus brodmann encouraging support writing article also thank referee careful reading suggested improvements references bass algebraic benjamin new york brodmann sharp local cohomology algebraic introduction geometric applications cambridge stud adv math cambridge university press cambridge cox homogeneous coordinate ring toric variety algebraic geom pardo topological aspects graded rings comm algebra pardo militaru homr equal homr category comm algebra grothendieck sur quelques points homologique tohoku math van oystaeyen graded ring theory math library publishing amsterdam quy rohrer bad behaviour injective modules preprint fred rohrer rentschler sur les modules tels que hom commute avec les sommes directes acad paris schenzel proregular sequences local cohomology completion math scand fachbereich mathematik auf der morgenstelle germany address
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rewriting logic semantics plan execution language gilles dowek polytechnique inria lix polytechnique palaiseau cedex france nasa langley research center hampton usa camilo rocha department computer science university illinois goodwin ave urbana usa plan execution interchange language plexil synchronous language developed nasa support autonomous spacecraft operations paper propose rewriting logic semantics plexil maude logical engine rewriting logic semantics formal interpreter language used semantic benchmark implementation plexil executives implementation maude additional benefit making available plexil designers developers formal analysis verification tools provided maude formalization plexil semantics rewriting logic poses interesting challenge due synchronous nature language prioritized rules defining semantics overcome difficulty propose general procedure simulating synchronous set relations rewriting logic sound deterministic relations complete also report two issues design level original plexil semantics identified help executable specification maude introduction synchronous languages introduced program reactive systems systems whose behavior determined continuous reaction environment deployed synchronous languages often used program embedded applications automatic control software family synchronous languages characterized synchronous hypothesis states reactive system arbitrarily fast able react immediately time stimuli external environment one main consequences synchronous hypothesis components running parallel perfectly synchronized arbitrarily interleave implementation synchronous language usually requires simulation synchronous semantics asynchronous computation model simulation must ensure validity synchronous hypothesis target asynchronous model plan execution interchange language plexil synchronous language developed nasa support autonomous spacecraft operations space mission operations require flexible efficient reliable plan execution computer system board spacecraft executes plans called executive component space mission universal executive authors alphabetical order klin eds workshop structural operational semantics sos eptcs dowek rocha work licensed creative commons attribution license rewriting logic semantics plan execution language open source plexil executive developed plexil used midsize applications robotic rovers prototype mars drill demonstrate automation international space station given critical nature spacecraft operations plexil operational semantics formally defined several properties language determinism compositionality mechanically verified prototype verification system pvs formal semantics defined using compositional layer five reduction relations sets nodes nodes building blocks plexil plan represent hierarchical decomposition tasks atomic relation describes execution individual node terms state transitions triggered changes environment micro relation describes synchronous reduction atomic relation respect maximal redexes strategy synchronous application atomic relation maximal set nodes plan remaining three relations quiescence relation macro relation execution relation describe reduction micro relation normalization interaction plan external environment macro relation corresponding respectively operational point view plexil complex general purpose synchronous languages esterel lustre plexil designed specifically flexible reliable command execution autonomy applications paper propose rewriting logic semantics plexil maude complements structural operational semantics written pvs contrast pvs logic specification rewriting logic semantics plexil executable interpreter language interpreter intended semantic benchmark validating implementation plexil executives universal executive testbed designers language study new features possible variants language additionally using graphical interface plexil developers able exploit formal analysis tools provided maude verify properties actual plans rewriting logic logic concurrent change wide range models computation logics faithfully represented rewriting semantics synchronous language plexil poses interesting practical challenges maude implements maximal concurrency rewrite rules interleaving concurrency although rewriting logic allows concurrent synchronous specifications mathematical level maude executes rewrite rules interleaving concurrency overcome situation develop serialization procedure allows simulation synchronous relation via set rewriting systems procedure presented library abstract set relations written pvs procedure sound complete synchronous closure deterministic relation maximal redexes strategy collaborating plexil development team nasa ames using rewriting logic semantics plexil validate intended semantics language wide variety plan examples report two issues plexil original semantics discovered help rewriting logic semantics plexil presented paper first found level atomic relation undesired interleaving semantics introduced computations second found level micro relation spurious infinite loops present computations solutions issues provided authors adopted latest version plexil semantics summarizing contributions presented paper rewriting logic specification plexil semantics http dowek rocha library abstract set relations suitable definition verification synchronous relations serialization procedure simulation synchronous relations rewriting equational version rewriting logic deterministic synchronous relations findings two issues design original plexil semantics corresponding solutions adopted updated version language semantics outline paper background rewriting logic connection logic structural operational semantics summarized section section present library set relations including soundness completeness proof serialization procedure section describes rewriting logic semantics plexil section discuss preliminary results related work concluding remarks presented section rewriting logic structural operational semantics rewriting logic general semantic framework unifies natural way wide range models concurrency language specifications executed maude rewriting logic implementation benefit wide set formal analysis tools available maude ltl model checker rewriting logic specification theory tuple membership equational logic theory signature set kinds family sets operators family disjoint sets sorts set universally quantified horn clauses atoms equations memberships terms sort set structural axioms typically associativity commutativity identity exists matching algorithm modulo producing finite number substitutions set universally quantified conditional rewrite rules form set sorted variables label terms variables among sorts intuitively specifies concurrent system whose states elements initial algebra specified theory whose concurrent transitions specified rules concurrent transitions deduced according set inference rules rewriting logic described detail together precise account general forms rewrite theories models using inference rules rewrite theory proves statement form written meaning state term transition state term finite number steps detailed discussion rewriting logic unified model concurrency inference system found rewrite iff find term rewritten using rule standard way see denoted furthermore arbitrary whether holds rewriting logic semantics plan execution language general undecidable even equations confluent terminating modulo therefore useful rewrite theories satisfy additional executability conditions reduce relation simpler forms rewriting modulo equality modulo matching modulo decidable first condition terminating ground confluent modulo means rewrite theory rewrite sequences terminate infinite sequences form unique class called form modulo exists terminating sequence zero one steps second condition rules coherent relative equations modulo precisely means decompose rewrite theory simpler theories decidable rewrite relations assumptions class always find corresponding rewrite intuitively coherence means always adopt strategy first simplifying term canonical form modulo apply rule modulo achieve effect rewriting modulo conceptual distinction equations rules important consequences giving rewriting logic semantics language rewrite theory rewriting logic abstraction dial captures precisely conceptual distinction one key features structural operational semantics provides formal description language evaluation mechanisms setting level abstraction interleaving behavior evaluations observable corresponds special case dial turned minimum position abstraction dial also turned maximal position special case thus obtaining equational semantics language general make specification abstract want identifying subset rewrite theory satisfies executability conditions aforementioned refer reader presentation relationship structural operational semantics rewriting logic semantics use equations rules capture rewriting logic dynamic behavior language semantics conceptual distinction equations rules also important practical consequences program analysis affords massive state space reduction make formal analyses search model checking enormously efficient explosion analyses could easily become infeasible use specification computation steps described rules rewriting library synchronous relations designing programming language useful able define semantic relation formally prove properties relation execute particular programs however defining semantic relation formally reasoning generally difficult time consuming would major endeavor done scratch language moreover since programming languages tend evolve constantly tools must allow reusing parts former developments support rapid yet correct prototyping fortunately operational semantic relations general built simple relations dowek rocha limited number operations extension reduction normal form parallel extension etc minimum framework include library containing definitions operations formal proofs properties considerably reduce amount work needed define semantic relation particular programming languages formally prove properties defining semantic relation synchronous languages requires defining synchronous extension atomic execution relation operation much less studied formally relation operations extension parallel extension present section first attempt design framework rapid yet correct prototyping semantic relations particular synchronous languages framework allows one define semantic relations execute particular programs formally prove properties using general theorems operations permit build relations relations experimenting framework using various versions plexil language see section definitions properties presented section developed pvs maude engine used executing semantic relations particular programs full development framework including formal semantics plexil available http set relations determinism let binary relation set say redex exists normal form otherwise denote identity relation composition closure respectively addition relations also define normalized reduction relation normalized reduction normal form henceforth assume relation defined sets abstract type define asynchronous extension denoted congruence closure parallel extension denoted parallel closure asynchronous extension exist sets parallel extension exist nonempty pairwise disjoint subsets sets definition synchronous reduction requires definition strategy selects redexes synchronously reduced strategy strategy function mapping elements nonempty pairwise disjoint subsets redexes synchronous extension let strategy exist natural way defining strategies via priorities priority function maps elements natural numbers rewriting logic semantics plan execution language maximal redex let let priority function nonempty subset said maximal redex redex nonempty subsets redex construction set maximal redexes set pairwise disjoint maximal redexes strategy function given priority function maps elements set maximal redexes addition definition relation operators presented library includes formal proofs properties related determinism compositionality abstract set relations paper focus determinism property fundamental specification synchronous relations rewriting logic determinism binary relation defined set said deterministic implies determinism stronger property confluence deterministic relation also confluent confluent relation necessarily deterministic proposition determinism relation deterministic relations contrast even relation deterministic relations always deterministic executing semantic relations executing semantic relation programming language desirable design phase language particular allows designer features experiment different semantic variants language implementing rewrite systems computational way defining binary relations since formalism based set relations consider rewrite systems algebra terms type modulo associativity commutativity identity idempotence basic axioms union sets denote equality terms algebra relation defined rewrite system defined follows relation defined rewrite system exists rewrite rule substitution remark previous definition uses substitution closure rewrite system rather traditional definition based congruence closure example consider rewrite system hand redex synchronous extension relation challenges standard asynchronous interpretation rewrite systems consider previous example asynchronous extension defined tion indeed encodes congruence closure relates however relate corresponds parallel tion particular case remark strategy however order select redexes reduced need additional machinery particular need keep log book redexes dowek rocha need reduced redexes already reduced propose following procedure implement asynchronous rewrite engine maude synchronous extension relation strategy serialization procedure let relation strategy given term compute term follows reduce pair normal form using following rewrite system hai term defined since strategy set redexes set finite procedure always terminates returns term however procedure necessarily deterministic previous example want apply procedure using maximal redexes strategy max assuming terms priority since max reduce pair normal form compute equal check theorem correctness serialization procedure serialization procedure sound procedure returns furthermore deterministic procedure complete procedure returns proof soundness assume procedure returns prove let definition strategy elements pairwise disjoint procedure let none subsets form hence definition completeness case suffices note proposition deterministic deterministic therefore normal form unique procedure returns unique term related relation rewriting logic semantics plexil framework presented section abstract respect elements set basic set relation consider set plexil nodes plexil atomic relation deduce proposition since plexil atomic relation deterministic plexil micro quiescence relations deterministic well therefore use serialization procedure presented section implement sound complete formal interpreter plexil maude section describe detail specification interpreter discuss atomic micro relations since interesting ones validating synchronous semantics plexil precisely present rewrite theory rpxl epxl apxl rpxl specifying rewriting logic semantics plexil atomic micro relations use determinism property plexil atomic relation encode computation rules epxl rewriting logic semantics plan execution language yields confluent equational specification consequently serialization procedure plexil synchronous semantics rewriting logic defined equationally thus avoiding interleaving semantics associated rewrite rules maude course due determinism property language one well turn abstraction dial maximum making rewrite rules rpxl computational rules result faster interpreter example nevertheless interested plexil semantics observable level micro relation therefore rewrite theory rpxl equational theory epxl apxl defines semantics atomic relation specifies serialization procedure synchronous semantics plexil rewrite rules rpxl define semantics micro relation section assume reader familiar syntax maude close standard mathematical notation plexil syntax plexil plan tree nodes representing hierarchical decomposition tasks interior nodes plan provide control structure leaf nodes represent primitive actions purpose node determines type list nodes group nodes provide scope local variables assignment nodes assign values variables also priority serves solve race conditions assignment nodes command nodes represent calls commands empty nodes nothing plexil node gate conditions check conditions former specify node start executing finish executing repeated skipped check conditions specify flags detect node execution fails due violations invariants declared variables nodes lexical scope accessible node descendants siblings ancestors execution status node given status inactive waiting executing etc execution state plan consists external state corresponding set environment variables accessed lookups environment variables internal state set nodes declared variables figure illustrates simple example standard syntax plexil particular example plan tasks represented root node safedrive interior node loop leaf nodes onemeter takepic counter onemeter takepic example nodes type command node counter two different conditions start gate condition constraining execution assignment start node takepic state finished pre check condition number pictures less internal state plan particular moment represented set nodes plan plus value variable pictures external state plan contains external variable wheelstuck external state plan defined functional module sort externalstate represents sets elements sort pair form name value assume sorts name value specifying names values respectively defined previously functional modules name value respectively fmod protecting name protecting value sort pair name value pair sort externalstate subsort pair externalstate mtstate externalstate externalstate externalstate externalstate assoc comm mtstate externalstate externalstate externalstate endfm dowek rocha list safedrive int pictures end lookuponchange wheelstuck true pictures list loop lookuponchange wheelstuck false command onemeter command drive command takepic start finished pictures command takepicture assignment counter start finished pre pictures assignment pictures pictures figure safedrive plexil plan example internal state plan represented help maude conf module supporting object based programming internal state structure set made objects messages called configurations maude objects represent nodes declared variables plan therefore view infrastructure internal state configuration built binary set union operator empty syntax juxtaposition configuration configuration configuration operator declared satisfy structural laws associativity commutativity identity mtconf objects messages singleton set configurations belong subsorts object msg configuration complex configurations generated set union object representing node declared variable given configuration represented term object name identifier sort oid class sort cid names object attribute identifiers corresponding values set pairs object state sort attribute formed repeated application binary union operator also obeys structural laws associativity commutativity identity order pairs object immaterial internal state plan defined functional module extending sort configuration sorts exp qualified assume defined used specify expressions qualified names respectively fmod extending configuration protecting exp protecting qualified subsort qualified oid qualified elements object identifiers ops list command assignment empty cid types nodes sort status ops inactive waiting executing finishing failing finished iterationended variable execstate sort outcome ops none success failure outcome status status attribute status execution outcome outcome attribute outcome execution ops start skip repeat end exp attribute gate conditions ops ops endfm rewriting logic semantics plan execution language pre post inv exp attribute check conditions command exp attribute command command node assignment exp attribute assignment assignment node initval actval exp attribute initial actual values variable node using infrastructure internal state safedrive figure represented configuration figure observe sort qualified provides qualified names means operator qualified qualified qualified use maintain hierarchical structure plans dots end object represent object attributes explicitly defined plan always present node status outcome compilation procedure plexil plans corresponding representation maude discuss paper includes implicit elements node attributes object representation node safedrive list end lookuponchange wheelstuck true pictures loop safedrive list repeat lookuponchange wheelstuck false onemetter loop safedrive command command drive takepic loop safedrive command start finished pictures command takepicture counter loop safedrive assignment pre pictures assignment pictures pictures pictures safedrive memory initval actval figure safedrive rpxl ready define sort state representing execution state plans functional module importing syntax external internal states fmod sort state externalstate configuration state endfm adopt syntax represent execution state plans external internal states respectively plexil semantics plexil execution driven external events set events includes events related lookup conditions changes value external state affects gate condition acknowledgments command initialized reception value returned command etc focus execution semantics plexil specified terms node states transitions node states triggered condition changes atomic relation synchronous closure maximal redexes strategy micro relation plexil atomic relation consists rules indexed type execution status nodes dozen groups group associates priority set rules defines linear order set rules atomic relation defined instance four atomic rules corresponding transitions executing nodes type assignment depicted dowek rocha figure rule updates status outcome node values iterationended success respectively variable value value expression state whenever expressions associated gate condition end check condition post node evaluate true rule ancinv predicate parametric name nodes stating none ancestors changed value associated invariant condition false value represents special value unknown use denote expression evaluates value state abuse notation write denote expression evaluate value ancinv false assignment executing node node status finished outcome failure false assignment executing node node status iterationended outcome failure true true assignment executing node node status iterationended outcome success true true assignment executing node node status iterationended outcome failure figure atomic rules corresponding transitions executing nodes type assignment relation labels two different rules specifies rule applied second rule applied binary relation rules defines order application deriving atomic transitions rule used derive atomic transition premises valid rule higher group applicable case plexil atomic relation binary relation rules linear ordering linearity key determinism plexil see micro relation synchronous closure atomic relation maximal redexes strategy defined icro set nodes variables affected micro relation two different processes say write variable update process higher priority considered assignment nodes associated priority always none otherwise rewriting logic semantics plan execution language order specify plexil semantics maude first define infrastructure serialization procedure functional module fmod inc oid cid attributeset object new syntactic sugar objects updatestatus qualified status msg update status message updateoutcome qualified outcome msg update outcome message updatevariable qualified value msg update variable message ops applyupdates unprime state state application updates unpriming var externalstate var internalstate var oid var cid var att attributeset vars status var state applyupdates status att updatestatus applyupdates status att applyupdates owise unprime att unprime att unprime owise endfm following idea serialization procedure distinguish unprimed primed redexes using syntactic sugar denoting objects maude specification unprimed redexes identified already defined syntax objects form primed redexes identified new syntax objects form use messages elements sort msg denote update actions associated reduction rules atomic relation accumulate messages internal state execution state plans also use internal state spirit log book serialization procedure example configuration updatestatus iterationended updateoutcome success updatevariable corresponds update actions conclusion rule figure functions applyupdates unprimes apply collected updates internal state unprimes primed nodes respectively specification shown status node updated primed nodes become unprimed give equational serialization procedure general setting consider linear ordering rules equational serialization procedure priorities let node node updatesi collection atomic rules horizontal notation defining transition relation nodes type status updatesi set update actions order update actions irrelevant conclusion set premises equational serialization procedure given following set equations maude notation defining function symbol say var var externalstate var oid var status configuration var cid var attr attributeset state state status attr true status attr messages else true else true status attr messages updatesn else status attr owise dowek rocha messages updadatesi represents message configuration associated update actions conclusion rule equational serialization procedure defines fresh function symbol say state state first equation tries apply atomic rules given order first evaluating condition marking affected node condition evaluates true update messages generated second equation removes function symbol possible atomic reductions rules atomic relation defined functional module instantiating equational serialization procedure one twelve groups atomic rules different function symbol one finally micro relation defined rule micro system module materializes rewrite theory rpxl maude mod micro unprime applyupdates endm function symbols defining serialization procedure one twelve groups rules preliminary results used rpxl validate semantics plexil wide variety plan examples report following two issues original plexil semantics discovered help rpxl atomic relation prior version atomic rules assignment nodes state executing presented figure introduced undesired interleaving semantics variable assignments invalidated synchronous nature language spurious plans due lack detail original specification predicates cases transitions list nodes state iterationended would lead spurious infinite loops although formal operational semantics plexil used prove several properties plexil neither one issues previously found matter fact issues compromise proven properties language solutions issues provided authors adopted latest version formal plexil semantics currently using rpxl formal interpreter plexil formal interactive visual environment prototype graphical environment enables execution plans scripted sequence external events validation language intended semantics also developed variant rpxl serialization procedure implemented rewrite rules instead equations rewrite strategies general rpxl outperforms variant two orders magnitude average three orders magnitude extreme cases rewrite theory rpxl approximately lines code lines correspond module rest corresponds syntax infrastructure specifications rewriting logic semantics plan execution language related work conclusion rewriting logic used previously testbed specifying animating semantics synchronous languages alturki meseguer studied rewriting logic semantics language orc includes synchronous reduction relation serbanuta define execution structured data continuations focus former use rewriting logic study mainly behavior orc programs focus latter study relationship existing continuation framework enriching strong features approach based exploiting determinism synchronous relation tackle problem associated interleaving semantics concurrency rewriting logic lucanu studies problem interleaving semantics concurrency rewriting logic synchronous systems perspective determinism property synchronous language esterel formally proven tardieu presented rewriting logic semantics plexil synchronous plan execution language developed nasa support autonomous spacecraft operations rewriting logic specification formal interpreter semantic benchmark validating semantics language relies determinism plexil atomic relation serialization procedure enables specification synchronous relation asynchronous computational model two issues original design plexil found help rewriting logic specification language atomic rule potential violate atomicity atomic relation thus voiding synchronous nature language set rules introducing spurious executions plans proposed solutions issues integrated current semantics language although focused plexil formal framework developed presented general setting abstract set relations particular think framework applied deterministic synchronous languages best knowledge mechanized library abstract set relations suitable definition verification synchronous relations neither soundness completeness proof serialization procedure simulation synchronous relations rewrite systems summarize view work step forward bringing use formal methods closer practice contribution modular mechanized study semantic relations iii yet another interesting contribution rewriting logic semantics project intend continue collaborative work plexil development team goal arriving formal environment validation plexil environment would provide rich formal tool plexil enthusiasts experimentation analysis verification plexil programs could extended towards plexil implementation associated analysis tools part future work also investigate modularity equational serialization procedure prioritized rules acknowledgments work supported national aeronautics space administration langley research center research cooperative agreement awarded national institute aerospace second author resident institute third author partially supported nsf grant iis authors would like thank members nasa automation operation project especially plexil development team led michael dalal nasa ames technical support dowek rocha references alturki meseguer reduction semantics formal analysis orc programs electr notes theor comput sci berry foundations esterel proof language interaction essays honour robin milner mit press cambridge usa bruni meseguer semantic foundations generalized rewrite theories theor comput sci available http caspi pilaud halbwachs plaice lustre declarative language programming popl proceedings acm symposium principles programming languages acm new york usa clavel eker meseguer lincoln talcott maude logical framework springer lncs vol edition dershowitz jouannaud rewrite systems handbook theoretical computer science volume formal models sematics mit press dowek formal analysis framework plexil proceedings workshop planning plan execution systems dowek semantics plexil technical report national institute aerospace hampton estlin simmons tso verna plan execution interchange language plexil technical memorandum nasa lucanu rewrite semantics membrane systems preserves maximal concurrency evolution rule actions electr notes theor comput sci meseguer conditional rewriting logic unified model concurrency theoretical computer science meseguer rosu rewriting logic semantics project theor comput sci available http owre rushby shankar pvs prototype verification system deepak kapur editor international conference automated deduction cade lecture notes artificial intelligence saratoga plotkin structural approach operational semantics log alg prog rocha cadavid graphical environment semantic validation plan execution language ieee international conference space mission challenges information technology serbanuta rosu meseguer rewriting logic approach operational semantics inf comput serbanuta stefanescu rosu defining executing systems structured data david corne pierluigi frisco gheorghe paun grzegorz rozenberg arto salomaa editors workshop membrane computing lecture notes computer science springer tardieu deterministic logical semantics pure esterel acm trans program lang syst verdejo executable structural operational semantics maude log algebr program verna latauro universal executive plexil engine language robust spacecraft control operations proceedings american institute aeronautics astronautics space conference viry equational rules rewriting logic theoretical computer science
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ketch sketching java jinseong xiaokang jeffrey armando university maryland college park mit csail jsjeon xkqiu jfoster asolar jul abstract jsketch synthesis epitomized ketch tool lets developers synthesize software starting partial program also called sketch template paper presents ketch tool brings synthesis java ketch input partial java program may include holes unknown constants expression generators range sets expressions class generators partial classes ketch translates synthesis problem ketch problem translation complex ketch finally ketch synthesizes executable java program interpreting output ketch parser ast encoder sketch solver figure ketch overview general terms design languages overloading overriding terms features available ketch section briefly explains several technical challenges addressed ketch section describes experience ketch ketch available http keywords program synthesis programming example java ketch examples introduction program synthesis attractive programming paradigm aims automate development complex pieces code deriving programs completely scratch given declarative specification challenging simplest algorithms recent work shown problem made tractable starting partial literature sketch scaffold constrains space possible programs synthesizer needs consider approach synthesis proven useful variety domains including program inversion program deobfuscation development concurrent even automated tutoring paper presents ketch tool makes synthesis directly available java programmers ketch built frontend top ketch synthesis system mature synthesis tool based simple imperative language generate code ketch allows java programmers use many ketch synthesis features ability write code unknown constants holes written unknown expressions described generator written addition jsketch provides new synthesis specifically tailored object oriented programs section walks ketch input output along running example illustrated figure ketch compiles java program unknowns partial program ketch language maps result ketch synthesis back java translation ketch challenging ketch object oriented translator must model complex features inheritance method categories subject descriptors automatic programming program synthesis specifying verifying reasoning programs assertions specification techniques decoder overview begin presentation two examples showing ketch key features usage basics input ketch ordinary java program may also contain unknowns synthesized two kinds unknowns holes written represent unknown integers booleans generators written range list expressions example consider following java similar example ketch manual class simplemath static int int return provided template implementation method method returns product hole either parameter notice even simple sketch possible instantiations bits hole one bit choice specify solution would like synthesize provide harness containing assertions method class test harness static void test assert https run ketch sketch harness result valid java source file holes generators replaced appropriate code cat class simplemath static public int int return interface token public int getid generator class automaton private int state static int num state harness static void min num state minimize num state public void transition token assert state state num state int minrepeat state state return public void transitions iterator token transition public boolean accept return state finite automata consider harder problem suppose want synthesize finite automaton given sample accepting rejecting many possible design choices finite automata language opt one efficient ones current automaton state simply integer series conditionals encode transition function figure shows automaton sketch input automaton sequence tokens getid method returning integer line automaton generator keyword fields current state line number states line notice fields initialized holes thus automaton start arbitrary state arbitrary yet minimal number states restricted ketch minimize function line class includes transition function asserts current state inbounds line updates state according current state input token value retrieved line face challenge however know number automaton states tokens bound number transitions solve problem use feature jsketch inherits ketch term minrepeat expands minimum length sequence satisfy harness case body minrepeat line conditional encodes arbitrary guard matches current state input token state updated method returns thus transition method synthesized include however many transitions necessary finally automaton class methods transitions accept first performs multiple transitions based sequence input tokens second one determines whether automaton accepting state notice inequality line means states bound accepting fully general exact state numbering matter synthesizer choose accepting states follow pattern class generators addition basic ketch generators like saw example ketch also supports class generators allow class instantiated differently different superclass contexts figure generator annotation line indicates automaton class class generators analogous function generators introduced ketch figure shows two classes inherit automaton first class dbconnection inner class monitor inherits automaton monitor class defines two tokens open close whose ids respectively outer class monitor instance transitions database opened line database closed line goal synthesize acts inline reference monitor check database never opened closed twice row course many better ways construct finite example expository purposes automaton sketch class dbconnection class monitor extends automaton final static token open new token public int getid return final static token close new token public int getid return public monitor monitor public dbconnection new monitor public boolean iserroneous return public void open public void close transition class cadsr extends automaton public boolean accept string str state init state backup transitions converttoiterator str return accept code using automaton sketch figure finite automata ketch closed opened harnesses testdbconnection figure describe good bad behaviors second class figure cadsr adds new overloaded accept string method converts input string token iterator details omitted brevity transitions according iterator returns whether string accepted goal synthesize automaton recognizes corresponding harness figure constructs cadsr instance makes various assertions behavior notice example relies critically class generators since monitor cadsr must encode different automata output figure shows output produced running jsketch code figures see generator instantiated inherited inherited cadsr automata equivalent would expect languages two things critical achieving result minimizing number states line sufficient harnesses figure experimented cadsr see changing sketch harness affects output first tried running smaller harness fewer examples case synthesized automaton covers examples full language example omit inputs figure resulting automaton accepts inputs whereas going inputs constrains problem enough ketch find full solution second omit state minimization line synthesizer chooses large widely separated indexes states also includes redundant states could merged textbook state minimization algorithm third manually bound number states small manually set num state synthesizer runs half hour fails since possible solution cases last two relatively easy deal since failure obvious first synthesis problem open research challenge however one good feature synthesis find cases handled current implementation simply add cases resynthesize rather manually fix code could quite difficult introduce bugs moreover ensure output program good heuristic avoid overfitting examples class testdbconnection harness static void scenario good dbconnection conn new dbconnection assert assert assert bad opening harness static void scenario dbconnection conn new dbconnection assert bad closing harness static void scenario dbconnection conn new dbconnection assert class testcadsr identifier harness static void examples cadsr new cadsr assert assert assert car assert cdr assert caar assert cadr assert cdar assert cddr implemented ketch series python scripts invokes ketch subroutine ketch comprises roughly lines code excluding parser regression testing code jsketch parses input files using python antlr standard java grammar extended grammar support holes generators minrepeat harness generator modifiers number technical challenges implementation ketch due space limitations discuss major ones figure automata use cases class int state static int num state public void transition token assert state state state state state state state state state state return return return return implementation open open init close public boolean accept return state class dbconnection class monitor extends class int state static int num state public void transition token assert state state state state return state state return state state return state state return public boolean accept return state class cadsr extends figure ketch output partial class hierarchy first issue face ketch language solve problem ketch follows similar approach encodes objects new type object defined struct containing possible fields plus integer identifier class precisely classes program define struct object int class gets unique ketch also assigns every method unique creates various constant arrays record type information method set belongsto class argnum number arguments argtype type ith argument model inheritance hierarchy using twodimensional array subcls subcls true class subclass class encoding names translate class hierarchy ketch also flatten namespace need avoid conflating overridden overloaded method names inner classes thus name inner classes inner outer inner name nested class outer name enclosing class also handle anonymous classes assigning distinct numbers cls support method overriding overloading methods named mtd cls params mtd name method cls name class declared params list parameter types example finite automaton example cadsr inherits method transition second variant class generator hence method named transition token object self object ketch first parameter represents callee method dynamic dispatch simulate dynamic dispatch mechanism java ketch method name suitably encoded introduce function dyn dispatch object self dispatches based class field callee void dyn dispatch object self int cid self class cid return self cid return self return note static self argument omitted java libraries perform synthesis need ketch equivalents java standard libraries used input sketch currently ketch supports following collections apis arraydeque iterator linkedlist list map queue stack treemap charsequence string stringbuilder stringbuffer library classes implemented using combination translation original source using ketch manual coding handle native methods cases efficiency issue note several classes include generics list naturally handled objects uniformly represented object limitations unsupported features java large language ketch currently supports core subset java leave several features java future versions jsketch including packages access control exceptions concurrency additionally ketch assumes input sketch type correct meaning standard parts program type correct holes used either integers booleans expression generators type correct assumption necessary although ketch includes static type checking distinctions different object types lost collapsing object using ketch translate ketch file composed template examples well supportive libraries necessary files input ketch example simplemath example section translated int int simplemath int int return harness void test test assert simplemath int refer reader elsewhere details ketch works solving synthesis problem ketch unparses java files unknowns resolved according ketch synthesis results use partial parsing make output process simpler experience ketch developed ketch part development another tool pasket aims construct framework models mock classes implement key functionality framework way much simpler actual framework code amenable static analysis pasket takes input log interaction real framework test application together description api framework design patterns framework uses pasket uses inputs automatically generate input ketch responsible actually synthesizing models pasket used ketch synthesize models key functionality swing android frameworks largest ketch inputs generated pasket contain classes lines code solve two minutes despite possible choices possible thanks new synthesis algorithm called adaptive concretization available ketch also developed part work acknowledgments research supported part nsf partnership umiacs laboratory telecommunication sciences references demaille levillain sigoure tweast simple effective technique implement ast rewriting using partial parsing sac pages jeon qiu foster synthesizing framework models symbolic execution unpublished manuscript jeon qiu foster adaptive concretization parallel program synthesis computer aided verification cav july jha gulwani seshia tiwari program synthesis icse pages manna waldinger deductive approach program synthesis acm transactions programming languages systems manna waldinger toward automatic program synthesis communications acm mar parr fisher foundation antlr parser generator pldi pages singh gulwani automated feedback generation introductory programming assignments pldi pages program sketching international journal software tools technology transfer sketch programmers manual version tancau bodik seshia saraswat combinatorial sketching finite programs pages jones bodik sketching concurrent data structures pldi pages srivastava gulwani foster program verification program synthesis popl pages srivastava gulwani chaudhuri foster inductive synthesis program inversion pldi pages june tool demonstration walkthrough mentioned introduction ketch available http tool fairly early stage development robust enough used wider research community demonstration generally follow overview section details plan present basics ketch performs program output program given input specification let begin small example cat class simplemath static int int return sketch also scaffold template partial java program part product listed expressions notice sketch possible instantiations addition template important input jsketch examples specify expected behavior template analogous unit tests provide harness containing assertions method states via distinct integers along series conditionals encode state transitions initial sketch interface token public int getid class automaton private int state static int num state harness static void min num state minimize num state public void transition token assert state state num state int minrepeat state state return public void transitions iterator token transition public boolean accept return state key things notice source code transitions taken based token interface initial state arbitrary line number states arbitrary line states dense line cat class test harness static void test assert use assertion check validity current state line transition arbitrary depends arbitrary current state arbitrary line minrepeat replicates body minimum necessary number run ketch sketch harness rewriting syntax sugar specializing generator rewriting exp hole semantics checking building class hierarchy encoding encoding sketch running sketch done replacing holes replacing generators semantics checking decoding synthesis done final result synthesized ketch valid java source file unknowns replaced appropriate code cat class simplemath static int int return times number transitions arbitrary line number accepting states arbitrary line packing states densely could use inequality define inline reference monitor follows monitor public dbconnection new monitor public boolean iserroneous return public void open public void close transition key idea database connection operation associated unique monitor maintains automaton keeps receiving operation ids point client check status connection asking monitor whether accepting state expected need provide harness database connection monitor consider harder problem suppose want synthesize inline reference monitor check basic properties database connection namely connection never opened closed twice row closed opened let use simple efficient implementation representing class dbconnection class monitor extends automaton final static token open new token public int getid return final static token close new token public int getid return public monitor class testdbconnection harness static void scenario good dbconnection conn new dbconnection assert assert assert bad opening harness static void scenario dbconnection conn new dbconnection assert examples illustrate one normal connection closing one abnormal connection twice given harnesses ketch finds solution public boolean accept return state class cadsr extends automaton public boolean accept string str state init state backup transitions converttoiterator str return accept class testcadsr harness static void examples cadsr new cadsr assert assert assert car assert cdr assert caar assert cadr assert cdar assert cddr class testdbconnection bad closing harness static void scenario dbconnection conn new dbconnection assert class automaton int state static int num state public void transition token assert state state state state state state state state state state return return return return open open init close public boolean accept return state resulting automaton exactly one write hand generator class automaton class dbconnection class int state static int num state public void transition token state state return state state return state state return state state return state state return state state return car cdr public boolean accept return state notice synthesizer picked fairly strange numbers states left lot states unused moreover automaton inefficient uses two different paths final states accept car cdr run synthesis using minimize get regular language identifier let consider synthesizing another automaton trying create finite automaton given sample accepting rejecting inputs one benefit java object oriented language code reuse via subclassing could make another class extends automaton assuming want part program subclassing quite work need different automata use case solve problem make automaton class generator instantiated differently different superclass provide less examples remove examples rejected strings synthesizer simply returns automaton make transitions initial state accepting state awkward automaton actually conforms accepted strings one easily figure necessity rejected strings see advantage using minimize let run synthesis without line get adding abnormal twice ketch finds solution note overloaded accept string method need specify sample strings accepted rejected synthesized automaton suppose want synthesize automaton recognizes identifier following harness constructs cadsr instance makes several assertions behavior sort looks okay odd transitions close operation monitor state given operation close fine monitor stay accepting state problematic close connection specified class cadsr extends automaton let use synthesize example automaton class automaton int state static int num state public void transition token assert state state state state return open state state return open class monitor extends automaton class int state static int num state public void transition token assert state state state state state state state state state state return return return return public boolean accept return state result better sense uses dense states encompasses one path final state accept valid strings whether indeed minimum number states test bounded number states class automaton static int num state case synthesizer runs half hour fails possible solution using two states internals ketch time permits show bit ketch translation ketch example translation example looks like cat int int int return cat harness void assert ketch extracts synthesis results looking hole solved ketch cat debug hole info replacing holes debug replaced debug replaced int int debug generator info replacing generators debug traverses original java sketch outputs plugging solved values holes
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inclusion graph subgroups group apr department mathematics gandhigram rural institute deemed university gandhigram tamil nadu india abstract finite group define inclusion graph subgroups denoted graph proper subgroups vertices two distinct vertices adjacent either paper classify finite groups whose inclusion graph subgroups one complete bipartite tree star path cycle disconnected also classify finite abelian groups whose inclusion graph subgroups planar given finite group estimate clique number chromatic number girth inclusion graph subgroups finite abelian group estimate diameter inclusion graph subgroups among results show groups determined inclusion graph subgroups keywords inclusion graph subgroups bipartite graph clique number girth diameter planar graph mathematics subject classification pdevigri rrajmaths introduction properties algebraic structure investigated several ways one ways associating suitable graph algebraic structure analyzing properties associated graph using graph theoretic methods subgroup lattice subgroup graph group well known graphs associated group intersection graph subgroups group another interesting graph associated group group intersection graph subgroups denoted graph proper subgroups vertices two distinct vertices adjacent corresponding subgroups intersects akbari assigned graph ideals ring follows ring unity inclusion ideal graph graph whose vertices left ideals two distinct left ideals adjacent motivated paper define following finite group inclusion graph subgroups denoted graph proper subgroups vertices two distinct vertices adjacent group subgroup lattice denoted height length longest chain partial order greatest least denote order element ordn number sylow group denoted recall basic definitions notations graph theory use standard terminology graphs see let simple graph vertex set edge set said partition disjoint subsets every edge joins vertex vertex graph said complete every vertex partition adjacent vertices remaining partitions denoted graph called star graph called claw graph two distinct vertices adjacent said complete graph whose edge set empty said totally disconnected two graphs isomorphic exists bijection preserving adjacency path connecting two vertices finite sequence distinct vertices except possibly adjacent path cycle length path cycle number edges path cycle length denoted respectively graph said connected two vertices connected path otherwise said disconnected connected graph cycle called tree connected graph diameter denoted diam maximum length shortest path two vertices disconnected define diam girth graph denoted girth length shortest cycle exist otherwise define girth clique set vertices two adjacent clique number cardinality largest clique chromatic number smallest number colors needed color vertices two adjacent vertices gets color graph said planar drawn plane two edges intersect except possibly end vertices define graph contain subgraph isomorphic given graph denotes complement graph two graphs denotes disjoint union paper classify finite groups whose inclusion graph subgroups one complete bipartite tree star path cycle disconnected theorems corollaries also give classification finite abelian groups whose inclusion graph subgroups planar theorem given finite group estimate clique number chromatic number girth inclusion graph subgroups theorem corollary finite abelian group estimate diameter inclusion graph subgroups theorem moreover show groups determined inclusion graph subgroups corollary sequel show interesting connections inclusion graph subgroups group subgroup lattice intersection graph subgroups theorems corollary main results theorem let groups proof group isomorphism map graph isomorphism remark converse theorem true example let easy see theorem let groups proof let lattice isomorphism define map since bijective suppose adjacent either since lattice isomorphism preserves meet order either follows adjacent following similar argument easy see adjacent adjacent thus graph isomorphism hence proof theorem let group subgroup subgraph addition normal subgroup isomorphic graph subgraph proof first result obviously true subgroup form subgroup containing two proper subgroups adjacent either implies either adjacent completes proof theorem let finite group complete prime moreover proof prime chain length chain exists least two subgroups follows complete theorem let finite group height proof let set maximal subgroups let set maximal subgroups subgroups partition vertex set also two vertices partition adjacent moreover minimal number vertex set property since height follows next result immediate consequence theorem definition clique number chromatic number graph corollary let finite group height next two results immediate consequences theorem definition subgroup lattice group theorem let group following equivalent totally disconnected every proper subgroups prime order iii height theorem let group identity element following equivalent bipartite iii height either corollary let finite group distinct primes totally disconnected one zpq bipartite one pqr consider product ordq distinct primes every product one types lemma note suppress subscript theorem let finite group distinct primes either zpqr tree one star either either proof first claim odd easy see suppose odd let cycle two possibilities suppose since adjacent either forms subgraph contradiction must next since adjacent either forms subgraph contradiction must proceed like get odd integer suffixes taken modulo possible suppose since adjacent either forms subgraph contradiction must next since adjacent either forms subgraph proceed like get even integer suffixes taken modulo implies forms subgraph contradiction thus odd claim proved next start prove main theorem since every tree star graph path even cycle bipartite classify finite groups whose inclusion graph subgroups one tree star graph path even cycle enough consider groups order pqr corollary case let theorem neither tree cycle case let use classification groups order theorem path cycle proper subgroups shown figure tree none star path cycle iii cbi contains subgraph hbi hci hai neither tree cycle hai hbi habi proper subgroups follows star neither path cycle hai abi hbi habi proper subgroups follows shown figure tree none star path cycle subgroup lattice isomorphic follows theorem figure tree none star path cycle vii abi contains subgraph hbi hci hai neither tree cycle case let use classification groups order given subcase let abelian easy see path neither star cycle zpq subgroups follows shown figure neither tree cycle subcase let subcase let sylow theorem groups two groups first one ordq hai habp hbp proper subgroups follows shown figure tree none star path cycle second one ordq contains proper subgraph hbi hci hai neither tree cycle iii groups mentioned subcase together group ordq subcase already dealt sylow theorem unique subgroup say order subgroups order say unique subgroup order unique subgroup say order subgroup subgroup subgroup two contained contained every follows shown figure neither tree cycle subcase let group case two groups first group sylow theorem unique subgroup order unique subgroup order sylow order say subgroups order say proper subgroups follows shown figure tree none star path cycle second family groups ordp isomorphism types family one one pair fpx refer groups order contains subgraph hbi hci hai neither tree cycle iii one group order given order unique subgroup order let subgroups order let subgroups order abi hbi hai habi habp hbp hai hbi figure zpq proper subgroups follows neither tree cycle note subcases mutually exclusive isomorphism three groups order already dealt subcase use argument iii subcase unique subgroup order say three subgroups order say four subgroups order say subgroup contained subgroups two remaining subgroups adjacent therefore neither tree cycle case let pqr zpqr let subgroups orders respectively subgroup subgroup subgroup turns tree sylow basis containing sylow let respectively proper subgroups follows contains proper subgraph neither tree cycle proof follows combining cases together next result characterize groups using inclusion graph subgroups corollary let group proof theorem trees figures trees uniquely determines corresponding group hence proof theorem let group connected connected proof since spanning subgraph connected let two adjacent vertices exactly one following holds neither one first two possibilities holds adjacent third condition holds path follows connected rulin shen classified finite groups whose intersection graphs subgroups disconnected consequence theorem following result corollary let finite group disconnected one primes frobenius group whose complement prime order group kernel minimal normal subgroup theorem let finite abelian group distinct primes planar one zpqr zpqrs zpq proof let distinct primes case let abelian subcase theorem planar subcase either chain length least four contains subgraph implies subgroups orders respectively subgraph bipartition implies iii subgroups order let respectively proper subgroups planar corresponding plane embedding shown figure subgraph planar subgroups order let respectively proper subgroups planar corresponding plane embedding shown figure subcase let subgroups orders respectively subgraph bipartition figure subdivision use similar argument show iii subgroups order let respectively proper subgroups planar corresponding plane embedding shown figure subgraph planar subcase let subgroups orders respectively subgraph bipartition implies subgroups order let respectively proper subgroups planar corresponding plane embedding shown figure subcase let subgroups orders respectively subgraph bipartition implies case let corollary planar figure planar iii zpq figure planar subgroup figure unique subgroup order say let subgroups order let subgroup order subgraph bipartition bai subgraph bipartition hap hap hap hbp hap subgroups subdivision subgraph shown figure vii primes least two equal since primes distinct integers one zpi zpi zpi zpi subgroup arguments follows proof follows putting together cases theorem finite abelian group diam proof disconnected diam assume nected prime theorem follows diam assume since connected order every proper subgroup prime let two proper subgroups prime proper subgroup path prime prime path proper subgroup primes path proper subgroups respectively primes path thus shown diam easy see diam diam diam zpq zpq shows diam takes values proof complete next aim prove following result describes girth intersection graph subgroups finite groups classifies finite groups whose inclusion graph subgroups theorem let finite group distinct primes girth one zpqr prove theorem start following result proposition let group order prime girth either proof proof divided two cases case let chain subgroups length least four contains subgraph hence girth theorem least two subgroups order let also since subgroup order let follows contains subgraph bipartition case isomorphic one cbi contains subgraph bipartition hai hbi habi case proof theorem proved contains subgraph also corollary bipartite girth suppose contains subgraph let either without loss generality may assume must implies possible girth abi contains subgraph bipartition hai hbi habi case proof theorem already proved contains subgraph also corollary bipartite girth part case one easily show girth iii theorem corollary figures see girth inclusion graph subgroups remaining groups infinity except contains subgraph proof follows two cases proposition let group order distinct primes girth either zpq proof proof divided several cases case zpq corollary totally disconnected girth case need consider following subcases subcase let abelian girth zpq figure subgraph girth subcase let proceed groups considered subcase proof theorem since subgroup together proper subgroups forms subgraph moreover already proved contains subgraph corollary bipartite girth suppose contains subgraph let either without loss generality may assume possible possible also get contradiction possible hence girth already proved contains subgraph also subgroup together proper subgroups forms subgraph corollary bipartite girth part subcase one easily show girth iii figures see girth inclusion graph subgroups remaining groups infinity contains subgraph case let chain subgroups length least four girth cyclic let subgroups orders respectively contains subgraph bipartition abelian zpq proper subgroup subcase contains subgraph assume let denote sylow shall prove contains subgraph first let sylow theorem group suppose contains also contains enough consider case proposition must contains subgraph bipartition hap hap hap hbi let consider case possible implies impossible since elements order however leaves elements sylow must normal case already considered therefore remaining possibility suppose contains enough consider case proposition must ordq contains subgraph bipartition hbp hbp hai states group order product also subgroup contains subgraph thus result true next let chain subgroups length least subgraph implies subgraph case let since solvable normal subgroup prime index say let let subgroups order respectively forms subgraph girth also contains subgraph bipartition proposition let solvable group whose order least three distinct prime factors girth zpqr distinct primes proof let distinct primes case let case proof theorem already proved contains subgraph also corollary bipartite girth using similar argument subcase proof proposition one easily see girth zpqr theorem subgroup composite order let follows together proper subgroups forms subgraph let since solvable subgroup order let let subgroups order respectively follows girth contains subgraph bipartition case since solvable subgroup order let let subgroups order respectively follows girth contains subgraph bipartition proof follows two cases well known group simple group every simple group minimal simple group show inclusion graph subgroups minimal simple group contains graph subgraph theorem inclusion graph subgroups group also contains recall slm group matrices determinant whose entries lie field elements slm prime suzuki group denoted integer dihedral group order given note subgraph bipartition hbi lemma contains subgraph proposition finite group subgraph girth proof show contains subgraph follows contains subgraph girth mentioned prove enough show inclusion graph subgroups minimal simple groups contains subgraph use thompson classification minimal simple groups given check list groups denote image matrix case group also subgroup together proper subgroups forms subgraph contains subgroup isomorphic namely subgroup matrices form fqp proposition contains subgraph together proper subgroups forms subgraph case note let consider subgroup consisting matrices form subgroup isomorphic group ozp proposition ozp contains subgraph together proper subgroups forms subgraph case note consider two subcases subcase mod subgroup isomorphic lemma together proper subgroups forms subgraph already dealt subcase mod subgroup isomorphic lemma together proper subgroups forms subgraph maximal subgroup since subgroup already dealt case case subgroup isomorphic proposition contains subgraph together proper subgroups forms subgraph proof follows putting together cases combining propositions obtain proof theorem references akbari habibi majidinya manaviyat inclusion ideal graph rings comm algebra akbari heydari maghasedi intersection graph group algebra appl article atlas finite group representations http version bohanon les reid finite groups planar subgroup lattices algebraic combin burnside theory groups finite order dover publications cambridge graph subgroups finite group russian czechoslovak math cole glover groups whose orders products three prime factors amer math harary graph theory philippines rulin shen enshi intersection graphs subgroups finite groups czechoslovak math schmidt subgroup lattices groups expositions vol gruyter schmidt planar subgroup lattices algebra universalis starr turner iii planar groups algebraic combin thompson nonsolvable finite groups whose local subgroups solvable bull amer math soc
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submodularity controllability complex dynamical networks sep tyler summers fabrizio cortesi john lygeros observability long recognized fundamental structural properties dynamical systems recently seen renewed interest context large complex networks dynamical systems basic problem sensor actuator placement choose subset finite set possible placements optimize controllability observability metrics network surprisingly little known structure combinatorial optimization problems paper show several important classes metrics based controllability observability gramians strong structural property allows either efficient global optimization approximation guarantee using simple greedy heuristic maximization particular mapping possible placements several scalar functions associated gramian either modular submodular set function results illustrated randomly generated systems problem power electronic actuator placement model european power grid ntroduction controllability observability recognized fundamental structural properties dynamical systems since seminal work kalman recently seen renewed interest context large complex networks power grids internet transportation networks social networks prominent example recent interest based kalman rank condition idea structural controllability presents graph theoretic maximum matching method efficiently identify minimal set driver nodes timevarying control inputs move system around entire state space render system controllable method applied across range technological social systems leading several interesting surprising conclusions using metric controllability given fraction driver nodes minimal set required complete controllability shown sparse inhomogeneous networks difficult control dense homogeneous networks easier also shown minimum number driver nodes determined mainly degree distribution network however implicit assumption diagonal entries dynamics matrix restrict application result many studies controllability complex networks followed including authors automatic control laboratory eth zurich switzerland email tsummers jlygeros work partially supported eth zurich postdoctoral fellowship program preliminary versions results paper appeared present work unified framework provide modified simplified proofs main results revise elaborate numerical examples one issue approach taken much work exclusive focus structural controllability associated quantitative notion controllability namely required driver nodes rather crude even misleading settings noted example response surprising result genetic regulatory networks seem require many driver nodes apparently contradicts findings biological literature cellular reprogramming suggests rather finding set driver nodes would render network completely controllable appropriate strategy might choose finite set possible actuator sensor placements subset optimizes controllability observability metrics network variety sophisticated metrics controllability observability proposed systems control literature sensor actuator placement selection problems dynamical systems see survey paper one important class metrics involves controllability observability gramians symmetric positive semidefinite matrices whose structure relates energy notions controllability observability use gramians quantitative metrics controllability networks studied important studies controllability networks include variety metrics proposed literature corresponding combinatorial optimization problems sensor actuator placement less wellunderstood solved brute force small problems testing possible placement combinations however problems arise large networks testing combinations quickly becomes infeasible recently context large networks researchers started investigate combinatorial properties sensor actuator placement problems optimizing system dynamics control metrics clark recently considered related different problem leader selection networks consensus dynamics set leader states selected act control inputs system shown minimum mean square error due link noise supermodular function leader set shown graph controllability index related structural controllability framework submodular function leader set discussed properties allow suboptimality guarantees using simple greedy algorithms present paper show one important class metrics controllability observability previously thought lead difficult combinatorial optimization problems fact easily optimized even large networks particular show mapping subsets possible placements linear function associated controllability observability gramian strong structural property modular set function furthermore show rank gramian log determinant gramian negative trace inverse gramian submodular set functions although maximization submodular functions difficult submodularity allows approximation guarantee one uses simple greedy heuristic maximization also describe observations define new dynamic network centrality measures networks whose dynamics described linear models assigning control importance value node network illustrate results randomly generated systems problem power electronic actuator placement model european power grid rest paper organized follows section reviews network model controllability metrics section iii introduces notions modular submodular supermodular set functions shows main results several set functions mapping possible actuator placements various functions controllability gramian either modular submodular section presents illustrative numerical examples finally section gives concluding remarks outlook future research inear odels etwork dynamics section defines linear model network dynamics reviews interprets metrics controllability based controllability gramian material section mostly standard found many texts linear system theory discuss material mostly set notation since controllability observability dual properties focus controllability actuator placement results analogous counterparts interpretations observability sensor placement literature controllability networks common start linear network models spirit focus linear dynamical network models dynamics given states network control inputs used influence network dynamics outputs constant matrices appropriate dimensions assume full row rank example might represent voltages currents frequencies devices power grid chemical species concentrations genetic regulatory network individual opinions propensities product adoption social network matrix typically interpreted set linear state measurements interpret weight matrix whose rows define important directions state space dynamics matrix induces graph network vertices correspond states edges correspond entries whenever aji entries input matrix describe actuator affects nodes network optimizing actuator placement one effectively designs network structure connecting sets inputs sets network nodes optimize metric controllability resulting network controllability definition controllability dynamical system called controllable time interval given states exists input drives system time time kalman rank condition states linear dynamical system controllable full rank since rank generic property matrix value almost values entries assuming nonzero entries independent suggests controllability cast structural property graph defined captured concept structural controllability described lin underpins recent results however informative practically relevant consider quantitative metrics controllability complex networks controllability metrics actuators real systems usually energy limited important class metrics controllability deals amount input energy required reach given state origin symmetric positive semidefinite matrix called controllability gramian time provides quantification controllability eigenvectors associated small eigenvalues large eigenvalues define directions state space less controllable require large input energy reach eigenvectors associated large eigenvalues small eigenvalues define directions state space controllable require small input energy reach stable systems state transition matrix eat comprises decaying exponentials finite positive definite limit controllability gramian always exists given controllability gramian computed solving lyapunov equation awc system linear equations therefore easily solvable specialized algorithms developed compute solution scale large networks focus gramian due ease computation however results also apply gramian disadvantage one must evaluate rather solve may difficult large networks advantage gramian used unstable systems alternative definition interpretation gramian unstable systems used quantify controllability discuss interpretation interest simplicity worth keeping mind though results stated asymptotically stable systems apply generally controllability gramian gives sophisticated quantitative picture controllability still need form scalar metric positive semidefinite matrix want large small requiring small amount input energy move around state space number possible metrics size several discuss trace gramian inversely related average energy interpreted average controllability directions state space also closely related system norm eat dtc cwc smallest eigenvalue gramian metric inversely related amount energy required move system direction state space difficult control rank rank gramian dimension controllable subspace remark main results much discussion generalize straightforwardly linear systems differences gramian depends initial final time rather difference must computed integrating rather solving lyapunov equation following section briefly review combinatorial notion submodularity consider controllability metrics submodularity property provides global optimality approximation guarantees associated actuator selection problems iii ptimal ensor actuator lacement etworks set functions modularity submodularity sensor actuator placement problems formulated set function optimization problems given finite set set function assigns real number subset setting elements represent potential locations placement actuators dynamical system function metric controllable system given set placements consider set function optimization problems form maximize system norm weighted trace controllability gramian trace inverse controllability gramian proportional energy needed average move system around state space note system uncontrollable inverse gramian exist average energy infinite least one direction impossible move system using inputs case one could consider trace pseudoinverse average energy required move system around controllable subspace log det determinant gramian related volume enclosed ellipse defines emin det gamma function means determinant volumetric measure set states reached one unit less input energy since determinant numerically problematic high dimensions logarithm monotone consider optimizing log det note uncontrollable systems ellipsoid volume zero log det case one could consider associated volume controllable subspace problem select subset maximizes finite combinatorial optimization problem one solve brute force simply enumerating possible subsets size evaluating subsets picking best subset however interested cases arising complex networks number possible subsets large number possible subsets grows factorially increases brute force approach quickly becomes infeasible becomes large focus instead structural properties set function make amenable optimization particular submodularity plays similar role combinatorial optimization convexity continuous optimization shares features concave functions occurs often applications though underexplored systems control theory preserved various operations allowing design flexibility supported elegant practically useful mathematical theory efficient methods minimizing approximation guarantees maximizing submodular functions definition submodularity set function called submodular subsets elements holds equivalently subsets holds intuitively submodularity diminishing returns property adding element smaller set gives larger gain adding one larger set made precise following result useful verifying submodularity set functions later definition set function called monotone increasing subsets holds called monotone decreasing subsets holds theorem set function submodular derived set functions monotone decreasing set function called supermodular reversed inequalities hold called modular submodular supermodular subsets modular function following simple equivalent characterization theorem modularity set function modular subset expressed weight function modular set functions analogous linear functions property element subset gives independent contribution function value consequently one see modular optimization problem easily solved simply evaluating set function individual element sorting result choosing top individual elements sorted list obtain best subset size maximization monotone increasing submodular functions greedy heuristic used obtain solution provably close optimal solution greedy algorithm starts empty set computes gain elements adds element highest gain arg max algorithm terminates iterations performance greedy algorithm guaranteed well known bound theorem let optimal value set function optimization problem let sgreedy value associated subset sgreedy obtained applying greedy algorithm submodular monotone increasing sgreedy best polynomial time algorithm achieve assuming note bound greedy algorithm often performs much better bound practice demonstrate modularity submodularity several classes controllability metrics involving functions controllability gramian recall space symmetric matrices partially ordered semidefinite partial order space symmetric positive definite matrices denoted space symmetric positive semidefinite matrices denoted trace gramian suppose stable system dynamics matrix set possible columns used form modify system input matrix problem choose subset maximize metric controllability consider linear function controllability gramian expressed weighted trace controllability gramian given form given possibly empty existing matrix using associated columns defined denote associated controllability gramian bst unique positive semidefinite solution lyapunov equation aws bst simplify notation write following result theorem set function mapping subsets linear function associated controllability gramian cws weighting matrix modular proof prove result directly using theorem easy see controllability gramian associated simply sum controllability gramians associated individual columns bst bts bts since trace linear matrix function weight matrix cws cws cws thus define weight function cws defining theorem implies cws modular set function theorem shows possible actuator placement gives independent contribution trace controllability gramian actuator placement problem using metric easily solved one needs compute metric individually possible actuator placement sort results choose best based interpretations previous section means placing actuators complex network maximize average controllability available move system around state space maximize energy system response unit impulse easily done since result holds weighted trace gives considerable design freedom actuator placement important directions state space weighted actuator placement done based weighted metric trace inverse gramian consider trace inverse controllability gramian assume subsection associated gramian invertible case example network already set actuators provide controllability would like add additional actuators improve controllability discuss deal gramian section theorem let set possible input matrix columns controllability gramian associated set function defined submodular monotone increasing proof use theorem prove result fix arbitrary consider derived set function defined take additivity property gramian noted theorem clear define clearly define note obtain second equality used matrix derivative formula obtain third equality used cyclic property trace since last inequality holds trace product positive negative semidefinite matrix since follows thus monotone decreasing submodular theorem finally seen additivity gramian monotone increasing means adding actuator system decrease controllability log determinant gramian consider log determinant controllability gramian assume associated gramian invertible following result theorem let set possible input matrix columns controllability gramian associated set function defined log det submodular monotone increasing proof proof uses idea namely showing derived set functions log det log det log det log det monotone decreasing arbitrary define log det log det log det log det used matrix derivative formula log det obtain second equality remainder proof follows previous proof set function defined rank submodular monotone increasing corollary related set function defined log det submodular monotone increasing proof log det thus theorem scaled version submodular monotone increasing function therefore also submodular monotone increasing directions state space may equal importance one might want use weight matrix additional design parameter actuator selection problem simple case weight matrix could diagonal matrix assigning relative weight every state following corollary proof follows exactly arguments previous theorems corollary let set possible input matrix columns controllability gramian associated set functions defined log det cws cws rank submodular monotone increasing remark interestingly combinatorial network design problems unrelated controllability strikingly similar mathematical structure specifically shown problems one chooses sets nodes edges optimize certain rigidity properties network relate formation control network localization objectives also submodular set function optimization problems greedy algorithms yield solutions suboptimality guarantees setting one define rigidity gramian quantify desirable rigidity properties results proofs techniques nearly identical present theorems furthermore problems one adds sets edges network optimize coherence resulting network relates robustness consensus process additive noise also similar structure rank gramian controllability metrics log det fail distinguish amongst subsets actuators yield fully controllable system particular functions undefined interpreted return gramian full rank one way handle cases controllability gramian invertible consider rank following result shows also submodular set function theorem let set possible input matrix columns controllability gramian associated proof two linear transformations rank rank rank dim range range form gain functions rank rank rank dim range range easy see monotone decreasing first term second line constant second term decreases dim range increases monotone increasing clear additivity gramian note olshevsky analyzed greedy algorithm maximizing rank controllability matrix though submodularity framework another way handle uncontrollable systems work related continuous metrics defined uncontrollable systems trace pseudoinverse corresponds average energy required move system around controllable subspace log product nonws zero eigenvalues log relates volume subspace reachable one unit input energy smallest eigenvalue gramian seen far trace gramian modular thus supermodular set function actuator subsets trace inverse gramian log determinant gramian rank gramian submodular set functions first two functions also concave matrix functions given connections submodular functions concave functions one might tempted conjecture concave function gramian corresponds submodular function actuator subsets however show counterexample false consider set function corresponds concave matrix function returns smallest eigenvalue gramian show example function violates diminishing gains property set function defined consider system see diminishing returns property holds cases violated others dynamic network centrality measures network centrality measures functions assign relative importance node within graph examples include degree betweenness closeness eigenvector centrality meaning importance relevance various metrics depends highly modeling context example pagerank variant eigenvector centrality turns much better indicator importance vertex degree context networks web pages one core factors leading google domination web search context complex dynamical networks controllability metrics described used define control centrality measures describing importance node terms ability move system around state space control input particular given system dynamics matrix imagine possible place actuator individual node network thus define standard unit basis vector ith entry zeros elsewhere define several control energy centrality measures complex dynamical network follows definition control energy centralities given complex network nodes associated stable linear dynamics matrix define following control energy centrality measures node average controllability centrality cac average control energy centrality cace volumetric control energy centrality log rank ywi controllability gramian satisfies awi eti measures provide relevant quantities centrality purely graph based measures context dynamical systems control give quite different view nodes important greedy algorithm choosing nodes inject control signals interpreted choosing central node iteration given current set controlled nodes interesting topic future work would explore distribution control energy centrality measure random networks networks various application domains pasqualetti also defined different network centrality measure based controllability gramian context networks consensus dynamics chapman mesbahi also defined related network centrality measure quantifies effectiveness agent tracking mean noisy signal possible define many network centrality measures related network dynamics control based leader selection metrics clark computational scaling large networks subsection discuss computational techniques used scale greedy algorithm described section large structured networks first specialized algorithms used exploit sparsity compute low rank solutions lyapunov equations particular computing gramian associated individual actuator cholesky direction implicit algorithm white allows one exploit rankone structure constant term lyapunov equation bbt sparsity structure network dynamics matrix moreover often case large networks gramian associated individual actuator low rank approximately low rank one obtain computational benefits computing low rank approximations gramians also using methods second several techniques used improve greedy algorithm iteration standard version trivially parallelizable gramians associated possible actuator using specialized methods mentioned independently parallel gramian additive actuators effectively one solve lyapunov equation set actuators solving parallel individual actuator summing results iteration marginal gain actuator also computed parallel adding gramian gramian current set added actuators evaluating metric trace logdet individual gramians low rank marginal gains computed efficiently using low rank update formulae formula trace inverse gramian matrix determinant lemma log determinant gramian alternatively also accelerated version greedy algorithm originally due minoux one exploit submodularity set functions significantly reduce number times marginal gains actuators need computed lead orders magnitude speedups practice see umerical xamples section illustrate results randomly generated systems problem power electronic actuator placement model european power grid problem data system dynamics matrix set possible input matrix columns integer number actuators choose set form input matrix maximizes controllability metric sgreedy sopt frequency histogram greedy result log det metric fig histogram displaying shifted log determinant metric possible selections actuators set result achieved greedy optimization displayed red line better selections greedy performance random systems evaluate performance greedy algorithm compare various controllability metrics first consider randomly generated data use matlab rss routine generate stable dynamics matrix random stable eigenvalues use ith unit vector assume one choose states control input injected figure shows result applying greedy algorithm maximize log determinant metric problem small enough evaluate every possible actuator subset result also shown histogram shifted mins support relatively narrow close optimal value hence greedy bound informative case optimum lower values achieved size subsets nevertheless algorithm finds good set sgreedy scoring log det optimum value sopt sopt optimal subset better choose possible choices repeated greedy algorithm randomly generated stable dynamic matrices found cases greedy algorithm returned selection better possible choices words example greedy algorithm provides selection also one performs much better worst case bound also compare example four continuous metrics log det figure shows eigenvalues gramians resulting applying greedy algorithm metric results averages random samples stable dynamics matrices trace metric theorem guarantees greedy algorithm finds globally optimal subset see metric tends focus making largest eigenvalues large expense smaller eigenvalues contrast trace inverse gramian log determinant strike compromise resulting similar eigenvalue distributions largest eigenvalues large ones achieved optimizing trace metric better average smaller eigenvalues although globally optimal subset guaranteed found submodularity metrics guarantees near optimal subset produced greedy algorithm proved theorems hand final metric focuses exclusively smallest eigenvalue example actually slightly worse average trace inverse gramian smallest eigenvalue may result fact smallest eigenvalue metric submodular thus always guaranteed produce near optimal selection however even using fig eigenvalues gramian averaged random samples stable dynamics matrices several continuous metrics resulting applying greedy algorithm select actuators set possible greedy algorithm metric much worse trace inverse gramian log determinant eigenvalues better trace gramian smaller eigenvalues power electronic actuator placement european power grid recently developed power electronic devices high voltage direct current hvdc links flexible alternating current transmission devices facts used improve transient stability properties power grids modulating active reactive power injections damp frequency oscillations prevent rotor angle instability section illustrate results using place power electronic actuators model european power grid emphasize section intended show practical relevant problems could studied using theory preceding sections however many important political economic issues neglected placements evaluated purely controllability perspective consider simplified model european grid derived buses connected ideally one would course want evaluate actuator placement nonlinear model even evaluating controllability metrics extremely difficult computationally even nonlinear systems section intended illustrate theory previous section focus linearized model though actuator placement problems nonlinear networks important topic future work fig network european grid model red dots show buses black lines buses show normal transmission lines best hvdc line placements according controllability gramian trace metric shown bold blue lines controllability gramian trace metric generator constant impedance load consider placement hvdc links modeled ideal current sources instantaneously inject currents terminal buses modeling details see system dynamics consider based swing equations nonlinear model time evolution rotor angles frequencies generator network hvdc link three degrees freedom allow influence frequency dynamics corresponding buses nonlinear model desired operating condition possible hvdc link placement placements evaluated based linearized model using controllability gramian principle one easily work finitehorizon gramian chose use due simplicity computation results qualitatively similar gramian used generator two associated states rotor angle frequency gives state space model always turns stable since hvdc link could placed principle two distinct nodes network possible locations consider problem finding best subset size gives approximately possible combinations far many brute force search figure shows network best placements according controllability gramian trace metric state space directions weighted equally best three relatively long lines connecting northeastsouthwest quadrants network respectively modal analysis dynamics matrix reveals choices correlate well directions associated lightly damped modes rotor angle dynamics next group placements concentrated southeast indicating room improve control authority increasing connectivity sparsely connected region also indicates potential weakness trace metric may cluster actuators get high controllability controllable directions expense controllability directions figure shows sorted values metric top placements giving substantial benefit placements figure shows placement obtained using greedy algorithm log determinant metric using rank metric system becomes controllable compared trace metric see lines general longer connecting buses apart evenly distributed network node part one hvdc line placements also seen align directions corresponding lightly damped modes rotor angle dynamics though different distribution across modes trace although metrics tend hvdc placement location index fig sorted values controllability gramian trace metric vertical axis shows amount particular actuator placement adds trace gramian optimal value sum first amounts top placement give substantial benefit placements produce placements similar qualitative function two sets obtained placements quite different onclusions utlook considered optimal actuator placement problems complex dynamical networks problems general difficult combinatorial optimization problems however shown important class metrics related controllability gramians yield modular submodular set functions allows globally optimal near optimal placements obtained simple greedy algorithm duality results hold corresponding sensor placement model discussed section eferences fig network european grid model best hvdc line placements achieved greedy algorithm maximizing log determinant controllability gramian shown bold blue lines problems using metrics observability gramian knowledge first investigation submodularity context controllability observability dynamical systems also defined several dynamic control energy centrality measures assigns importance value node dynamical network based ability move system around state space control input results illustrated via placement power electronic actuators model european power grid many open problems involving structure combinatorial optimization problems optimal placement sensors actuators complex networks example many quantitative metrics controllability observability associated optimal control filtering design problems may appropriate certain settings nothing known modularity submodularity metrics future work involves exploring case studies power networks biological networks social networks discretized models infinitedimensional systems complicated system models constrained nonlinear hybrid corresponding controllability questions much complicated available tools scale well computationally one could explore efficient methods could used obtain approximate metrics types systems finally important interesting topic future work investigate various graphical properties network structure affect actuator placement results may lead insights controllability complex dynamical networks acknowledgements authors would like thank alex fuchs providing details helpful discussion power system bartels stewart solution matrix equation communications acm boykov jolly interactive graph cuts optimal boundary region segmentation objects images proceedings eighth ieee international conference computer vision volume pages ieee callier desoer linear system theory chapman mesbahi system theoretic aspects influenced consensus single input case ieee transactions automatic control chapman mesbahi consensus network measures adaptive trees ieee transactions automatic control clark bushnell poovendran leader selection performance controllability systems ieee conference decision control pages ieee clark bushnell poovendran supermodular optimization framework leader selection link noise linear systems ieee transactions automatic control clark poovendran submodular optimization framework leader selection linear systems ieee conference decision control pages ieee cortesi summers lygeros submodularity energy related controllability metrics ieee conference decision control los angeles usa cowan chastain vilhena freudenberg bergstrom nodal dynamics degree distributions determine structural controllability complex networks plos one feige threshold approximating set cover journal acm fuchs morari grid stabilization using wide area measurements ieee powertech conference trondheim norway ieee fuchs morari actuator performance evaluation using lmis optimal hvdc placement european control conference zurich switzerland ieee fuchs morari placement hvdc links power grid stabilization transients ieee powertech conference grenoble france ieee hammarling numerical solution stable definite lyapunov equation ima journal numerical analysis hasse anforderung eine durch erneuerbare energien energieversorgung untersuchung des regelverhaltens von kraftwerken und verbundnetzen phd thesis university rostock kailath linear systems volume englewood cliffs kalman general theory control systems ire transactions automatic control kalman contributions theory optimal control boletin sociedad matematica mexicana kempe kleinberg tardos maximizing spread influence social network proceedings ninth acm sigkdd international conference knowledge discovery data mining pages acm krause golovin submodular function maximization tractability practical approaches hard problems krause singh guestrin sensor placements gaussian processes theory efficient algorithms empirical studies journal machine learning research kundur power system stability control white low rank solution lyapunov equations siam journal matrix analysis applications lin structural controllability ieee transactions automatic control liu slotine controllability complex networks nature submodular functions convexity mathematical programming state art pages minoux accelerated greedy algorithms maximizing submodular set functions optimization techniques pages springer schuppert inputs reprogram biological networks nature nemhauser wolsey fisher analysis approximations maximizing submodular set mathematical programming nepusz vicsek controlling edge dynamics complex networks nature physics newman networks introduction oxford university press olshevsky minimal controllability problems ieee transactions control network systems pasqualetti zampieri bullo controllability metrics limitations algorithms complex networks ieee transactions control network systems petersen pedersen matrix cookbook technical university denmark rahmani mesbahi egerstedt controllability systems perspective siam journal control optimization rajapakse groudine mesbahi dynamics control networks probing genomic organization proceedings national academy sciences shames summers rigid network design via submodular set function optimization submitted ieee transactions network science engineering sorrentino bernardo garofalo chen controllability complex networks via pinning physical summers lygeros optimal sensor actuator placement complex dynamical networks ifac world congress cape town south africa pages summers shames lygeros topology design optimal network coherence european control conference sun motter controllability transition nonlocality network control physical review letters tang gao zou kurths identifying controlling nodes neuronal networks different scales plos one van wal jager review methods selection automatica wang lai grebogi optimizing controllability complex networks minimum structural perturbations physical review yan ren lai lai controlling complex networks much energy needed physical review letters yang huijun kurths evolutionary pinning control application uav coordination industrial informatics ieee transactions zhou salomon balanced realization model reduction unstable systems international journal robust nonlinear control
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jun irreducible characters even degree normal sylow nguyen ngoc hung pham huu tiep abstract classical theorem character degrees groups asserts degree every complex irreducible character group coprime given prime normal sylow propose new direction generalize theorem introducing invariant concerning character degrees show average degree linear irreducible characters less normal sylow well corresponding analogues characters strongly real characters results improve several earlier results concerning theorem introduction celebrated theorem one deep fundamental results relation character degrees local structure finite groups asserts prime divide degree every complex irreducible character finite group normal abelian sylow past decades several variations refinements result considering brauer characters nonvanishing elements fields character values indicator one primary direction weaken condition irreducible characters degree coprime assume instead subset characters specified field values property see characters characters paper introduce invariant concerning character degrees propose generalize theorem completely new direction date june mathematics subject classification primary key words phrases groups character degrees normal subgroups sylow subgroups real characters strongly real characters hung partially supported nsa young investigator grant faculty scholarship award buchtel college arts sciences university akron tiep partially supported nsf grant simons foundation fellowship nguyen ngoc hung pham huu tiep given finite group let irr denote set complex irreducible characters write irrp irr acdp acdp average degree linear characters irreducible characters degree divisible theorem reformulated following way acdp normal abelian sylow first result significantly improves prime theorem let finite group normal sylow theorem basically says even group irreducible characters even degree still normal sylow long number linear characters group large enough course implies theorem required group irreducible characters even degree one key steps proof theorem establish solvability groups consideration fact theorem let finite group solvable remark bounds theorems optimal shown groups furthermore get close wish consider extraspecial therefore one get commutativity sylow theorem theorem fact also improve main results restricting attention characters even strongly real characters character irr called strongly real indicator equivalently afforded real representation let irrp irrp acdp irrp irrp strongly real acdp characters even degree normal sylow theorem let finite group solvable normal sylow theorem immediately implies theorem theorem furthermore since character degree automatically strongly real following consequence corollary let finite group solvable normal sylow example shows bounds theorem corollary optimal prove theorems use classification finite simple groups show every nonabelian finite simple group possesses irreducible character even large enough degree extendible stabilizer aut theorem together proposition result allows bound number strongly real irreducible characters small degrees finite group nonabelian minimal normal subgroup control invariant group see section proposition hope techniques developed useful future study problems involving average degree certain set characters invariants concerning character degrees like largest character degree character degree ratio one obvious question one may ask analogue theorem odd primes although ideas proof prime carry smoothly odd primes believe following true conjecture let prime finite group acdp normal sylow bound conjecture perhaps optimal primes cyclic group order act nontrivially abelian group order bound clearly lower instance mersenne prime example act trivially abelian group order think best possible bound theorems respectively proved sections extendibility characters even degree throughout paper finite group positive integer write denote number irreducible complex characters degree denote number strongly real irreducible complex characters nguyen ngoc hung pham huu tiep degree furthermore normal subgroup denotes number irreducible characters degree whose kernels contain similarly character normal subgroup write denote stabilizer inertia subgroup notation standard follows defined needed need following result whose proof relies classification finite simple groups theorem every nonabelian finite simple group irreducible character even degree extendible strongly real character iaut furthermore chosen proof cases one sporadic finite simple groups checked directly using note cases always find extension iaut follows may therefore assume isomorphic listed groups particular follows main results degree nontrivial complex irreducible character least certainly one find many different choices desired character follows mind applications try construct way extension iaut possible assume consider irreducible characters irr labeled partitions degree respectively since given partitions restrict irreducibly furthermore exactly one even degree aut done case choosing even degree next consider case finite simple group lie type characteristic shown steinberg character degree extends character rational representation aut whence done iii may assume finite simple group lie type defined finite field odd characteristic consider cases psun shown proof theorem aut rank permutation character characters restrict nontrivially irreducibly furthermore exactly one even degree afforded rational representations hence done choosing even degree argument applied rank permutation character aut see proof theorem handles cases characters even degree normal sylow suppose psln shown proof proposition aut permutation representation whose character contains irreducible character even degree mod mod multiplicity one restricts irreducibly follows also afforded rational representation choose remaining cases choice yields necessarily rational still admits strongly real extension iaut first unique character irr degree extends aut lemma next proof proposition yields strongly real character irr even degree proof proposition choose degree suppose simple group type view derived subgroup finite group adjoint type respectively shown proof proposition types case iib proof proposition proof proposition type contains strongly real character even degree restricts irreducible character irr done cases well finite group nonabelian minimal normal subgroup using theorem produce irreducible character even degree extendible strongly real character stabilizer theorem let finite group nonabelian minimal normal subgroup exists irr even degree extendible strongly real character proof since nonabelian minimal normal subgroup direct product copies nonabelian simple group replacing necessary may assume aut aut let irreducible character found theorem let orbit action aut irr consider character irr orbit action aut irr nguyen ngoc hung pham huu tiep clearly invariant iaut hand iaut iaut therefore deduce iaut iaut assume extends strongly real character iaut say afforded iaut acts naturally acts character follows extendible strongly real character iaut afforded since general observe long done theorem used bound number irreducible characters degrees finite groups nonabelian minimal normal subgroup shown next proposition compared proposition proposition let finite group nonabelian minimal normal subgroup assume irr extendible let following hold extends strongly real character iii moreover invariant proof simplicity write first since therefore wish show normal subgroup contained kernel every linear character recall irr extendible let irr extension using gallagher theorem clifford theorem see corollary theorem see linear character produces irreducible character degree character turns produces irreducible character degree maps injective follows therefore desired group let denote subgroup generated note whence characters even degree normal sylow furthermore irr strongly real linear character indicator particular strongly real furthermore strongly real character subgroup induced character argue complete proof iii first claim let irr take irreducible constituent frobenius reciprocity implies turn irreducible constituent irreducible constituents degree deduce irreducible characters degree arise constituents hand irreducible constituents degree deduce irreducible characters degree arise constituents claim proved since already proved prove proposition iii suffices show claim words contained kernel every irreducible character degree let irr since irreducible character degree one linear character trivial one follows ker claimed recall irr extension using gallagher theorem clifford theorem obtain irreducible character irr degree produces character irr degree follows thus desired invariant yielding immediately solvability theorem section use results section prove theorem next proposition handles important case theorem proposition let finite group nonabelian minimal normal subgroup nguyen ngoc hung pham huu tiep proof let nonabelian minimal normal subgroup first assume theorem guarantees irr even degree extendible inertia subgroup using proposition follows since hence check follows thus therefore knk desired remains consider irreducible character degree extendible aut see hence extendible well follows proposition thus yields proof completed ready prove theorem restate theorem let finite group solvable proof assume contrary theorem false let minimal counterexample particular nonsolvable let minimal nonsolvable clearly perfect contained last term derived series choose minimal normal subgroup choose denotes largest normal solvable subgroup characters even degree normal sylow view proposition assume abelian follows quotient nonsolvable since nonsolvable minimality must since every positive integer follows thus exists irr ker shown proof theorem central product amalgamated subgroup order ker ker since central product central amalgamated subgroup bijection irr irr irr bijection must since three possibilities namely irr extension unique linear character using gallagher theorem bijection irr irr particular since employing arguments proof theorem evaluate estimate terms follows iii putting things together follows contradiction normal sylow theorem next lemma crucial proof theorem lemma let abelian group assume nonprincipal irreducible character invariant nguyen ngoc hung pham huu tiep orbit even size action set irreducible characters proof let set representatives action irr let inertia subgroup since nonprincipal irreducible character invariant observe every proper subgroup assume contrary orbit even size action irr exists index even set even even since splits clear every also splits follows extends linear character say linear gallagher theorem implies mapping bijection irr set irreducible characters lying using clifford theorem obtain bijection irr set irreducible characters lying note hence even either even even even even even even even hand even even therefore deduce even even even since every follows even characters even degree normal sylow hence even observe therefore follows inequality thus impossible since proof complete prove main theorem restated theorem let finite group normal sylow proof argue induction order therefore theorem solvable abelian nothing prove assume nonabelian choose minimal normal subgroup solvable abelian since irreducible character kernel containing degree must least therefore deduce follows induction hypothesis normal sylow say normal sylow done assume elementary abelian group odd order schurzassenhaus theorem implies sylow well frattini argument qng nng follows done assume implies abelian normal subgroup deduce minimality conclude would done assume noncentral thus minimality follows nonprincipal irreducible character invariant situation lemma therefore conclude orbit even size action set irreducible characters particular orbit even size action set irreducible characters means acts trivially since odd order deduce completes proof theorem nguyen ngoc hung pham huu tiep strongly real characters theorem section prove theorem first prove theorem theorem let finite group solvable proof since nothing prove abelian assume nontrivial let minimal normal subgroup strongly real character irr kernel containing degree must least average therefore deduce working induction assume nonabelian theorem exists irr even degree extends strongly real character note assumed theorem one choose irreducible character degree character extendible strongly real character apply proposition since follows therefore completes proof knk prove theorem begin two known observations strongly real characters lemma let normal subgroup finite group odd order every strongly real character irr lies unique strongly real irreducible character proof lemma lemma let assume irr exists strongly real character irr proof lemma characters even degree normal sylow also need following observation lemma let acting abelian group odd order unique minimal normal subgroup frattini subgroup proof suppose product copies cyclic groups prime since minimal normal subgroup action induces faithful irreducible representation field extend representation representation xfq algebraically closure since even odd maschke theorem xfq completely reducible moreover faithful since faithful using fongswan theorem theorem lifts irreducible brauer characters solvable groups conclude complex faithful character say degree apply theorem deduce number generators minimal generating set say number linear constituents particular follows easy check every positive integer lemma follows lemmas allow control following special situation proposition let split extension elementary abelian group odd order nontrivial assume unique minimal normal subgroup proof well known groups odd order real irreducible character therefore real irreducible character follows every strongly real linear character inside kernel particular together lemma implies hence find disjoint irr size hypotheses know acts irreducibly faithfully therefore irr well since two actions permutationally isomorphic therefore nontrivial character irr inverted central involution nguyen ngoc hung pham huu tiep words nontrivial irr lemma guarantees strongly real character irr lying since splits extendible hence even number since applying argument get psa strongly real character irr even degree particular follows average degree characters strongly real linear characters least since strongly real irreducible character degree least conclude proof theorem completed theorem let finite group normal sylow proof assume statement false let minimal counterexample particular let minimal normal subgroup follows degree every strongly real irreducible character whose kernel contain least hence minimality deduce normal sylow thus sylow since theorem guarantees solvable elementary abelian even order would done may assume elementary abelian group odd order moreover normal observe strongly real linear character restricts strongly real linear character moreover lemma strongly real linear character lies unique strongly real linear character follows lemma imply thus minimality hence contradiction may assume since normal see unique minimal normal subgroup hypotheses proposition therefore conclude contradiction completes proof characters even degree normal sylow acknowledgement grateful referee helpful comments suggestions significantly improved exposition paper references conway curtis norton parker wilson atlas finite groups clarendon press oxford cossey nguyen controlling composition factors group character degree ratio algebra cossey halasi nguyen conjecture gluck math navarro pham huu tiep primes dividing degrees real characters math sanus spiga orders zeros irreducible characters algebra feit extending steinberg characters linear algebraic groups representations contemp math gluck largest irreducible character degree group canad math halasi hannusch nguyen largest character degrees symmetric alternating groups proc amer math soc hung characters thompson character degree theorem rev mat appear http hung lewis fry finite groups irreducible character large degree manuscripta math isaacs character theory finite groups ams chelsea publishing providence rhode island isaacs number generators linear canad math isaacs navarro groups whose real irreducible characters degree coprime algebra isaacs loukaki average degree irreducible character group israel math studies group characters nagoya math lewis nguyen character degree ratio composition factors group monatsh math manz modular version ito theorem character degrees groups odd order nagoya math manz wolf brauer characters groups algebra marinelli pham huu tiep zeros real irreducible characters groups algebra number theory michler brauer conjectures simple groups lecture notes math springer berlin michler simple group lie type defects algebra nguyen ngoc hung pham huu tiep nguyen average character degree groups bull lond math soc navarro characters blocks finite groups london mathematical society lecture note series cambridge university press cambridge navarro pham huu tiep degrees rational characters groups adv math navarro pham huu tiep degrees characters bull lond math soc rasala minimal degrees characters algebra seitz zalesskii minimal degrees projective representations chevalley groups algebra pham huu tiep real ordinary characters real brauer characters trans amer math soc department mathematics university akron akron ohio usa address hungnguyen department mathematics university arizona tucson arizona usa address tiep
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jul note itoh rings ideal youngsu kim louis ratliff david rush february abstract let regular proper ideal noetherian ring let integer let indeterminate let itoh rings rings varies height one associated prime ideals unique minimal prime ideal contained show among things radical ideal common multiple rees integers integer correspondence itoh valuation rings rees valuation rings namely quotient field integral closure integer corresponding valuation rings finite integral extension domain satisfy fundamental equality splitting also greatest common divisor integer multiple exists unit finite free integral extension domain rees integers equal simple free integral extension domain introduction rings paper commutative identity element terminology mainly nagata thus basis ideal generating set ideal term altitude refers often called dimension krull dimension pair local rings dominates case write shiroh itoh proved following interesting useful theorem lines terminology defined section theorem let regular proper ideal noetherian ring let rees ring respect let rees valuation rings let uwj rees integers let arbitrary common multiple also let let let radical ideal rees integers equal one correspondence rees valuation rings rees valuation rings namely quotient field integral closure let corresponding rees valuation rings ree spectively ramification index relative equal actually part theorem itoh specifically stated radical ideal least common multiple proof essentially shows hold goals prove several nice applications radicality ideal find additional properties rees valuation rings ideal however turns rees valuation rings ideals like additional nice properties goal present paper derive properties facilitate discussing valuation rings make following definition definition arbitrary integer itoh rings rees valuation rings definitions known results section recall needed definitions mention needed known results concerning definition let ideal ring denotes integral closure total quotient ring denotes integral closure ideal root equation form ideal integrally closed case rees ring resect graded subring indeterminate assume noetherian regular proper ideal contains regular element let regular elements generate let let height one associated prime ideals see let unique see remark minimal prime ideal contained possibly let dvr set set rees valuation rings rees valuation rings well defined depend basis rees valuation ring see rees integer respect positive integer ivi called rees integers dvrs localization integral extension domain ramification index relative positive integer remark let regular proper ideal noetherian ring concerning shown definition propositions regular nonunit integral closure noetherian ring finite primary decomposition associated prime ideal height one contains exactly one associated prime ideal dvr shown proposition rees valuation rings denotes set positive integers readily follows definition set rees valuation rings disjoint union sets runs minimal prime ideals also rees valuation ring rees integer respect rees integer respect follows definition proved example regular proper principal ideal rees valuation rings rings varies height one associated prime ideals unique minimal prime ideal contained readily checked correspondence associated prime ideals associated prime ideals namely follows definitions ring rees valuation rings rees integers also follows positive integers rees valuation rings well known readily proved much proof noetherian integral domain integral extension domain rees valuation rings extensions rees valuation rings quotient field following theorem special case theorems terminology fundamental inequality due endler theorem fundamental inequality let dvr quotient field let finite algebraic extension field let let valuation rings extensions integral closure exactly maximal ideals let equality holds integral closure finite terminology theorem equality holds said satisfy fundamental equality splitting notation integral domain denotes quotient field therefore finite algebraic extension domain denotes dimension quotient field quotient field finite free integral extension domain clear rank equal often write place next three propositions known know specific references sketch proofs proposition let dvr let let integer let let dvr simple free integral extension domain maximal ideal therefore satisfy fundamental equality splitting see terminology proof maximal ideal since field principal ideal since generated also integral unique maximal ideal every maximal ideal contains follows maximal ideal hence dvr therefore follows therefore since follows fundamental inequality hence simple free integral extension domain thus last statement clear proposition let maximal ideal noetherian ring let monic polynomial image irreducible irreducible maximal ideal simple free integral extension ring rank equal deg proof considering maps root irreducible hypothesis polynomial follows irreducible maximal ideal maximal ideal simple free integral extension ring rank equal deg proposition let regular proper ideal noetherian ring let regular elements basis let rees ring respect exists correspondence rees valuation rings rees valuation rings namely associated prime ideal minimal prime ideal contained tbi tbi tbi note tbi tbi corresponds isomorphism see proof since transcendental correspondence minimal prime ideals minimal prime ideals namely thus follows remark suffices prove proposition case noetherian integral domain therefore assume noetherian domain fix let let also since transcendental exists correspondence height one associated prime ideals height one associated prime ideals namely therefore exists correspondence dvrs dvrs corresponding follows rees valuation ring form ring rees valuation ring finally let rees valuation ring say height one associated prime ideal see remark complete proof correspondence suffices show exists rees valuation ring lemma assumption analytically unramified used proof lemma therefore height one associated prime ideal let height one associated prime ideal follows second preceding paragraph properties itoh rings section show itoh rings several nice properties need following proposition essentially corollary proposition proposition let regular proper ideal noetherian ring let regular elements generate let rees ring respect let corresponding proposition rees valuation rings respectively say height one associated prime ideal minimal prime ideal contained let integer let let let root fixed algebraic closure quotient field dvr simple free integral extension domain maximal ideal therefore satisfy fundamental equality splitting see terminology proof since transcendental extended ideal transcendental follows image irreducible hence therefore follows proposition irreducible ufd principal maximal ideal integral unique maximal ideal every maximal ideal contains maximal ideal hence dvr simple free integral extension domain maximal ideal therefore last statement clear first theorem theorem expanded version itoh theorem see theorem note proved theorem let regular proper ideal noetherian ring let regular elements basis let rees valuation rings let rees integers let arbitrary common multiple let also let rees ring respect let let let radical ideal correspondence itoh rings rees valuation rings namely given quotient field integral closure let corresponding valuation rings let rees valuation ring corresponds proposition assume exists unit itoh valuation ring simple free integral extension domain maximal ideal ramification index relative equal one see definition also therefore satisfy fundamental equality splitting let exists nonunit simple free integral extension domain ramification index relative equal also satisfy fundamental equality splitting finite free integral extension domain also ramification index relative equal satisfy fundamental equality splitting assume let simple free integral extension domain maximal ideal therefore satisfy fundamental equality splitting radical ideal proof first prove fix itoh ring definitions exists height one associated prime ideal minimal prime ideal since total quotient ring follows minimal prime ideal also minimal prime ideal let quotient field quotient field also definition rees valuation ring also rees valuation ring remark also rees valuation ring remark therefore rees valuation ring ideal remark also remark hence extension let rees valuation ring corresponds proposition assume integral closure total quotient ring height one associated prime ideal unique minimal prime ideal contained let maximal ideals respectively proposition transcendental let hypothesis therefore qej exist units since see last part proposition follows let fixed algebraic closure let follows proposition dvr simple free integral extension domain maximal ideal satisfy fundamental equality splitting continuing proof next show first show hence quotient field moreover integral unit integrally closed unit therefore maximal ideal thus complete proof remains show itoh ring tej tej height one associated prime ideal tej say height one associated prime ideal tej since therefore tej tej itoh ring tej dvrs quotient field tej itoh ring thus holds let let since let maximal ideal follows proposition dvr simple free integral extension domain maximal ideal satisfy fundamental equality splitting since quotient field thus dvrs follows therefore holds follows finite free integral extension domain also since ramification index equal one ramification index equal follows ramification index equal thus follows satisfy fundamental equality splitting hence holds moreover since finite free integral extension domain since integrally closed follows integral closure quotient field therefore shown itoh ring height one associated prime ideal minimal prime ideal contained minimal prime ideals respectively quotient field integral closure rees valuation ring follows itoh ring quotient field corresponds rees valuation ring hand rees valuation ring height one associated prime ideal minimal prime ideal contained minimal prime ideals respectively quotient field integral closure maximal ideal rees valuation ring remarks rees valuation ring remark hence itoh ring definition therefore follows first part paragraph exactly one maximal ideal correspondence holds therefore since arbitrary itoh ring follows definitions rees integers equal one also since remarks follows therefore follows remark radical ideal thus holds hold finally let rees valuation ring let corresponding itoh ring let exists hypothesis simple free integral extension domain also since holds itoh rings radical ideal hence holds concerning theorem shown corollary always exists finite integral extension ring rees integers equal arbitrary common multiple rees integers theorem directly applies place remark let rees valuation ring assume let rees valuation ring corresponds proposition let arbitrary positive integer follows theorem dvrs finite free integral extension domains maximal ideal ramification index relative relative equal therefore follows satisfy fundamental equality splitting also follows theorem itoh ring remark following result called theorem independence valuations proved let valuation rings quotient field assume containment relations among let let maximal ideals vpj corollary let nonzero proper ideal noetherian integral domain let quotient field let let rees valuation rings let positive common multiple rees integers let let fixed algebraic closure dedekind domain exactly maximal ideals simple free integral extension domain rank exactly maximal ideals exist distinct elements quotient field intersection itoh rings dedekind domain finite integral extension domain exactly maximal ideals jacobson radical assume exists bwj iwj unit jacobson radical proof part follow immediately independence valuations theorem see remark first part clear also maximal ideal since jacobson radical since free follows exactly maximal ideals holds let rees valuation ring corresponds proposition exists height one prime ideal let let qnv therefore follows independence valuations theorem see remark dedekind domain exactly maximal ideals thus principal ideal domain theorem exists vqj wqj since rees integers hypothesis exists unit hence exists unit wpj also since rees integer respect ivj exists unit therefore since follows hence transcendental therefore mej irreducible let fixed algebraic closure shown theorem itoh ring unit also integral closure theorem since follows hence see let therefore wnj dnj enj enj enj also exactly maximal ideals follows integral dependence exactly maximal ideals integral domain maximal ideal therefore follows thus hold follows immediately theorem finally let rees valuation ring corresponds proposition follows hypothesis let remark dedekind domain exactly maximal ideals thus principal ideal domain theorem exists let let let since rees integers equal follows follows exist units since since follows transcendental therefore irreducible let fixed algebraic closure follows proposition together irreducible ufd principal maximal ideal also integral follows nonzero prime ideals hence dedekind domain simple free integral extension domain jacobson radical therefore since follows unit unit next remark lists several well known facts concerning finite field extensions ramification remark let dvrs finite integral extension also let denote ramification index relative relative relative satisfy fundamental equality splitting follows fundamental inequality hence satisfy fundamental equality splitting satisfy fundamental equality splitting next proposition shows theorem holds integers greater one proposition notation theorem let arbitrary integer let let exists correspondence itoh rings rees valuation rings namely given quotient field integral closure let corresponding finite integral extension domain satisfy fundamental equality splitting proof let least common multiple let clear integral also theorem exists correspondence rees valuation rings itoh rings integral closure quotient field follows integral dependence holds follows proof exists itoh ring also finite free integral extension domain theorem hence finite integral extension domain satisfy fundamental equality splitting theorem follows remark holds terminology rees valuation ring corresponding proposition itoh ring integral closure quotient field say itoh ring corresponds also rees valuation ring corresponds proposition say itoh ring corresponds proposition let regular proper ideal noetherian ring let rees valuation ring let rees valuation ring corresponds proposition let itoh ring corresponds see arbitrary integer assume following hold multiple relatively prime greatest common divisor proof let itoh ring corresponds follows remark also clear item readily follows together remark let uke pke itoh ring corresponds follows remark pke uke uke uke also clear uke since relatively prime readily follows together remark let itoh ring corresponds since follows since multiple follows qud since relatively prime follows quc follows remark holds proposition gives several equivalences property theorem also shows hypothesis theorem necessary proposition let regular proper ideal noetherian ring let rees ring respect let arbitrary integer let let following statements equivalent radical ideal rees integers equal one rees integers equal one common multiple rees integers proof primary decomposition regular proper principal radical ideal together remark shows next last paragraph proof theorem shows also remark since assume holds let itoh rings let uwj suppose multiple let greatest common divisor let integers follows proposition place since uwj follows cej however contradicts therefore supposition multiple leads contradiction hence finally theorem corollary let regular proper ideal noetherian ring let rees ring respect integer let let following statements equivalent rees integers equal one integers ideal radical ideal proof follows immediately proposition related theorem section first prove expanded version corollary theorem prove closely related general theorem next theorem expanded version corollary theorem one main reasons including theorem show displays except separability realization see definition powerful classical method explained somewhat fully remark theorem let nonzero proper ideal noetherian integral domain let quotient field let let rees valuation rings let positive common multiple rees integers let let let dedekind domain jacobson radical ideals maximal ideals exists integral domain ideal dedekind domain finite integral extension domain fixed algebraic closure jacobson radical exactly maximal ideals rees integers equal exists rees valuation ring exists unit simple free integral extension domain rank height one associated prime ideal proof shown corollary dedekind domain jacobson radical ideals maximal ideals follows corollary corollary note frequently denoted corollary proof follows corollary proved corollary corollary shown corollary exist therefore also therefore follows theorem since follows corollary follows definition proved corollary finally intersection itoh rings corollary also intersection itoh rings definition holds next consider powerful classical theorem krull state theorem use following terminology gilmer definition let distinct dvrs field let denote residue field let positive integer system mean collection sets satisfying following conditions simple algebraic field extension sum definition system definition said realizable exists separable algebraic extension field exactly extensions residue field ramification index relative equal say field realizes realization remark definition mind notation theorem dedekind domain quotient field theorem except separability realization system rees valuation rings arbitrary positive common multiple rees integers eej root playing role proof corollary generally let nonzero proper ideal noetherian integral domain let rees ring respect let rees valuation rings let let arbitrary integer let rees integer respect let greatest common divisor also let quotient field let algebraic closure let integral closure realization system valuation rings dkj root playing roll proof corollary proof item follows immediately theorem follows proposition prime ideal item follows proposition theorem krull let distinct dvrs field let positive integer let system realizable one following conditions satisfied least one least one dvr distinct monic polynomial exists irreducible separable polynomial observation condition theorem property system condition theorem property family dvrs field condition theorem property family remark let noetherian integral domain quotient field assume altitude exist infinitely many height one prime ideals infinitely many height one prime ideals theorem therefore since krull domain exist infinitely many distinct dvrs quotient field hence theorem always satisfied fields state prove first new result section closely related theorem also considerably general theorem let noetherian integral domain quotient field assume altitude let nonzero proper ideal let rees valuation rings let dedekind domain exactly maximal ideals let irredundant primary decomposition rees integers let least common multiple let let positive multiple exists integral domain ideal dedekind domain simple free separable integral extension domain jacobson radical exactly maximal ideals kej associated prime ideals rees integers equal proof let kej observe system dmn realizable system dmn theorem remark therefore integral closure realization dmn simple free separable integral extension domain dedekind domain theorem also since kej exactly kej extensions ken kej kej therefore hold qkej kej kej jacobson radical thus holds since domain jacobson radical follows immediately definition remark remark proof proposition many cases simpler econsistent system kej used place kej resulting realization dmn number maximal ideals however always used since example algebraically closed kej extension fields kej remark proof proposition kej replaced kdj conclusions hold replace exactly kej extensions exactly extensions rees integers equal rees integers equal kej kdj qke qke kdj first corollary theorem complete detailed version theorem corollary let nonzero proper ideal noetherian domain let rees valuation rings let rees integer respect let positive multiple least common multiple exists integral domain dedekind domain simple free separable integral extension domain rees integers ibe equal proof let intersection rees valuation rings let dedekind domain exactly maximal ideals therefore proposition exists simple free separable extension field quotient field integral closure finite free separability integral extension domain dedekind domain exactly kej maximal ideals lying jacobson radical separability exists integral let simple free separable integral extension domain intersection rees valuation rings ibe remark since ibe since jacobson radical follows rees integers ibe equal remark notation corollary exists integral domain dedekind domain simple free separable integral extension domain rees integers ice equal let regular proper ideal noetherian ring let rees valuation rings let rees integer respect let positive multiple least common multiple assume rad prime say rad exists ring simple free integral extension ring rees integers ibe equal zbe rad zbe minimal prime ideal notation exists ring simple free integral extension ring rees integers ice equal zce rad zce minimal prime ideal proof proof proof corollary use remark place proposition since rad minimal prime ideal rees valuation rings rees valuation rings see remark also corollary exists integral domain dedekind domain simple free separable integral extension domain rees integers ibe equal let irreducible monic polynomial degree let since integral domain follows zbe prime ideal also hypothesis exists follows zbe minimal prime ideal holds therefore rees valuation rings rees integers ibe rees valuation rings rees integers ibe remark follow immediately proof proof use remark place corollary next corollary theorem extends corollary noetherian rings corollary let regular proper ideal noetherian ring let rees integers let least common multiple let positive multiple exist rings finite integral extension ring ibe regular proper ideals correspondence minimal prime ideals ibe minimal prime ideals minimal prime ideals namely rees integers ibe equal proof let irredundant primary decomposition zero ideal let rad assume minimal prime ideals remaining minimal prime ideals rewrite intersection let let unique minimal prime ideal therefore rad let regular proper ideal remark set rees valuation rings disjoint union sets rees valuation rings ideals rees integers among rees integers hence multiple least common multiple rees integers ideals therefore remark exists simple free integral extension ring rees integers equal minimal prime ideal let let let finite integral extension ring regular proper ideal minimal prime ideals also rees valuation rings rees valuation rings ideals rees integers let notational convenience let also let finite also finite integral extension ring ibe regular proper ideal minimal prime ideals since ideals minimal prime ideals since minimal prime ideal follows minimal prime ideals ibe set rees valuation rings disjoint union sets rees valuation rings ideals therefore since unique minimal prime ideal follows rees valuation rings rees integers finally rees integers equal last sentence second preceding paragraph follows rees integers ibe equal state additional corollary need following definition definition let ideals ring projectively equivalent case exist see projectively full case ideal projectively equivalent see remark definition mind noted theorem shows jacobson radical projectively equivalent since dedekind domain follows projectively full radical ideal whose rees integers equal one next corollary theorem proved theorem case noetherian domain altitude one corollary noetherian ring altitude one regular proper ideal exists finite integral extension ring ideal projectively equivalent projectively full radical ideal rees integers equal one proof shown theorem result holds nonzero proper ideals noetherian domains altitude one proof similar proof corollary shows continues hold regular proper ideals noetherian rings altitude one references heinzer ratliff rush projectively full ideals noetherian rings algebra endler valuation theory new york gilmer domains rings polynomials algebra heinzer ratliff rush projective equivalence ideals noetherian integral domains algebra itoh integral closures ideals generated regular sequences algebra krull einen existensatz der bewertungstheorie abh math sem univ hamburg nagata local rings interscience john wiley new york ratliff characterization analytically unramified rings applications pacific math ratliff prime divisors integral closure principal ideal reine angew math rees note form rings ideals mathematika swanson huneke integral closure ideals rings modules cambridge univ press cambridge zariski samuel commutative algebra vol van nostrand new york zariski samuel commutative algebra vol van nostrand new york department mathematics university california riverside california address department mathematics university california riverside california address ratliff department mathematics university california riverside california address rush
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advanced signal processing techniqes study normal epileptic eeg debadatta dept electrical electronics engineering veer surendra sai university technology vssut burla india paper human normal epileptic electroencephalogram signals analyzed popular efficient signal processing techniques like fourier wavelet transform delta theta alpha beta gamma sub bands eeg obtained studied detection seizure epilepsy extracted feature applied ann classification eeg signals epileptogenic focus hippocampal formation set contains single channel eeg segments duration sampled hence data segments contain samples collected intervals exemplary eegs plotted epilepsy fouriertransform dwt introduction english physician richard canton discovered electrical currents brain german psychiatrist hans berger recorded currents named eeg berger electroencephalogram record time series evoked potentials caused systematic neural activities brain epileptic seizures result transient unexpected electrical disturbance brain approximately one every persons experience seizure time life monitoring important milestone provide valuable information candidates suffer epilepsy days however magnetic resonance brain tomography used diagnosis structural disorders brain observe special illness like epilepsy measurement human eeg signals done recording voltage difference scalp placing electrodes inside brain specified internationally recognized electrode placement system five major brain waves distinguished different frequency ranges delta theta alpha beta greater gamma waveforms types epileptic seizures epileptic syndromes classified commission classification terminology international league epilepsy fig normal eeg signal epileptic eeg signal fig seen epilectic signal higher amplitude distinct transients complete different normal eeg iii fourier analysis compare frequency distribution eeg signals plotted plot periodogram power frequency plot power explained square absolute value fourier transform plot called fluctuation seen function frequency data used two sets eeg data healthy volunteer eyes open epileptic subjects seizure publicly available given andrzejak type epilepsy diagnosed temporal lobe epilepsy fig comparison periodograms eeg signals peaks indicate alpha beta rythms clearly visible normal eeg periodogram seen epileptic signal epileptic eeg shows abrupt increase frequency different ranges wavelet analysis basic idea underlying wavelet analysis consists expressing signal linear combination particular set functions wavelet transform obtained shifting dilating one single function called mother wavelet since sampling frequency eeg according nyquist sampling theorem maximum useful frequency half sampling frequency correlate wavelet decomposition frequency changes physiological sub bands eeg band convolved low pass fir filter band limited basic methodology shown eeg lpf feature epilepsy detection extraction band limited eeg decomposed wavelet level components retained reconstructions five components using inverse wavelet transform approximately correspond five physiological eeg sub bands delta theta alpha beta gamma fig comparison features normal epileptic eeg signal two exemplary data given table table eeg signal normal epileptic max mean median mode std conclusion detecting epileptic seizures long eeg recordings time consuming costly task done visually trained professional paper proposes technique detection feature extraction epilepsy using fourier wavelet transform matlab local low frequency components normal epileptic eeg clearly detected wavelet transform fourier transform conclude wavelet transform wellsuited method frequency analysis eeg signals delta wave identified measured features like mean max min median calculated ann used classification accuracy level decomposition eeg five eeg sub bands using daubechies wavelet detail parts shows high frequency components shows low frequency components original signal reconstructed adding components local low frequency waves concerned epileptic seizures eeg clearly detected dwt analysis done sets data healthy epileptic database feature extracted delta waves decomposed eeg signals helped classification eeg signals epileptic normal category feature extracted category mean median mode min max standard deviation std showed accuracy classification eeg signals min references detection epilepsy disorder eeg signal international journal emerging trends engineering development issue iasemidis shiau sackellares pardalos principe adaptive epileptic seizure prediction system ieee transactions biomedical engineering vol alexandros tzallas use distributions epileptic seizuredetection eeg recordings ieee maan shaker eeg waves classifier using wavelet transform fourier transform international journal biological life sciences alexandros tzallas markos tsipouras dimitrios tsalikakis evaggelos karvounis loukas astrakas spiros konitsiotis margaret tzaphlidou automated epileptic seizure detectionmethods review study andrzejak lehnertz rieke david elger epileptic process nonlinear deterministic dynamics stochastic environment evaluation mesial temporal lobe epilepsy epilepsy res vol husain epileptic seizures classification eeg signals using neural networksinternational conference information network technology ipcsit vol singapore chiri yamaguchi fourier wavelet analysis normal epileptic electroencephalogram memoirs fukui university technology vol
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facets tiers gems ontology patterns hypernormalisation phillip lord robert nov school computing science newcastle university school computer science university manchester manchester abstract many methodologies techniques easing task ontology building describe intersection two ontology normalisation fully programmatic ontology development first describes standardized organisation ontology singly inherited entities number small taxonomies refining entities former described defined terms latter used manage polyhierarchy entities fully programmatic development technique ontology developed using language within programming language meaning well defining ontological entities possible add arbitrary patterns new syntax within environment describe new patterns used enable new style ontology development call hypernormalisation introduction building ontologies difficult business number reasons abstract knowledge domain difficult gather understand represent ontologically immediately ontologies especially complex representation taxing describe define consistently update expand change representation needs change numerous attempts simplify clarify process including development methodologies ontoclean defines set inform ontological modelling guarino welty upper ontologies dolce bfo grenon provide upper classification another approach leverage techniques ontology normalisation rector originally intended mechanism untangling existing hierarchies classifications reused basis ontology also significant use pattern building ontologies novo broadly normalised ontology defined using skeleton strict tree acyclic graph concepts differentiated using inheritance partonomy relationship split set entities children disjoint cover parent partitioning correspondence addressed refining concepts form closed covering disjoint hierarchies building ontology way allows ontology developer exploit reasoner build polyhierarchy using classes define entity terms refining partitions polyhierarchies difficult build manually human ontology developers matter good domain knowledge find hard ensure possible parents entity taken account normalisation approach uses defined classes reasoning remove chore creating tree entities still however remains task developer normalisation approach significantly increase robustness reduce work manual maintenance wroe latter form ontological normalisation widely implicitly used term ontology normalisation borrowed somewhat metaphorically database engineering process building ontologies using set standard design patterns rather direct relationship software engineering equivalent reusing standard set patterns possible build ontology rapidly consistently manifested number different ways number different tools termgenie dietze populus jupp generate ontologies according pattern previously described fully programmatic methodology ontology development lord using environment built around programming language clojure enables ontology take advantage features programming language environment including unit testing warrender lord build evaluation course development simple use functions warrender lord respect patterns environment several advantages first unlike tools populous oppl egana aranguren patterns developed environment syntax simple ontology concepts therefore easy define pattern define class second based clojure language homoiconic little syntax possible build arbitrary syntactic constructions represent patterns way convenient attractive developer paper describe extension normalisation technique call hypernormalisation technique typified near complete absence asserted hierarchy among entities describe allows construction exemplar ontology stevens lord move describe recent developments lord stevens top level entities body substance protein steroid person organic ion doctor role refining type role care role nurse role value types patient role adult age type sex child male female fig normalised ontology slightly modified rector graph necessarily reflect subsumption see text details environment including definition two new design patterns tier facet one syntactic abstraction gem used enable hypernormalised ontology development finally discuss application approach ontologies hypernormalisation amino acids normalisation methodology aims disentangle ontological structure process managing maintainability utility expressivity ontology generated achieve ontology split two main hierarchies entities refining types see figure example hierarchy contains entities central hierarchy skeleton part ontology would expect hierarchy contains levels children cover parents parents closed new children contrasted refining hierarchy consists classes exhaustive many cases children nonoverlapping therefore disjoint say refining types hierarchy necessarily complete figure example representation sex simple many medical uses might sufficient customer relations system general entities defined terms refining types polyhierarchical relationships entities determined use reasoner form ontology development quite different upper ontology agnostic choice upper ontology none rector suggests entities refining types made clear mechanism owl could upper ontological term annotation next introduce used exemplar defines biological terms physiochemical properties relevant biological role structurally interesting ontology normalised clear clean separation entities five refining concepts rather though entities split three sets aminoacids alanine large set defined classes describing refined types small neutral amino acid finally single class amino acid stated alternatively contains skeleton hierarchy relationships classes arrived reasoning particularly relevant amino acid ontology contains defined classes subsumption relationships amino acids maintaining form ontology hand would impractical call style ontology development hypernormalised believe natural extension normalisation rector notes example choice aspect form skeleton degree arbitrary rigid ontoclean guarino welty pragmatically stable unlikely change evolution ontology however true refining concepts aminoacids short choice skeleton arbitrary actually unnecessary brings utility ontology achieved use reasoning note distinction normalisation hypernormalisation absolute one degree simply describing tendency toward ontology flat asserted hierarchy introduced notions hypernormalised ontology next consider set new patterns enable style ontology development patternising environment lord ability support patterns warrender lord described elsewhere detail provide quick overview rest paper clear implemented dsl language clojure language implemented java running java virtual machine facets tiers gems top level entities defined class small neutral amino acid aliphatic alanine arginine refining type size charge tiny small large hydro polarity sidechain fig hypernormalised ontology representing using terminology figure labels abbreviated wraps owl api horridge bechhofer library underpins protg gains much functionality simple sections ontology generated using syntax based lispified version manchester owl notation example following code defclass super declares new class class superclass manchester owl notation would expressed class subclassof code entirely valid clojure evaluated clojure environment also possible define new patterns example following pattern definition defn property clazzes list property clazzes property clazzes defines pattern generates set existential restrictions one universal union existential fillers filler implements ontological closure pattern function definition clojure terms defn introduces function property clazzes argument list functions provided list returns prosaically critically possible define pattern environment file simple class definition easy define class define use new pattern ontologies karyotype ontology make extensive use facility moving freely ontology pattern definitions well literal data structures utility functions unit tests warrender lord mature used software product first alpha release nov first full release nov followed four point releases paper describes mostly upcoming release although features described available earlier versions value partition common pattern building normalised ontology called value partition pattern rector addresses problem ontological modelling continuous range example modelling consider concept size could described directly using molecular weight however purpose easy general practice split size three categories tiny small large achieved straightforwardly using defpartition defpartition size tiny small large domain aminoacid super physiochemicalproperty axiomatically expands class size three subclasses tiny small large property hassize see lord explanation super used rather subclass function shown slightly simplified version one provided knowledge lisp actually macro main implementation function provides support implementing syntactic macros whose function simply allow use bare symbols without knowledge lisp distinction important lord stevens property functional range size domain aminoacid expanded would expressed class large subclassof size class size equivalentto large small tiny subclassof physiochemicalproperty class small subclassof size class tiny subclassof size disjointclasses large small tiny subclasses disjoint cover parent following terminology rector value partition useful defining partitioning refining concepts tier value partition pattern aimed specific purpose segmenting continuous range practice though found axiomatization pattern generally useful example considering ontology natural model chemistry defpartition sidechainstructure aromatic aliphatic domain aminoacid super physicochemicalproperty intuitive ontologically sidechainstructure actually different form size reflect spectrum either contains benzene ring making aromatic form partition also noted rector includes classes male female spectrum least simplified representation introduce therefore general notion tier small set concepts hierarchy tier function supports range options deftier charge positive neutral negative domain aminoacid super physiochemicalproperty suffix true use suffix true causes simple change naming entities positive become positivecharge would expanded follows class positivecharge subclassof charge names modified equivalently default manifest referring class environment iri concept serialized owl value annotation addition naming also possible optionalise whether subclasses disjoint covering whether property functional whether created tier general pattern fact current version latter defined terms former value partition tier introduce new object property named tier range limited classes defined within tier converse also true use one tier classes positivecharge likely wish use hascharge property defined part taken together describe combination classes property facet facets well known technique first proposed library classification colon classification ranganathan named use separator seen facetted browsers used many websites navigation complex product catalogues provides explicit support facets allowing association property set classes demonstrated following code hascharge positive neutral negative practical implication use facet function return existential restriction providing class express programmatically example might use assert function provided clojure unit test framework assert hascharge positive facet positive ability slightly succinct however used multiple facetted classes advantages become considerably clearer shown following assertion assert list hascharge also adds annotations elided facet duplication annotation iri fragment iri schemes numeric style obo ids annotations elided brevity facets tiers gems neutral hashydrophobicity hydrophobic haspolarity nonpolar hassidechainstructure aliphatic hassize tiny facet neutral hydrophobic nonpolar aliphatic tiny addition succinctness pattern also reduces risk errors class tiny always used correct property without use facets ontology developer must achieve hand would also possible detect error using reasoning although succeed appropriate range disjoint restrictions ontology defpartition deftier functions course add range disjoint restrictions declare classes facets properties gem finally define gem provides syntactic abstraction class composed entirely mainly facets following terminology rector abstraction would useful mostly concepts example could define amino acid alanine using following defgem statement defgem alanine comment amino acid single methyl group facet neutral hydrophobic nonpolar aliphatic tiny likewise defined series gems fact amino acids regular five facets use syntactic abstract specific ontology form pattern describe localized warrender gem represents generalised syntax useful developing ontology annotation previously discussed relationship design methodology normalisation use upper ontology patterns described orthogonal agnostic choice upper ontology none place entities particular part class hierarchy define classes outside required domain ontology although could easily extended ontology developer require however agree rector use patterns made clear explicit within ontology reason patterns described also make use annotations using annotation properties defined using internal annotation ontology example entities generated result pattern deftier explicitly annotated means use patterns informally explicit owl serialization actually uses annotations internally example enable facet functionality providing relationship classes appropriate object property strictly implementation detail could achieved without annotations however believe shows value knowledge explicit owl discussion paper describe used provide patterns applied ontology development patterns provide functionality syntactic abstraction underlying owl implementation process enable easy accurate construction ontologies specifically demonstrate two new patterns tier facet tier extension existing value partition pattern used generation many small hierarchies used refining properties facet borrows library sciences notion facetted classification used associate set classes specific set values form classification common web majority web stores example offer facetted browsing often facets changing different subsections catalogue taken together two patterns enable new form ontology development hypernormalisation extreme form normalisation form normalisation away creation tree entities instead rely reasoner build hierarchy well making ontologist task easier makes characteristic would used create tree entities explicit form refining characteristic described application methodology exemplar ontology course dangerous extrapolate generality exemplar also started apply hypernormalisation ontologies real domains including clouds meterological sense cell lines reworking gene ontology tier made generic require example refining types closed possibilities known advance disjoint clearly forms ontology naturally represented hypernormalised form example karyotype ontology warrender lord far form define concepts use reasoning set defined classes effectively operate facets warrender lord however popularity facetted browsers shows possible use form classification many areas believe introduction concept hypernormalisation implementation could significant implications future development ontologies references dietze berardini foulger hill lomax osumisutherland roncaglia mungall termgenie web application ontology class generation journal biomedical semantics egana aranguren stevens antezana transforming axiomisation ontologies ontology preprocessor language nature precedings lord stevens grenon smith goldberg biodynamic ontology applying bfo biomedical domain stud health technol inform guarino welty evaluating ontological decisions ontoclean commun acm horridge bechhofer owl api java api owl ontologies semantic web journal jupp horridge iannone klein owen schanstra wolstencroft stevens populous tool building owl ontologies templates bmc bioinformatics suppl lord semantic web takes wing programming ontologies owled lord manchester syntax bit backward http ranganathan colon classification rector representing specified values owl value partitions value sets working group note rector normalisation ontology implementations towards modularity maintainability proceedings workshop ontologies multiagent systems omas conjunction european knowledge acquisition workshops siguenza spain stevens lord semantic publishing knowledge amino acids http warrender consistent representation scientific knowledge investigations ontology karyotypes mitochondria thesis school computing science newcastle university warrender lord approach biomedical ontology engineering warrender lord karyotype ontology computational representation human cytogenetic patterns warrender lord test ontology wroe stevens goble ashburner methodology migrate gene ontology description logic environment using pacific symposium biocomputing
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killed albert einstein open data murder mystery games feb gabriella michael cerny antonios julian tandon school engineering new york university new york usa institute digital games university malta msida malta gabbbarros mcgreentn julian paper presents framework generating adventure games open data focusing murder mystery type adventure games generator able transform open data wikipedia articles openstreetmap images wikimedia commons wikimysteries every wikimystery game revolves around murder person wikipedia article populates game suspects must arrested player guilty murder absolved innocent starting one person victim extensive generative pipeline finds suspects alibis paths connecting open data transforms open data cities buildings characters locks keys dialog options paper describes detail generative step provides specific playthrough one wikimystery albert einstein murdered evaluates outcomes games generated influential people century index games open data murder mystery adventure games data adventures game generation ntroduction games cast player role detective gameplay main challenge revolve around solving crime mystery popular many decades games world carmen sandiego software task player finding fugitive criminal games indiana jones fate atlantis lucasarts series tomb raider eidos uncharted naughty dog see player embark adventure solve ancient mysteries face opposition shadowy goons common games feature frequent travel exotic locales around world interact colorful people gather clues solve puzzles overcome resistance games often make heavy use locations stories items characters build narrative authoring games complex requires considerable skill however fact games depend much information information freely available structured form resources wikipedia openstreetmap suggests would possible somehow automatically incorporate realworld information games furthermore murder mysteries similar adventure games often highly structured suggesting possibility generating game could practical manner research projects game generation rogue dream create simple games respectively murder mystery requires much larger volume variety content locations people dialog clues solving mystery moreover consistency content necessary internally within game narrative externally fidelity real world contribution generator murder mysteries open data explores disparate data connected together create represent identifies design formula structure murder mysteries constituent elements dialog used generate mystery games wikipedia entry tests limits autonomous game generation issues arise absurd incomplete source data algorithmic combination paper presents wikimystery framework generating complete playable adventure games minimal human input case name person wikipedia page wikimystery generative system featured paper builds upon earlier work extending significantly description full generative pipeline sophisticated dialog system broader evaluation nearly games solving murder influential people century wikimystery game generator built previous project data adventures reuses much technology discover paths victim suspects murder current framework however offers much engaging coherent complete experience clear goal arrest culprit murder facilitated extensive story branching towards several suspects enhanced ludic elements game objects unlock certain locations enriched dialog elements allow nonplayer characters npcs share facts mystery based open data paper starts brief survey section game plot dialog generation paper provides overview generative pipeline wikimystery section iii specifics culprit evidence selection section path generation section location npc item generation section finally npc dialog generation section assess generated games sample playthrough described section vii section viii analyzes games created murders influential people century paper leads discussion section concludes section background wikimystery system framework transforming open data adventure games section discusses domains data games game plot dialog generation data games world ubiquitous technology amount data consume daily rapidly increasing one data category growing exponentially open data information freely used redistributed anyone creating games data seen form visualization instead using charts figures make information easier grasp one creates playable media data games use real world information open data automatically generate game content players interact content gameplay often must learn understand data order play game well typically use data game content one must select parts data useful content generation structurally transform applicable game content example data games discussed open trumps data game cards content based entirely published governmental data generator creates balanced top trumps deck using evolutionary algorithms required learn data helps playing open trumps content selected user playing museum curator must theme museum based interests barchartball physics game uses census data transform playable level necessary infer data would affect playfield modified based upon high low selected attribute rogue dream uses results google queries using templates choose names player abilities enemies healing items names given visual found via google image search finally geographical data openstreetmap used generate maps players initial positions freeciv story quest dialog generation games research story generation tends focus textual form brutus generator creates dark stories characters backgrounds narratives minstrel generator uses problem solving write short stories based king arthur knights bardiche acts collaborative tool create good stories based user input improvising based user input creating games stories stories games part broader subject recontextualizing data one medium another examples include transformation museumville https portal accessing digitised cultural heritage material paintings books institutions across europe freciv open souce version civilization http levels sonancia text audio metaphor soundscapes news articles games angelina adventure games stories quests crucial progression needs taken account generating game charbitat procedurally generates environment player explores lacks sense progression order anchor player quest generator introduced charbitat uses mechanisms advance game world similarly procedural generator built mystery solaris constructing maps missions via two separate grammars first mission graph constructed containing quest information mission graph taken input map generator builds map around mission symon adventure game uses procedural content generation create meaningful puzzles generation symon expanded puzzle dice system generating puzzles adventure games since murder mystery games largely rely interaction characters quality npc dialog important factor gameplay experience dialog generation hardly new topic research going back way eliza another example system transforms monological text dialog agents act dialog uses textual coherence relationships map text pairs able create fairly believable dialog however less interested creating realistic sequences chat responding accommodatingly player request rather driving npc interactions towards specific direction providing clues player interactive storytelling studied dialog generation delivery extensively combining implicit forms character expression overall narrative goals emotional relationships characters generate realistic dialog iii overview ame enerator wikimystery procedurally generated adventure game uses data wikipedia wikimedia commons automatically create different game content plot progressing images strongly inspired classic adventure games world carmen sandiego software osterweil describe main characteristics adventure game gameplay driven story core mechanics interaction game world object manipulation character motivated explore interact surroundings game player assumes role detective trying solve murder case victim central point story suspects based people related use wikipedia identify possible suspects five selected game plot victim root suspect leaf path europeana openstreetmap open source project attempts mapping world https selecting victim dbpedia wikimedia commons open street maps allen shenstone rosa beddington evolving set suspects generating paths suspects generating items generating cities buildings generating npcs adding blocks solutions adding dialogue received royal society award david mackay received royal society award died princeton new jersey albert einstein field physics born german empire jakob meisenheimer william rankine playable adventure game fig flowchart wikimystery open data sources representation hyperlinks victim suspect wikipedia articles initially location available victim house player talk people related victim player also becomes aware five suspects murder interact people inside house new locations objects npcs become available player explores interacts world collect information suspects characteristics year death occupation every suspect except culprit value one characteristic died suspect birth place cleveland suspect acts evidence innocence identified dialog note done player receives information culprit culprit also share value characteristic evidence innocence suspects culprit die born cleveland game ends player issues arrest warrant identifying culprit specifying values acting evidence innocence suspects player correctly finds culprit provides correct evidence suspects game player specify right culprit evidence innocence suspect incorrect game lost game generation involves several steps shown figure first selecting victim victim system uses find set five suspects via artificial evolution presented section generate paths suspect victim via constructive algorithm presented section paths generated system creates locations items npcs constructive processes covered section finally generates puzzles accessing locations dialog options learning clues general information npcs discuss latter section rawling pedia plot wikimystery created series hyperlinks wikipedia generated using several consecutive queries dbpedia victim introduced system tries find suspects pinpoint culprit among searches paths victim suspects dbpedia project extracts information wikipedia structured manner http fig selecting suspects dbpedia use game finding related victim suspects initially system single node victim black node suspects related victim selected genetic algorithm white nodes paths victim suspects created dbpedia see fig suspects share direct connections victim shown arrows finding suspects culprit selection set suspects involves identifying related victim subset interesting given wikipedia article person system queries dbpedia find anyone something common victim common living place specific band list pool suspects one query dbpedia find everything known point list suspects containing list characteristics characteristic multiple values example suspect could albert einstein would characteristic field values physics philosophy figure shows simplified selection suspects victim black node system finds suspects white nodes related somehow victim arrows system must also find set characteristics single culprit among suspects characteristic value characteristic together form evidence innocence used identify culprit issue arrest warrant list possible subjects characteristics values large times characteristics may multiple values available use one per characteristic suspects multiple characteristics example publication date paper distinct people related albert einstein way constituting possible suspect pool one average least five characteristics may may multiple values selecting five suspects four characteristics values therefore challenging select subset list interesting turn evolutionary algorithm goal finite set suspects typically finite set characteristics pair characteristic person leftover person culprit characteristics evidence innocence suspects allows player eliminate innocent suspects finding clue paired remaining suspect value evidence innocence must killer fitness function evaluates solvability diversity solvability favors complete solutions player identify culprit excluding characteristics knows killer search applied every chromosome marks one suspects chromosome killer search states characteristics chromosome paired one suspect visited valid states three properties killer least one value characteristic one suspects value characteristic least one suspect one value different killer algorithm tries pair suspect one characteristic suspect least one value different killer specific characteristic match pair backtracks tries different suspect characteristic paired one suspect optimal solution leaf characteristics paired successfully suspects fitness depth leaf diversity evaluates different characteristics values game outputs one value per pair necessary optimize value use example game suspects characteristics solution values one per pair better one one suspect value characteristic one matched additionally solution suspects job live city less diverse solution different jobs live different cities even though still use characteristics job residence second one diversity values actual fitness value given pij pij root tree crime scene branch leads possible suspect tree represents plot points player able unravel game order move plot forward locations npcs clues clues present tree evidence innocence suspect system queries dbpedia multiple times searching possible paths victim said suspect rates path based diverse type articles links path example path articles locations less diverse one even number articles people locations process computationally expensive since necessary create one query per node path per direction edge practice using paths longer nodes proven time consuming bypass divided search two steps first finds path length longer nodes described call major path consecutive pair nodes major path system searches minor path nodes minor path replaces edge two nodes major path figure shows example path victim albert einstein suspect william rankine identifying major minor paths system measures path quality based length uniqueness longer paths preferred extend game node transformed city npc item story uniqueness calculated entropy path compared nodes edges found paths particular search thus path type edge found possible paths better typical edges scientists typical edges type influenced influenced number characteristics number values characteristic number people characteristic multiply reward suspects sharing characteristic pij calculated number people value characteristic divided system uses cascading elitism population individuals generations mutation chance cascading elitism uses fitness functions sorts population using solvability fitness removes worse individuals sorts remaining using diversity fitness highest population duplicated mutated new population filled far important games solvable playability sake rather diverse solvability applied first cascading elitism introduces stronger genetic bias finding paths suspects system victim suspects clues evidence innocence weaves plot searching dbpedia path hyperlinks victim suspect path consists nodes wikipedia articles edges links set paths seen tree merge initial node paths victim therefore nriching data system transforms set paths tree obtained wikipedia gameplay objects player interact node tree becomes location item npc creates necessary game objects generates dialogs links verifies objects appear correct order add puzzles nodes tree roughly categorized places london canada people albert einstein everything else mathematicians century system begins transforming nodes simplest objects possible locations npcs items node based article place generates city place contains geographic coordinate building game logic world contains cities buildings places inside cities buildings also contain items characters system generates building tries place respective city find city related building randomly pick place wikipedia generate city placing building buildings cities created system takes nodes based real people generates one npc npc gets original person name small description node person location influenced albert einstein nce lue inf doctoral advisor nathan rose lac thp bir brooklyn ntr cou united states nat ity nal field john slater ora oct visor william shockley ced uen nfl field clinton davisson physics fiel eld william rankine fig major minor paths albert einstein william rankine major paths dotted arrows minor paths black arrows locations represented hexagons npcs circles items books photographs squares transformed item either book list letter photograph depending type item different text templates generated explain possible transform tree root npc supposed murdered wikimystery attempts solve adding people related victim instead suspect searches person directly connected victim transforms npc find enough people generates random npcs whose sole purpose give clue following node objects needed plot generated necessary create logical sequence steps victim suspect system traverses branch tree adds clues conditions one node next current node location npc item generated placed person dialog created directing player next node discuss dialog generation detail section otherwise clue added item text description additionally random times game may generate fake npc sole purpose provide red herring given random name description dialog less helpful condition manager guarantees game objects available triggered another object finally system adds puzzles one wellknown puzzles adventure games location inaccessible unless player uses specific item unlock wikimystery generates kind puzzles creating items able unlock buildings flashlights dark places crowbars chained gates puzzle objects placed via variation search algorithm first nodes tree separated depth depth location root npcs depth would contain locations available talking root npcs simulate playthrough perform said separation also maintains array possible keys keys crowbars etc initially empty stack locks every depth randomly chooses whether put key building depth adds respective lock stack locks additionally may randomly pop lock stack add another building example depth may chose put flashlight key root building automatically add lock darkness stack victim house building depth already chosen algorithm goes depth randomly chooses put key lock skips straight depth finds church house randomly decides put keys decides pop lock stack darkness lock adds church adding key lock guarantees puzzle solvable ialog eneration game dialog two goals advance game giving hints evidence needed win provide sense depth immersion hard capture data game npc dialog tree lines dialog player npc use interacting along dialog options player root tree simple hello choices follow called dialog branches two types branches main branch containing information necessary complete game side branch several subtypes contains information necessary complete game increases immersion main branch main dialog branch contains hints allow player advance game every npc generator parses data stores information person dbpedia page anything places persons items concepts associated person well personal information like birthplace birthday stored dialog generator takes sentence templates replaces placeholder text example rosa beddington person object might jamaica stored within associated place player talks another npc associated rosa beddington might dialog node telling player think rosa beddington saw person place probably look become saw rosa beddington jamaica probably look packaging sentence dialog dialog node generator adds node child dialog root dialog choices hidden default unless parent visited thus player must select root hello option branches revealed side branches beyond main dialog branch generator also selects randomly set side branches effect overall story branches provide extra information npc player speaking future plans use educational purpose players fig example dialog side branch player speaking hermann einstein albert einstein game learn characters backgrounds talking currently possible side branches current residence lifetime achievement data originally parsed dbpedia birth current residency overview information stored creating side branch data replaces placeholder words templates main branch see fig example side branch main branch created generator randomly selects two topics generate side branches none generated system shows option dialog screen leads user branches initial world map initial city map switzerland house albert einstein clue given dialog travel cities icon photograph item photograph item dialog suspect finding evidence innocence issuing warrant vii xample laythrough indicative playthrough describe first minutes wikimystery gameplay game uses input text albert einstein identified time magazine person century game launches user load influential people century purposes analysis section viii game starts world map see fig one point visited switzerland chosen birthplace albert einstein clicking point interest user moves map location switzerland collected single location titled house albert einstein see fig visited player also access backpack screen bottom right fig currently empty store items used access locked locations player clicks house albert einstein move building screen shows background house coupled informative text switzerland bottom area see fig six different game icons interacted right first five icons npcs last icon displays crowbar stored inventory clicking hand button crowbar icon noted house albert einstein five npcs player option observing eye button icon fig talking dialog button icon fig npcs leo szilard david joseph bohm jean gebser riazuddin james last npc randomly generated given random name remaining npcs physicists except gebser philosopher clicking eye button gives information npcs random npc james text apparently shown fig open area near melchtal valley dbpedia entry places coordinates country switzerland center fig screenshots mystery around murder albert einstein says information available character note images chosen npcs images people case szilard bohm instead images related atomic bomb images correct people case gebser riazuddin clicking dialog button npc icon moves user dialog overlay see fig npc name followed response text followed turn set dialog options general dialog sequence npc house albert einstein revolves around first asking help asking name information might player look responses npc depend path towards suspect guide player indicatively player initiates dialog leo szilard npc respond player question please state maybe call leo szilard one case leo szilard influence albert einstein subtly explains character game player asks something think know leo szilard responds talked hermann einstein player ask hermann einstein leo szilard responds hermann einstein went united states long see fig immediately adds united states location world map puts house hermann einstein city map npc named hermann einstein within player talked five npcs house albert einstein five locations world map visit clicking point small plane shown traveling player current location selected one see fig locations world map united states containing house hermann einstein princeton new jersey containing tunnel building israel containing house nathan rosen wrttemberg containing stadium building swiss federal institute technology zurich containing building name placed zurich map similarly house albert einstein one npcs clues building listed instance tunnel building princeton new jersey random npc named vlad photograph icon fig clicking photograph shows image jewish people see fig description says photograph jews corner says jews also known jewish people ethnoreligious group originating israelites hebrews ancient near east jewish ethnicity nationhood religion strongly interrelated judaism traditional faith jewish nation observance varies strict observance complete nonobservance names written behind canada israel information based abstract wikipedia regarding category jews stored dbpedia used link different npcs mystery together npcs nathan rosen israel random npc also named vlad university building world map location named canada vlad reveals allen goodrich shenstone located house princeton new jersey extensive investigation taking player many different cities around globe slowly revealing buildings npcs clues previously visited cities player finds five suspects mystery suspects sir david mackay whose dialog pleas innocence shown fig allen shenstone william john macquorn rankine jakob meisenheimer rosa beddington five names also provided questioning one five npcs player starting location house albert einstein suspect npcs rosa beddington linked einstein fellow scientist awarded royal society culprit marker based dbpedia information placed center https accessed march suspects sir david mackay absolved finding evidence innocence case provided chemist jacob meisenheimer see fig player confident collected enough evidence click cellphone bottom left corner fig choose guilty person fig player chooses guilty person remaining suspects placed another window bottom half fig player must specify one characteristic correct value person make incapable committed murder characteristics values absolve suspects except rosa beddington included table player selects culprit chooses values remaining suspects click arrest button bottom right fig point game ends message success failure viii valuation playthrough section vii provides glimpse means play generated murder mystery section evaluates content generated broader set murdered wikipedia persons goal estimate number interactions afforded game dialogs npcs visits cities item pickups assess sensitivity system different inputs wikipedia persons former several metrics regarding instances specific elements cities npcs dialog lines per generated game listed latter describe wikipedia persons murdered games highest lowest values metrics paper perform user playtest generated games assess intuitive connections npcs provided evaluation vital understanding complex generated games generated gameplay elements contribute complexity evaluation thus first step prior playtest assess instance minimum number player clicks via tree size metric combined dialoge nodes metric game completed metrics compared actual metrics derived playtests also inform changes generative algorithms playtests take place assess broad range games based persons strong presence wikipedia used list time magazine influential people century input person list became victim procedurally generated game preprocessing excluding two american unknown rebel system able generate games first represents whole category could choose single person represented category latter represents unknown person contain tag person dbpedia page additionally system process groups people inputs kennedy family transformed single individual entries groups transformed one known people group example beatles became john lennon kennedy political family became john kennedy system generated total games one per input table shows quantitative results table average metrics generated adventure games influential people location metrics cities buildings average buildings per city item puzzle metrics items books photographs torn torn photographs key items locked buildings npc metrics npcs npcs based real people average ratio real npcs npcs average npcs per building dialog metrics dialog nodes average dialog nodes per npc achievement residence birth complexity average length paths tree size game content based table average tree size generated games nodes game smallest tree size robert goddard input nodes marlon brando martin luther king richard rodgers willis carrier tied nodes tree average length paths victim suspect nodes six games lowest path length nodes highest path length nodes game average around cities buildings approximately buildings per city common cities amongst games united states appearing games followed new york city district columbia north american locations dominated top common cities locations remaining two london germany note game represents locations cities buildings city category may include countries united states states actual cities average items generated per game mostly books games fewest books created respectively corbusier books theodore roosevelt key items locked buildings tend appear together average key items buildings every game least one key one locked building three keys three locked buildings single game number keys always equal higher locked buildings ensuring solvability average npcs created per game average based real people ratio ratio optimal believe improved future versions lenient npc generation look people distance article originated node person could expand search degrees distances believe improve ratio believe increasing percentage npcs based real people random npcs would provide interesting characters interactions ratio npcs based real people ranged npcs game real npcs generated lech real npcs one least walter reuther based table dialog nodes average wikimystery game distributed across npcs game results show average dialog nodes per person every person main branch dialog tree number main branches equal number npcs addition average sidebranches game average refer npc personal achievements concern npc current residence associated person birth nearly twice many birth two types since generator creates two branches birth date birth place selecting birth suspects direct connections evidence generated game must set suspects evidence innocence direct connections suspect victim reason selecting suspects table shows set suspects evidence innocence direct connection victim suspect three influential people time list albert einstein franklin roosevelt mahatma gandhi values italics used evidence innocent suspects allowing player differentiate culprit notice culprit value game generated franklin roosevelt daniel poulter value party characteristic value would fit differentiate suspect game check characteristic specific paired suspect game generated franklin roosevelt gwendolyn garcia paired party characteristic additionally two innocent suspects share value characteristic used evidence one long different culprit merely means evidence one suspect innocent absolve suspect example shown mahatma gandhi game see table tex avery jhunnilal verma died evidence innocence avery note eddie lyons culprit died cases actual value appeared consequence wikipedia organization mahatma gandhi game primary reason selecting tex avery eddie lyons suspects belonging like gandhi category articles containing video clips indicating appear wikipedia list articles containing video clips secondary reason set five suspect allowed solvable somewhat diverse game according direct connections relations victim suspect since depend hyperlinks victim table solution games generated top influential people albert einstein franklin roosevelt mahatma gandhi innocent suspect paired one characteristic blue italics differentiate killer whose name shown asterisk last list empty values appear game unknown direct connection column shows primary criteria choosing suspect relationship victim politicians means victim suspect politicians thus suspect directly connected victim albert einstein suspects jakob meisenheimer death place field subject deaths nazi germany sir david mackay information theory living people thermodynamicists fellows royal society women scientists glasgow physics allen shenstone united states physics rosa beddington great tew developmental biology suspects term end kevin cahill gwendolyn garcia william rankine almamater ludwig maximilian university munich california institute technology university edinburgh brasenose college oxford direct connection born german empire received royal society award physicists died princeton new jersey received royal society award almamater direct connection state university new york part democratic party princeton university franklin roosevelt party democratic party district office member new york assembly new paltz one cebu governor cebu johnny ellis democratic party jane griffiths labour party daniel poulter majority leader alaska senate member parliament member parliament suspects death year university philippines diliman claremont mckenna college part democratic party politicians durham university politicians university bristol politicians direct connection appear wikipedia category articles containing video clips politicians mahatma gandhi birth place subject occupation articles containing video clips animator cartoonist voice actor director tex avery taylor texas stanley rosen cleveland jhunnilal verma eddie lyons writer century philosophers damoh india jewish american writers philosophers people damoh lawyer beardstown illinois usa articles containing video clips actor director screenwriter producer indian lawyers appear wikipedia category articles containing video clips landau volker zotz wikipedia page varied article therefore games usually emergent underlying theme game generated albert einstein allen shenston william rankine field physics shenston also died princeton new jersey einstein david mackay rosa beddington received royal society award einstein jakob meisenheimer born german empire franklin roosevelt game roosevelt suspects except kevin cahill roosevelt cahill johnny ellis part democratic party finally mahatma gandhi game jhunnilal verma indian lawyers gandhi volker zotz stanley rosen century philosophers eddie lyons tex avery appear wikipedia category articles containing video clips last connection arguably poorer others demonstrating source data difficult tailor needs discussed extensively section cahill actually politician tagged one dbpedia century philosophers iscussion sample playthrough section vii numerical evaluations section viii provide overview types games generated current wikimystery prototype contrary early attempts adventure generation created one path two people murder mystery far less linear includes dialog gameplay options fact paths traversed nonsequentially inevitably difficult keep track npc object forms path towards suspect increases exploration branching factor terms part player turn leads interesting gameplay gives greater sense player agency gameplay improved branching better visual presentation results interesting dialog options concrete winning condition priority authors improve gameplay quality broader data adventures project biggest appeal remains link data accessed darling house early colonial image confucius chosen australia highlighted placed hermann einstein ask israel jews dialog unfortunate fig absurd potentially offensive combinations data occur wikimystery via open data repositories based metrics table aspect wikimystery strengthened well game containing multitude cities placed locations city map showing street view based openstreetmap ratio random npcs real npcs based wikipedia articles also kept balance introduction photograph objects increases modes open data experienced images rather text information found book objects importantly improved npc dialog allows engaging intuitive way solve mystery also allows yet another way present open data player choose questions ask npc regarding life achievements rather presented data large chunk text observing npc example although substantial improvements presentation content earlier iterations data adventures nature generating games open data hinges uncontrollable nature data allows expressivity person wikipedia presence potentially star generated game murdered lack control lead unexpected unintended even unwanted outcomes one hand ongoing efforts data adventures line research focus controlling vast repository data transforming intuitive playable objects instance attempting find unique connections people rather trivial ones human however impossible ever fully control constrain experience would obfuscate origins living vast knowledge base rooted deeply real world absurdity makes outcomes appealing way user another game titled rogue dream states feels like playing videogame internet least case wikimystery intentional absurdity however causes hilarious sometimes appalling outcomes noted playthrough section vii npcs images correct either due lack appropriate images people wikimedia commons flaws image parsers currently hand cases random search image man male npcs woman female npcs used instead buildings moreover image search based name building without context geographical location lead results fig building background old photograph actual highlighted building red circle closer inspection chosen building result search house nathan rose darling house holds historical significance early colonial australia game used domicile nathan rosen israel choice using freely available sources wikimedia commons complicates retrieval specific images source google images could improve results contradicts scope freely available solutions therefore future work improve search appropriate images possibly increasing breadth searches repositories performing computer vision verification image one person additionally problematic instances unforeseen combination content transformation lead insensitive offensive results example fig shows dialog hermann einstein part playthrough section vii player seeking culprit albert einstein murder image unsurprisingly hermann einstein instead random search image man serendipitously ended drawing confucius hand dialog chosen highlight connection person next along path category jews category also cued photograph fig discussed extensively section vii case player interacts hermann einstein dialog line hask jewsi certainly true actual story albert einstein deeply affected jewish events world war category path found accurate perhaps desirable however random choice dialog random assignment image confucius avatar unfortunate insensitive likely offensive combination difficult envision instances could avoided largely issue simple transformations data combination going awry case one also underestimate nature open online data often tainted popular belief misconception stereotype prejudice opposed purely factual information thus unfortunate instances may actually occur due prejudice source data even transformed still important directions future work order improve usability game narrative consistency example interface additions travel diary could help player keep track clues connections npcs objects locations characteristics moreover current dialog format uses fixed templates sequences perhaps approach tracery could result diverse dialogs furthermore important missing component narrative murder culprit motive possible motives suspects unlikely motive jealousy consider inoffensive similar line saying hask physicsi would would based real facts data although could generated based relationships people siblings spouses exploring relations npcs personalities goals seems promising using data trivial even sentiment analysis wikipedia article person would express writer feelings actual subject hand yet fully understand players interact view data presented wikimystery one priorities release playable version game online setting logging system perform user studies also intend investigate possibility using wikimystery gain insight correctness data dbpedia wikipedia onclusion paper presented latest installment wikimystery game detailed complex generation pipeline name person wikipedia article full interactive murder mystery game open data used multitude ways order find npc suspects murder specified person find paths linking npcs place locations around globe provide way player absolve innocents deduce culprit moreover open data used create levels cities buildings npcs found create objects photographs books act clues enhance dialog options npcs beyond merely functional needs completing game many directions future work order increase gameplay intuitiveness provide better link visuals content reduce absurdity combinations current wikimystery generator first create fully playable adventure games minimal human authorship curation acknowledgments npcs discussed generated adventures instantiated real people similarities end npcs actions game victims culprits way reflect people based generator output way accuses misrepresents individuals wikimystery creates fictional counterparts public figures presence wikipedia similarity fictional npcs game people therefore due data available open online freely accessible editable repositories thank ahmed khalifa scott lee helpful insight gabriella barros acknowledges financial support capes science without borders program bex eferences treanor blackford mateas bogost generating videogames represent ideas proceedings fdg workshop procedural content generation cook colton rogue dream automatically generating meaningful content games proceedings aiide workshop experimental games barros liapis togelius murder mystery generation open data proceedings seventh international conference computational creativity data adventures proceedings fdg workshop procedural content generation games friberger togelius cardona ermacora mousten jensen tanase data games workshop procedural content generation cardona hansen togelius gustafsson friberger open trumps data game proceedings foundations digital games togelius gustafsson friberger bar chart ball data game proceedings foundations digital games barros togelius balanced civilization map generation based open data ieee congress evolutionary computation cec ieee bringsjord ferrucci artificial intelligence literary creativity inside mind brutus storytelling machine computational linguistics vol turner minstrel computer model creativity storytelling dissertation los angeles usa vink bardiche interactive online narrative generator thesis lopes liapis yannakakis sonancia sonification procedurally generated game levels proceedings computational creativity games workshop thorogood pasquier eigenfeldt audio metaphor audio information retrieval soundscape composition proceedings sound music computing cong smc cook colton gow automating game design three dimensions proceedings aisb symposium games alderman ashmore compton shapiro nitsche many worlds charbitat game set match ashmore nitsche quest generated world proceedings digra conference lavender thompson adventures hyrule generating missions maps action adventure games thomson procedural generation narrative puzzles adventure games system proceedings third workshop procedural content generation games acm creating dreamlike game worlds procedural content generation seventh intelligent narrative technologies workshop weizenbaum eliza computer program study natural language communication man machine vol hernault piwek prendinger ishizuka generating dialogues virtual agents using nested textual coherence relations international workshop intelligent virtual agents springer cavazza charles dialogue generation interactive proceedings interactive digital entertainment conference osterweil key adventure game design insight proceedings meaningful play togelius nardi lucas towards automatic personalised content creation racing games ieee symposium computational intelligence games ieee barros liapis togelius playing data procedural generation adventures open data proceedings international joint conference digra fdg golden albert einstein vol compton kybartas mateas tracery generative text tool international conference interactive digital storytelling springer stockdale cluegen exploration procedural storytelling format murder mystery games proceedings aiide 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nov ensemble sampling xiuyuan stanford university lxy benjamin van roy stanford university bvr abstract thompson sampling emerged effective heuristic broad range online decision problems basic form algorithm requires computing sampling posterior distribution models tractable simple special cases paper develops ensemble sampling aims approximate thompson sampling maintaining tractability even face complex models neural networks ensemble sampling dramatically expands range applications thompson sampling viable establish theoretical basis supports approach present computational results offer insight introduction thompson sampling emerged effective heuristic trading exploration exploitation broad range online decision problems select action algorithm samples model system prevailing posterior distribution determines action maximizes expected immediate reward according sampled model basic form algorithm requires computing sampling posterior distribution models tractable simple special cases complex models neural networks exact computation posterior distributions becomes intractable one resort laplace approximation discussed example approach suitable posterior distributions unimodal computations become obstacle complex models like neural networks compute time requirements grow quadratically number parameters alternative leverage markov chain monte carlo methods computationally onerous especially model complex practical approximation thompson sampling address complex models problems requiring frequent decisions facilitate fast incremental updating time required per time period learn new data generate new sample model small grow time fast incremental method builds laplace approximation concept presented paper study fast incremental method applies broadly without relying unimodality sanity check offer theoretical assurances apply special case linear bandits also present computational results involving simple bandit problems well complex neural network models demonstrate efficacy approach approach inspired applies similar concept complex context deep reinforcement learning without theoretical analysis essential idea maintain incrementally update ensemble statistically plausible models sample uniformly set time period approximation sampling posterior distribution model initially sampled prior updated manner incorporates data random perturbations diversify models intention ensemble approximate posterior distribution variance among models diminish posterior concentrates refine methodology bound incremental regret relative exact thompson sampling conference neural information processing systems nips long beach usa broad class online decision problems bound indicates suffices maintain number models grows logarithmically horizon decision problem ensuring computational tractability approach problem formulation consider broad class online decision problems thompson sampling could principle applied though would typically hindered intractable computational requirements define random variables respect probability space endowed filtration convention random variables index use denote probabilities expectations conditioned decisionmaker chooses actions observes outcomes random variable represents model index conditioned independent depend thought bayesian formulation randomness reflects prior uncertainty model corresponds true nature system assume finite action chosen randomized policy realization probability mass function actions sampled independently agent associates reward outcome reward function fixed known let denote reward realized time let uncertainty induces uncertainty true optimal action denote arg max let conditional regret actions chosen according defined regret expectation taken randomness actions outcomes conditioned illustrate couple examples fit formulation example linear bandit let drawn distributed according prior set actions time action selected reward observed example neural network let denote mapping induced neural network weights suppose actions serve inputs neural network goal select inputs yield desirable outputs time action selected observed reward associated observation let distributed according prior idea data pairs used fit neural network model actions selected trade generating data pairs reduce uncertainty neural network weights offer desirable immediate outcomes algorithms thompson sampling offers heuristic policy selecting actions time period algorithm samples action posterior distribution optimal action words thompson sampling uses policy easy see equivalent sampling model index posterior distribution models selecting action arg max optimizes sampled model thompson sampling computationally tractable problem classes like linear bandit problem posterior distribution gaussian parameters updated incrementally efficiently via kalman filtering outcomes observed however algorithm ensemblesampling sample sample unif act arg max observe update end ing complex models like neural networks computing posterior distribution becomes intractable ensemble sampling serves approximation thompson sampling contexts posterior interpreted distribution statistically plausible models mean models sufficiently consistent prior beliefs history observations interpretation mind thompson sampling thought randomly drawing range statistically plausible models ensemble sampling aims maintain incrementally update sample finite set models spirit particle filtering set models approximates posterior distribution workings ensemble sampling ways intricate conventional uses particle filtering however interactions ensemble models selected actions skew distribution elements ensemble sampling require customization general template presented algorithm algorithm begins sampling models prior distribution time period model sampled uniformly ensemble action selected maximize expected reward sampled model resulting outcome observed models updated produce explicit algorithm must specify model class prior distribution algorithms sampling prior updating models concrete illustration let consider linear bandit example though ensemble sampling unwarranted case since thompson sampling efficient linear bandit serves useful context understanding approach standard algorithms used sample models prior one possible procedure updating models maintains covariance matrix updating according generates model parameters incrementally according independent random samples drawn updating algorithm easy show resulting parameter vectors satisfy arg min admits intuitive interpretation model fit randomly perturbed prior randomly perturbed observations establish appendix deterministic sequence conditioned models independent identically distributed according posterior distribution sense ensemble approximates posterior new observation deterministic action sequences scheme generates exact samples posterior distribution see however stochastic action sequences selected algorithm immediately clear well ensemble approximates posterior distribution provide bound next section establishes number models increases regret ensemble sampling quickly approaches thompson sampling ensemble sampling algorithm described linear bandit problem motivates analogous approach neural network model example approach would begin models connection weights sampled prior could natural let variance chosen range probable models spans plausible outcomes incrementally update parameters time model applies number stochastic gradient descent iterations reduce loss function form present computational results section demonstrate viability approach analysis ensemble sampling linear bandit past analyses thompson sampling relied independence models sampled time periods ensemble sampling introduces dependencies may adversely impact performance immediately clear whether degree degradation tolerable depends number models ensemble section establish bound linear bandit context result serves sanity check ensemble sampling offers insight extend broader model classes though leave formal analysis beyond linear bandit future work consider linear bandit problem described example let denote thompson ensemble sampling policies problem latter based ensemble models generated updated according procedure described section let denote worst mean reward let denote gap maximal minimal mean rewards following result bounds difference regret function gap ensemble size number actions theorem log regret regret inequality bounds regret realized ensemble sampling sum regret realized thompson sampling error term since talking cumulative regret error term bounds degradation relative thompson sampling value made arbitrarily small increasing hence sufficiently large ensemble loss small supports viability ensemble sampling important implication result suffices ensemble size grow logarithmically horizon since thompson sampling requires independence models sampled time sense relies models one per time period useful ensemble sampling operate effectively much smaller number logarithmic dependence suitable bound also grows log manageable modest number actions conjecture similar bound holds depends instead multiple log linear dimension would offer stronger guarantee number actions becomes large infinite though leave proof alternative bound future work bound theorem notion regret conditioned realization bayesian regret bound removes dependence realization obtained taking expectation integrating regret regret provide complete proof theorem appendix due space constraints offer sketch sketch proof let denote action process procedure generating updating models ensemble sampling designed deterministic conditioned history rewards models comprise ensemble independent identically distributed according posterior distribution verified via algebra done appendix recall denotes posterior probability explicitly indicate dependence action process use superscript let denote approximation given arg note given action process time thompson sampling would sample next action ensemble sampling would sample next action deterministic since conditioned history rewards distributed represents empirical distribution samples drawn follows sanov theorem deterministic dkl kpt naive application union bound deterministic action sequences would establish deterministic stochastic max dkl however proof takes advantage fact deterministic depend ordering past actions observations make precise encode sequence actions terms action counts particular let number times action selected time apply coupling argument introduces dependencies noise terms action counts without changing distributions observable variables let random variables let zct similarly let random variables let make explicit dependence use superscript write denote action counts time action process given hard verify done appendix two deterministic action sequences pat pat allows apply union bound action counts instead action sequences get deterministic stochastic dkl kpt max dkl kpt specialize action process action sequence aes selected ensemble sampling omit superscripts decompose regret ensemble sampling dkl kpt dkl kpt first term bounded dkl kpt dkl kpt bound second term use another coupling argument couples actions would selected ensemble sampling would selected thompson sampling let ats denote thompson sampling would select time dkl kpt ktv pinsker inequality conditioning dkl kpt construct random variables distributions ats respectively using maximal coupling make probability least ktv second term sum decomposed dkl kpt dkl kpt algebraic manipulations leads dkl kpt ats result follows straightforward algebra computational results section present computational results demonstrate viability ensemble sampling start simple case independent gaussian bandits section move complex models neural networks section section serves sanity check empirical performance ensemble sampling thompson sampling efficiently applied case able compare performances two algorithms addition provide simulation results demonstrate ensemble size grows number actions section goes beyond theoretical analysis section gives computational evidence efficacy ensemble sampling applied complex models neural networks show ensemble sampling even models achieves efficient learning outperforms dropout example neural networks gaussian bandits independent arms consider gaussian bandit actions action mean reward drawn time step select action observe reward note special case example since posterior distribution explicitly computed case use sanity check performance ensemble sampling figure shows regret thompson sampling ensemble sampling applied gaussian bandit independent arms see number models increases ensemble sampling better approximates thompson sampling results averaged realizations figure shows minimum number models required expected regret ensemble sampling plus expected regret thompson sampling large time horizon across different numbers actions results averaged realizations chose plot shows number models needed seems grow sublinearly number actions stronger bound proved section neural networks section follow example show computational results ensemble sampling applied neural networks figure shows ensemble sampling applied bandit problem mapping actions expected rewards represented neuron specifically set actions mean reward selecting action given max weights drawn time period select action observe reward set input dimension number actions prior variance noise variance dimension action sampled uniformly except last dimension set figure consider bandit problem mapping actions expected rewards represented neural network weights entry weight matrices drawn independently set actions mean reward choosing action max ensemble sampling independent gaussian bandit arms number models regret ensemble sampling independent gaussian bandits thompson sampling models models models models number actions figure ensemble sampling compared thompson sampling gaussian bandit independent arms minimum number models required expected regret ensemble sampling plus expected regret thompson sampling gaussian bandits across different numbers arms time period select action observe reward used input dimension dimension hidden layer number actions prior variance noise variance dimension action sampled uniformly except last dimension set ensemble sampling models starts sampling prior distribution independently model time step pick model uniformly random apply greedy action respect model update ensemble incrementally time period apply steps stochastic gradient descent model respect loss function perturbations drawn besides ensemble sampling heuristics sampling approximate posterior distribution neural networks may used develop approximate thompson sampling gal ghahramani proposed approach based dropout approximately sample posterior neural networks figure include results using dropout approximate thompson sampling neural network bandit facilitate gradient flow used leaky relus form max internally agents target neural nets still use regular relus described took stochastic gradient steps minibatch size model update used learning rate ensemble sampling learning rate dropout dropping probabilities respectively results averaged around realizations figure plots regret ensemble sampling single neuron bandit see ensemble sampling even models performs better best tuned parameters increasing size ensemble improves performance ensemble size achieves orders magnitude lower regret figure show different versions applied neural network model see annealing schedule tends perform better fixed figure plots regret dropout approach different dropping probabilities seems perform worse figure plots regret ensemble sampling neural net bandit see ensemble sampling moderate number models outperforms approaches significant amount ensemble sampling agent name instant regret figure ensemble sampling applied single neuron bandit fixed epsilon annealing epsilon agent name instant regret dropout ensemble sampling figure fixed annealing dropout ensemble sampling applied neural network bandit conclusion ensemble sampling offers potentially efficient means approximate thompson sampling using complex models neural networks provided analysis offers theoretical assurances case linear bandit models computational results demonstrate efficacy complex neural network models motivated largely need effective exploration methods efficiently applied conjunction complex models neural networks ensemble sampling offers one approach representing uncertainty neural network models others might also brought bear developing approximate versions thompson sampling analysis various forms approximate thompson sampling remains open ensemble sampling loosely relates ensemble learning methods though important difference motivation lies fact latter learns multiple models purpose generating accurate model combination former learns multiple models reflect uncertainty posterior distribution models said combining two related approaches may fruitful particular may practical benefit learning many forms models neural networks models etc viewing ensemble representing uncertainty one sample acknowledgments work generously supported research grant boeing marketing research award adobe references charles blundell julien cornebise koray kavukcuoglu daan wierstra weight uncertainty neural networks proceedings international conference international conference machine learning volume icml pages olivier chapelle lihong empirical evaluation thompson sampling shawetaylor zemel bartlett pereira weinberger editors advances neural information processing systems pages curran associates thomas dietterich ensemble learning handbook brain theory neural networks yarin gal zoubin ghahramani dropout bayesian approximation representing model uncertainty deep learning maria florina balcan kilian weinberger editors proceedings international conference machine learning volume proceedings machine learning research pages new york new york usa jun pmlr carlos online algorithms parameter mean variance estimation dynamic regression arxiv preprint ian osband charles blundell alexander pritzel benjamin van roy deep exploration via bootstrapped dqn lee sugiyama luxburg guyon garnett editors advances neural information processing systems pages curran associates george papandreou alan yuille gaussian sampling local perturbations lafferty williams zemel culotta editors advances neural information processing systems pages curran associates thompson likelihood one unknown probability exceeds another view evidence two samples biometrika proof theorem recall section procedure generwithout loss generality assume ating updating models ensemble sampling first sampled vectors adapted according arg min note yet specified actions selected formulation put forth could random variable denote process say deterministic exist probability one lemma deterministic conditioned random variables proof say let matrix row equal let arg min arg min first observe conditioned follows normal distribution since affine next check mean covariance since independently sampled cov therefore deterministic random variable conditioned since independent independent recall denotes posterior probability explicitly indicate dependence action process use superscript let denote approximation given arg note given action process time thompson sampling would sample next action ensemble sampling would sample next action following lemma shows deterministic action sequence conditioned action distribution ensemble sampling would sample close action distribution thompson sampling would sample high probability lemma deterministic action sequence dkl kpat proof deterministic conditioned pat independent thus dkl kpat dkl kpat lemma conditioned distributed posterior thus represents empirical distribution samples drawn pat sanov theorem implies dkl kpat result follows next establish results action process deterministic stochastic useful introduce notion action counts one way encoding sequence actions terms counts particular let number times action selected time takes values set since components component takes value sometimes use superscript write explicitly denote dependence action process introduce dependencies noise terms action counts without changing distributions observable variables turn useful take union bound later let random variables let zct similarly let random variables let following lemma establishes deterministic action sequence pat depend action counts words pat depend ordering past actions observations lemma deterministic sequences pat pat proof recall means observe reward first time take action regardless action appears action sequence similarly action sequences observe reward second time take action therefore implies pat pat reasoning since action sequences would yield model parameters follows lemma process dkl kpt proof dkl kpt max dkl kpat max dkl kpat dkl kpat follows lemma follows union bound follows lemma fact total number counts specialize action process action sequence aes selected ensemble sampling omit superscripts expected cumulative regret ensemble sampling conditioned decomposed regret aes aes dkl kpt aes dkl kpt bound regret case divergence dkl kpt large case divergence small respectively lemma aes dkl kpt proof follows directly definition lemma assumption simplicity assume lemma size ensemble satisfies log aes dkl kpt proof show satisfies condition equivalently log log log log log log log log log log log log log since assumption implies log result follows lemma lemma let denote thompson sampling policy aes dkl kpt regret proof apply coupling argument couples actions would selected ensemble sampling would selected thompson sampling let ats denote action thompson sampling would select time dkl kpt pinsker inequality implies ktv conditioning dkl kpt construct random variables distribution respectively using maximal coupling make probability least dkl kpt aes dkl kpt dkl kpt dkl kpt dkl kpt first part sum dkl kpt ats ats inequality follows nonnegativity second part sum dkl kpt dkl kpt last inequality follows way couple thus result follows combining lemma lemma delivers proof main result particular regret aes dkl kpt aes dkl kpt regret regret inequality follows lemma lemma
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published conference paper iclr eep iaffine attention eural ependency parsing christopher manning stanford university manning mar timothy dozat stanford university tdozat bstract paper builds recent work kiperwasser goldberg using neural attention simple dependency parser use larger thoroughly regularized parser recent approaches biaffine classifiers predict arcs labels parser gets state art near state art performance standard treebanks six different languages achieving uas las popular english ptb dataset makes parser outperforming kiperwasser goldberg comparable highest performing parser kuncoro achieves uas las also show hyperparameter choices significant effect parsing accuracy allowing achieve large gains approaches ntroduction dependency annotate sentences way designed easy humans computers alike found extremely useful sizable number nlp tasks especially involving natural language understanding way bowman angeli levy goldberg toutanova parikh however frequent incorrect parses severely inhibit final performance improving quality dependency parsers needed improvement success downstream tasks current neural dependency parser kuncoro substantially outperforms many much simpler neural parsers modify neural graphbased approach first proposed kiperwasser goldberg ways achieve competitive performance build network larger uses regularization replace traditional attention mechanism affine label classifier biaffine ones rather using top recurrent states lstm biaffine transformations first put mlp operations reduce dimensionality furthermore compare models trained different architectures hyperparameters motivate approach empirically resulting parser maintains simplicity neural approaches approaching performance sota one background elated work sentences left right maintaining buffer words yet parsed stack words whose head seen whose dependents fully parsed step parsers access manipulate stack buffer assign arcs one word another one train machine learning classifier features extracted stack buffer previous arc actions order predict next action chen manning make first successful attempt incorporating deep learning dependency parser step feedforward network assigns probability action parser take based word tag label embeddings certain words published conference paper iclr root nsubj dobj figure dependency tree parse casey hugged kim including tags special root token directed edges arcs labels relations connect verb root arguments verb head stack buffer number researchers attempted address limitations chen manning chen manning parser augmenting additional complexity weiss andor augment beam search conditional random field loss objective allow parser undo previous actions finds evidence may incorrect dyer kuncoro instead use lstms represent stack buffer getting performance building way composing parsed phrases together parsing processes sentence sequentially build parse tree one arc time consequently parsers use machine learning directly predicting edges use predicting operations transition algorithm parsers contrast use machine learning assign weight probability possible edge construct maximum spaning tree mst weighted edges kiperwasser goldberg present neural parser addition one uses kind attention mechanism bahdanau machine translation kiperwasser goldberg model bidirectional lstm recurrent output vector word concatenated possible head recurrent vector result used input mlp scores resulting arc predicted tree structure training time one word depends highestscoring head labels generated analogously word recurrent output vector gold predicted head word recurrent vector used mlp similarly hashimoto include dependency parser neural model addition training model multiple distinct objectives replace traditional attention mechanism kiperwasser goldberg use bilinear one still using mlp label classifier makes analogous luong proposed attention mechanism neural machine translation cheng likewise propose neural dependency parser way attempts circumvent limitation neural parsers unable condition scores possible arc previous parsing decisions addition one bidirectional recurrent network computes recurrent hidden vector word additional unidirectional recurrent networks keep track probabilities previous arc use together predict scores next arc roposed ependency parser eep biaffine attention make modifications architectures kiperwasser goldberg hashimoto cheng shown figure use biaffine attention instead bilinear traditional attention use biaffine dependency label classifier apply mlps recurrent output vector applying biaffine choice biaffine rather bilinear mlp mechanisms makes classifiers model analogous traditional affine classifiers use affine transformation single lstm output state vector input predict vector scores classes think proposed biaffine attention mechanism traditional affine paper follow convention using lowercase italic letters scalars indices lowercase bold letters vectors uppercase italic letters matrices uppercase bold letters higher order tensors also maintain notation indexing row matrix would represented published conference paper iclr arc arc mlp bilstm embeddings root root kim nnp figure bilstm deep biaffine attention score possible head dependent applied sentence casey hugged kim reverse order biaffine transformation clarity classifier using linear transformation stacked lstm output place weight matrix transformation bias term affine classifier arc biaffine classifier addition arguably simpler approach involving one bilinear layer rather two linear layers nonlinearity conceptual advantage directly mod eling prior probability word receiving dependents term likelihood receiving specific dependent term analogously also use biaffine classifier predict dependency labels given gold predicted head label biaffine classifier ryi likewise directly models prior probability class likelihood class given word probable word take particular label likelihood class given head word probable word take dependents particular label likelihood class given word head probable word take particular label given word head applying smaller mlps recurrent output states biaffine classifier advantage stripping away information relevant current decision every top recurrent state need carry enough information identify word head find dependents exclude assign correct label assign dependents correct labels well transfer relevant information recurrent states words thus necessarily contains significantly information needed compute individual score training superfluous information needlessly reduces parsing speed increases risk overfitting reducing dimensionality applying nonlinearity addresses problems call deep bilinear attention mechanism opposed shallow bilinear attention uses recurrent states directly arc mlp mlp apply mlps recurrent states using label classifier well models predicted tree training time one word dependent highest scoring head although test time ensure parse tree via mst algorithm published conference paper iclr yperparameter configuration param embedding size lstm size arc mlp size label mlp size lstm depth annealing value param embedding dropout lstm dropout arc mlp dropout label mlp dropout mlp depth tmax value table model hyperparameters aside architectural differences parsers make number hyperparameter choices allow outperform laid table use uncased word pos tag vectors three bilstm layers dimensions direction relu mlp layers also apply dropout every stage model drop words tags independently drop nodes lstm layers input recurrent connections applying dropout mask every recurrent timestep bayesian dropout gal ghahramani drop nodes mlp layers classifiers likewise applying dropout mask every timestep optimize network annealed adam kingma steps rounded nearest epoch xperiments esults datasets show test results proposed model english penn treebank converted stanford dependencies using version version stanford dependency converter chinese penn treebank conll shared task following standard practices dataset omit punctuation evaluation ctb english datasets use pos tags generated stanford pos tagger toutanova chinese ptb dataset use gold tags conll dataset use provided predicted tags hyperparameter search done validation dataset order minimize overfitting popular benchmark hyperparameter analysis following section report performance test set shown tables yperparameter choices attention mechanism examined effect different classifier architectures accuracy performance see deep bilinear model outperforms others respect speed accuracy model shallow bilinear arc label classifiers gets unlabeled performance deep model settings label classifier much larger opposed runs much slower overfits one way decrease overfitting increasing mlp dropout course change parsing speed another way decrease recurrent size hinders unlabeled accuracy without increasing parsing speed levels deeper model also implemented approach attention classification used kiperwasser goldberg found version compute trained embedding matrix composed words occur least twice training dataset add embeddings corresponding pretrained embeddings words occur either embedding matrix replaced separate oov token exclude japanese dataset evaluation access version tensorflow used model memory requirements training exceeded available memory single gpu default settings used reduced mlp hidden size published conference paper iclr classifier uas las recurrent cell uas las model deep shallow shallow drop shallow mlp model lstm gru model layers layers layers layers layers size uas las table test accuracy speed statistically significant differences marked asterisk input dropout model uas default word dropout tag dropout tags las model adam uas las table test accuracy statistically significant differences marked asterisk likewise somewhat slower significantly underperform deep biaffine approach labeled unlabeled accuracy etwork size also examine closely network size influences speed accuracy kiperwasser goldberg model network uses layers bidirectional lstms hashimoto model one layer bidirectional lstms dedicated parsing two lower layers also trained objectives cheng model one layer gru cells find using three four layers gets significantly better performance two layers increasing lstm sizes dimensions likewise signficantly improves ecurrent cell gru cells promoted faster simpler alternative lstm cells used approach cheng however model drastically underperformed lstm cells also implemented coupled gate lstm cells suggested greff finding resulting model still slightly underperforms popular lstm cells difference two much smaller additionally gate candidate cell activations computed simultaneously one matrix multiplication model faster gru version even though number parameters hypothesize output gate model allows maintain sparse recurrent output state helps adapt high levels dropout needed prevent overfitting way gru cells unable model recurrent states significantly outperforms one validation set test set addition using coupled gate remove first tanh nonlinearity longer needed using coupled gate published conference paper iclr type model english uas las chinese ptb uas las transition ballesteros andor kuncoro graph kiperwasser goldberg cheng hashimoto deep biaffine table results english ptb chinese ptb parsing datasets model catalan uas las chinese uas las czech uas las andor deep biaffine model english uas las german uas las spanish uas las andor deep biaffine table results conll shared task datasets mbedding ropout increase parser power also increase regularization addition using relatively extreme dropout recurrent mlp layers mentioned table also regularize input layer drop words tags training one dropped scaled factor two compensate dropped together model simply gets input zeros models trained word tag dropout wind signficantly overfitting hindering label accuracy latter accuracy interestingly using tags actually results better performance using tags without dropout ptimizer choose optimize adam kingma among things keeps moving average norm gradient parameter throughout training divides gradient parameter moving average ensuring magnitude gradients average close one however find value recommended kingma controls decay rate moving high task suspect generally value large magnitude current update heavily influenced larger magnitude gradients far past effect optimizer adapt quickly recent changes model thus find setting instead makes large positive impact final performance esults model gets nearly uas performance current sota model kuncoro spite substantially simpler architecture gets sota uas performance ctb well sota performance conll languages worth noting conll datasets contain many dependencies difficult impossible predict may account large consistent difference model andor model applied datasets like thank zhiyang teng finding bug original code affected ctb dataset published conference paper iclr model appears lag behind sota model las indicating one possibilities firstly may result inefficiencies errors glove embeddings pos tagger case using alternative pretrained embeddings accurate tagger might improve label classification secondly sota model specifically designed capture phrasal compositionality another possibility capture compositionality effectively results worse label score similarly may result general limitation parsers access less explicit syntactic information parsers making decisions addressing latter two limitations would require innovative architecture relatively simple one used current neural parsers onclusion paper proposed using modified version bilinear attention neural dependency parser increases parsing speed without hurting performance showed larger regularized network outperforms neural parsers gets comparable performance current sota parser also provided empirical motivation proposed architecture configuration similar ones existing literature future work involve exploring ways bridging gap labeled unlabeled accuracy augment parser smarter way handling tokens morphologically richer languages eferences daniel andor chris alberti david weiss aliaksei severyn alessandro presta kuzman ganchev slav petrov michael collins globally normalized transitionbased neural networks association computational linguistics url https gabor angeli melvin johnson premkumar christopher manning leveraging linguistic structure open domain information extraction proceedings annual meeting association computational linguistics acl dzmitry bahdanau kyunghyun cho yoshua bengio neural machine translation jointly learning align translate international conference learning representations miguel ballesteros yoav goldberg chris dyer noah smith training exploration improves greedy parser proceedings conference empirical methods natural language processing samuel bowman jon gauthier abhinav rastogi raghav gupta christopher manning christopher potts fast unified model parsing sentence understanding acl danqi chen christopher manning fast accurate dependency parser using neural networks proceedings conference empirical methods natural language processing hao cheng hao fang xiaodong jianfeng gao deng attention agreement dependency parsing arxiv preprint chris dyer miguel ballesteros wang ling austin matthews noah smith transitionbased dependency parsing stack long memory proceedings conference empirical methods natural language processing yarin gal zoubin ghahramani dropout bayesian approximation representing model uncertainty deep learning international conference machine learning klaus greff rupesh kumar srivastava jan bas steunebrink schmidhuber lstm search space odyssey ieee transactions neural networks learning systems published conference paper iclr kazuma hashimoto caiming xiong yoshimasa tsuruoka richard socher joint model growing neural network multiple nlp tasks arxiv preprint diederik kingma jimmy adam method stochastic optimization international conference learning representations eliyahu kiperwasser yoav goldberg simple accurate dependency parsing using bidirectional lstm feature representations transactions association computational linguistics adhiguna kuncoro miguel ballesteros lingpeng kong chris dyer graham neubig noah smith recurrent neural network grammars learn syntax corr url http omer levy yoav goldberg word embeddings acl luong hieu pham christopher manning effective approaches attentionbased neural machine translation empirical methods natural language processing ankur parikh hoifung poon kristina toutanova grounded semantic parsing complex knowledge extraction proceedings north american chapter association computational linguistics kristina toutanova dan klein christopher manning yoram singer tagging cyclic dependency network proceedings conference north american chapter association computational linguistics human language association computational linguistics kristina toutanova victoria lin yih compositional learning embeddings relation paths knowledge bases text acl david weiss chris alberti michael collins slav petrov structured training neural network parsing annual meeting association computational linguistics
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minimax design nonlinear phase fir filters optimality certificates aug sefa demirtas algorithm provides efficient method designing linear phase fir filter weight function approximation error given filter order specified weight function filter designed algorithm unique optimal frequency response approximates desired filter response certified alternation theorem paper nonlinear phase fir filter design algorithm provided allows specification piecewise constant weight function approximation error analogous manner linear phase fir filters given filter order weight function resulting filter provably unique optimal magnitude response approximates desired filter response certification optimality given also based alternations weighted error function exhibits furthermore method applicable designing filters coefficients turn determines number required alternations index phase fir design minimax optimality alternation theorem ntroduction despite desirable properties fir filters certain disadvantages compared iir counterparts example minimum order required fir filter approximate desired filter response within bounds usually much higher order iir filter task translates multiplications additions per input sample hardware implementation requires larger power memory furthermore even though linear phase fir filters introduce dispersion uniform latency amount latency samples half filter order may become unacceptably high applications therefore crucial optimize fir filter given order keeping order low possible optimality stated respect particular norm several design methods exist meet different optimality criteria example windowing methods used minimize sum absolute squares approximation error filter response desired response given order hand method aims minimize maximum absolute error also known minimax chebyshev error since minimizing norm equivalent minimizing energy error prevent large narrow deviations filter frequency response desired response result small therefore even though much easier design optimality author analog devices lyric labs email work personal sefa adopted widely signal processing community due superior performance focus paper design method yields unique linear phase fir filter given order exploiting theorem provides necessary sufficient conditions global known alternation theorem even though desirable asserting linear phase restricts coefficients filter realvalued restriction exhibits halving number degrees freedom available approximate desired response choice filter coefficient already determines another coefficient generate symmetric pairs variety applications linear phase crucial becomes unnecessary constraint filters magnitude responses provide much better approximation desired response obtained removing linear phase constraint however renders characterization optimality alternation theorem inapplicable currently known form turn prevents direct utilization method paper first provide characterization method global fir filters restrictions exist phase words state necessary sufficient conditions magnitude response fir filter unique best approximation desired response course due restrictions imposed linear phase design optimal filter unrestricted phase always least good approximation desired response optimal linear phase solution since characterization stated terms magnitude response fir filters sharing order magnitude response optimal therefore although magnitude response unique optimal response finite number distinct optimal fir filters related cascade filter secondly paper arguments characterization shown naturally lead design method involving computation autocorrelation sequence intermediate step end designer able choose variety options phase including minimum phase maximum phase design without compromising global optimality magnitude response since originally introduced hermann schuessler work designing nonlinear phase fir filter first designing autocorrelation sequence finding filter admits autocorrelation widely known technique spectral factorization obvious first choice obtain filter coefficients designed autocorrelation sequence requires finding roots polynomial coefficients autocorrelation sequence since highly impractical approach designing high order filters minimum phase designs several algorithms proposed alternative polynomial root finding however earlier methods nonlinear phase fir filter design autocorrelation sequence filter impulse response designed minimax optimality approximating desired response necessarily imply optimality filter furthermore since magnitude response filter autocorrelation sequence related squaring weight function applied design autocorrelation match desired weight function approximation error attained final design paper first characterize optimality nonlinear phase fir filter instead autocorrelation sequence provide method compute correct weight applied computation autocorrelation sequence resulting filter exhibits desired ratio passband stopband deviations spectral factorization using polynomial root finding alternative methods current literature used obtain final design autocorrelation sequence leading fir design desired error weights global optimality certified characterization optimality inear hase onlinear hase fir ilters linear phase fir filters coefficients expressed amplitude function multiplied linear phase term since linear phase corresponds time delay samples usual approach designing linear phase fir filters first design zerophase filter response necessarily symmetric time domain zero phase afterwards filter time delayed causal corresponds multiplying phase form parks mcclellan exploited fact frequency response symmetric filter even order expressed terms real sinusoids specifically example cos words expressed linear combination basis functions cos order approximate ideal filter response search optimal set linear combination coefficients basis functions maximum absolute error minimized alternation theorem asserts need alternations optimal filter remez exchange algorithm used efficiently find set coefficients since nonlinear phase filters amount exhibit response since alternation theorem remez exchange algorithm apply functions used characterize design minimax optimal nonlinear phase fir filters directly iii haracterization heorem onlinear hase ilters section describe straightforward method characterize global minimax optimality given nonlinear phase fir filter optimality implied magnitude response filter compared desired filter response unity passband zero stopband words examining magnitude response able tell whether filter magnitude response best approximation magnitude response achievable order attain smaller infinity norm weighted approximation error able characterize optimality complexvalued filter coefficients require symmetry coefficients therefore applies general case next section arguments characterization enable find efficient algorithm design nonlinear phase fir filters cases magnitude response specified phase restricted however still able choose among different available phase characteristics including example minimum phase design without compromising global optimality respect magnitude assume fir filter coefficients frequency response provided passband stopband closed subsets approximate desired magnitude response assume desired weight function wdes provided expresses relative emphasis error stopband compared passband specifically wdes kdes kdes scalar given part filter specifications weighted error function bounds passband stopband errors defined wdes max kdes respectively theorem unique magnitude response attained fir filter order order approximate ideal filter magnitude response desired weight function wdes adjusted weighted error function wdes exhibits least alternations filter coefficients restricted least alternations restricted wdes defined wdes subscripts refer real imaginary parts respectively frequency response conjugatesymmetric autocorreation sequence represented cos note context linear phase fir filters alternation theorem used characterization optimality characterized counting alternations weighted error function computed using desired response weight function wdes characterization given nonlinear phase fir filters based adjusted weighted error function computed using wdes equation formal proof characterization theorem theorem given excluded brevity however intuitively optimality related optimality autocorrelation sequence follows due specific choice values stopband wdes number alternations points alternations occur specifically attains extreme value specific frequency hence form alternation fourier transform autocorrelation function also attain extremal value form alternation related weighted error function frequency therefore number required alternations magnitude response filter related autocorrelation sequence turn characterized optimality using traditional form alternation theorem number required alternations filters coefficients larger filters coefficients also consistent intuition reflects additional degrees freedom choosing filter coefficients relaxing constraint realvalued formally autocorrelation function filters also coefficients general exhibit instead real case means autocorrelation sequence filter coefficients satisfy equivalently pre pre pim sin implies frequency response represented linear combination basis functions given cos sin means flexibility choosing coefficients results addition basis functions set available functions represent potentially leading smaller approximation errors intuitively expected furthermore basis set also satisfies haar condition therefore leads unique optimal solution additional basis functions manifest increase number required alternations satisfy traditional form alternation theorem design autocorrelation therefore theroem applies filters alternations opposed alternations example proceeding design procedure close section example fir filter globally minimax optimal illustrate computation adjusted desired response adjusted weight function wdes alternation counting process characterization nonlinear phase fir filters optimality example design coefficients figure smaller ripple sizes one designed matlab firpm function based design filter order passband stopband specified weight kdes chosen example meaning degrees freedom chosen suppress stopband error passband error factor compute equation provided figure computation computed respectively therefore wdes become wdes leads adjusted weight function illustrated figure obtained error indeed exhibits points necessary sufficient condition unique global optimality magnitude response asserted theorem ptimal onlinear hase fir ilter esign lgorithm magnitude response section describe design algorithm filters restricted coefficients therefore require alternations arguments apply filters coefficients simply requiring alternations including sines basis functions computation optimal squared response design constraints weighted error adjusted weighted error adjusted weighted error fig magnitude responses two order fir filters coefficients one designed algorithm proposed section using firpm function matlab weighted function design design adjusted weighted error adjusted weighted error firpm design firpm design exhibits alternations adjusted weighted error computed similarly illustrated figure clearly suboptimal formal steps provided testing optimality bypassed practical observation setting maximum error passband maximum error stopband one directly verify whether filter satisfies desired weight checking equals kdes case alternations also counted directly magnitude response points function reaches extreme points alternating fashion including band edges since wdes tailored turn points alternation points adjusted weighted error approach verified figure extremal points indeed alternations lead alternations including occur band edges designing sequence approximates ideal filter response lifting frequency response nonnegative treating lifted sequence autocorrelation fir filter used nonlinear phase fir filter design method least since however since design specifications relative weight stopband versus passband deviation autocorrelation domain remain filter due squaring relationship resulting filter necessarily reflect desired weight furthermore optimality arguments available final design optimality autocorrelation sequence one set metrics make corresponding filter optimal metrics provide design method correctly accounts relationship computes weight applied design autocorrelation sequence final design exhibits desired ratio passband stopband deviations furthermore characterization theorem section iii certify optimality filter opposed optimality autocorrelation sequence since alternation frequencies number alternations due specific choice wdes characterization theorem designing autocorrelation sequence instead filter correct number alternations satisfies conditions characterization theorem optimality therefore design autocorrelation function satisfies required number alternations recover filter coefficients accept function autocorrelation function using either spectral factorization minimum phase filter particularly desired methods autocorrelation sequence least alternations designed first computing coefficients optimal sequence length fourier transform approximates ideal filter scaling shifting obtain following constraints satisfied swings symmetrically around unity passband extremal values become positive minimum value zero iii maximum value stopband satisfies desired weight constraint kdes first condition guarantees effects squaring properly taken account autocorrelation domain second constraint ensures proper autocorrelation sequence third constraint ensures desired weight compromised squaring relationship illustrated example figure referring passband stopband deviations respectively three constraints represented mathematically terms scale shift coefficients weight applied design specifically choose scaling coefficient shifting coefficient midpoints passband stopband ranges match yields relative weight passband stopband change scaling therefore weights identical since kdes write order match upper bound filter response stopband autocorrelation scale shift inserting values equations inserting equation des since obtain des obtain solving yields finally appropriate weight satisfies equation found scaling shifting coefficients fig example fourier transform symmetric sequence approximates ideal filter response autocorrelation obtained scaling shifting magnitude response nonlinear phase filter autocorrelation example kdes computed directly parameters filter using equation obtain equation equation implicit nonlinear equation expresses correct weight needs applied design terms actual passband deviation obtained using remez exchange algorithm solved efficiently example using iterative procedure cut search space time binary search fashion using method appropriate found design weight scale shift obtain autocorrelation response recover filter coefficients using spectral factorization choose filter minimum phase maximum phase anything desired filter minimum phase efficient methods used instead spectral factorization overall algorithm assume want compute magnitude response order therefore coefficients given kdes start initial guess kdes physically meaningful design see appendix derivation lower bound weight well justification following iterations converge compute coefficients evensymmetric therefore filter order approximate target function weight function done directly using remez exchange algorithm modifying algorithm etc compute frequency response compute maximum value passband error also equivalent maximum value absolute weighted error resulting step satisfies equality equation step otherwise smaller expression equation increase value step greater expression equation decrease value step amount increased decreased bounds search space decided several ways including methods binary search bisection method method appropriate numerical method choice must satisfy kdes physically meaningful design compute scale shift coefficients using equations compute function unit impulse function confused passband stopband ripples namely autocorrelation looking using method including limited spectral factorization obtain coefficients autocorrelation sequence one filter autocorrelation sequence related one another cascade allpass filter choosing zeros unit circle well one pair zeros located unit circle leads minimum phase design choosing zeros outside unit circle one pair pair unit circle leads maximum phase design minimum phase design desired filter order high perform spectral factorization using polynomial root finding methods used find minimum phase solution solutions found finding roots lower order polynomial reflecting zeros inside unit circle outside unit circle desired example figure illustrates design high pass filter kdes particular design desired relative ratio kdes passband stopband deviations weight applied design zero phase sequence computed corresponding iterative procedure section pair verified satisfy equation obtained iterations using bisection method figure illustrates minimum phase impulse response obtained using design algorithm globally optimal solution approximates desired response restrictions phase exist certified theorem designed linear phase filter using firpm function matlab based algorithm figure illustrate comparisons magnitude responses entire frequency range passband stopband respectively removing restrictions phase clearly led sharper magnitude response characteristic furthermore since minimum phase design group delay entire passband much less linear phase design expense delay profile since high order filter polynomial root finding based spectral factorization impractical exploited algorithm instead compute computed used matlab implementation algorithm provided roof design firpm design impulse response samples design firpm design magnitude response ppendix onvergence terations first show resulting passband ripple res obtained applying weight increasing function design filter passband stopband specifications larger res also higher subscript res stands resulting prove res increasing function contradiction assume maximum weighted error filter designed weight function given max design firpm design magnitude response similarly define another filter weight scalar greater since frequency responses designed using remez exchange algorithm respective unique designs since satisfy alternation theorem least alternations order obtain contradiction assume compute another weighted error function using weight along maximum value weighted error becomes design firpm design magnitude response samples design firpm design max max since filter least good approximating desired function weight function contradicts unique optimality latter established alternation theorem therefore passband error equal smaller implies strictly increasing function consider range weights results physically meaningful designs given weight kdes always design trivial filter des leads kdes stopband error des passband error des satisfy desired weight constraint kdes sum errors exactly unity therefore optimal design coefficients sum errors never larger unity fig comparison optimal design linear phase design filter kdes impulse responses magnitude responses magnitude response zooming passband magnitude response zooming stopband group delays samples passband kdes inserting equation obtain kdes res fig example resulting passband deviation res function weight function given equation yields physically meaningful lower bound weight kdes need find weight yields resulting value weighted error res equal expression equation already established res increasing function always lower hand stated equation equals lower bound found equation monotonically decreasing weights greater lower bound asymptotically approaching zero means two functions intersect regime weights weight looking also another proof optimum exists unique figure illustrates example res curve wdes physically meaningful values start iterations res current value right intersection therefore needs decreased otherwise increased iterations continue two values meet search particularly efficient search space halved time corresponding binary search bisection scheme eferences parks mcclellan chebyshev approximation nonrecursive digital filters linear phase circuit theory ieee transactions vol mar cheney introduction approximation theory herrmann schuessler design nonrecursive digital filters minimum phase electronics letters vol may boite leich new procedure design high order minimum phase fir digital ccd signal processing vol chen parks design optimal minimum phase fir filters direct factorization signal vol jun evans optimal design real complex minimum phase digital fir filters acoustics speech signal processing ieee international conference vol mar kamp wellekens optimal design fir filters ieee transactions acoustics speech signal processing vol aug mian nainer fast procedure design equiripple minimumphase fir filters ieee transactions circuits systems vol may samueli design optimal equiripple fir digital filters data transmission applications ieee transactions circuits systems vol dec boyd vandenberghe fir filter design via semidefinite programming spectral factorization decision control proceedings ieee conference vol dec demirtas characterization design nonlinear phase fir filters global preparation smith introduction digital filters audio applications http accessed may online book
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statement approved release distribution unlimited presented partial fulfillment requirements master science degree penn state department mechanical engineering february applying artificial neural networks predict nominal vehicle performance adam last applied research laboratory box state college abstract paper investigates use artificial neural networks anns replace traditional algorithms manual review identifying anomalies vehicle run data specific data used study undersea vehicle qualification tests data highly therefore traditional algorithms adequate manual review time consuming using anns predict nominal vehicle performance based solely information available vehicle deviation expected performance automatically identified data capability becoming available due rapid increase understanding ann framework available computing power past decade ann trained purpose investigation relatively simple keep computing requirements within parameters modern desktop ann showed potential predicting vehicle performance particularly transient events within run data however also several performance cases steady state operation cases sufficient training data ann showed deficiencies expected computational power becomes readily available ann understanding matures training data acquired real world tests performance predictions ann surpass traditional algorithms manual human review timothy miller applied research laboratory box state college nfn service lives ensure proper functionality results qualification runs include hundreds channels data including speeds pressures temperatures flow rates vibrations many others qualification run usually unique mission profile progression commanded speeds throughout run makes direct comparison qualification runs difficult therefore identifying anomalous behavior data requires many highly trained experienced engineers manually reviewing data channel even due human error subtle anomalies overlooked conversely normal behavior flagged anomalous counter creating algorithm method predict results based solely mission profile highly desirable current best method average measured values given data channel given speed previous runs assume normal operating condition speed however response data channel highly nonlinear current speed command also cumulative effect prior operation determines current performance result traditional algorithms become unwieldy unreliable application problem undersea vehicles like high performance systems require many qualification runs prior throughout introduction overcoming breakdown traditional algorithms large data sets focus many industries recent years data center optimization handwriting recognition language translation image classification examples problems traditional algorithms effectively solve decades statement approved release distribution unlimited exponential growth available computing power along recent developments mathematical methods enabled new approach handling large data artificial neural networks anns anns deployed last years enable everything real time language translation instant handwriting recognition financial market analysis paper outlines approach using existing undersea vehicle qualification run data train ann predict future hypothetical run results input mission profile hypothetical qualification run actually performed real world results fed back computer ann identify differences exist predicted result actual results automatically flag anomalous behavior technical background first work artificial neural networks began century concept perceptrons perceptron essentially logic gate multiple inputs taken individually weighted summed sum weighted inputs greater threshold value output perceptron otherwise output shown graphically figure graphic adapted nielson figure perceptron logic perceptron later refined allow nonbinary output output accomplished replacing step function logic sigmoid function reason new model perceptron named sigmoid neuron neuron portion name derives sigmoid neuron similarity operation biological neuron basic level activates output given sufficient input sigmoid neurons become useful linked together perform complex logic renamed hidden neurons scope neural networks figure graphic adapted nielson shows artificial neural network containing inputs hidden neurons one output statement approved release distribution unlimited figure simple ann orient figure current problem hand input layer would known values commanded speed output layer would whatever data channel trying simulate intuitive actual vehicle speed obviously need far hidden neurons meaningful results neural network useful weighting factors input hidden layer sigmoid neuron must defined training process must undertaken training process requires large set already known pairings case known set available previous vehicle qualification runs known inputs commanded speed measured output actual speed initially training process random set weights defined output calculated compared actual output error neural network current evolution weights adjusted shown figure ann resolved process repeated error acceptable value preset number evolutions achieved several methods determining amounts adjust individual weight one easiest implement computationally efficient optimization method stochastic gradient decent gradient descent mathematical model finding minima maxima stochastic methods employed introduce element randomness optimization routine allows optimization escape local minima maxima solution continue improving solution many evolutions stochastic gradient decent efficient still computationally intensive solving weights many hundreds hidden layer sigmoid neurons recent sustained exponential growth available computing power critical making anns practical ann weights solved takes little computational resources run new inputs ann find output several advances mathematical methods recent years made anns much powerful extensible applications however purpose paper using methods already outlined sufficient ann setup simple ann set capable run single desktop computer ram ann built using netlab matlab toolbox developed nabney training data taken qualification runs undersea vehicle gave unique time steps time step represents discrete amount runtime time speeds time step input values must provided ann engine vehicle like many vehicles true control input commanded speed engine parameters adjusted internally goal matching commanded speed closely possible though strictly control parameter engine input amount energy reserves remaining extension amount reserves already used various systems could take form percentage combustible fuel remaining tank remaining battery power etc many systems one study energy reserves input also functions state health parameter electric systems motors may perform optimally batteries drop threshold voltage general inputs specific input data deemed pertinent provided ann speed command current commanded speed time current commanded speed previous commanded speed energy reserves depleted termed utilization purposes paper also provided training ann actual measured vehicle speed data point success ann training would determined comparing actual vehicle measured speed anns predicted vehicle speed ann initialized hidden sigmoid neurons training set terminate evolutions hours computation time intel core cpu running ghz assist visualization training process specific ann notation figure used figure depicting ann setup routine figure would run time step evolution times every evolution data set figure ann setup training visualization statement approved release distribution unlimited nondimensionalized speed nondimensionalized time figure full ann results statement approved release distribution unlimited consistently low prediction high speed command performance nondimensionalized speed ann results figure shows commanded speed dark blue line ann prediction actual speed red line also given true actual speed measured qualification tests light blue line inspection figure shows ann largely succeeded matching general speed response however several spans time ann fails predict accurately simply guessing actual speed equal commanded speed obvious example long periods constant speed known steady state conversely several spans ann prediction much accurate traditional algorithm could speed changes transients critical attempt understand reasons ann accurate accurate various situations reasons well understood adjustments made ann training routine address deficiencies promote strengths examples would providing additional input information working obtain test data situations ann shows deficiencies changing number hidden neurons adjusting evolutions simulation completion end remainder section paper examines three specific performance cases ann shows interesting prediction behavior upcoming figures show magnified subsets data figure allow analysis nondimensionalized time figure ann results low predictions high speed operation figure shows magnified portion ann data figure nondimensional time added plot present figure nondimensionalized utilization energetics used shown magenta note three separate runs shown utilization resetting zero indicative new run timeframe three instances vehicle commanded highest speed three instances ann predicts actual speed achieve full speed value however true one three occurrences two vehicle achieve high speed operation probable ann chronically predicting actual high speed operation safe prediction speed sometimes undershoots almost never overshoots high speed command likelihood vehicle undershoot high speed command increased high vehicle utilization near end individual effective control difficult one instance undershooting figure ann shows slight sensitivity high speed command figure furthest right occurs lowest utilization ann displays highest predicted speed however prediction still low ann clearly fully incorporate link utilization potential undershooting run data contains high speed command operation various utilizations would likely remedy issue data overfitting staircase speed increase accurate prediction transient performance figure shows brief speed command nondimensionalized speed ann predicted speed appears match actual speed vehicle closely time however predicted speed slowly increasing entire time anticipating next speed command reason speed change part staircase pattern speed changes speed goes minimum maximum series small changes speed rapid succession staircase patterns common training run data therefore ann learned speed changes moderate speed slightly higher speed likelihood middle staircase speed profile anticipate another small speed step however obviously always true small speed steps must part staircase sequence therefore anticipation ann another immediate speed change unwarranted indicative overfitting data would remedied additional run data showed small speed steps part staircase pattern nondimensionalized speed nondimensionalized time figure ann results transient performance figure shows one speed command changes data transient ann predicted speed showed sharp drop gradual step final steady state speed consistent actual vehicle operation feature traditional algorithm would trouble matching particularly data speed temperatures pressures etc may take much longer time speed reach steady state considered statement approved release distribution unlimited nondimensionalized speed nondimensionalized time figure ann results overfit solution conclusions results show even simple anns show promise outperforming traditional prediction methods times data highly speed transients however simple ann adequate performance times notably steady state operation highly probably ann would meet exceed performance traditional algorithms complex ann solved powerful computer fundamental limitation ann performance may potential overfitting data result sufficiently large set training data specifically sequence speed steps present data sets ann highly accurate predicting results exact sequence sequence altered slightly therefore possible even advanced ann solved powerful computer size training data set could prove limitation ann effectiveness mitigation problem inclusion additional data sets training data especially diverse mission profiles future work size training data set continues increase qualification runs vehicle ongoing additionally computational power continues become readily available therefore prudent develop advanced ann framework run existing data set understanding results may yet outperform traditional algorithms manual data analysis however framework place retraining ann new data becomes trivial exercise data computational power becomes available expected point ann provide best predictive capability point additional data channels temperatures pressures etc predicted using ann framework anns proven reliably identify anomalies post run data possible extend use identifying anomalies run instance vehicle behaves margin ann alert vehicle control evaluate whether issue early termination run even future anns capable assuming control adjusting vehicle run parameters flow rates pressures etc real time optimize vehicle performance closing remarks conclusions paper requiring computational power better mathematical techniques importantly additional training data consistent needs nearly ann projects currently development computational side chipmakers rushing provide hardware designed specifically ann training deployment hardware leverages architecture traditionally used intensive graphics processing brand new architecture development even optimized machine learning neural networks training data side companies using devices owned used individuals collect data improve algorithms relevant paper might tesla motors receives data every one cars road improve autopilot autonomous driving feature interestingly type training data collection driving force recent public debate privacy personal electronic devices instance google efforts track cell phone activity stems desire expand training sets algorithms control search result prioritization traffic congestion prediction language translation nomenclature ann training data input ann sigmoid neuron weight statement approved release distribution unlimited acknowledgments data paper provided arl psu experiments performed navsea contract thank entirety arl energy science power systems division support resources netlab matlab toolbox utilized perform ann training developed aston university ian nabney christopher bishop references gao jim machine learning applications data center optimization mountain view google web https cun handwritten character recognition using neural network architectures usps advanced technology conference pag bell laboratories web http auli michael joint language translation modeling recurrent neural networks microsoft research pag microsoft web http szegedy christian building deeper understanding images research blog google web http bergen mark google neural nets translate text instantly practically everywhere recode recode july web http martendale jon computers forge letters checks digital trends digital trends july web http neural networks applications neural networks applications web http nielson michael neural networks deep learning determination determination press web http bishop christopher neural networks pattern recognition oxford clarendon print bottou leon stochastic gradient descent tricks microsoft esearch pag web http nabney ian netlab algorithms pattern recognition london springer print documentation improve neural network generalization avoid overfitting mathworks web http ferrucci david magazine building watson overview deepqa project aaai web http colaner seth movidius google team leverage deep learning mobile devices tom hardware tom hardware web http fehrenbacher katie tesla ushering age learning car tesla ushering age learning car comments fortune web http lafrance adrienne tradeoff atlantic atlantic media company web http annex input file netlab toolbox matlab code running script requires installation netlab matlab toolbox netlab toolbox available http script derived standard netlab input file ian nabney clear clc input arrays must initialized variable follows first column current commanded speed second column previous commanded speed run started third column time since last speed command change fourth column current utilization set network parameters nin number inputs see input array comment nhidden number hidden sigmoid neurons nout number outputs alpha coefficient prior create initialize network weight vector net mlp nin nhidden nout alpha set vector options optimiser options zeros options provides display error values options number training cycles run netlab train neural network net options netopt net options use neural network generate predicted speeds mlpfwd net statement approved release distribution unlimited annex validation set several months ann described paper trained new field test data became available new data used validation set order test efficacy ann without results shown figure nondimensionalized speed nondimensionalized time figure validation set results observations ann results given time ann unable capture speed change startup likely relatively startups training data startup per field test field tests used training data time speed step ann predicts speed average speed transient likely due insufficiently advance algorithm ann settled minimizing error picking speed rather accurately modeling transient time ann anticipated speed overshoot transient occur analysis training data shows several occasions overshoot happen speed therefore ann error results insufficient number training sets overshoot similar speed step time ann correctly predicts speed undershoot transient time end run ann predicts speed much less actual speed end run likely due previous runs training set failed maintain speed late run therefore training data necessary ann able distinguish characteristics runs able maintain speed results calculate accumulated error using two speed prediction methods first method simply presuming actual speed equal setpoint referred method noted method could improved simple algorithms assuming time constant speed changes focus analysis second method prediction ann error statement approved release distribution unlimited methods calculated absolute value prediction minus actual speed graph accumulated error shown figure accumulated error speed actual nondimensionalized time figure accumulated error ann speed prediction methods validation set figure shows ann methods speed prediction nearly error accumulation rate late data set second last speed change encouraging sign additional data advanced ann methods neural networks practical tools predicting vehicle performance near future summary results validation set reinforce assertions made conclusion paper ann performance improved advanced methods computing power training data needed well statement approved release distribution unlimited
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feature engineering predictive modeling using reinforcement learning udayan khurana horst samulowitz deepak turaga abstract sin feature engineering crucial step process predictive modeling involves transformation given feature space typically using mathematical functions objective reducing modeling error given target however basis performing effective feature engineering involves domain knowledge intuition lengthy process trial error human attention involved overseeing process significantly influences cost model generation present new framework automate feature engineering based performance driven exploration transformation graph systematically compactly enumerates space given options highly efficient exploration strategy derived reinforcement learning past examples sep ukhurana samulowitz turaga ibm watson research center original data engineered data figure illustration different representation choices introduction predictive analytics widely used support decision making across variety domains including fraud detection marketing drug discovery advertising risk management amongst several others predictive models constructed using supervised learning algorithms classification regression models trained historical data predict future outcomes underlying representation data crucial learning algorithm work effectively cases appropriate transformation data essential prerequisite step model construction instance figure depicts two different representations points belonging classification problem dataset left one see instances corresponding two classes present alternating small clusters machine learning algorithms hard draw reasonable classifier representation separates two classes hand feature replaced sine seen image right makes two classes reasonably separable classifiers task process altering feature representation predictive modeling problem better fit training algorithm called feature engineering sine function instance transformation used perform consider schema dataset forecasting hourly bike rental figure deriving several features figure dramatically reduces modeling error instance extracting https hour day given timestamp feature helps capture certain trends peak versus demand note valuable features derived composition multiple simpler functions perhaps central task improving predictive modeling performance documented detailed account top performers various kaggle competitions wind practice orchestrated data scientist using hunch intuition domain knowledge based continuously observing reacting model performance trial error result often timeconsuming prone bias error due inherent dependence human decision making colloquially referred making difficult automate existing approaches automate either computationally expensive lack capability discover complex features present novel approach automate based reinforcement learning involves training agent examples learn effective strategy exploring available choices given budget learning application exploration strategy performed http https original features target count additionally engineered features using technique figure kaggle biking rental count prediction dataset technique reduced relative absolute error retaining interpretability features transformation graph directed acyclic graph representing relationships different transformed versions data best knowledge first work learns strategy effective feature transformation historical instances also work space provides adaptive budget constrained solution finally output features compositions mathematical functions make human readable usable insights predictive analytics problem one illustrated figure related work given supervised learning dataset ficus markovitch rosenstein performs beam search space possible features constructing new features applying constructor functions inserting original feature composition transformations ficus search better features guided heuristic measures based information gain decision tree surrogate measures performance constrast approach optimizes prediction performance criterion directly rather surrogate criteria require constructor functions note ficus general number less recent approaches ragavan bagallo haussler yang rendell blix matheus rendell kibler fan fan propose fctree uses decision tree partition data using original constructed features splitting points nodes tree ficus markovitch rosenstein fctree uses surrogate criteria guide search opposed true prediction performance fctree capable generating simple features capable composing transformations search smaller space approach also propose weight update mechanism helps identify good transformations dataset used frequently deep feature synthesis component data science machine dsm kanter veeramachaneni relies applying transformations features combinations transformations performing feature selection model optimization combined augmented dataset similar approach adopted one button machine lam call category approach approach suffers performance performance scalability bottleneck due performing feature selection large number features explicitly generated simultaneous application transforms spite expansion explicit expansion feature space consider composition transformations feadis dor reich relies combination random feature generation feature selection adds constructed features greedily requires many expensive performance evaluations related work explorekit katz expands feature space explicitly employs learning rank newly constructed features evaluating promising ones approach scalable type still limited due explicit expansion feature space hence instance reported results obtained running days moderately sized datasets due complex nature method consider compositions transformations refer approach cognito khurana introduces notion exploration transform space present simple handcrafted heuristics traversal strategies search capture several factors adapting budget constraints paper generalizes concepts introduced lfe nargesian proposes learning based method predict likely useful transformation feature considers features independent demonstrated work classification far allow composition transformations plausible approaches optimization bergstra transformation choice could parameter optimization strategies bayesian optimization ones feurer best knowledge approaches employed solving smith bull employ genetic algorithm determine suitable transformation given data set limited single transformations certain methods perform level recent survey topic appears found storcheus rostamizadeh kumar dimensionality reduction methods principal component analysis pca variants kernel pca fodor aim mapping input dataset space fewer methods also known embedding methods storcheus rostamizadeh kumar kernel methods shawetaylor cristianini support vector machines svm class learning algorithms use kernel functions implicitly map input feature space space neural networks allow useful features learned automatically minimize training loss function deep learning methods made remarkable successes various data video image speech manual tedious bengio courville vincent however deep learning methods require massive amounts data avoid overfitting suitable problems instances small medium sizes quite common additionally deep learning mostly successful video image speech natural language data whereas general numerical types data encompasses wide variety domains need technique domain model independent works generally irrespective scale data also features learned deep network may always easily explained limiting application domains healthcare che contrary features generated algorithm compositions mathematical functions analyzed domain expert overview automation challenging computationally well terms first number possible features constructed unbounded since transformations composed applied repeatedly features generated previous transformations order confirm whether new feature provides value requires training validation new model upon including feature expensive step infeasible perform respect newly constructed feature approaches described related work section operate manner take days complete even datasets unfortunately reusability results one evaluation trial another hand approach performs fewer one attempts first explicitly applying transformations followed feature selection large pool features presents scalability speed bottleneck practice restricts number new features considered cases lack performance oriented search insights proposed framework performs systematic enumeration space choices given dataset transformation graph nodes represent different versions given datasets obtained application transformation functions edges transformation applied dataset applies function possible features sets features case functions produces multiple additional features followed optional feature selection therefore batches creation new features transformation function lies somewhat middle approaches provides computation advantage also logical unit measuring performance various transforms used composing different functions manner translates problem finding node dataset transformation graph highest performance exploring graph little possible also allows composition transformation functions secondly decision making manual exploration involves initiation complex associations based variety factors examples prioritizing transformations based performance given dataset even based past experience whether explore different transformations exploit combinations ones shown promise thus far dataset hard articulate notions set rules basis decisions hence recognize factors involved learn strategy function factors order perform exploration automatically use reinforcement learning examples variety datasets find optimal strategy based transformation graph resultant strategy policy maps instance transformation graph action applying transformation particular node graph notation problem description consider predictive modeling task consisting set features target vector pair two specified dataset nature whether categorical continuous describes classification regression problem respectively applicable choice learning algorithm random forest linear regression measure performance auroc use simply signify performance model constructed given data using algorithm performance measure additionally consider set transformation functions disposal application transformation set features suggests application corresponding function valid input feature subsets applicable instance square transformation applied set features eight numerical two categorical features produce eight new output features square extends functions work input features derived feature recognized hat entire open set derived features denoted operator two feature sets associated target union two feature sets preserving row order generally transformations add features hand feature selection operator transformation algebraic notation removes features note operations specified feature set exchangeably written corresponding dataset implied operation applied corresponding feature set also binary sum implied target common across operands result goal feature engineering stated follows given set features target find set features feature original derived maximize modeling accuracy given algorithm measure arg max even height bounded tree modest number transforms computing performance across entire graph computationally expensive log square sum sum square square log graph exploration budget constraint log sum figure example transformation graph dag start node corresponds given dataset hierarchical nodes circular sum nodes rectangular example see three transformations log sum square well feature selection operator transformation graph transformation graph given dataset finite set transformations directed acyclic graph node corresponds either dataset derived transformation path every node dataset contains target number rows nodes divided three categories start root node corresponding given dataset hierarchical nodes one one incoming node parent node connecting edge corresponds transform including feature selection sum nodes result dataset sum similarly edges correspond either transforms operations children type type nodes respectively direction edge represents application transform source target dataset node height transformation graph refers maximum distance root node transformation graph illustrated figure node transformation graph candidate solution problem equation also complete transformation graph must contain node solution problem certain combination transforms including feature selection operator signifies nodes graph signifies hierarchical nodes also signifies transformation application created child alternatively sum node one parents complete transformation graph unbounded nonempty transformation set constrained bounded height complete transformation graph transformations hierarchical nodes equal number corresponding edges sum nodes times corresponding edges seen emphasized exhaustive exploration transformation graph option given massive potential size instance transformations height complete graph contains million nodes exhaustive search would imply many model training testing iterations hand known property allows deterministically verify optimal solution subset trials hence focus work find performance driven exploration policy maximizes chances improvement accuracy within limited time budget exploration transformation graph begins single node grows one node time current state graph beginning reasonable perform exploration environment stumble upon transforms signal improvement time elapsed budget desirable reduce amount exploration focus exploitation algorithm transformation graph exploration input dataset budget bmax initialize root bmax bratio bmax arg max bratio apply output argmax algorithm outlines general methodology exploration step estimated reward posi sible move bmax used rank options actions available given state transformation graph bmax bmax overall allocated budget number note algorithm allows different exploration strategies left definition function defines relative importance different steps step parameters function suggest depends various aspects graph point remaining budget specifically attributes action characterized briefly discuss factors influence exploration choice step factors compared across choices pairs budget considered terms quantity monotonically increasing time elapsed simplicity work number steps node accuracy higher accuracy node incentives exploration node compared others transformation average immediate reward till number times transform already used path root node accuracy gain node parent gain parent testing gains recent node depth higher value used penalize relative complexity transformation sequence fraction budget exhausted till ratio feature counts original dataset indicates bloated factor dataset transformation feature selector whether dataset contain numerical features datetime features string features simple graph traversal strategies handcrafted strategy essentially translates design reward estimation function line cognito khurana strategy perhaps mix described simplistic strategies work suitably specific circumstances seems hard handcraft unified strategy works well various circumstances instead turn machine learning learn complex strategy several historical runs traversal policy learning far discussed hierarchical organization choices transformation graph general algorithm explore graph budget allowance heart algorithm function estimate reward action state design reward estimation function determines strategy exploration strategies could handcrafted however section try learn optimal strategy examples several datasets transformation graph exploration behavioral nature problem perceived continuous decision making transforms apply node interacting environment data model etc discrete steps observing reward accuracy improvement notion final optimization target final improvement accuracy modeled problem interested learning function satisfy expected reward function algorithm absence explicit model environment employ function approximation due large number states recall millions nodes graph small depth infeasible learn transitions explicitly consider graph exploration process markov decision process mdp state step combination two components transformation graph node additions consists root node corresponding given dataset contains nodes remaini ing budge step bratio bmax let entire set states hand action step pair existing tree node transformation signifies application one transform already applied one exiting nodes graph let entire set actions policy determines action taken given state note objective learn optimal policy exploration strategy learning function elaborate later section formulation uniquely identifies state considering remaining budget factor state mdp helps address runtime exploration versus exploitation given dataset note runtime identical commonly referred training context selecting actions balance reward getting stuck local optimum step occurrence action results new node hence new dataset model trained tested accuracy obtained step attribute immediate scalar reward max max definition cumulative reward time state onwards defined max discount factor prioritizes early rewards later ones goal find optimal policy maximizes cumulative reward use watkins dayan function approximation learn state action respect policy defined hypothetical transition function cumulative reward following state optimal policy arg max however given size infeasible learn directly instead linear approximation qfunction used follows weight vector action vector state characteristics described previous section remaining budget ratio therefore approximate linear combinations characteristics state mdp note heuristic rule strategy used subset state characteristics manner however based approach select entire set characteristics let find appropriate weights characteristics different actions hence approach generalizes handcrafted approaches update rule follows wcj wcj max state graph step learning rate parameter proof follows irodova sloan variation linear approximation coefficient vector independent action follows handcrafted method reduces space coefficients learnt factor makes faster learn weights important note still independent action one factors actually average immediate reward transform present dataset hence equation based approximation still distinguishes various actions based performance transformation graph exploration far however learn bias different transformations general based feature types factor refer type strategy experiments efficiency somewhat inferior strategy strategy learned equation refer experiments training used datasets overlapping test datasets select training examples using different values maximum budget bmax dataset random order used discount factor learning rate parameter weight vectors size initialized training example steps drawn randomly probability current policy probability used following transformation functions general except specified smaller number log square square root product zscore timebinning aggregation using min max mean count std temporal window aggregate spatial aggregation spatio temporal aggregation frequency sum difference division sigmoid binningu binningd nominalexpansion sin cos tanh comparison tested impact publicly available datasets different datasets used training variety domains various sizes report accuracy base dataset routine bmax implementation transformations first applied separately add original columns followed feature selection routine random randomly applying transform function random feature adding result original dataset measuring performance repeated times finally consider new features whose cases showed improvement performance along original features train model implementation cognito khurana global search heuristic nodes used random forest default weka parameters learning algorithm comparisons gave strongest baseline average cross validation using random stratified sampling used results representative captured table seen learned figure comparing efficiencies exploration policies outperforms others cases one better tied two cognito global search technique reduces error relative abs error mean unweighted fscore median datasets presented table reference time taken took bikeshare dataset minutes seconds run nodes single thread processor times taken random cognito similar datasets took times time different datasets traversal policy comparison figure see average datasets strategies times efficient handcrafted strategy breadthfirst global described khurana finding optimal dataset given graph transformations bounded height hmax also figure tells eqn takes data train efficient eqn demonstrating learning general bias transformations one conditioned data types makes exploration efficient figure policy effectiveness training dataset sources dataset higgs boson amazon employee pimaindian spectf german credit bikeshare housing boston airfoil lymphography ionosphere openml openml openml openml openml openml openml credit default messidor features wine quality red wine quality white spambase source ucirvine kaggle ucirvine ucirvine libsvm ucirvine kaggle ucirvine ucirvine openml ucirvine ucirvine openml openml openml openml openml openml openml ucirvine ucirvine ucirvine ucirvine ucirvine rows features base random table comparing performance base dataset style random tree heuristic using datasets performance fscore classification absolute error regression exploration cost due higher depth also using feature selection compared none transform improves final gain performance measured datasets aforementioned another finally use different models learning algorithms lead different optimal features engineered dataset even similar improvements performance conclusion future work figure performance hmax internal system comparisons additionally performed experimentation test tune internals system figure shows maximum accuracy node representative datasets found height constrained different numbers using bmax nodes hmax signifies base dataset majority datasets find maxima hmax find hmax hmax tiny fraction shows deterioration interpreted unsuccessful paper presented novel technique efficiently perform feature engineering supervised learning problems cornerstone framework transformation graph enumerates space feature options exploration available choices find valuable features models produced using proposed technique considerably reduce error rate median across variety datasets relatively small computational budget methodology potentially save data analyst hours weeks worth time one direction improve efficiency system complex modeling state variables additionally extending described framework aspects predictive modeling missing value imputation model selection potential interest well since optimal features depend model type learning algorithm joint optimization two particularly interesting references bagallo haussler bagallo haussler boolean feature discovery empirical learning machine learning bengio courville vincent bengio courville vincent representation learning review new perspectives ieee transactions pattern analysis machine intelligence bergstra bergstra bardenet bengio algorithms optimization zemel bartlett pereira weinberger advances neural information processing systems curran associates che che purushotham khemani liu distilling knowledge deep networks applications healthcare domain arxiv preprint dor reich dor reich strengthening learning algorithms feature discovery information sciences fan fan zhong peng verscheure zhang ren yan yang generalized feature construction improved accuracy feurer feurer klein eggensperger springenberg blum hutter efficient robust automated machine learning nips fodor fodor survey dimension reduction techniques kibler kibler generation attributes learning algorithms aaai irodova sloan irodova sloan reinforcement learning function approximation flairs conference kanter veeramachaneni kanter veeramachaneni deep feature synthesis towards automating data science endeavors ieee data science advanced analytics katz katz chul shin song explorekit automatic feature generation selection ieee icdm khurana khurana turaga samulowitz parthasarathy cognito automated feature engineering supervised learning ieee icdm demo lam lam thiebaut sinn chen mai alkan one button machine automating feature engineering relational databases arxiv preprint jamieson desalvo rostamizadeh talwalkar efficient hyperparameter optimization infinitely many armed bandits corr markovitch rosenstein markovitch rosenstein feature generation using general constructor functions machine learning matheus rendell matheus rendell constructive induction decision trees ijcai nargesian nargesian samulowitz khurana khalil turaga learning feature engineering classification ijcai ragavan ragavan rendell shaw tessmer complex concept acquisition directed search feature caching ijcai cristianini cristianini kernel methods pattern analysis cambridge university press smith bull smith bull feature construction selection using genetic programming genetic algorithm berlin heidelberg springer berlin heidelberg storcheus rostamizadeh kumar storcheus rostamizadeh kumar survey modern questions challenges feature extraction proceedings international workshop feature extraction nips watkins dayan watkins dayan machine learning wind wind concepts predictive machine learning master thesis technical university denmark yang rendell blix yang rendell blix feature construction comparative study unifying scheme icml
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finite blocklength rates fading channel csit csir nov deekshith vinod sharma ece indian institute science bangalore india email deeks vinod work obtain lower upper bounds maximal transmission rate given codeword length average probability error power constraint finite valued block fading additive white gaussian noise awgn channel channel state information csi transmitter receiver bounds characterize deviation finite blocklength coding rates channel capacity turn achieved water filling power allocation across time bounds obtained also characterize rate enhancement possible due csi transmitter finite blocklength regime results elucidated via numerical examples ntroduction next generation cellular networks ought handle mission critical data delay requirements far stringent present day cellular networks refined engineering insights build delay critical systems obtained using analytical methods pioneered work characterize data rate enhancement wireless system delay constraints means power adaptation transmitter certain side information channel restrict attention delay incurred physical layer sending codeword receiver cellular system instantaneous channel gain fed back transmitter transmitter use knowledge perform power control increase overall data rate particular assumption perfect csi transmitter csit receiver csir optimal power allocation delay constraints flat fading awgn channel well known interpretation water filling time delay constraints imposed physical layer traditional approach study rate enhancement due knowledge csit first characterize either delay limited outage average capacity see details obtain power allocation strategy maximizes required one quantities regard obtains optimal power allocation maximises outage capacity assumption csit authors obtain optimal power allocation scheme maximizing average capacity causal csit nonetheless mentioned schemes provide realistic metric evaluate performance actual delay sensitive systems various notions capacity used therein inherently asymptotic work provide lower upper bounds maximal channel coding rate block fading awgn channel csit csir finite blocklength regime two kinds constraints transmitted codewords consequently characterize rate enhancement possible due power adaptation transmitter assumption perfect csit idealistic nevertheless rates obtained assumption provide upper bounds rates achievable without csit imperfect csit commonly made literature also knowledge power control strategies suitable delay constrained systems sheds insights system energy used systems efficient usage system energy beneficial energy constrained transmitters ought become prominent future wireless networks overview preceeding works presented next scalar coherent fading channel stationary fading generalization block fading without csit considered dispersion term characterized authors show second order optimal power allocation scheme quasi static fading channel csit csir truncated channel inversion quasi static fading model special case block fading model consider however bounds involve asymptotic terms asymptotic number blocks derived assumption finite valued fading process mimo rayleigh block fading channel csit csir considered achievability converse bounds derived short packet communication regime authors consider discrete memoryless channel csit csir obtains second order coding rates general assumptions state process contrast consider cost constrained setting real valued channel inputs renders direct translation techniques used therein infeasible normal approximation maximal coding rate block fading rayleigh channel without csit csir obtained contribution finer characterization delay limited performance wireless link channel state feedback two kinds power constraints wireless transmitters normally subjected bounds obtained maximal coding rate characterize rate enhancement due power adaptation given codeword length error probability deriving bounds csit assumption makes analysis involved non trivial particular obtaining upper bounds dependence channel input fading states makes corresponding optimization problems difficult solve circumvent derive alternate bounds utilizing properties asymptotically optimal power allocation scheme viz water filling scheme paper organized follows section introduce system model notation provide lower bounds maximal channel coding rate section iii next section provide upper bounds maximal coding rate section compare bounds numerically exemplify utility bounds derived conclude section proofs delegated appendices odel otation consider point point discrete time memoryless block fading channel subject awgn noise density denotes gaussian density mean variance noise independent identically distributed across channel uses let denote channel coherence time duration gain underlying physical channel remains constant let appropriate time units denote delay constraint imposed communication tdc denotes number blocks communication spans dxe denotes smallest integer let denote number times channel used within block number channel uses whole communication equivalently codeword length bnc channel gain fading coefficient block denoted random variable taking values finite set min cardinality set denotes number fading states let denote probability taking value channel gains across blocks independent additive noise process instantaneous channel gains assumed known transmitter well receiver transmitter gets know causally refer full csit csir assumption let denote channel input corresponding channel use bth block convenience let distributed denote corresponding noise variable channel output respectively delay tends infinity hence number blocks tends infinity well known channel capacity given pwf log pwf max denotes expectation respect distribution random variable obtained solving equation pwf pwf water filling solution average power constraint make note certain notations use throughout let log set positive integers denoted wherever required pnwe denote sum numbers choose represent channel input output vectors conveniently collection vectors length thus channel input bnc similarly noise vector channel output vector also vector channel fading gains collection scalars corresponding random variables denoted let min qmin min similarly corresponding maximum values denoted qmax respectively cartesian product two sets denoted fold cartesian product sets denoted set integers denoted set positive integers real line denoted positive real line dimensional euclidean space times given vectors denotes euclidean norm denotes inner product variance random variable denoted function denotes cumulative distribution function cdf standard gaussian random variable denotes corresponding density function denotes inverse cdf notation equivalent lim also equivalent constant sufficiently large notation lim lim sup indicate relations hold almost surely use abbreviation use notation mean equivalence distribution denote indicator function event exponential function denoted exp set denotes complement logarithms taken natural base iii aximal oding ate ower bounds section obtain lower bounds maximal coding rate given codeword length average probability error two different kinds power constraints transmitted codewords following definitions let denote message set let uniformly distributed random variable corresponding message transmitted taking values given fading coefficients instance channel use block output encoder denoted decoder obtaining pair outputs estimate message encoding decoding done average probability error prefixed throughout work adhere average probability error formalism work consider two types power constraints bnc bnc constraint equation referred short term power constraint long term power constraint studying communication system constraint motivated peak power limitations circuitry involved whereas imposing constraint captures requirement power utilization efficiency communication device instance battery powered mobile radio transmitters addition wireless communication setting studying systems allocate resources rate power dynamically constraint natural metric consider though reality constraints simultaneously present work study isolation constraint goal characterize maximum size cardinality codebook block length average probability error denoted maximal coding rate log first result gives lower bound log constraint theorem block fading channel input subject short term power constraint average probability error maximal codebook size satisfies log nvbf pwf vbf proof see appendix following result provides lower bound log subject constraint theorem block fading channel input subject long term power constraint average probability error maximal codebook size given blocklength satisfies log nvbf notation theorem proof see appendix aximal oding ate pper bounds section provide upper bounds maximal coding rate constraints deriving upper bounds assume csit known first obtain upper bound case next proceed provide upper bound case theorem block fading channel input subject short term power constraint average probability error maximal codebook size satisfies log log nvbf theorem vbf proof see appendix theorem block fading channel input subject long term power constraint average probability error maximal codebook size satisfies log nvbf log vbf theorem proof see appendix umerical xamples section compare numerically various bounds obtained thus far end need refined characterization lower bounds particular terms need explicitly identified terms contributing expression lower bound constraint csit case constraint hence terms identified revisiting analysis computing constants therein block fading case obtain small computation fix next consider lower bound constraint case note interested error probability assumption right hand side readily simplified using taylor theorem fact vbf therein monotonically increasing function virtue choice mentioned appendix thus obtain log nvbf term case coefficient log upper bound case compute terms excluding terms lower bound term upper bound thus approximate bound compute akin normal approximation awgn channels proceed compute bounds consider following discrete version rayleigh distribution fix define rayleigh distribution parameter unity fix corresponds fast fading case fixing plot convergence various bounds channel capacity use equation increases figure acronyms refer lower bound upper bound also plotted rates csit constraint figure fix compare various bounds different values comparing lower bound constraint rate csit use conjunction power controller next explain power control scheme choose chosen appropriately later let pwf beginning block transmitter checks constraint cst fig rate versus blocklength met cst met constraint violated transmission halted error declared short error message sent receiver receive without error else transmission blockq channel input chosen pwf transmitter sends codeword symbols successfully blocks without violating cst pwf bnc constraint met transmission error occur channel output fig rate versus input power case constraint rate enhancement possible due knowledge csit via power control observed onclusion paper obtained upper lower bounds maximal coding rate block fading channel short term long term average power constraints transmitted codeword bounds obtained shed light rate enhancement possible due availability csit bounds also characterize performance water filling power allocation finite block length regime acknowledgement authors would like thank gautam konchady shenoy ece iisc bangalore wei yang princeton university valuable discussions useful comments ppendix lower bound constraint let let codebook codewords codeword length average probability error codewords belonging block fading channel consideration csir csit codewords satisfy constraint pwf let pwf rate achievable block fading channel fading process csir csit satisfying constraint unity also achievable channel subject block fading process csit csir satisfying constraint henceforth consider channel block fading process csir csit denote upper bound probability let event cst violated specific choice made later let immediate see following bound theorem holds auxiliary output distribution defined corresponds channel transition probability analysis follows log nvbf vbf vbf note difference vbf vbf defined statement theorem vbf evaluated according distribution rather defined theorem per definition evaluated pwf first order term lower bound obtained match channel capacity hence proceed rectify shortcoming convenience let note defined theorem follows following reason pwf pwf function taylor theorem applied functional certain function given probability violating constraint cst following way pwf pwf exp applying taylor theorem functional yields nbc pwf follows hoeffding inequality set random variables zero mean constant follows choosing arbitrary small specified choice taylor theorem next apply taylor theorem obtain functional vbf denotes first derivative respect vbf gathering terms obtainpthat log upper bounded nvbf since chosen arbitrarily small result follows vbf ppendix vbf evaluated way vbf vbf also given lower bound constraint appendix fix constraint set let codebook codewords codeword length average probability error codewords belonging block fading channel one consideration csir csit virtue belonging codewords inherently satisfy constraint denotes covariance function finally use codebook conjunction power controller apply taylor theorem function following way beginning block channel gain next channel uses available transmitter fix water filling power allocation pwf equation transmission block channel notational convenience let input chosen pwf note substituting values obtain nvbf log nvbf next fix choice get exact expression end upper bound joint fading state scheme generates codebook satisfying constraint channel csit csir fly causal manner channel output pwf nconsider block fading channel fading process pwf assume csit full csir rate achievable channel constraint achievable original channel full csit csir subject constraint well follows analysis log nvbf optimizing choice input distributions auxiliary channel consideration log log inf log prescribed choice auxiliary channel hxb ppendix upper bound constraint pwf next seen let denote maximal codebook phb size channel average probability error satisfying constraint particular assume constraint satisfied equality note make assumption without loss optimality see instance lemma next assume csit known phb defined length bound derived assumption give valid upper vector note independent conditional bound causal case consideration corresponding random variables let denote distribution given codeword satisfying constraint obtained non causally identify power allotment vector log dfh log dfh inf log dfh also since assume perfect csit csir let denote equivalent channel input output respectively theorem minimum false alarm probability deciding subject minimum detection probability denotes auxiliary channel average probability error auxiliary channel choose auxiliary channel channel output distribution pwf denotes support next note inf log inf phb log inf inf inf min max pwf since message independent output auxiliary channel independent hence next bounded equation lower log log sup follows fact definition phbp log mean noted alternate notation summation phb similarly variance phb next follows berry esseen theorem independent identically random variables chapter constant depending term defined proof lemma provided also definition therein together fact taking infimum first term supremum second term right hand side separately lower value follows infimum supremum replaced minimum maximum respectively noting following functions definition lemma appendix continuous also compact set hence minimum maximum respectively attained also note minimum due equality constraint definition finally denote power allocations attain minimum maximum respectively follows lemma stated proved appendix universal positive constant next proceed obtain tractable lower bound end make following two facts first since pwf also using fact pwf pwf seen pwf using two relationships seen pwf define let log inequality follows fact monotonically increasing function using lemma provided appendix obtain log vbf vbf definedqas statement theorem vbf log choose prescribed choice combining applying taylor theorem function obtain log nvbf log ppendix upper bound constraint let denote maximal codebook size channel average probability error satisfying constraint particular case assume constraint satisfied equality mentioned assumption made without loss optimality see instance lemma next assume csit known bound derived assumption give valid upper bound causal case consideration constraints insists every fading realization channel input vector constraint requires function fix deriving upper bound first restrict class policies note min belongs converges original class policies function realizations random vector inequalities hold sense hence particular satisfies bounded difference condition chapter let mcdiarmid inequality exp note event enc channel input vector satisfies constraint power hence deriving upper bound case lower bound probability term enc follows derivative analysis case let denote summation right hand side equation note hence log upper bounded vbf using fact events obtain proceed obtain tractable upper bound log note hence invoke vbf lines arguments end observe taylor theorem pwf choose noting fact denotes power allocated according vbf monotonically increasing adopting pwf lines arguments case obtain log pwf thus log vbf exceed pwf since would applying taylor theorem small appendix obtain virtue choice exp thus lognm upper bounded log vbf note monotonically increasing since bound holds good take infimum obtain tightest bound next taylor theorem finally ing taylor expansion function exp around vbf yields required result ppendix lemma let defined statement theorem vbf log vbf positive constant proof proof follows along similar lines arguments lemma lemma however contrast case therein function sum random variables due dependence power allocation policy non causally show circumvent problem adapt proof ton case end define high probability set max logbb hoeffding inequality follows notational convenience let taylor expansion function around yields like upper bound event noting fact obtain log pwf log let note summation inequality upper bounded following way pwf pwf pwf pwf obtaining used definition getting made use facts pwf satisfies constraint equality finally note pwf bounded pwf pwf follows fact event follows fact pwf using see log obtaining similar upper bound let function obtained replacing pwf function defined earlier using taylor theorem follows pwf note thus logbb pwf pwf used fact definition pwf pwf proof since interested asymptotic function prove case avoid unnecessary notation cluttering result proved identical way first consider term defined note let defined cmin cmin min log qmin qmin used fact definition follows fact pwf made use definition used fact satisfies constraint equality follows fact qmin holds fixed enough log let max thus shown log also theorem note pwf first inequality follows definition equality follows invoking last bound follows jensen inequality thus obtained lower bound hence obtained log follows analysis logn constant next note function sum random variables taking values hence along bound directly invoking result lemma obtain log vbf cmin min cmin thus also note cmin consider term defined note made use first term thus seen constants depending next show end since lim lim implies lim hence ppendix lemma given realization fading vector lim lim vector corresponds power allocated state occurs block let constant less qmin qmin cmin defined section next denote constants depending min qmin also power allocation vector let note hence since event obtain using upper bounded using union bound hoeffding inequality thus next note fact qmin using bounded convergence theorem follows follows hence well combining bound get required result eferences agiwal roy saxena next generation wireless networks comprehensive survey ieee communications surveys tutorials vol strassen asymptotische shannon informationstheorie trans third prague conf inf theory polyanskiy poor channel coding rate finite blocklength regime ieee transactions information theory vol hayashi information spectrum approach coding rate channel coding ieee transactions information theory vol goldsmith varaiya capacity fading channels channel side information ieee transactions information theory vol gamal kim network information theory cambridge university press caire taricco biglieri optimum power control fading channels ieee transactions information theory vol negi cioffi capacity causal feedback ieee transactions information theory vol mahapatra nijsure kaddoum hassan yuen energy efficiency tradeoff mechanism towards wireless green communication ieee communications surveys tutorials vol polyanskiy scalar coherent fading channel dispersion analysis information theory proceedings isit ieee international symposium ieee yang caire durisi polyanskiy optimum power control finite blocklength ieee transactions information theory vol durisi koch polyanskiy yang shortpacket communications channels ieee transactions communications vol tomamichel tan coding rates channels state ieee transactions information theory vol lancho koch durisi normal approximation rayleigh channels information theory isit ieee international symposium ieee boucheron lugosi massart concentration inequalities nonasymptotic theory independence oxford university press polyanskiy channel coding fundamental limits dissertation princeton princeton usa feller introduction probability theory applications john wyley vol
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genetically modified hoare logic university antipolis laboratory umr cnrs les algorithmes euclide sophia antipolis cedex france jun irccyn umr cnrs rue nantes cedex france abstract important problem modeling gene networks lies identification parameters even consider purely discrete framework one thomas interested exhaustive search parameter values consistent observed behaviors gene network present article new approach based hoare logic weakest precondition calculus generate constraints possible parameter values observed behaviors play role programs classical hoare logic computed weakest preconditions represent sets compatible parameterizations expressed constraints parameters finally give proof correctness hoare logic gene networks well proof completeness based computation weakest precondition introduction gene regulation complex process expression level gene time depends large amount interactions related genes hence regulations genes seen gene network different methods studying behavior gene networks systematic way proposed among ordinary differential equations played important role however mostly lead numerical simulations moreover nonlinear nature gene regulations makes analytic solutions hard obtain besides abstraction procedure thomas approximating sigmoid functions step functions makes possible describe qualitative dynamics gene networks paths finite state space nevertheless qualitative description dynamics governed set parameters remain difficult deduced classical experimental knowledge therefore even modeling discrete approach thomas main difficulty lies identification parameters context interested exhaustive search parameter values consistent specifications given observed behavior gene regulatory networks exponential number parameterizations consider two main kinds approaches emerged one hand information cooperation concurrence two regulators target taken account order reduce number parameterizations consider see example also notion cooperation treated via grouping states hand using constraints helpful represent set consistent parameterizations see example paper present new approach based hoare logic weakest precondition calculus generate constraints parameters feature approach lies fact specifications partially described set paths seen since method avoids building complete state graph results powerful tool find constraints representing set consistent parameterizations tangible gain cpu time indeed weakest precondition computation builds constraints goes program independent size gene network works undertaken objectives application temporal logic biological regulatory networks presented constraint programming used biological systems ideas continued specifically genetic regulatory networks paper organized follows basic concepts hoare logic dijkstra weakest precondition quickly reminded section formal definitions discrete gene regulatory networks given section section gives way describe properties states presents path language finally introduces notion hoare triplet semantics extended hoare triples given section previous material section extended hoare logic gene networks defined thomas discrete models section example incoherent feedforward loop type made popular uri alon highlights whole process approach find suitable parameter values section contains proof correctness hoare logic gene networks well proof completeness based computation weakest precondition conclude section reminders standard hoare logic hoare logic formal system reasoning correctness imperative programs hoare introduced notation pgm mean assertion precondition satisfied performing program pgm program terminates assertion postcondition satisfied constitutes facto specification program form triple called hoare triple dijkstra defined algorithm taking postcondition program pgm input computing weakest precondition ensures pgm terminates words hoare triple pgm satisfied precondition pgm satisfied hoare logic weakest preconditions widely known teached world basic idea stamp sequential phases program assertions infered according instruction surround several equivalent versions hoare logic prefered one following offers simple proof strategy compute weakest precondition via proof tree stand programs stand assertions stands declared variable imperative program expr means expr substituted free occurrence assignment rule sequential composition rule alternative rule else iteration rule empty program rule stands empty program standard additional rules first order logic establish introduced empty program rule practice reasonnings data structures program integers order simplify expressions much possible fly proof trees iteration rule requires comments assertion called loop invariant well known finding weakest loop invariant undecidable included within programming language reason ask programmer give loop invariant explicitely keyword although may appear redundant also precondition hoare triple within program instruction carries sub specification consequently proved apart rest program using hoare logic rules following proof strategy builds proof tree performs proof computing weakest precondition statement within hoare triple perform independent according iteration rule first step strategy leads proofs subprograms contain instruction apply sequential composition rule program rule reduced instruction leads perform proof starting postcondition end treat instructions backward never apply empty program rule except leftmost instruction treated instructions treated since assignment rule central makes possible precisely define one precondition postcondition since rules relate evaluate conditions proof tree done end beginning program computes unique assertion precondition obtained applying last empty program rule actually weakest precondition assuming programmer given weakest loop invariants remainder article call strategy backward strategy section always follow backward strategy striking feature hoare logic weakest precondition proofs according backward strategy consist simple sequences syntactic formula substitutions end first order logic proofs nevertheless worth noticing question partial correctness since hoare logic give proof termination analyzed program instructions may induce infinite loops discrete gene regulatory networks multiplexes section presents modeling framework based general discrete method thomas introduced starting point consists labeled directed graph vertices either variables multiplexes variables abstract genes products multiplexes contain propositional formulas encode situations group variables inputs multiplexes influence evolution variables outputs multiplexes hence multiplexes represent biological phenomena formation complexes activate genes next definition labeled directed graph formally defined associated family integers see later integers correspond parameters drive dynamics network definition gene regulatory network multiplexes grn short tuple satisfying following conditions disjoint sets whose elements called variables multiplexes respectively labeled directed graph edges start variable end multiplex edges start multiplex end either variable multiplex every directed cycle contains least one variable every variable labeled positive integer called bound every multiplex labeled formula belonging language inductively defined belongs belongs interval atom belongs atom belong also belong family integers indexed set predecessors set multiplexes edge must satisfy notation flaten version formula denoted obtained applying following algorithm formula contains multiplex atom substitute associated formula formula exists since directed cycle multiplexes result atoms form state assignment integer values variables assignment allows natural evaluation formula replacing variables values becomes propositional formula whose atoms integer inequalities definition states satisfaction relation resources let grn set variables state function let set propositional formula whose atoms form positive integer formula every multiplex satisfaction relation state formula inductively defined reduced atom form proceed similarly connectives given variable multiplex resource state set resources state defined dynamical point view given state variable supposed evolve direction specific level depends set focal level given logical parameter hence state increase decrease stable suppose instance two input multiplexes mab mcd formula respectively mab mcd may seen complexes dimers regulating level suppose addition mab mcd mab mcd complexes mab mcd specify activator complexes individual effect less cumulated effect focal level presence single complex less focal level presence complexes example illustrates fact multiplexes encode combinations variables regulate given variable parameters giving weight possible combinations multiplexes indicate multiplexes regulate given variable thomas method assumed variables evolve asynchronously unit steps toward respective target levels dynamics gene regulatory network described following asynchronous state graph definition state graph let grn state graph directed graph defined follows set vertices set states exists edge transition one following conditions satisfied exists hence state stable state successor every variable stable state every variable stable state least one outgoing transition precisely variable transition allowing evolve toward focal level every outgoing transition supposed possible indeterminism soon several outgoing transitions example given figure see also section another example path sets order formalize known information gene network introduce section language express properties states assertion language language express properties state transitions path language combining properties state transitions properties states beginning end sequences state transitions leads notion hoare triplet path programs assertion language discrete models gene networks describe properties states meaningful way need terms allow check compare manipulate variable values taking parameter values account following definitions define language suitable needs extends definition terms assertion language let grn well formed terms assertion language inductively defined integer constitutes well formed constant term variable name variable considered symbol constitutes well formed constant term similarly subset symbol constitutes well formed constant term well formed terms also well formed terms definition assertion language semantics let grn assertion language inductively defined follows well formed terms atoms assertion language belong assertion language also belong assertion language state network satisfies assertion interpretation valid substituting variable symbol value according family note path language discrete models gene networks assertion language introduced subset first order logic well suited describe properties sets states express dynamical aspects since dynamics system encoded transitions state graph description dynamical properties equates precise formulation properties paths language proposed suitable encoding properties definition path language path program let grn path language language inductively defined expressions belong path language respectively increase decrease assignment variable value formula belonging assertion language assert also belongs path language belong path language also belongs path language sequential composition moreover sequential composition associative write without intermediate parentheses belong path language formula belonging assertion language else also belongs path language belongs path language formulas belonging assertion language also belongs path language assertion called invariant loop belong path language also belong path language quantifiers moreover quantifiers associative commutative write useful abbreviations technical purposes also consider empty program outside inductive definition well formed expression path language called path program intuitivelly resp means level variable increasing one unit resp decreasing one unit set particular value assert allows one express property current state without change state sequential composition allows one concatenate two path programs whereas statement allows one choose two programs according evaluation formula finally becomes possible express properties several paths thanks quantifiers lastly notice appears path program path reduced empty program intuitions formalized section syntax path programs next step combine properties state transitions path program properties states assertions begining end considered path program done via notion hoare triplet path programs notation grn given hoare triple path programs expression form well formed assertions called respectively path program intuitively precondition describes set states states variable value zero path program describes dynamical processes increase variable postcondition describes set states states variable value one small example encodes process variable changing value zero one whether expression satisfied given gene network depends state transition graph thus depends corresponding parameter values semantics hoare triples path programs firstly define semantics path programs via binary relation general ideas motivate definition following starting initial state sequences instructions without existential universal quantifier either transform another state feasible undefined example simple instruction transforms exists contrary transition exist instruction feasible existential quantifiers induce sort non determinism according chosen path existential quantifier one may get differents resulting states consequently one define semantics partial function associates unique binary relation suited mathematical object universal quantifiers induce sort solidarity states obtained according chosen path universal quantifier satisfy postcondition later reason define binary relation associates set states initial state set understood grouping together states contains scope universal quantifier path program contains existential universal quantifiers may consequently get several sets possibility existential quantifiers states belonging given together universal quantifiers contrary feasible set notation state variable define state definition path program relation let grn let state graph whose set vertices denoted let path program binary relation smallest subset state reduced instruction resp let consider resp transition reduced instruction reduced instruction assert form form form family state sets form else form empty program definition calls several comments relation exists set relations satisfy properties definition empty relation links states sets states satisfies properties intersection relations satisfy properties also satisfies properties simple instruction feasible state transition case set situation happens program assertion evaluated false current state universal quantifiers propagate non feasible paths one feasible feasible case existential quantifiers one even one feasible loop terminate exist set due minimality binary relation contrary loop terminates equivalent program containing finite number sequence starting semantics sequential composition may seem unclear familiar commutations quantifiers better take example explain construction see figure gives figure example semantics sequential composition let assume starting state two sets states reachable via intuitively means permits choice existential quantifier paths path leading contains universal quantifier grouping together let also assume starting state two sets states reachable via starting state two sets states reachable via set focusing paths encounter paths leads must grouped together ones leads nevertheless permits choice consequently grouping together possible futures one needs consider four possible combinations lastly focusing paths encounter since future via family indexed mentioned definition consequently adds relation lastly let remark empty always contains least one state proof easy structural induction program using fact loop terminates equivalent program containing finite number definition semantics hoare triple let grn let state graph whose set vertices denoted hoare triple satisfied satisfying exists satisfies previous definition implies consistency paths described path program state graph path program feasible one states satisfying precondition hoare triplet satisfied instance required figure left graphical representation grn bounds respectivelly finally family integers right representation state graph path program increasing possible according state graph hoare triple satisfied generally path language plays role programming language classical hoare logic hoare triplet satisfied iff states satisfying precondition program feasible leads set states postcondition satisfied path program viewed sequence actions one use order modify state memory system nevertheless similarly classical hoare logic reflects partial correctness imperative programming language previous definition imply termination loops path language also define infinite paths notice non terminating loop end program biological meaning represents periodic behaviours circadian cycle instance examples let consider grn figure state graph hoare triplet satisfied unique state satisfying precondition state path program possible leads state state satisfies postcondition opposite hoare triplet satisfied state satisfying precondition first instruction possible leads state next instruction consistant state graph following hoare triplet contains two existantial quantifiers universal one clearly since trivially moreover deduce considered program lead differents set states postcondition satisfied states see last set states relation one deduce hoare triplet satisfied hoare logic discrete models gene networks section define genetically modified hoare logic giving rules instruction path language definition first let introduce notations intensively used formulas notation let grn let variable subset set predecessors network denote following formula stands complementary subset definition states variables set resources state consequently exists unique denote following formula definition transition state graph variable increase denote following formula similarly variable decrease state state graph way practice assertion assert often biological point view obviously defined hoare logic discrete models gene networks defined following rules variable grn rules encoding thomas discrete dynamics incrementation rule decrementation rule rules coming hoare logic rules similar ones given section obvious rules instruction assert quantifiers added assert rule assert universal quantifier rule existential quantifier rule assignment rule sequential composition rule alternative rule else iteration rule empty program rule axioms axioms assert values stay bounds variable grn boundary axioms remark derived previous rules indeed implies corresponding current set resources using boundary axiom get similarly implications used section prove section modified hoare logic correct complete provided path program consideration contains weakest loop invariants statements precisely proof strategy called backward strategy already described end section also applies computes weakest precondition giving two proofs let show next section usefulness genetically modified hoare logic via formal study possible biological functions simple network example uri alon studied common vivo patterns involving three genes among enlightened incoherent feedforward loop type composed transcription factor activates second transcription factor regulate gene activator whereas inhibitor short positive action long negative action via activates inhibits left hand side figure shows feedforward loop considering thresholds actions equal leads boolean network since case variable take value action activates right figure boolean incoherent feedforward type right graphical representation grn bounds equal finally family integers hand side figure shows corresponding grn multiplexes encodes short action whilst followed constitute long action several authors like uri alon consider equal sufficiently long time also equal need resource order reach state also consider function feedforward loop ensure transitory activity signals switched idea activates productions stops production take look question via four different path programs prove formally affirmation valid constraints parameters network assumption starts activity transitory production possible already stated function classically associated feedforward loop ensure transitory activity signals switched interesting question conditions previous property true example path program together possible formalization previous property behaviour feedforward loop backward strategy using genetically modified hoare logic example gives following successive conditions weakest precondition obtained last instruction following conjunction simplifies conjunction weakest precondition obtained instruction simplifies owing boundary axioms remarks lastly weakest precondition obtained first program simplifies using empty program rule comes simplification get correctness proves whatever values parameters system exhibit transitory production response switch transitory production possible without increasing previous program one reflecting transitory production may realisations property example one consider path program respect path program weakest precondition obtained last instruction course previously weakest precondition obtained course satisfiable implies parameter associated equal path program feasible inconsistent weakest precondition indeed retrieve obvious property thomas approach negative action sequence arise resources must change order switch direction evolution another possible path compatible path program let notice even system exhibit transitory production via prevent paths exhibit behaviour example simple path leaves constantly equal hoare triplet satisfied corresponding weakest precondition clearly implied precondition constantly equal production impossible even worst constantly equal reached level impossible increase prove property showing following triplet inconsistent whatever loop invariant subprogram reflects fact stays constant evolves statement allows evolve freely becomes equal applying iteration rule satisfy property trivially satisfied whatever assertion due boundary axioms apply existential quantifier rule gives following weakest precondition consequently assertion let denote precondition path program applying empty program rule comes must also satisfy turn implies moreover let remark consequently hoare triple correct impossible satisfied false indeed implies false implies false implies false implies formally proved constantly equal reached level impossible increase partial correctness completeness partial understood assuming loops terminate usual hoare logic correctness correctness modified hoare logic means according inference rules section hoare triple semantically satisfied according definition set states exists proof made usual induction proof tree hence prove rule section correct develop incrementation rule sequential composition rule since correctness inference rules either similar decrementation rule trivial assert rule quantifier rules assignment rule empty program rule standard hoare logic alternative rule iteration rule let note correctness sequential composition rule neither trivial standard semantics enriched cope quantifiers let grn let state associated state space incrementation rule definition hypothesis variable grn prove conclusion exists let choose notation hypotesis equivalent turn according definition implies hence remains prove results hypothesis sequential composition rule definition consider following three hypotheses exists exists prove conclusion exists let arbitrarily choose set know exists similarly choose set know family exists fact let definition way union built weakest precondition completeness hoare logic would course defined follows hoare triple satisfied according definition using inference rules section well first order logic proofs integers obviously hoare logics complete already mentioned classical hoare logic finding weakest loop invariants undecidable complete logic integers following dijkstra prove completeness assumptions loop invariants statements weakest invariants needed properties integers admitted adopt strategy computes weakest precondition prove following theorem theorem dijkstra theorem genetically modified hoare logic grn hoare triple given backward strategy defined end section inference rules section computes weakest precondition last inference uses empty program rule means satisfied satisfied theorem obvious corrolary corollary grn given modified hoare logic complete assumption given loop invariants weakest ones needed properties integers established proof corollary satisfied dijkstra theorem proof tree infers hoare triple proof tree property semantically satisfied weakest precondition first order logic complete properties integers axiomatically assumed proof tree exists proof dijkstra theorem hypotheses loop invariants minimal hoare triple satisfied hypotheses satisfying exists satisfies statements corresponding loop invariant weakest one one prove conclusion satisfied precondition computed proof according backward strategy inference rules section proof done structural induction according backward strategy form construction backward strategy applying iteration rule get conclusion results immediately form set hypothesis becomes satisfying satisfies transition incrementation rule conclusion becomes satisfied straightforwardly results definition notation use form firstly inherit two structural induction hypotheses assertions satisfied satisfied precondition computed via backward strategy assertions satisfied satisfied precondition computed via backward strategy moreover hypothesis becomes definition satisfying exists family state sets satisfies lastly sequential composition rule conclusion becomes satisfied weakest precondition weakest precondition results states hypothesis satisfy consequently satisfied thus comes proves conclusion similarly correctness proof develop cases structural induction either similar already developed cases decrementation rule trivial assert rule quantifier rules assignment rule standard hoare logic alternative rule ends proof discussion cornerstone modeling process lies whatever application domain determination parameters paper proposed approach exhibiting constraints parameters gene network models relies adaptation hoare logic initially designed proofs imperative programs leads several questions usability implementations language issues path language way describe formally specification correct models gene networks classically specifications expressed temporal logics like ctl ltl also allows modeler take account behavioral information even exists links path language temporal logics formal languages temporal logics path language comparable properties expressed path language classical temporal logics converselly one hand path program assignment instruction allows one express gene gene expressible ctl ltl hand ctl ltl able express properties infinite cyclic traces properties infinite traces would expressed path language program terminate consequently would make sense nevertheless succession instructions corresponds property expressed ctl language example one knows starting point say path program corresponds formula correctness program path precondition becomes equivalent verify ctl formula true possible states generally path language well suited sequential properties whereas ctl express non sequential ones initializing evolutions figure path program top corresponding ctl formula bottom path language invariants loops mandatory hoare logic able prove program statements invariants given words entire information hoare logic needs perform proof given invariants unfortunately invariants difficult devise thus statements often used proofs refutation proof done possible invariant see example section plateform issues hoare logic gene networks designed order support software aims helping determination parameters models gene networks already done proof feasability indeed developped prototype named smbionet enumerates possible valuations parameters retains coherent specified temporal logic formula developped new prototype called uses path program weakest precondition calculus backward strategy produce constraints parameters order compare approaches ctl formulae versus path programs consider property expressed temporal logic ctl path program modeling biological system triggering tail resorption metamorphose tadpole see references therein expression profiles translated path program see fig turn translated equivalent ctl formula example whereas smbionet needs hours selects among possible parameterizations lead dynamics coherent ctl formula needs seconds computer construct constraints parameters ask enumeration parameters satisfying constraints using choco total search time minutes example shows hoare logic speed determination coherent parameterizations notice complexity weakest precondition calculus linear number instructions path program depend size gene regulatory networks node syntaxic tree program visited opposite ctl model checking algorithm depends size network thus use path program instead ctl formula leads postpone enumeration step use constraints parameters cut drastically set parameterizations consider software plateform dedicated analysis gene regulatory networks combine different technics indeed constraints solving technics necessary enumerate parameters give theorem prover also useful get strategies proofs refutation model checking technics hoare logic precondition calculus combined order give efficient algorithm already noted seems natural use hoare logic behavioural specification focuses finite time horizon whereas model checking natural temporal specification expresses global properties infinite traces would interesting complete plateform improved features theoretical point view one could also develop approaches help finding loop invariants build seems possible adapt iterative approach adopted astree another context abstract interpretation pragmatically one begins simple invariant one tries make proof completes iterativelly partially invariant application point view specifications often stem dna profiles would valuable develop program automatically produces path programs dna chips data two questions emerge choice thresholds based discretization expression levels determination good time steps questions scope article mainly rely biological expertise experimental conditions acknowledgment grateful alexander bockmayr heike siebert fruitful discussions comments paper authors thank french national agency research biotempo project support work also supported cnrs pepii project circlock references bernot comet richard guespin application formal methods biological regulatory networks extending thomas asynchronous logical approach temporal logic journal theoretical biology blass gurevich inadequacy computable loop invariants acm transactions computational logic boileau prados geiselmann trilling using constraint programming learning experiments transcriptional activation geometry dna schiex duret gaspin editor actes des journes ouvertes biologie informatique mathmatiques jobim toulouse cousot cousot building information society chapter basic concepts abstract pages kluwer academic publishers cousot cousot feret mauborgne min monniaux rival astre analyser sagiv editor esop european symposium programming number lncs pages springer fabien corblin conception mise oeuvre outil pour analyse des discrets phd thesis grenoble choco team choco open source java constraint programming library research report ecole des mines nantes corblin tripodi fanchon ropers trilling declarative method analyzing discrete genetic regulatory networks biosystems edsger dijkstra guarded commands nondeterminacy formal derivation programs commun acm august fanchon corblin trilling hermant gulino modeling molecular network controlling adhesion human endothelial cells inference simulation using constraint logic programming cmsb pages hatcher semantic basis program verification cybernetics hoare axiomatic basis computer programming communications acm oct khalis comet richard bernot smbionet method discovering models gene regulatory networks genes genomes genomics special issue khalis logique hoare identification formelle des paramtres rseau gntique phd thesis university essonne leloup buscaglia triiodothyronine hormone mtamorphose des amphibiens acad mateus gallois comet gall symbolic modeling genetic regulatory networks journal bioinformatics computational biology milo itzkovitz kashtan chklovskii alon network motifs simple building blocks complex networks science milo mangan alon network motifs transcriptional regulation network escherichia coli nature genetics thomas ari biological feedback crc press thomas regulatory networks seen asynchronous automata logical description theor thomas kaufman multistationarity basis cell differentiation memory logical analysis regulatory networks terms feedback circuits chaos troncale thuret ben pollet comet bernot modelling regulation tadpole tail resorption journal biological physics chemistry
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parsimonious random vector functional link network data streams mahardhika pratama plamen angelov edwin lughofer abstract theory random vector functional link network rvfln provided breakthrough design neural networks nns since conveys solid theoretical justification randomized learning existing works rvfln hardly scalable data stream analytics inherent issue complexity result absence structural learning scenarios novel class rvlfn namely parsimonious random vector functional link network prvfln proposed paper prvfln adopts fully flexible adaptive working principle network structure configured scratch automatically generated accordance nonlinearity property system modelled prvfln equipped complexity reduction scenarios inconsequential hidden nodes pruned input features dynamically selected prvfln puts perspective online active learning mechanism expedites training process relieves operator labelling efforts addition prvfln introduces type hidden node developed using data cloud hidden node completely reflects real data distribution constrained specific shape cluster learning procedures prvfln follow strictly learning mode applicable online applications advantage prvfln verified numerous simulations data streams benchmarked recently published algorithms demonstrated comparable even higher predictive accuracies imposing lowest complexities furthermore robustness prvfln investigated new conclusion made scope random parameters plays vital role success randomized learning keywords random vector functional link evolving intelligent system online learning randomized neural networks introduction decades research artificial neural networks mainly investigated best way determine network free parameters reduces error close possible zero various approaches proposed large volume work based first derivative approach respect loss function due rapid technological progress data storage capture transmission machine learning community encountered information explosion calls scalable data analytics significant growth problem space led scalability issue conventional machine learning approaches require iterating entire batches data multiple epochs phenomenon results strong demand simple fast machine learning algorithm deployment numerous applications provides strong case research area randomized neural networks rnns popular late early rnns offer algorithmic framework allows generate network parameters randomly still retaining reasonable performance one prominent rnns literature random vector functional link rvfl network features solid universal approximation theory strict conditions due simple sound working principle rnn regained popularity current literature nonetheless vast majority rnns literature suffer issue complexity make computational complexity memory burden prohibitive data stream analytics since complexities manually determined rely heavily expert domain knowledge presents structure lacks adaptive mechanism encounter changing training patterns data streams random selection network parameters often leads network complexity beyond necessary due existence superfluous hidden nodes contribute little generalization performance although universal approximation capability rnn assured sufficient complexity selected choosing suitable complexity given problems entails knowledge problemdependent novel rvfln namely parsimonious random vector functional link network prvfln proposed prvfln combines simple fast working principles rfvln network parameters output weights randomly generated tuning mechanism hidden nodes characterises online adaptive nature evolving intelligent systems network components automatically generated fly prvfln capable following variations data streams matter slow rapid gradual sudden temporal drifts data streams initiate learning structure scratch initial structure structure data streams learning mode automatically adding pruning recall hidden nodes furthermore compatible online deployment data streams handled without revising previously seen samples prvfln equipped hidden node pruning mechanism guarantees low structural burdens rule recall mechanism aims address cyclic concept drift prvfln incorporates dynamic input selection scenario makes possible activation deactivation input attributes fly online active learning scenario rules inconsequential samples training process moreover prvfln scenario single training process encompasses learning scenarios manner without steps novelties prvfln elaborated follows network architecture unlike majority existing rvflns prvfln structured locally recurrent neural network loop hidden node capture temporal system dynamic recurrent network architecture puts perspective activation degree takes account compatibility previous current data points without compromising local learning property local recurrent connection prvfln introduces new type hidden node namely data cloud inspired notion recursive density estimation anya angelov hidden node requires parameterization per scalar variable constrained specific shape firing strength defined local density calculated accumulated distances local mean data points cloud seen thus far version paper distinguishes predecessors fact interval uncertainty incorporated per local region targets imprecision uncertainties sensory data data cloud still satisfies universal approximation condition rvfln since derived using cauchy kernel asymptotically function moreover output layer consists collection output nodes created second order chebyshev polynomial increases degree freedom output weight improve approximation power online active learning mechanism prvfln possesses online active learning mechanism meant extract data samples training process mechanism capable finding important data streams training process discarding inconsequential samples strategy expedites training mechanism improves model generalization since prevents redundant samples learned scenario underpinned sequential entropy method sem forms modified version neighbourhood probability method data streams sem quantifies entropy neighbourhood probability recursively differs integrates concept data cloud concept data cloud simplifies working principle sem since activation degree already abstracts density local region key attribute neighbourhood probability realm classification problems sem always call ground truth leads significant reduction operator annotation efforts hidden node growing mechanism prvfln capable automatically generating hidden nodes fly help clustering mechanism originally designed incremental feature clustering mechanism method relies correlation measure input space target space carried learning mode ease original version directly implemented original version yet designed data cloud rule growing process also differs rnns dynamic structure hidden nodes randomly generated rather created rule growing condition considers locations data samples input space argue randomly choosing centers focal points hidden nodes independently original training data risks performance deterioration may hold completeness principle prvfln relies data hidden node require parameterization thus offering simple fast working framework rnns hidden node pruning recall mechanism rule pruning scenario integrated prvfln rule pruning scenario plays vital role assure modest network structures since capable detecting superfluous neurons removed training process prvfln employs socalled relative mutual information method extends rmi method working principle hidden node method examines relevance hidden nodes target concept thus captures outdated hidden nodes longer compatible portray current target concept addition method applied recall previously pruned rules strategy important deal recurring drift recurring drift refers situation old concept future absence rule recall scenario risks catastrophic forgetting previously valid knowledge cyclic drift imposes introduction new rule without memorization learning history differs rule recall mechanism separated rule growing scenario online feature selection mechanism prvfln capable carrying online feature selection process borrowing several concepts online feature selection ofs note although feature selection offline situation online feature selection remains challenging unsolved problem feature contribution must measured absence complete dataset notwithstanding online feature reduction scenarios literature issue stability still open input feature permanently forgotten pruned ofs delivers flexible online feature selection scenario makes possible select deselect input attributes every training observation assigning crisp weights input features another prominent characteristic ofs method lies capability deal partial input features cost extracting input features may expensive certain applications convergence ofs full partial input attributes also proven nevertheless original version originally devised linear regression calls modification perfectly fit prvlfn generalized ofs gofs proposed incorporates refinements adapt working principle prvfln paper puts forward following contributions presents novel random vector functional link network termed prvfln prvfln aims handle issue data streams remains uncharted territory current rvfln works puts perspective data cloud requires parameterization per scalar variable makes use local density information data cloud strategy aims hinder complete randomization hidden node parameters still maintaining tuning principle rvfln randomization hidden node parameters blindly without bearing mind true data distribution lead common pitfall rvfl hidden node matrix tend compromises completeness network structure confirmed four learning components namely sem gofs proposed efficacy prvfln thoroughly evaluated using numerous data streams benchmark recently published algorithms literature prvfln demonstrated highly scalable approach data stream analytics trivial cost generalization performance supplemental document containing additional numerical studies analysis predefined threshold effect learning components also provided moreover analysis robustness random intervals performed concluded random regions carefully selected chosen close enough true operating regions system modelled matlab codes prvfln made publicly available help study rest paper structured follows network architecture prvfln outlined section algorithmic development prvfln detailed section proof concepts outlined section conclusions drawn last section paper network architecture prvfln prvfln utilises local recurrent connection hidden node generates spatiotemporal property literature exist least three types recurrent network structures referring recurrent connections global interactive local local recurrent connection deemed compatible recurrent type case harm local property assures stability adding pruning hidden nodes prvfln utilises notion neural network presents enhancement layer furthermore hidden layer prvfln built upon data cloud require parametrization granulation hidden node shape fully evolves shape respect real data distribution derived recursive local density estimation bring idea data cloud integrating principle suppose data tuple streams time instant input vector target vector respectively numbers input output variables prvfln works strictly online learning environment access previously seen samples data point simply discarded learned due online learner total number data assumed unknown output prvfln defined follows temporal gtemporal denotes number hidden nodes stands output node produced weighting weight vector extended input vector extended input vector resulting functional link neural network based chebyshev function connective weight output node input weight vector randomly generated bias fixed simplicity temporal data cloud triggered upper lower data cloud temporal temporal expanding data cloud following obtained temporal temporal design factor reduce function crisp one worth noting upper lower activation functions temporal temporal deliver spatiotemporal characteristics result local recurrent connection hidden node combines spatial temporal firing strength hidden node temporal activation functions output following temporal spatial temporal temporal spatial temporal weight vector recurrent link local feedback connection feeds spatiotemporal firing strength previous time step git back consistent local learning principle trait happens useful coping temporal system dynamic functions internal memory component memorizes previously generated spatiotemporal activation function also recurrent network capable overcoming overdependency input features lessens strong temporal dependencies subsequent patterns trait desired practise since may lower input dimension prediction done based recent measurement conversely feedforward network assumes problem function current past input outputs strategy least entails expert knowledge system order determine number delayed components hidden node prvfln extension hidden node embeds concept address problem uncertainty instead computing activation degree hidden node sample hidden node enumerates activation degree sample intervals local region results local density information fully reflects real data distributions concept defined anya patented concept also underlying component autoclass come angelov sound work rde paper aims modify prominent works intervalvalued case suppose denotes support data cloud activation degree cloudbased hidden node refers local density estimated recursively using cauchy function spatial spatial spatial spatial interval data cloud data sample observed requires presence data points seen far impossible dealing data streams recursive form formalised generalized problem spatial spatial signify upper lower local means cloud uncertainty factor cloud determines degree tolerance uncertainty uncertainty factor creates interval data cloud controls degree tolerance uncertainty worth noting data sample considered population cloud resulting highest density moreover upper lower mean square lengths data vector cloud follows observed hidden node specific shape evolves naturally according supports furthermore parameters centroid width etc encountered conventional hidden node need assigned although concept hidden node generalized tedaclass introducing eccentricity typicality criteria idea uncharted note cauchy function asymptotically function satisfying integreable requirement rvfln universal approximator unlike conventional rnns prvfln puts perspective nonlinear mapping input vector chebyshev polynomial second order concept realises enhancement layer linking input layer output layer consistent original concept rvfln note recently developed rvflns literature mostly neglect direct connection designed output node direct connection expands output node higher degree freedom aims improve local mapping aptitude output node direct connection produces extended input vector making use second order chebyshev polynomial suppose three input attributes given extended input vector expressed chebyshev polynomial second order note term represents intercept output node avoids output node going origin may risk untypical gradient exist functions well neural networks trigonometric polynomial power chebyshev function however scatters fewer parameters stored memory trigonometric function chebyshev function better mapping capability polynomial functions order addition polynomial power function robust extrapolation case prvfln implements random learning concept rvfln parameters namely input weight design factor recurrent link weight uncertainty factor randomly generated weight vector left parameter learning scenario since hidden node randomization takes place hidden node parameters network structure prvfln data cloud depicted fig respectively fig network architecture prvfln iii learning policy prvfln section discusses learning policy prvfln section outlines online active learning strategy deletes inconsequential samples samples selected sample selection mechanism fed learning process prvfln section deliberates hidden node growing strategy prvfln section elaborates hidden node pruning recall strategy section concerns online feature selection mechanism section explains parameter learning scenario prvfln algorithm shows prvfln learning procedure online active learning strategy active learning component prvfln built extended sequential entropy esem method derived sem method esem method makes use entropy neighborhood probability estimate sample contribution exist least three salient facets distinguish esem predecessor forms online version sem combined concept data cloud accurately represents local density handles regression well classification sample contribution enumerated without presence true class label one may agree vast majority sample selection variants designed classification problems delve sample location respect decision surface best knowledge das address regression problem still shares principle predecessors exploiting hinge cost function evaluate sample contribution concept neighborhood probability refers probability incoming data stream sitting existing data clouds written follows algorithm learning architecture prvfln algorithm training procedure prvfln predefined parameters define training data step online active learning calculate neighborhood probability spatial firing strength end calculate entropy neighborhood probability esem step online feature execute algorithm else execute algorithm end step data cloud growing compute end calculate input coherence calculate compute output coherence end associate sample data cloud data cloud update local mean square length data cloud else create new data cloud take next sample phase end step data cloud pruning calculate temp end discard data cloud end end step adaptation output update output weights using fwgrls end end newly arriving data point data sample associated rule stands similarity measure defined similarity measure bottleneck problem however caused requirement revisit already seen samples issue tackled formulating recursive expression instead formulating recursive definition spatial firing strength data cloud suffices alternative derived idea local density computed based local mean summarizes characteristic data streams written follows activation degree spatial spatial determined entropy formulated follows neighbourhood probability log entropy neighbourhood probability measures uncertainty induced training pattern sample high uncertainty admitted model update existing network structure learning sample beneficial minimises uncertainty sample accepted model updates provided following condition met thres thres uncertainty threshold parameter fixed training process rather dynamically adjusted suit learning context strategy necessary compensate potential increase training samples accepted presence concept drift threshold set thresn thresn inc augments thresn thresn inc sample admitted training process whereas decreases thresn thresn inc sample ruled training process inc step size set simply follows default setting fig interval valued data cloud hidden node growing strategy prvfln relies method grow data clouds demands notion extended scc method hidden node working framework significance hidden nodes prvfln evaluated checking input output coherence analysis correlation existing data clouds target concept let local mean data cloud input vector target vector input output coherence written follows correlation measure linear correlation measures applicable however correlation measure rather hard deploy online environment usually executed using discretization parzen window method often leads assumption uniform data distribution implemented differential entropy pearson correlation measure widely used correlation measure insensitive scaling translation variables well sensitive rotation maximal information compression index mci one attempts tackle problems used perform correlation measure defined follows var var var var var var cov var var substituted var cov respectively stand variance covariance pearson correlation index local mean data cloud used represent data cloud captures influence intervals data cloud essence mci method indicates amount information compression ignoring newly observed sample principal component direction referred signifies maximum information compression resulting maximum cost imposed ignoring datum mci method features following properties var var maximum correlation given symmetric property mean expression discounted makes invariant translation dataset also robust rotation verifiable perpendicular distance point line unaffected rotation input features input coherence explores similarity new data existing data clouds directly output coherence focusses dissimilarity indirectly target vector reference input output coherence formulates test determines degree confidence current hypothesis predefined thresholds hypothesis meets conditions new training sample assigned data cloud highest input coherence accordingly number intervals local mean square length updated respectively well new data cloud introduced provided none existing hypotheses pass test one conditions violated situation reflects fact new training pattern conveys significant novelty incorporated enrich scope current hypotheses note larger specified fewer data clouds generated vice versa whereas larger specified larger data clouds added vice versa sensitivity two parameters studied section paper data cloud model parameterization committed new data cloud output node new data cloud initialised large positive constant output node set data cloud highest input coherence data cloud closest one new data cloud furthermore setting covariance matrix produce close approximation global minimum solution batched learning proven mathematically hidden node pruning recall strategy prvfln incorporates data cloud pruning scenario termed relative mutual information method method firstly developed fuzzy system extended adapt working principle method convenient use prvfln estimates mutual information data cloud target concept analysing correlation hence mci method applied measure correlation although method effectiveness handling data clouds recurrent structure implemented prvfln date open question unlike rmi method method apply classic symmetrical uncertainty method method formulated using mci method follows temp temp temp temp lower temporal activation function rule temporal activation function included rather spatial activation function order dependency recurrent structure considered mci method chosen offers good tradeoff accuracy simplicity possesses significantly lower computational burden symmetrical uncertainty method even implemented differential entropy robust widely used pearson correlation index data cloud deemed inconsequential thus able removed negligible impact accuracy met mean mean std respectively mean standard deviation mci lifespan criterion aims capture obsolete data cloud keep current data distribution due possible concept drift computes downtrend mci lifespan worth mentioning mutual information hidden nodes target variable reliable indicator changing data distributions line definition concept drift concept drift refers situation posterior probability changes overtime method also functions rule recall mechanism capable coping cyclic concept drift cyclic concept drifts frequently happen weather customer preference electricity power consumption problems etc seasonal change comes picture points situation previous data distribution reappears current training step pruned data cloud forgotten permanently inserted list pruned data clouds case local mean square length population output node output covariance matrix retained memory data clouds reactivated future whenever validity confirmed data trend worth noting adding completely new data cloud observing previously learned concept violates notion evolving learner catastrophically erases learning history data cloud recalled subject following condition max max situation reveals previously pruned data cloud relevant existing ones condition pinpoints previously learned concept may reappear previously pruned data cloud regenerated follows note although previously pruned data clouds stored memory data cloud pruning module still contributes lowering computational load previously pruned data clouds excluded training scenarios except unlike predecessors rule recall scenario involved data cloud growing process please refer algorithm plays role another data cloud generation mechanism mechanism also developed method represent change posterior probability concept drift accurately density concept online feature selection strategy although feature selection extraction problems attracted considerable research attention little effort paid toward online feature selection two common approaches tackling issue soft hard dimensionality reduction techniques soft dimensionality reduction minimizes effect inconsequential features assigning low weights still retains complete set input attributes memory whereas hard dimensionality reduction lowers input dimension cutting spurious input features nonetheless hard dimensionality reduction method undermines stability input feature retrieved pruned date existing works always start input selection process full set input attributes gradually reduces number observation come across prominent work namely online feature selection ofs developed covers partial full input conditions appealing trait ofs lies aptitude flexible feature selection enables provision different combinations input attributes episode activating deactivating input features adapts data trends furthermore technique also capable handling partial input attributes happens fruitful cost feature extraction expensive ofs originally devised generalized fit context prvfln start discussion condition learner provided full input variables suppose input attributes selected training process simplest approach discard input features marginal accumulated output weights maintain input features largest output weights note second term required extended input vector rule consequent informs tendency rule used alternate gradient information changes point although straightforward use ensure stability pruning process due lack sensitivity analysis feature contribution correct problem sparsity property norm analyzed examine whether values input features concentrated ball allows distribution input values checked determine whether concentrated largest elements pruning smallest elements harm model accuracy concept actualized first inspecting accuracy prvfln input pruning process carried system error large enough realm classification problem misclassification made nevertheless system error large case underfitting also case overfitting modify condition taking account evolution system error constant predefined parameter fixed simplicity output nodes updated using gradient descent approach projected ball guarantee bounded norm algorithm details algorithmic development prvfln algorithm gofs using full input attributes input learning rate regularization factor number features retained output selected input features selected make prediction regression max classification min prune input attributes except largest else end end respectively learning rate regularization factor assign following setting optimization procedure relies standard mean square error mse objective function leads following gradient term temporal temporal furthermore system error theoretically proven bounded upper bound also found one also notice gofs enables different feature subsets elicited training observation relatively unexplored area existing online feature selection situation limited number features accessible training process actualise scenario assume input variables extracted training process strategy however done simply acquiring input features scenario results subset input features training process problem addressed using bernaoulli distribution confidence level sample input attributes input attributes algorithm displays feature selection procedure algorithm gofs using partial input attributes input learning rate regularization factor number features retained confidence level output selected input features selected sample bernaoulli distribution confidence level randomly select input attributes end make prediction regression max classification min prune input attributes except largest else end end algorithm convergence scenario theoretically proven upper bound derived one must bear mind pruning process algorithm carried assigning crisp weights fully reflects importance input features random learning strategy prvfln adopts random parameter learning scenario rvfln leaving output nodes analytically tuned online learning scenario whereas others namely randomly generated without tuning process begin discussion recall output expression prvfln follows temporal note prvfln possesses enhancement layer expressed nonlinear mapping using chebyshev function original rvfln referring rvfln theory hidden node spatial derivative must integrable furthermore large number hidden nodes usually selected ensure adequate coverage hidden nodes data space chosen random nevertheless condition relaxed prvfln data cloud growing mechanism namely method partitions input region respect real data distributions concept data neurons features concept local density adapts variation data streams furthermore concept thus require parameterization calls approximation complicated optimization procedure parameters namely randomly chosen region randomisation carefully selected referring ingelnik pao random parameters sampled randomly following probability measures nevertheless strategy impossible implement online situations often entails rigorous process determine parameters rvfl works simply follow schmidt strategy setting region random parameters range assuming complete dataset observable solution defined determine output weights although original rvfln adjusts output weight conjugate gradient method solution still utilised ease mere obstacle use original work issue limited computational resources note regularization technique needs undertaken hidden node matrix although easy use ensures globally optimum solution parameter learning scenario however imposes revisiting preceding training patterns intractable online learning scenarios prvfln employs fwgrls method adjust output weight underpinned generalized weight decay function fwgrls approach detailed recounted robustness rvfln network parameters usually sampled uniformly within range literature new finding wang exhibits process randomly generating network parameters fixed scope ensure theoretically feasible network often hidden node matrix full rank surprisingly hidden node matrix invertible cases randomly sampling network parameters range far better numerical results achieved choosing scope trend consistent different numbers hidden nodes properly select scopes random parameters corresponding distribution still require investigation practice process normally required arrive decent local mapping note range random parameters ingelnik pao still theoretical level touch implementation issue study different random regions section see prvfln behaves variations scope random parameters numerical examples section presents numerical validations proposed algorithm using case studies comparisons prominent algorithms literature two numerical examples namely modelling nox emissions car engine tool condition monitoring end milling process presented section three numerical examples placed supplemental keep paper compact numerical studies carried two scenarios scenario scenario procedure orderly executes data streams according arrival time partitions data streams two parts namely training testing mode prvfln compared evolving algorithms bartfis panfis genefis ets dfnn gdfnn faospfnn anfis scenarios taken place experiment order follow commonly adopted simulation environment rvflns literature prvfln benchmarked prominent rvflns literature decorelated neural network ensemble dnne centralized rvfl matlab codes algorithms available online comparisons performed five evaluation criteria accuracy data clouds input attribute runtime network parameters scope random parameters followed schmidt suggestion scope random parameters insert analysis robustness part provides additional results different random regions effect individual learning component end results influence predefined thresholds analysed supplemental allow fair comparison consolidated algorithms executed computational resources matlab environment table prediction nox emissions using mode model prvfln prvfln panfis genefis bartfis dfnn gdfnn ets anfis rmse node input runtime network samples result obtained different computer platform table prediction nox emissions using mode model prvfln prvfln dnne nrmse node input runtime network samples modeling nox emissions car engine section demonstrates efficacy prvfln modeling nox emissions car engine problem relevant validate learning performance features noisy uncertain characteristics nature car engine also characterizes high dimensionality containing input attributes physical variables captured consecutive measurements furthermore different engine parameters applied induce changing system dynamic simulate real driving conditions procedure data points streamed consolidated algorithms samples https http http set training samples remainder fed testing purposes procedure experiment run scheme fold repeated five times similar scenario adopted strategy checked consistency rvfln learning performance adopts random learning scenario avoids data order dependency table exhibit consolidated numerical results benchmarked algorithms table tool wear prediction using time series mode model prvfln prvfln ets bartfis panfis genefis dfnn gdfnn anfis rmse node input runtime network samples table tool wear prediction using mode model prvfln prvfln dnne nrmse node input runtime network samples evident prvfln outperformed counterparts evaluation criteria except genefis number input attributes network parameters worth mentioning however three criteria predictive accuracy execution time number training samples genefis inferior prvfln equipped online active learning strategy discarded superfluous samples learning module significant effect predictive accuracy furthermore prvfln gofs method capable coping curse dimensionality note unique feature gofs method allows different feature subsets picked every training episode case avoids catastrophic forgetting obsolete input attributes temporarily inactive due changing data distributions gofs handle partial input attributes training process resulted level accuracy full input attributes use full input attributes slowed execution time needed deal input variables first reducing input dimension case study selected five input attributes kept training process experiment showed number selected input attributes able set desirable number cases significant performance difference using either full input mode partial input mode hand consistent numerical results achieved prvfln although prvfln built random vector functional link algorithm observed experimental scenario addition prvfln produced encouraging performance almost evaluation criteria computational speed rvflns implement less comprehensive training procedure prvfln output weights without structural learning feature selection mechanisms tool condition monitoring machining process section presents problem taken complex manufacturing problem courtesy xiang singapore objective case study perform predictive analytics tool wear end milling process frequently found metal removal process aerospace industry total features extracted force signal samples collected experiment concept drift case study resulted changing surface integrity tool wear degradation well varying machining configurations experimental procedure consolidated algorithms trained using data cutter testing phase exploited data cutter experimental procedure process undertaken fold undertaken five times arrive consistent finding tables report average numerical results across folds fig depicts many times input attributes selected one fold process fig frequency input features observed table prvfln evolved lowest structural complexities retaining high level accuracy worth noting although dnne exceeded prvfln accuracy imposed considerable complexity offline algorithm revisiting previously seen data samples adopts ensemble learning paradigm efficacy online sample selection strategy seen led significant reduction training samples learned experiment using partial input information led subtle differences full input information seen fig gofs selected different feature subsets every training episode reflected importance input variables training process additional numerical examples sensitivity analysis predefined thresholds analysis learning modules found analysis robustness section aims numerically validate claim section iii range always assure reliable model additional numerical results different intervals random parameters presented four intervals namely tried two case studies section experiments undertaken procedure fold repeated five times prevent effect random sampling experiment made use feature selection scenario full number input attributes table displays numerical results scope table analysis robustness criteria tool wear rmse node input runtime network samples rmse node input runtime network samples rmse node input runtime network samples rmse node input unstable runtime network samples nox emission unstable tool wear case study model generated range higher range model inferior model went point model longer stable range side range induced model highest accuracy evolving comparable network complexity nox emission case study higher scope led deterioration numerical results moreover range deliver better accuracy range either since range generate diverse enough random values numerical results interpreted nature prvfln algorithm success prvfln mainly determined compatibility zone influence hidden nodes real data distribution performance worsens scope remote true data distribution finding complementary wang relies sigmoidbased rvfl network scope random parameters outside applicable operating intervals predictive performance set approximation capability output space conclusions novel random vector functional link network namely parsimonious random vector functional link network prvfln proposed prvfln aims provide concrete solution issue data stream putting perspective synergy adaptive evolving characteristics fast characteristics rvfln prvfln fully evolving algorithm hidden nodes automatically added pruned recalled dynamically network parameters except output weights randomly generated absence tuning mechanism prvfln fitted online feature selection mechanism online active learning scenario strengthens aptitude processing data streams unlike conventional rvflns concept data clouds introduced concept simplifies working principle prvfln neither requires parameterization per scalar variables follows cluster shape features intervalvalued spatiotemporal firing strength provides degree tolerance uncertainty rigorous case studies carried numerically validate efficacy prvfln prvfln delivered encouraging performance ensemble version prvfln subject future investigation aims improve predictive performance prvfln acknowledgement third author acknowledges support austrian program linz center mechatronics lcm funded austrian federal government federal state upper austria thank wang suggestion pertaining robustness issue rvfln vii references angelov yager new type simplified fuzzy system international journal general systems anomalous system state identification patent 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jan characters equivariant spaces matrices claudiu raicu dedicated jerzy weyman occasion birthday abstract compute characters simple holonomic vector spaces general symmetric matrices realize explicitly subquotients pole order filtration associated generic matrix others local cohomology modules give direct proof conjecture levasseur case general matrices provide counterexamples case symmetric matrices character calculations used subsequent work weyman describe composition factors local cohomology modules determinantal pfaffian support introduction algebraic group acting smooth algebraic variety natural problem describe simple holonomic acts finitely many orbits regular singularities classified via correspondence simple local systems orbits group action describing explicitly however difficult problem see open problem section paper consider case vector space matrices general symmetric natural rank preserving group symmetries cases reductive group representations decompose direct sum irreducible representations appearing finite multiplicity purpose paper describe representations refer characters equivariant realize explicitly motivation work computing local cohomology describe characters composition factors local cohomology modules case space matrices general symmetric orbit closure natural group action expect combination commutative algebra techniques employ study local cohomology case matrices apply cases interest appendix note character calculations context analyzing local cohomology modules appear also cases representations maximal torus equivariant study paper general large levasseur conjecture class levasseur conjectured conjecture equivalence category equivariant holonomic whose characteristic variety union conormal varieties orbits group action module category admitting nice quiver description formulation equivalent fact date january mathematics subject classification primary key words phrases equivariant rank stratifications local cohomology claudiu raicu simple contains sections invariant action derived subgroup character description provides direct proof conjecture general matrices yields counterexamples symmetric matrices work complements existing literature studies categories rank stratifications see also corresponding categories perverse sheaves realize concretely simple objects categories discuss applications filling gaps arguments generally painting transparent picture give flavor level concreteness seek begin following zndom denotes set dominant weights denotes schur functor associated throughout paper use convention theorem let vector space matrices let coordinate ring write det det let sdet localization det filtration sdet denotes sdet generated successive quotients simple gln holonomic natural action row column operations characters given dom case symmetric matrices obtained theorem cover roughly half simple equivariant remaining half mysterious provide counterexamples conjecture case matrices well case matrices odd size simple equivariant arise local cohomology modules case matrices even size simple equivariant arise theorem pole order filtration associated pfaffian generic matrix simple irreducible characteristic variety roughly half ones arising symmetric matrices characteristic variety two connected components deduced remark consequence character information suggested theorem one motivation behind investigation simple building blocks many interest one would like understand precisely every holonomic finite length finite filtration composition series whose successive quotients composition factors simple holonomic connected composition factors also prop mainly interested two types holonomic local cohomology modules subset local cohomology modules smooth irreducible write codimx codimension inside hyc unique local cohomology module simple general irreducible subvariety one define intersection homology simple inclusion hyc whose cokernel suported proper subset case subvariety singular matrices implicitly described theorem sdet cokernel composition factors general local cohomology modules hyi may supported proper subsets characters equivariant spaces matrices interesting problem decide non vanishing refined level understand composition factors generated polynomial complex number define holonomic generated see recent survey strict inclusion implies root polynomial happen rational negative interesting question decide whether root gives rise strict inclusion question question section generally one may interested composition factors det completely answered theorem simple see proof theorem similar conclusions obtained symmetric determinant pfaffian matrix even size stating results detail give simple example illustrate character calculations alone allow one determine composition factors example let affine space let torus orbits indexed subsets stabilizer connected correspondence orbits simple gequivariant holonomic theorem remark given corresponding orbit closure since affine space codimension particular smooth therefore local cohomology module hyi write coordinate ring defined ideal generated variables using complex description local cohomology get decomposition irreducible take get torus weights appearing form partition appearing appears composition factor multiplicity one using similar argument obtain proof theorem see section symmetric matrices results run parallel three spaces matrices general symmetric made effort apply uniform strategy three cases able treat combinatorial details uniformly sake brevity chosen treat case symmetric matrices full detail indicate changes required cases two features make case symmetric matrices interesting presence equivariant local systems existence counterexamples levasseur conjecture positive integer consider collections dominant weights zndom zndom mod mod claudiu raicu note positive integer identify vector space symm symmetric matrices squares correspond matrices rank one write misymm subvariety matrices rank define cjs theorem equivariant symmetric matrices section exist simple gln holonomic symm whose characters cjs precisely denote csj character cjs symm simple holonomic supported origin symm symm mod symm symm mod symm symm symm symm usual intersection homology intersection homology associated irreducible gln local system orbit rank matrices let coordinate ring symm write sdet det determinant generic symmetric matrix let ssdet localization sdet consider ssdet ssdet generated remark contain sln sections provide counterexamples conjecture may interesting note among symm counterexamples intersection homology symm even failure levasseur conjecture solely explained presence local systems remark give quick derivation polynomial sdet appendix bsdet follows cayley identity bsdet divides strict inclusion shows root bsdet enough conclude equality remark interesting note character calculation allows determine characteristic varieties csj fourier transform see section permutes csj rotates characteristic varieties note rotating conormal variety orbit rank matrices yields conormal variety rank matrices formula together shows symm symm since support support follows characteristic variety two components namely conormal varieties orbits rank rank matrices symm since support support mssymm follows characteristic variety irreducible namely conormal variety orbit rank matrices similar considerations show general matrices characteristic varieties simple equivariant irreducible calculation characteristic varieties also deduced characters equivariant spaces matrices strategy computing characters equivariant approach computing characters equivariant based performing euler characteristic calculations using functoriality together combinatorial geometric methods precisely inclusion orbit direct image object derived category whose cohomology groups cases study representations analyzing inclusion directly complicated make use resolution singularities orbit closure variety vector bundle grassmannian product grassmannians inclusion affine open immersion map factors sris regular embedding projection onto first factor compute euler characteristic virtual admissible using factorization pretend correspondence simple equivariant orbits true general matrices write corresponding matrices rank euler characteristic calculations together general considerations regarding structure direct images allow write matrix ones diagonal represents change coordinates grothendieck group admissible representations appropriately defined linearly independent characters fourier transform one hand preserves matrix hand makes allows conclude matrix fact identity therefore see section process computing euler characteristics led following combinatorial problem let grassmannian quotients denoting line bundle denoting sheaf differential define virtual gln problem compute zndom corresponds power sum symmetric function answer given exercise relevance formula computing euler characteristics follows write simple holonomic supported origin orbit rank matrices inclusion ambient space minor adjustments depending space matrices analyze ook ook lim limit taken grothendieck group admissible representations see section precise formulation section calculations organization section establish notation basic results concerning representation theory general linear groups used throughout rest paper section compute relevant euler characteristics limits grothendieck group representations sections prove main results characters equivariant finally claudiu raicu section discuss simple arise local systems orbits prove levasseur conjecture general matrices preliminaries representation theory let complex vector space dimension dim denote group invertible linear transformations irreducible finite dimensional denoted indexed dominant weights dominant weight said partition parts nonnegative size conjugate partition defined transposing associated young diagram number example write set given subset integer let sequence simply write instead exterior power let det denote top exterior power admissible representations given reductive algebraic group write set isomorphism classes finite dimensional irreducible mainly interested case general linear group write zndom also consider dim dim write dom zdom admissible decomposes say finite finitely many define grothendieck group admissible representations direct product copies indexed set call elements virtual representations write typical element define multiplicity inside sequence virtual representations said convergent every sequence integers hur eventually constant convergent write hur define limit write lim combinatorics weights convenient make sense even dominant order let consider write sort sequence obtained rearranging entries order entries let sgn denote sign unique permutation realizing sorting sequence define sort let element defined sgn dominant entries otherwise characters equivariant spaces matrices example note particular denote collection subsets size write set partitions correspondence sets partitions given write conjugate partition complement given every define follows write elements increasing order let define via concatenation particular define permutation via sgn notation obtain note dominant sufficiently large also dominant define sets partitions mod mod quick counting argument yields following lemma cardinality computed even odd odd odd even otherwise claudiu raicu generalized pieri rule grothendieck group module representation ring finite dimensional ring generated exterior powers inverse det det det following lemma generalizes describing multiplicative action exterior powers since multiplication continuous commutes limits suffices determine action indecomposables lemma pieri rule every following equality proof may assume without loss generality dominant follows terms appearing right hand side dominant vertical strip size follows usual pieri formula corollary define elements every note following generalization pieri rule lemma every following equality proof identity element conclusion trivial may thus assume also assume dominant multiplication det invertible operation proving equivalent proving particular may assume partition moreover consider ordering partitions parts induced graded reverse lexicographic order conjugates precisely say largest index one prove partitions induction respect said ordering empty partition coincides assume consider parititon obtained removing last column young diagram conjugate given let denote size column removed using induction hypothesis lemma get consider collection partitions vertical strip size characters equivariant spaces matrices note every rewrite left hand side order prove sufficient show right hand side equal since dominant dominant check note way fail dominant index fix index note follows show enough prove correspondence collection subsets subsets given moreover pair follows obtained switching part part proves concludes proof lemma bott theorem grassmannians consider grassmannian quotients subspaces tautological sequence tautological rank quotient bundle tautological rank bott theorem grassmannians corollary computes cohomology large class bundles need weaker version computes euler characteristics suppose sheaf say admissible resp finite cohomology cohomology groups admissible resp finite dim therefore make sense euler characteristic element resp define euler characteristic virtual representation theorem bott let zkdom zdom dominant weights let concatenation euler characteristic given convention give alternative interpretation elements introduced claudiu raicu lemma let denote sheaf forms write det line bundle proof cauchy formula cor yields twisting det taking euler characteristics get using theorem using get smooth algebraic variety let denote sheaf differential operators section sheaf left module action definition let algebraic group acting let differentiating action yields map lie derx lie algebra vector fields operation composed yields action lie admits action compatible see def precise meaning compatibility action lie obtained differentiating action coincides one induced lie derx discussed introduction examples holonomic subset local cohomology modules well intersection homology vector space origin let dim unique simple supported origin vector space det sym following theorem gives classification simple equivariant holonomic group action finitely many orbits see section theorem let algebraic group acting finitely many orbits smooth algebraic variety correspondence characters equivariant spaces matrices simple holonomic pairs irreducible local system pairs irreducible representation component group isotropy group isotropy group mean stabilizer element isomorphic algebraic group denote connected component identity normal subgroup quotient called component group remark representation theorem trivial corresponding part closure follows case isotropy groups gaction connected correspondence simple orbits group action let positive integers consider complex vector spaces general matrices symm symmetric skew matrices respectively spaces admit natural action group via row column operations glm gln acts gln acts symm skew write resp mssymm subvariety resp symm consisting matrices rank msskew subvariety skew consisting skewsymmetric matrices rank following theorem classification simple holonomic spaces matrices general matrices simple vector space matrices namely intersection homology symmetric matrices simple vector space symm symmetric matrices intersection homology mssymm symm remaining ones intersection homology mssymm symm corresponding irreducible equivariant local systems orbits matrices simple vector space skew matrices namely msskew skew proof theorem follows theorem remark since isotropy groups general matrices connected symmetric matrices isotropy groups orbits two connected components computing euler characteristics let smooth complex projective algebraic variety denote dimension consider finite dimensional vector space short exact sequence locally free sheaves think affine space linear forms let totx denote total space bundle define morphism via commutative diagram totx ppp ppp pppp top map inclusion trivial bundle vertical map projection onto factor interested understanding euler characteristic pushforward claudiu raicu along map certain affine morphisms identify freely sheaves exercise let symox specx consider locally free sheaf rank one symi define toty total space line bundle write natural map inclusion symi defines section exercise define spec affine space ring polynomial functions corresponds vector space spanned polynomial degree vanishing locus consider complement let denote inclusion since affine open immersion thought sheaf algebras lim lim case spec localization affine space define sheaf graded dual det symox proposition notation assume admits action reductive group finite dimensional locally free sheaves assume isomorphism sheaves lim let isomorphic sheaf line bundle denote sheaf forms assume every sheaves det symox cohomology define sequence via lim det symc remark apply proposition case grassmann variety line bundle square follows lemma follows dim dim proof proposition since sheaves det symox admissible cohomology follows corollary det symox computing euler characteristics commutes colimits associated graded constructions get filtration yields filtration symox characters equivariant spaces matrices symox symox symox also get det det det therefore det symox det symox trivial bundle det symc multiplying equality summing taking limit using identification tensored get weyl algebra fourier transform positive integer weyl algebra ring differential operators section give coordinate independent description weyl algebra use describe fourier transform given finite dimensional space dimension write natural pairing let define form otherwise write tensor product let denote tensor algebra natural inclusion define weyl algebra quotient tensor algebra bilateral ideal generated differences note ring differential operators vector space choose basis dual basis coincides lemma fourier transform left det structure left example basic example sym coordinate ring case det sym equal simple holonomic supported origin see proof lemma using identification coming natural isomorphism easy see duop denotes opposite ring since left also right duop identified right canonical sheaf vector space free rank one module generated det prop association det gives equivalence categories right left motivated lemma define fourier transform relative denoted grothendieck group admissible follows det claudiu raicu context apply fourier transform follows constructions functorial certain admissible representations group way lemma fourier transform character equal slightly imprecisely refer character fourier transform little linear algebra consider finite partially ordered set let denote free abelian group basis write indicate strictly larger respect partial order allow equality assume order reversing bijection abuse notation also write induced automorphism given following lemma suppose collection elements exist relations apq integers apq automorphism permutes elements hence apq proof write permutation applying get apq necessarily permutation relations since follows one also apq get equality implies easy induction height defined shows concludes proof lemma limit calculations grothendieck group admissible representations recall terminology sections using freely throughout section particular recall notation grothendieck group admissible group definition also lemma vector space write dual section compute three cases limits type lim sequence finite virtual det sym character simple supported origin finite dimensional limit exist arbitrary instead consider cases even resp odd separately even mentioned introduction explained section limits correspond euler characteristic calculations certain direct images essential character calculations sections reader interested details limit calculations may wish record results propositions skip section characters equivariant spaces matrices symmetric matrices let vector space dimension define elements cjs via cjs defined proposition det sym even odd lim mod odd lim mod even odd even even even odd odd equalities easy verify trivial representation left hand side reduces regardless parity right hand side either therefore fix rest section begin notation preliminary results proving proposition let zkdom mod mod zdom convention define zndom zndom let note sets form partition mod comparing get lemma mod claudiu raicu proof consider unique let maximal element assume using therefore contradicts follows hence implies get using fact yields mod mod concluding proof lemma lemma assume exists index mod consider collection using notation sgn proof show exactly one contained moreover show assignment establishes bijection sgn since sgn conclusion follows assume find mod contradicting mod contradicting choose set choose let since mod get mod mod since way correspondence could fail induce bijection get case however inequality would imply equivalently since hypothesis get contradicting fact lemma correspondence elements set defined moreover every sgn empty mod characters equivariant spaces matrices proof correspondence sets resp complements partitions resp conjugates given resp follows lemma largest elements namely set determined since condition equivalent mod condition equivalent mod turn equivalent mod follows establishes desired bijection moreover sgn last equality follows fact even mod lemma proof proposition sgn since det det get using cauchy formula prop det sym dom mod using notation obtain otherwise follows using notation lim mod dom sgn lemmas need consider mod mod multiplying sides using lemma get lim mod separate contributions right hand side according two cases claudiu raicu terms lemma consider terms mod case get odd lemma even otherwise comparing coefficient proposition cases even odd see agree terms terms mod contribute coefficient terms mod contribute efficient get appears observing using conclude terms contribute mod lemma even odd otherwise comparing coefficient proposition conclude proof proposition general matrices positive integers let zndom define dominant weight dom vector spaces dim dim define characters equivariant spaces matrices proposition write det sym lim proof proposition consider dominant weights dom zdom let obtain using easy manipulations equals sgn sgn sym using writing sequence obtained appending zeros get sym otherwise let unique index condition equivalent inclusion implies last entries forces elements contained modify follows consider zndom defined conditions equivalent since dominant weights equalities hold note freedom choosing choice increasing sequence inside choices fix writing get moreover follows using since sgn sgn follows sgn sgn putting everything together using obtain otherwise claudiu raicu multiplying taking limit yields desired conclusion matrices positive integer let dom dom vector space dim define via proposition det sym lim define following collections dominant weights dom zdom odd partition zndom following collections dominat weights defined zndom analogy lemma one prove lemma conditions equivalent odd odd lemma assume satisfy equivalent conditions lemma even odd write moreover proof conclusions follow fact give permutation odd odd odd follows moreover follows assume even follows even write since get since taking maximal index find characters equivariant spaces matrices lemma let define collection partitions even even every partition even size cardinality set given odd even proof since even even compute size first note condition implies determined since must condition equivalent using odd even lemma number choices odd respectively even lemma assume collection subsets corresponds via odd even proof consider satisfying conditions lemma odd contains corresponding even set coincides differences even fact assume next even use write previous paragraph implies differences even even shows since get contradiction verification yields subset follows easily tracing back arguments proof proposition sgn using notation get otherwise claudiu raicu follows lim lemmas sgn dom even desired equality follows odd similarly even get equivariant symmetric matrices section compute characters vector space symm symmetric matrices let denote complex vector space dimension identify symm squares correspond matrices rank one write let mssymm denote subvariety matrices rank main result section theorem exist simple holonomic symm namely symm symm mod csj symm symm mod symm character csj cjs defined remaining assertion theorem equivariant symmetric matrices described introduction identification proof follows closely proof theorem next section leave details interested reader classification holonomic simple explained theorem need check cjs character csj consider situation section let write becomes totxk locally trivializes vector space dimension gets identified space symmetric matrices take det consider inclusion symk note locally generated symmetric determinant let open set defined locally determinant dyk note maps isomorphically via orbit symmetric matrices rank sheaf det det sym lim even mod characters equivariant spaces matrices condition satisfied context euler characteristic pushforward easily computed consequence proposition remark det sym lim mod evaluated explicitly proposition next explain det also structure dyk consider double cover defined locally symmetric determinant structure sheaf naturally hence also dyk contains define cokernel inclusion sheaf given det odd follows satisfies setting proposition det compute euler characteristic direct image via lim det sym sym sym mod evaluated proposition ready prove main result section proof theorem classification simple follows theorem remains check equalities csj cjs equalities together proposition yield asi bsi integers asi bsi since cnj det sym character cjn cauchy formula prop equation also satisfied fourier transform permutes modules csj takes apply lemma poset lexicographic ordering given let cjs csj conclude using lemma csj cjs equivariant matrices section compute characters vector space matrices consider vector spaces dimension dim dim let identify tensor products correspond matrices rank one write let denote subvariety matrices rank recall notation characters main result section claudiu raicu theorem equivariant general matrices simple holonomic character described theorem expressed terms local cohomology codim need show character prove theorem classification holonomic simple explained theorem follows comparing characters local cohomology modules thm thm proof theorem let assume character write using cauchy formula cor get equality sdet dom example shows composition factors sdet appearing multiplicity one remains check prove induction composition factors clearly true assume induction hypothesis valid sdet must inclusion sdet using character description must contain class inside quotient sdet therefore must also contain classes contradicts formula character conclude inclusion sdet since simple generated class image note remark strict inclusions theorem combined cayley identity show generic determinant bdet conclude showing character consider situation section let write note locally trivialize vector spaces dimension gets identified space matrices take line bundle det det consider inclusion symk note locally generated function assigns matrix determinant let open set defined locally determinant sheaf given lim det sym condition satisfied context euler characteristic pushforward easily computed consequence propositions remark lim det sym since maps isomorphically via orbit rank matrices conclusion character follows proof theorem linear algebra trick section characters equivariant spaces matrices equivariant matrices section compute characters vector space skewsymmetric complex vector space dimension matrices let denote skew vector space matrices exterior products identify correspond matrices rank two write let msskew denote subvariety matrices rank recall notation characters theorem equivariant matrices simple skew skew character holonomic skew odd described terms local cohomology skew codim skew skew skew skew skew skew skew let denote coordinate even let equation defining hypersurface ring skew consider hpf localization spf generated classification holonomic simple explained theorem equality follows theorem get proof theorem note cayley identity shows bpf divides hpf turn implies hpf strict inclusions force root bpf fact bpf prove theorem remains check character consider situation section let write locally trivializes vector space dimension gets identified space matrices take line bundle det line bundle consider inclusion symk note locally generated function assigns matrix pfaffian let open set defined locally pfaffian get using cauchy formula prop condition satisfied consequence propositions remark obtain lim det sym since maps isomorphically via orbit rank matrices skew conclude proof theorem character simple regular holonomic rank stratifications let denote vector spaces general symmetric matrices natural group action row column operations corresponding group considered previous sections denote union conormal varieties orbits consider category modrh regular holonomic whose characteristic variety contained goal section describe explicitly simple objects obtain corollary direct proof levasseur conjecture conj case general matrices via correspondence simple objects classified irreducible local systems local systems corresponding claudiu raicu described previous sections orbits irreducible local systems orbits rank matrices vector space general symmetric matrices vector space matrices cases complement defined single polynomial determinant generic symmetric matrix first two cases pfaffian generic matrix last case fundamental group equal monodromy corresponding local system given complex number let denote coordinate ring consider depends class theorem notation consider irreducible local system whose monodromy given corresponding simple object modrh proof restriction rank one integrable connection whose corresponding local system monodromy given follows order prove theorem need check simple condition equivalent see theorems space general matrices space symmetric matrices assume follows cayley identity symmetric versions generated order prove simple sufficient show contains fix write lie algebra note particular true since multiplicity free decomposition irreducible form irreducible integral may assume contains one replacing may assume since generates ideal invariant action defines proper closed subset necessarily contained zero locus complement dense orbit obtain ideal generated contains large enough powers therefore contains concludes proof theorem end remarking theorem yields proof levasseur conjecture case general matrices already seen irreducible local systems orbits group action give rise simple containing hence generated sections invariant action derived subgroup theorem remaining simple objects form since true contains sections acknowledgments grateful nero budur david eisenbud sam evens mircea uli walther jerzy weyman interesting conversations helpful advice well anonymous referee suggesting many improvements presentation experiments computer algebra software provided numerous valuable insights work supported national science foundation grant references bgk borel grivel kaup haefliger malgrange ehlers algebraic perspectives mathematics vol academic press boston characters equivariant spaces matrices tom braden mikhail grinberg perverse sheaves rank stratifications duke math doi coutinho levcovitz morphisms comm algebra doi daniel grayson michael stillman macaulay software system research algebraic geometry available http robin hartshorne algebraic geometry new york graduate texts mathematics ryoshi hotta kiyoshi takeuchi toshiyuki tanisaki perverse sheaves representation theory progress mathematics vol boston boston translated japanese edition takeuchi masaki kashiwara holonomic systems rationality roots invent math george kempf complex induced representation adv math doi tatsuo kimura introduction prehomogeneous vector spaces translations mathematical monographs vol american mathematical society providence translated japanese original makoto nagura tsuyoshi niitani revised author thierry levasseur radial components prehomogeneous vector spaces rational cherednik algebras int math res imrn doi macdonald symmetric functions hall polynomials oxford mathematical monographs clarendon press oxford university press new york contributions zelevinsky oxford science publications robert macpherson kari vilonen elementary construction perverse sheaves invent math doi philibert nang class holonomic related action gln gln adv math doi classification regular holonomic matrices algebra doi claudiu raicu characters equivariant veronese cones arxiv appear trans ams claudiu raicu jerzy weyman local cohomology support generic determinantal ideals algebra number theory doi claudiu raicu jerzy weyman emily witt local cohomology support ideals maximal minors pfaffians adv math doi claudiu raicu jerzy weyman local cohomology support ideals symmetric minors pfaffians arxiv morihiko saito generated rational powers holomorphic functions arxiv michel van den bergh local cohomology modules covariants adv math doi vilonen intersection homology local complete intersections isolated singularities invent math doi uli walther survey arxiv jerzy weyman cohomology vector bundles syzygies cambridge tracts mathematics vol cambridge university press cambridge department mathematics university notre dame hurley notre dame institute mathematics simion stoilow romanian academy address craicu
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cam theory comparison statistical models may ester mariucci abstract recall main concepts cam theory statistical experiments especially notion cam distance properties also review classical tools bounding distance presenting examples proof classical equivalence result density estimation problems gaussian white noise models analyzed keywords statistical experiments cam distance deficiency density estimation model ams classification primary secondary introduction theory mathematical statistics based notion statistical model also called statistical experiment experiment statistical model original formulation due blackwell triple sample space set called parameter space family probability measures definition mathematical abstraction intended represent concrete experiment consider example following situation taken book cam yang physicist decides estimate half life carbon supposes life atom exponential distribution parameter order develop investigation takes sample atoms physicist fixes advance duration experiment say hours counts number disintegrations formally leads definition statistical model represents law random variable counting number disintegrations observed hours way proceed want estimate half life carbon indeed physicist could choose consider first random time fixed number disintegrations say occurred case represent experiment via statistical model law random variable natural question much statistical information considered experiments contain precisely experiment informative conversely quest comparison statistical experiments initiated paper bohnenblust shapley sherman followed papers blackwell following definition introduced informative bounded loss function decision procedure experiment exists decision procedure experiment date may research leading results received funding european research council erc grant agreement ester mariucci denote statistical risk experiments respectively however lead two models issue solved cam introduced notion deficiency give precise definition forthcoming sections remark two interesting properties well defined real number every two given statistical models sharing parameter space every loss function every decision procedure available using exists decision procedure solves issue mentioned could strictly positive case comparable according first definition nevertheless still say much information lose passing one model one cam theory found applications several problem statistical decision theory developed example nonparametric regression nonparametric density estimation problems generalized linear models diffusion models models spectral density estimation problem historically first results asymptotic equivalence nonparametric context date due brown low nussbaum first two authors shown asymptotic equivalence nonparametric regression gaussian white noise model third one density estimation problems gaussian white noise models years many generalizations results proposed brown carter grama nussbaum meister rohde nonparametric regression brown carter nussbaum mariucci appear nonparametric density estimation models another active field study diffusion experiments first result equivalence diffusion models euler scheme established see milstein nussbaum later papers generalizations result considered see laredo mariucci well different statistical problems always linked diffusion processes see dalalyan delattre hoffmann laredo nussbaum among others also cite equivalence results generalized linear models see grama nussbaum time series see grama neumann milstein nussbaum garch model see buchmann functional linear regression see meister spectral density estimation see golubev nussbaum zhou volatility estimation see jump models see mariucci negative results somewhat harder come notable among brown zhang efromovich samarov wang another new research direction explored involves quantum statistical experiments see buscemi aim survey paper present basic concepts cam theory asymptotic equivalences statistical models aim review give accessible introduction subject therefore follow general approach theory also approach already available literature see cam cam yang van der vaart order achieve goal paper organized follows section recall definition cam distance statistical meaning particular attention payed interpretation cam distance terms decision theory section collect classical tools control cam cam theory comparison statistical models distance passing examples described section section devoted show details proof classical result cam theory namely asymptotic equivalence density estimation problems gaussian white noise models deficiency cam distance already pointed possible way compare two given statistical models parameter space could compare corresponding risk functions ask much information lose passing one model one saying loss disposal mechanism able convert observations distribution observations adopt latter point view natural formalization mechanism notion markov kernel definition let two measurable spaces markov kernel source target map following properties map every map probability measure every denote markov kernel source target starting markov kernel probability measure one construct probability measure following way roughly speaking think two models contain amount information exist two markov kernels depending idea formalized sixties lucien cam led notion deficiency hence introduction class statistical experiments parameter space definition deficiency general form involves notion transition generalization concept markov kernel paper however prefer keep things simpler focus case one deal dominated statistical models polish sample spaces see definition advantage case definition deficiency simplifies abstract concept transition coincides markov kernel see proposition nussbaum definition statistical model called polish sample space separable completely metrizable topological space said dominated exists measure absolutely continuous respect measure called dominating measure example typical examples polish spaces probability theory spaces space continuous functions equipped supremum norm space functions equipped skorokhod metric definition let two probability measures defined measurable space total variation distance defined quantity sup ester mariucci denotes norm definition let two experiments deficiency respect number inf sup infimum taken markov kernels denotes total variation distance definition cam distance defined max space statistical models satisfies triangle inequality equality imply actually coincide concerning glossary experiment reconstructed experiment markov kernel say less informative better informative models said equivalent two sequences statistical models called asymptotically equivalent way interpret cam distance experiments see numerical indicator cost needed reconstruct one model one viceversa via markov kernels said introduction way compare statistical models seems natural compare respective risk functions let highlight definition deficiency clear interpretation terms statistical decision theory aim start recalling standard framework statistical model indexed set probability measures defined measurable space set equipped elements sometimes called states nature space possible actions decisions statistician take observing example estimation problems take make sense notion integral need equipped loss function interpretation action incurs loss true state nature randomized decision rule markov kernel risk precisely standard interpretation risk follows statistician observes value obtained probability measure know value must take decision choosing probability measure picking point random according chosen true distribution suffers loss average loss observed rdz loss picked according average integral important result allowing translate notion deficiency described decision theory language following cam theory comparison statistical models theorem see cam theorem page cam let fixed decision rule bounded loss function exists decision rule words inf sup sup sup last supremum taken set loss functions belongs set randomised decision procedures experiment remark important consequence previous theorem two sequences experiments asymptotically equivalent cam sense asymptotic properties inference problem experiments means two sequences statistical experiments proven asymptotically equivalent enough choose simplest one study inference problems one interested transfer knowledge inference problems complicated sequence via markov kernels transfer decision rules via randomisations let two sequences statistical models sharing parameter space polish sample spaces suppose exist markov kernels kkn uniformly parameter space given decision rule estimator define decision rule asymptotically statistical risk show let start considering easier case deterministic precisely suppose form functions suppressing index shorten notations particular assuming loss function bounded defining one finds decision rule asymptotically risk kind computations work general case deterministic sequence decision rule action spaces case one show randomized sequence decision rules asymptotically risk ester mariucci remark let probability measure eiq markov kernel one define markov kernel following way clearly control cam distance even definition deficiency perfectly reasonable statistical meaning easy compute explicit computations appeared rare see hansen torgersen torgersen section shiryaev spokoiny generally one may hope find easily upper bounds collect useful techniques purpose property let two statistical models sample space define particular property allows bound statistical models sharing sample space means classical bounds total variation distance aim collect useful classical results fact see cam let two probability measures dominated common measure densities define important property following property product measures defined measurable space proof see zolotarev thus one express distance distributions vectors independent components terms distances consequence property property product measures defined measurable space proof see strasser lemma cam theory comparison statistical models property hellinger distance two normal distributions exp proof see mariucci fact likelihood process another way control cam distance lies deep relation linking equivalence experiments proximity distributions related likelihood ratios let statistical model dominated dpj let dpj density respect particular one see real random variable defined probability space one see stochastic process reason introduce notation call likelihood process key result theory cam following property let two experiments family dominated equivalent likelihood processes dominating measures coincide proof see strasser corollary let suppose two processes defined probability space law equal law following holds see cam yang lemma property likelihood processes associated experiments sup sufficiency cam distance useful tool comparing statistical models different sample spaces look sufficient statistic introduction term sufficient statistic usually attributed fisher gave several definitions concept cite presentation subject cam fisher relevant statement seems requirement statistic chosen summarize whole relevant information supplied requirement may made precise various ways following three interpretations common classical operational definition sufficiency claims statistic sufficient given value one proceed randomization reproducing variables distributions originally observable variables bayesian interpretation statistic sufficient every priori distribution parameter posteriori distributions parameter given entire result experiment given iii decision theoretical concept statistic sufficient every decision problem every decision procedure made available experiment decision procedure depending performance characteristics ester mariucci study sufficiency abstract way found halmos savage last section work named value sufficient statistics statistical methodology starts following observation gather conversations able prominent mathematical statisticians doubt disagreement sufficient statistic sufficient particular sense contains information sample bahadur continuation work halmos savage found particular effort done highlight interest using sufficient statistics statistical methodology one main results bahadur theorem establishing equivalence decision theoretical concept sufficiency operational concept terms conditional probabilities mention fact similarity result cam stated theorem core theory comparison statistical experiments formally let statistical model statistic measurable map measurable space another measurable one denote image law definition sufficient statistic exists function arbitrary subalgebra said sufficient exists function set called subalgebra induced statistic property see bahadur statistic sufficient subalgebra induced sufficient accordance notation introduced section state theorem bahadur follows recall denotes space theorem see theorem bahadur subalgebra sufficient every decision rule exists decision rule focusing relation notion sufficient statistic one equivalence statistical models let recall factorization theorem powerful tool identifying sufficient statistics given dominated family probabilities let family probabilities absolutely continuous respect measure denote density theorem statistic sufficient exists function function important result linking cam distance existence sufficient statistic following cam theory comparison statistical models property let two statistical models let sufficient statistic distribution equal proof order prove enough consider markov kernel defined conversely show one consider markov kernel defined since sufficient statistic markov kernel depend denoting distribution one asymptotic arguments one also needs appropriate version notion sufficiency definition let sequence statistical models sequence subalgebras asymptotically sufficient denotes restriction experiment stronger notion asymptotic equivalence indeed let two sequences experiments parameter space triangle inequality clear exist two sequences asymptotically sufficient statistics respectively taking values measurable space sup sequences asymptotically equivalent also recall important generalization notion sufficiency notion insufficiency discussion concept beyond purposes paper reader referred cam chapter cam exhaustive treatment subject examples better understand typical form asymptotic equivalence result let analyze examples toy example let start considering following parametric case example let statistical model associated observation vector independent gaussian random variables inference concerns parameter space interval formally law let denote experiment associated observation empirical mean relative previous random variables law gaussian random variable means factorization theorem easy see application sufficient statistic immediate application property implies ester mariucci passing examples nonparametric framework let recall result due carter concerning asymptotic equivalence multinomial gaussian multivariate experiment parameter space subset reason focus result lies useful tool establishing global asymptotic equivalence results density estimation problems example let random vector multinomial distrip bution parameters denote statistical model associated multinomial distribution parameters belong set consisting vectors probabilities maxi mini main result carter bound cam distance statistical models associated multinomial distributions multivariate normal distributions means covariances multinomial ones precisely let denote statistical model associated family multivariate normal distributions theorem see carter notations constant depends another interesting result contained carter approximation gaussian experiment independent coordinates let denote statistical model associated independent gaussian random variables theorem see carter notations constant depends let consider examples nonparametric framework precisely recall results brown low nussbaum first asymptotic equivalence results nonparametric experiments example brown low authors consider problem estimating function continuously observed gaussian process satisfies sde dyt dwt dwt gaussian white noise find statistical model associated continuous observation asymptotically equivalent statistical model associated discrete counterpart nonparametric regression time grid uniform standard normal variables assume varies nonparametric subset defined smoothness conditions tends infinity slowly precisely drift function unknown positive constant one sup cam theory comparison statistical models moreover defining one asks lim sup diffusion coefficient supposed known absolutely continuous function positive constant example nussbaum author establishes global asymptotic equivalence problem density estimation sample gaussian white noise model precisely let random variables density respect lebesgue measure densities unknown parameters supposed belong certain nonparametric class subject restriction positivity restriction let denote statistical model associated observation furthermore let experiment one observes stochastic process dyt dwt standard brownian motion main result nussbaum done first showing result holds certain subsets class described shown one estimate rapidly enough fit various pieces together without entering detail let mention key steps poissonization technique use functional kmt inequality last years asymptotic equivalence results also established discretely observed stochastic processes example let present result mariucci close spirit one brown low example let sequence time inhomogeneous processes defined dws random initial condition standard brownian motion inhomogeneous poisson process intensity function independent sequence real random variables distribution independent supposed known horizon observation finite belongs class also unknown belong classes respectively ester mariucci mariucci problem estimating high frequency observations considered precisely suppose observe discrete times goes infinity let statistical model associated continuous observation one associated observations xti finally let gaussian white noise model associated continuous observation gaussian process dyt dwt suppose subclass uniformly bounded functions nuisance parameters satisfy following conditions exist two constants derivable derivative exists constant assumption three models asymptotically equivalent bound given subclass discrete distributions support subclass absolutely continuous distributions respect lebesgue measure uniformly bounded densities fixed neighborhood particular result tells jumps process ignored goal estimation drift function moreover proof constructive explicit markov kernel constructed filter jumps density estimation problems gaussian white noise models constructive proof section following carter see detail one prove constructive way asymptotic equivalence density estimation problem gaussian white noise model presented example however respect work nussbaum ask stronger hypotheses parameter space order simplify proofs precisely fixed strictly positive constants consider functional parameter space form example density estimation problem density gaussian noise model dyt dwt idea carter use bound distance multinomial gaussian normal variables presented example make assertions density estimation experiments intuition see multinomial experiment result grouping independent observations continuous density subsets say using square root transformation multinomial variables asymptotically approximated normal variables constant variances normal variables turn approximations increments process sets subsection analyze cam theory comparison statistical models obtain asymptotically equivalent multinomial experiment starting assuming results carter stated theorems obtain bound multinomial experiment one associated independent gaussian random variables subsection explain show asymptotic equivalence adequate normal approximation independent coordinates density estimation problems multinomial experiments let consider partition intervals denote application mapping writing stands number belonging interval let law law multinomial distribution particular means appropriate multinomial experiment informative precisely proven statistical model associated multinomial distribution denoted let investigate quantity trivial observation total variation distance multinomial distribution law always hence order prove need construct non trivial markov kernel divide proof three main steps step denote midpoints intervals introduce discrete random variable concentrated points masses let denote statistical model associated observation independent realizations means sufficient statistic argument get indeed consider application mapping observe density law independent realizations respect counting measure given means factorization theorem conclude sufficient statistic thus step starting realizations want obtain something close independent realizations law aim define approximation follows piecewise linear functions interpolating values points figure figure definition functions ester mariucci particular piecewise linear function consider markov kernel written denoting law random variable let statistical model associated observation random variables density respect lebesgue measure applying remark get step left check actually case show indeed total variation distance family probabilities associated experiments bounded since one write order control distance split follows application mean theorem gives control term let consider taylor expansion points denotes point smoothness condition allows bound error follows certain point linear character write cam theory comparison statistical models denotes left right derivative depending whether let observe considering right derivatives left ones would indeed applying mean theorem function fact get allows exploit condition indeed exists using fact get collecting pieces together find hence conclude independent gaussian random variables gaussian white noise experiments subsection seen reduce density estimation problem adequate multinomial experiment application results carter recalled example allows obtain asymptotic equivalence statistical model associated observation random variables density respect lebesgue measure one observes gaussian independent random variables course gaussian experiment equivalent nmq istical model associated independent gaussian random variables claim asymptotically equivalent white noise model associated continuous observation trajectory gaussian process solution sde dyt dwt standard brownian motion divide proof two steps denote statistical model associated observation intervals associated independent mpis experiment firstly show gaussian random variables asymptotically equivalent observing asymptotically equivalent observing increments ester mariucci step means property get triangular inequality bound denote using trick step subsection bound kri used fact denotes remainder taylor expansion hand remainder taylor expansion observe belongs functional class still bounded away zero infinity continuous derivative precisely particular deduce magnitude kri thanks know kri kri hence quantities order find step since model associated observation increments process defined clear let discuss bound start introducing new stochastic process functions defined figure independent centered gaussian processes independent variances var processes constructed standard brownian bridge independent via references construction gaussian process mean variance given respectively var var var one compute way covariance deduce standard brownian motion applying fact get total variation distance process constructed random variables gaussian process bounded since implies kind computations made step subsection allows conclude choice subsection proven cost needed pass model associated observation random variables unknown density adequate multinomial approximation order using theorem take step obtaining gaussian approximation independent coordinates starting multinomial one comes price finally subsection found appropriate choices asymptotic equivalence gaussian approximation gaussian noise model bound rate convergence constants given particular deduce log log choice references bahadur sufficiency statistical decision functions ann math statistics blackwell comparison experiments proceedings second berkeley symposium mathematical statistics probability berkeley los angeles university california press references blackwell equivalent comparisons experiments ann math statistics bohnenblust shapley sherman reconnaissance game theory rand research memorandum brown low asymptotic equivalence nonparametric regression white noise ann statist brown zhang asymptotic nonequivalence nonparametric experiments smoothness index ann statist brown cai low zhang asymptotic equivalence theory nonparametric regression random design ann statist dedicated memory lucien cam brown carter low zhang equivalence theory density estimation poisson processes gaussian white noise drift ann statist buchmann limit experiments garch bernoulli buscemi comparison quantum statistical models equivalent conditions sufficiency comm math phys carter deficiency distance multinomial multivariate normal experiments ann statist dedicated memory lucien cam continuous gaussian approximation nonparametric regression two dimensions bernoulli asymptotic approximation nonparametric regression experiments unknown variances ann statist asymptotically sufficient statistics nonparametric regression experiments correlated noise probab stat art dalalyan asymptotic statistical equivalence scalar ergodic diffusions probab theory related fields asymptotic statistical equivalence ergodic diffusions multidimensional case probab theory related fields delattre hoffmann asymptotic equivalence null recurrent diffusion bernoulli efromovich samarov asymptotic equivalence nonparametric regression white noise model limits statist probab lett laredo asymptotic equivalence nonparametric diffusion euler scheme experiments annals statistics laredo nussbaum asymptotic equivalence estimating poisson intensity positive diffusion drift ann statist dedicated memory lucien cam golubev nussbaum zhou asymptotic equivalence spectral density estimation gaussian white noise ann statist grama nussbaum asymptotic equivalence nonparametric generalized linear models probab theory related fields asymptotic equivalence nonparametric regression math methods statist grama neumann asymptotic equivalence nonparametric autoregression nonparametric regression ann statist references halmos savage application theorem theory sufficient statistics ann math statistics hansen torgersen comparison linear normal experiments annals statistics nussbaum asymptotic equivalence model independent non identically distributed observations statist decisions cam sufficiency approximate sufficiency ann math statist cam asymptotique statistique les presses montreal cam information contained additional observations ann statist cam asymptotic methods statistical decision theory springer series statistics new york cam yang asymptotics statistics second springer series statistics basic concepts new york mariucci asymptotic equivalence inhomogeneous jump diffusion processes white noise esaim probab stat asymptotic equivalence pure jump processes unknown density gaussian white noise stochastic process appl asymptotic equivalence discretely observed diffusion processes euler scheme small variance case stat inference stoch process appear asymptotic equivalence density estimation gaussian white noise extension annales isup meister asymptotic equivalence functional linear regression white noise inverse problem ann statist meister asymptotic equivalence nonparametric regression errors probab theory related fields milstein nussbaum diffusion approximation nonparametric autoregression probab theory related fields nussbaum asymptotic equivalence density estimation gaussian white noise ann statist asymptotic equivalence nonparametric regression multivariate random design ann statist asymptotic equivalence inference volatility noisy observations ann statist rohde asymptotic equivalence rate convergence nonparametric regression gaussian white noise statist decisions asymptotic equivalence regression fractional noise annals statistics shiryaev spokoiny statistical experiments decisions vol advanced series statistical science applied probability asymptotic theory river edge world scientific publishing strasser mathematical theory statistics vol gruyter studies mathematics statistical experiments asymptotic decision theory berlin walter gruyter torgersen comparison translation experiments annals mathematical statistics comparison experiments factorization univ oslo references van der vaart statistical work lucien cam ann statist dedicated memory lucien cam wang asymptotic nonequivalence garch models diffusions ann statist dedicated memory lucien cam zolotarev probability metrics teoriya veroyatnostei primeneniya leiden university address
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adaptive algorithm precise pupil boundary detection using entropy contour gradientsi cihan topala halil ibrahim cakirb cuneyt akinlarb department electrical electronics engineering anadolu university eskisehir turkey computer engineering anadolu university eskisehir turkey sep department abstract eye tracking spreads vast area applications ophthalmology assistive technologies gaming virtual reality detection pupil critical step many tasks hence needs performed accurately although detection pupil smooth task clear sight possible occlusions odd viewpoints complicate problem present adaptive pupil boundary detection method able infer whether entire pupil clearly visible modal heuristic thus faster detection performed assumption occlusions heuristic fails deeper search among extracted image features executed maintain accuracy furthermore algorithm find pupil aidful information many applications prepare dataset containing high resolution eye images collected five subjects perform extensive set experiments obtain quantitative results terms accuracy localization timing proposed method outperforms three state art algorithms run standard laptop computer keywords pupil detection eye tracking elliptical arc detection ellipse detection gaze estimation shape recognition introduction eye tracking emerged important research area diverse set applications including human computer interaction diagnosis psychological neurological ophthalmologic individuals assistive systems drivers disabled people marketing research biometrics addition efforts integrate technology virtual reality studies increase feeling immersion via rendering virtual environment depth field effect similar human vision pupil boundary detection center estimation essential step eye tracking systems work supported scientific technological research council turkey tubitak anadolu university commission scientific research projects bap grant numbers respectively cuneyt akinlar department computer engineering anadolu university time study email addresses cihant cihan topal halilibrahimcakir halil ibrahim cakir cuneytakinlar cuneyt akinlar preprint submitted elsevier performed precisely pog detection extraction pupil center required estimate location gaze applications even loss single pixel precision pupil center may cause error degrees gaze direction vector would result significant drift estimated gaze point pupil boundary detection difficult problem performed accurately biometric applications medical studies another emerging application area eye tracking virtual reality technologies recently significant leap popularity technology renders scene two different point views views left right eyes user prevent problems like motions sickness rendering locations match interpupillary distance user moreover developers integrate eye tracking systems better simulate human visual system render locations user focuses sharper blur regions boost immersion effect enhance experience september study propose novel adaptive pupil boundary detection method eye images works extracting arcs edge segments image joining find pupil boundary hence center organization paper follows give comprehensive related work pupil boundary center detection section explain method fine detail section analysis method terms accuracy localization running time presented section finalize paper concluding remarks boundaries aiming maximize contour integral value smoothed image derivative max arvacheh tizhoosh developed iterative algorithm based active counter model also capable detecting shapes methods work fine however require full search image plane order find parameters maximize response given model search approach computationally expensive therefore employed eye tracking applications authors use curvature pupil contour sort boundary pixels belong prospective occlusions detect blobs binarized image extract contour biggest blob finally edge pixels pupil boundary selected employing set heuristics eyelids positive curvature etc ellipse fit applied chosen pixels another interesting approach pupil detection proposed utilized eyeseecam project algorithm extracts edge segments removes glints unfavourable artefacts sequence morphological operations based several assumptions finally delaunay triangulation applied remaining pixels pupil boundary detected assuming convex hull starburst algorithm estimates pupil center iterative radial feature detection technique instead finding edges starts locating removing glints exists rays cast initial point within radial step ray stops image derivative greater threshold value sharp intensity change occurs operation iterated updated starting point set feature points collected step finally ellipse fit applied collected points ransac another study authors aim adapt starburst algorithm elliptical iris segmentation problem approximately detects pupil region feature secondly apply segmentation determine proper pupil threshold modified ellipse fitting method employed utilizes gradient information well related work literature pupil detection rich many different techniques proposed section goal give picture proposed solutions pupil boundary detection pupil center estimation many early methods literature utilize discriminative visual structure human eye detect pupil dark intensity pupil region high contrast bright sclera region offers relatively easy way solve problem manner many algorithms extracts pupil iris studies center combinations several methods like thresholding morphological operations connected component analysis center mass algorithms various additional steps addition methods also benefit model fitting approaches find pupil iris boundary circle ellipse studies edge contour extraction employed followed hough transform ransac algorithm accurately estimate boundary pupil differencing another approach roughly detect eye locations remotely taken image works differencing two successive frames captured illumination respectively due physical structure human eye illumination causes significant brightness inside pupil therefore pupil regions become salient difference image along methods also purely approaches mostly utilized iris recognition studies literature daugman proposes integrodifferential operator detecting pupil iris lin wang sun dey samanta vertical ellipse hough edges ellipse hiley iterative morimoto fuhl ellipse edges radial ellipse iterative ellipse circle kumar ebisawa zhu parallelogram symmetric glint adaptive agustin modified comments use temporal information keil center mass algorithm circle ellipse fitting long blob connected comp analysis edge detection morphological operations pupil differencing adaptive algorithms downsampling image thresholding binarization table brief taxonomy pupil boundary detection center estimation algorithms detect pupil boundary estimates center performs iris detection performs roi detection applies histogram power transform image make pupil salient ellipse fitting tries determine false pupil contour pixels curvature values set heuristics requires removal glints assumes initial point ray casting inside pupil iterative algorithm performs fast radial symmetry detection delaunay triangulation removes glints artefacts set morphological assumptions tries find ellipse matches edge image points orthogonal gradients image detects filter edges uses two different approaches algorithmic morphological rescales image fails first attempt spatial coordinates find pupil boundary recent study fuhl detect edges eye image filter respect several morphological criteria later edge segments constructed remaining edge pixels edge segments straight lines eliminated various heuristics finally remaining contours evaluated ellipse fitting best ellipse selected cost function utilize inner gray value roundest shape table gives brief taxonomy abovementioned pupil detection algorithms seen table thresholding common technique literature despite thresholding quickly discriminate image regions different intensity values highly vulnerable lighting conditions parameter configuration consequently fails finding exact location intensity change occurs easily causes decrease curacy another frequently employed technique morphological operations applied thresholded binary image suppress remaining undesired pixel sets improve modal structure image however morphological operations may also degrade actual information image cause significant errors result similarly algorithms utilize thresholding blob detection find center point pupil obviously capable detecting boundary hence applied biometrics medical studies requires precise detection boundaries pupil iris downsampling image save computational time obvious cost decreases accuracy wasting spatial resolution pupil differencing requires little amount computation eases roughly locating pupil however important drawbacks first needs segment yes input frame roi detection edge segment detection segment search corner detection segment ellipse fitting pupil candidate generation pupil detection elliptical arc segments extraction figure processing pipeline proposed algorithm image aidful information many applications proposed method simple flow consists processing pipeline shown fig processing starts detection region interest roi convolving eye image feature extract edge segments contiguous array pixels next step determine whether segment exists traces entire boundary pupil edge segment would found pupil clearly visible little occlusion find whether edge segment circular geometry devise fast heuristic method utilizes gradient distribution edge segments condition segment found extract elliptical arcs segment segment found would case pupil severely occluded eyelids eyelashes arcs edge segments roi extracted following extraction elliptical arcs join every possible combination generate set ellipse candidates least one traces pupil boundary candidates finally evaluated relevance actual pupil contour best one exists chosen among candidate ellipses following subsections elaborate step proposed algorithm fine detail make discussion clear tional hardware obtain bright dark pupil images consecutively synchronous manner furthermore reduces temporal resolution since needs two frames perform single detection due reason sensitive motion fails large pupil displacement occurs two consecutive images ebisawa specifically addresses problem proposes various methods positional compensation section presented overview related studies covering biometrics eye tracking areas viewpoint pupil detection problem interested readers also comprehensive surveys review gaze estimation literature particular proposed method study propose adaptive method pupil boundary detection able save computation inferring whether occlusion case manner computation takes little time pupil clear sight camera contrary algorithm infers pupil severely occluded spends effort detect pupil without compromising applicability main strategy improves algorithm occlusions extracting elliptical arcs input image finding one arc group arcs representing pupil contour way relevant features partially visible pupil extracted detection performed fusion separate features besides detecting pupil boundary center precisely algorithm also identify pupil roi estimation first step proposed method roughly estimate pupil area entire eye image purpose utilize pupil factor external internal figure sample eye image detected roi indicated left edge segments obtained edpf algorithm within roi indicated different colors right edge localization outputs binary edge map similar output conventional edge detectors also outputs result set edge segments contiguous connected pixel chain property extremely eases application algorithm detection recognition problems similar edge detectors several parameters must tuned user different tasks ideally one would want edge detector runs fixed set parameters type image achieve goal incorporated contrario edge validation mechanism due helmholtz principle obtained fast edge segment detector edpf works running parameters extremes detects possible edge segments given image many false positives validates extracted edge segments helmholtz principle eliminates false detections leaving perceptually meaningful segments respect contrario approach little overhead computation fig illustrates detected edge segments example eye image figure color represents different edge segment width contiguous array pixels figure roi detection means convolution operation feature detected rois two eye images different size pupils ric intensity attributes similar vein method pupil described dark compact blob since consists darker intensity levels surrounding iris usually elliptical shape locate pupil region convolve input image feature find maximum response image shown fig pupil size vary real life due reasons like physiological differences pupil dilation light changes reason feature kernel applied various aperture sizes maximum response per unit picked end fig results roi estimation process two eye images two individuals different pupil sizes presented edge segment detection detect edge segments inside roi employ edge drawing edge segment unlike traditional edge detectors work identifying set potential edge pixels image eliminating pixels operations morphological operations suppression hysteresis thresholding follows proactive approach algorithm works first identifying set points image called anchors joins anchors way maximizes gradient response edge paths hence ensures good online segment search main goal step detecting pupil easy computation efficient way circumference entirely visible case occlusion edge segments detected need find one traverses pupil demo http page http values dominate thus distinguish edge segments circular shapes picking ones result plain gradient distribution achieve use entropy function quantized gradient distributions segments log gradient operator tan since entropy function maximizes flat distributions frequency symbol equal compute entropy gradient distribution separate edge segments follows figure gradient computation image derivatives quantization computed gradient directions arg max fgi log fgi boundary intuitive solution apply brute force search follows fit ellipse edge segment compute fitting error pick edge segment smallest fitting error method might work pupil clearly visible occluded glints eyelashes however fitting ellipse calculating fitting error segment requires much computation reduce computational burden devise faster method based analysis gradient directions find segment one exists gradients segments contain substantial information geometrical structure used shape matching retrieval recognition problems since already vertical horizontal derivatives eye image computed edge detection scheme find gradient directions little amount computation see fig arctan function obviously results angle values interval providing angle range examining distribution gradients quantize angles obtain discrete symbols thus divide unit circle regions different directions see fig get quantized gradient directions pixels segment infer modal characteristics segment analyzing gradient distributions easy observe segment form would even gradient distribution tangential gradients perimeter sampled fixed angular step intuitively circular edge segments would relatively uniform gradient distribution whereas straight edge segments unbalanced distribution fgi frequency ith gradient direction entropy values edge segments maximized perfect circle gets lower segment shape differs circle elliptic etc finally entropy becomes zero straight lines since straight line one gradient direction along trajectory since quantize unit circle direction see fig number different symbols maximum entropy value case fig shows edge segments input eye image gradient distributions lengths entropy values sample segments easy observe circular edge segments higher gradient entropy values regardless size whereas straight edge segments lower values expected heuristic discard segments producing small entropy values certain threshold extremely fast manner examining speed method measure computing gradient entropy edge segment available image derivatives faster ellipse fitting error computation times way avoid spending computation time segments irrelevant geometries ellipticals following computation gradient entropies edge segments one segment chosen segment extract elliptical arcs satisfies following three criteria must high gradient entropy theoretical entropy different gradient directions accordingly choose segments gradient entropy due small occlusions glints consider pixels threshold distance start end points segment edge segment gradient distr length entropy along second criterion ellipse fitting essential tool employed various steps proposed method crowd literature two renowned ellipse fitting methods known fast robust among two algorithms taubin method results better ellipse contour slightly lower error however guarantee resulted conic ellipse rather return hyperbola well addition fitzgibbon method always ensures resulted conic ellipse tends extract eccentric ellipses higher fitting errors benefit advantages methods follow simple procedure following first use taubin method examine coefficients resulted conic understand whether ellipse hyperbola turns get hyperbola use fitzgibbon method get ellipse due fact apply ellipse fit consecutive edge elements rather scattered pixel data usually end valid ellipse taubin method ellipse fitting methods need compute fitting error quantitatively evaluate success purpose straightforward method literature except numerical approximations since inaccurate approximations easily cause misjudgements elliptical features developed quantitative fitting error computation method based estimate distance ellipse point solving equations described newtonraphson method couple iterations estimate distance values point ellipse calculate normalized rms distance obtain single scalar represent fitting error ellipse perimeter calculation also exact solution hence employ ramanujan second approximation event one edge segment satisfies three conditions given one minimum ellipse fitting error chosen segment existence segment speeds computation compulsory detection pupil important note shape segment figure two sample eye roi without occlusion detected edge segments top list gradient distributions lengths ans entropy value several edge segments selected sample images bottom must small ellipse fitting error pixels ellipse fit pixels form segment iii must closed segment avoid problems figure roi detection arc extraction pupil candidate generation pupil detection steps first two rows consist examples pupil completely clear sight examples occlusions rows pupil last one input image detected roi detected edge segments within roi segment indicated red exists high entropy segments subjected arc extraction segment could found indicated green short low gradient entropy segments omitted indicated blue detected corners green boxes ellipses fit pixels lying two consecutive corners successful low fitting error ellipses indicated extracted elliptical arcs pupil candidates generated joining possible arc combinations selected ellipse representing pupil contour using argument best viewed color necessarily give high entropy values addition segments geometry gradient distributions segments concave shapes follow complex trajectories also end high entropy values therefore use entropy test prerequisite accelerate algorithm make final decision segment ellipse fitting segments high gradient entropy subjected arc extraction process manner algorithm adapts requires less computation occlusions pupil contour entirely visible due fact straight geometry rarely contains elliptical arcs omit segments low gradient entropy short segments pixels save computation time previous work extract circular arcs combining consecutive line segments detect circles however pupil projection onto camera plane elliptic hence need detect elliptical arcs study elliptical arc extraction next step algorithm extracting elliptical arcs referred arc hereafter edge segments obtained previous step segment could found previous step arcs extracted segment segment found demo page http cost function frames cost function threshold figure images apparatus collect database video sequences figure output cost function set example frames seen result cost function increases proportional pupil occlusion occlusion dramatic pupil image value function overshoots note plot logarithmic scale try fit ellipse subset extracted arcs excluding empty set different arc combinations arcs fig shows generated pupil candidates generated extracted arcs fig since pupil candidate generation process considers subsets selected arcs groups unrelated arcs form valid elliptic structure also subjected eliminated ellipse fit therefore eliminate candidates result high fitting error due fact hat belong pupil boundary eliminate candidates result high fitting error one remaining candidates going selected final pupil utilization cost function final step algorithm ends decision pupil image output cost function diverges solve problem devise another strategy finds start end points potential elliptical arc within edge segment locating corners along segment detect corners segments fast curvature css method utilize image gradient information compute turning angle curvature afterwards apply ellipse fit points lying two consecutive corners along segment obtain elliptical arcs fig present results arc extraction process several test images first two rows pupil completely visible hence segment indicated red detected therefore arc extraction applied segment segment found due occlusions arcs extracted segments high gradient entropy avoid missing critical information see rows fig note even pupil clear sight may appear highly elliptical view angle cases pupil segment may result low gradient entropy segment would detected consequence elliptical arc extraction would applied segments even though occlusion detection pupil previous step get number pupil candidates subset elliptical arcs consisting several arcs accordingly still need select one candidate ellipse final pupil contour make decision define cost function considers following properties candidate ellipse ellipse fitting error eccentricity iii ratio arc pixels perimeter resulting ellipse generation pupil candidates step generate candidate ellipses grouping extracted arcs thus aim detect pupil boundary completely even boundary partially visible generate pupil candidates pupil candidates formed one arcs pupil boundary detected multiple arcs fitting error reasonable expect arcs parts elliptic contour thus need minimize fitting error eccentricity indicates compactness ellipse words diversity ellipse circle computed axes respectively eccentricity circle parabola among pupil candidates subset elliptical arcs also diverse ellipses whose eccentricities get close however pupil projection onto image plane usually closer circle rather skewed ellipse majority applications therefore tend select candidate eccentricity close parameter ratio number pixels involved ellipse fitting perimeter resulting ellipse circumstances one single short arc may result large pupil candidate ellipse may lead inconsistency therefore look pupil candidates formed consensus arc pixels greater experiments observed effect eccentricity less effect due possibility true pupil compact ellipse among candidates accordingly take squares increase effect cost function finally need minimize maximize formulation select candidate minimizes following argument arg min figure snapshot pupil annotation tool two different ellipse fitting algorithms utilized find best conic represent pupil clicking location inside pupil guide displayed order help user equally sample contour points gorithm still possible obtain arcs pupil candidates although actually pupil observe cost function overshoots circumstances due large small values therefore easily find pupil quantifying output fig present plot cost function versus number frames sampled eye blink operation clearly seen output rapidly increases visible part pupil periphery gets smaller due occlusions similarly algorithm ends pupil image cost function overshoots pupil candidates last row fig examining several frames find stable threshold value provide promising results deciding whether pupil provide detail topic next section quantitative experimental results experimental results ith pupil candidate constant fig shows pupil detection results sample images among pupil candidates shown fig one minimizes selected pupil section present results comprehensive set experiments quantitative qualitative manner compare proposed algorithm three state art pupil detection algorithms starburst else quantitatively assess algorithms terms pupil detection accuracy means fmeasure pupil localization running time also provide qualitative results provide useful insight readers performance algorithms addition content present paper also provide supplementary detection true negatives many applications information pupil image important much detecting information provide useful extensions eye tracking applications blink detection proposed figure illustration localization test input image pupil ground truth pupil detected pupil det swirski overlapping indicated blue indicated red green pixels material codes videos etc webeven though points sufficient fit candidates ellipse hence degree freedom picked site points average pupil boundary better reduce effect perspective distortion note pupil detection dataset circle projection onto image may perfect ellipse due perspective distortion order perform experiments first prelens distortion points set along pare dataset containing high resolution pupil boundary fit ellipse eye frames collected subjects starburst two different algorithms select pawe used simple eye tracking apparameters provides lower fitting error thereratus see fig consisting two cameras fore obtain best possible conic represent scene eye built gaze estimation pupil eye image study knowledge available pupil detection dataset resolution higher vga literature localization assessment collection frames ask subthe first quantitative test perform localiza starburst jects move eyes different certion assessment pupil detection algorithms tain order way obtain eye images evaluation quantify success algorithms pupil viewed diverse angles without precisely detect pupils respect occlusions camera furthermore also want ground truth data apply test users blink several times obtain negative imframes pupil truly detected age samples pupil exist eventually algorithm dataset source codes algoin frames pupil clear sight rithms downloaded websites authors severe occlusions provided corresponding papers pupil dataset count set parameters algorithms according corpupil positive sample half peresponding publications use best performing riphery visible otherwise considered values explicitly indicated paper negative sample code algorithm used single parameafter collect test frames implement ter set images dataset provides efficient annotation tool eases rigorous annothe best overall result tation procedure see fig annotation tool order quantify localization performance overlays grid polar form ease selection compute overlap ratio detected pixels pupil boundary equal angular pupil ground truth counting number corresolution addition ensure localization responding pixels follows ground truth conics collect exact pixel coordinates users click instead area edet area egt edet egt searches local pixel neighborhood clicked locaarea edet area egt tion find maximum image gradient response edet egt ellipse detected pupil picks location way guarantee selection exact edge pixels pupil ground truth ellipse respectively manner calculate ratio number overand iris high resolution images precisely localization subject subject starburst subject subject subject swirski else overall proposed figure localization test results algorithm subject lapping pixels total number overlapping pixels seen fig number overlapping nonoverlapping pixels determined calculate take average images algorithm higher indicates better localization hence provides higher accuracy application pupil detection utilized obviously fig present average localization results individual subjects overall algorithm although else also give prospering results proposed algorithm performs best improvement runner overall results count samples calculate precision recall values order compute present results fig respect range corresponds varying perfectly aligned ellipses overlap evaluate pupils detected lower sketches fig clearly seen accuracy tests less contentious localization experiments competition among algorithms tighter experiment proposed method outperforms others accuracy rapidly increases even small errors follows stable path regardless subject also see else algorithms performs closely significant success starburst algorithm accuracy assessment previous experiment assess localization performance algorithms considering images detect pupil correctly step evaluate accuracy algorithms counting number images pupil correctly detected entire frame sequences consider pupil image correct detection calculate overlap error compare result threshold value edet egt precision recall count pupils cellipsest pupils pupils count pupils count pupils pupils range value obviously increases intersection area detected ellipse decreases compare threshold value make decision detected pupil whether true positive false positive likewise also evaluate images algorithms detect pupil true negative actual pupil image false negative vice versa precision recall precision recall qualitative results along quantitative accuracy localization results also present qualitative results fig figure provide two results five subjects top bottom also clearly shown algorithm successfully subject subject overlap error overall subject overlap error overlap error overlap error subject subject overlap error starburst swirski else overlap error proposed figure results algorithm subject determine true negatives rows fig present several examples proposed algorithm fails common reason fail cases motion blur algorithm extract edges pupil contour therefore elliptical arcs hence pupil contour detected besides images presented also provide video sequences algorithms interested readers consideration algorithm implementations except starburst matlab according study typical execution matlab times slower based application therefore divide timing results starburst implementation implemented order benefit parallel computing libraries order utilize cpus able make fair comparison assign application specific core measure running times running times algorithms average images summarized table according average running times table proposed method fastest one among algorithms seen proposed algorithm run single thread images resolution per subject analysis see proposed method slightly slower else algorithm subject investigate reason behind longer execution subject see many occlusions cause algorithm fail detecting segment extract arcs edge segments table gives dissection running times proposed algorithm separate steps roi detection obviously computation demanding running time assessment run experiments laptop computer intel ghz cpu able make fair comparison take implementation platforms table average running times algorithms subject milliseconds best timings indicated bold algorithm subject subject subject subject subject subject average starburst else proposed figure qualitative results algorithms subjects images every beginning belong different subject step algorithm takes roughly half entire execution due computation tegral images convolution features several scales another step figure several examples proposed algorithm fails algorithm could detect pupil last two images especially segment could last step pupil detection main reason behind algorithm fall back subject since step contains many computationally expensive ellipse fitting error calculation routines significantly stretchs execution time absence segment case analysis boost execution algorithm pave way applications high resolution images performed comprehensive set experiments comparing proposed method three state art algorithms provided quantitative qualitative results experimental evaluations show proposed algorithm detect pupil even tough occlusive cases without compromising applicability constraints conclusions pupil detection indispensable step many eye tracking applications performed precisely studies pupil detection handled straightforward methods lack accuracy fail occlusive cases study focused developing efficient algorithm pupil boundary detection using entropy edge segments basically find elliptical arcs input image try obtain final ellipse encircling pupil consensus obtained features edge segment detection method employed provide optimum localization elliptical arcs extract edge segments accurately encircle pupil boundary estimates center moreover means gradient distribution references references goni echeto villanueva cabeza robust algorithm vector detection videooculography eyetracking system int conf pattern recognition icpr iterative algorithm fast iris detection advances biometric person authentication springer berlin heidelberg long tonguz kiderman high speed eye tracking system robust pupil center estimation algorithm ieee int conf engineering medicine biology keil albuquerque berger magnor gaze tracking video camera agency lin pan wei robust accurate detection pupil images int conf biomedical engineering informatics vol wang sung study eye gaze estimation ieee trans systems man cybernetics part cybernetics tan wang zhang efficient iris recognition characterizing key local variations ieee transactions image processing dey samanta efficient approach pupil detection iris images int conf advanced computing communications san agustin skovsgaard mollenbach barret tall hansen hansen evaluation gaze tracker acm int symposium research applications etra new york usa ballard generalizing hough transform detect arbitrary shapes pattern recognition table detailed average timing results proposed algorithm different subjects dataset roi detection edge segment detection gradient entropy comp corner detection arc extraction pupil detection total step subject subject subject subject subject overall fischler bolles random sample consensus paradigm model fitting applications image analysis automated cartography communications acm ebisawa improved detection method ieee transactions instrumentation measurement hiley redekopp low cost human computer interface based eye tracking int conf ieee engineering medicine biology society embs morimoto koons amir flickner pupil detection tracking using multiple light sources image vision computing daugman iris recognition works ieee int conf image processing icip vol arvacheh tizhoosh iris segmentation detecting pupil limbus eyelids ieee int conf image processing icip zhu moore raphan robust pupil center detection using curvature algorithm computer methods programs biomedicine kumar kohlbecher schneider novel approach pupil tracking ieee int conf systems man cybernetics smc winfield parkhurst starburst hybrid algorithm eye tracking combining approaches ieee conf computer vision pattern recognition cvpr workshops ryan woodard duchowski birchfield adapting starburst elliptical iris segmentation ieee int conf biometrics theory applications systems bulling dodgson robust pupil tracking highly images acm int symposium eye tracking research applications etra fuhl santini kuebler kasneci else ellipse selection robust pupil detection environments acm int symp eye tracking research applications etra viola jones rapid object detection using boosted cascade simple features ieee conf computer vision pattern recognition cvpr vol ebisawa robust pupil detection image difference positional compensation ieee int conf virtual environments interfaces measurements systems hansen eye beholder survey models eyes gaze ieee trans pattern analysis machine intelligence pami morimoto mimica eye gaze tracking techniques interactive applications computer vision image understanding akinlar topal edcircles circle detector false detection control pattern recognition topal akinlar edge drawing combined realtime edge segment detector journal visual munication image representation topal ozsen akinlar edge segment detection edge drawing algorithm int symp image signal processing analysis ispa desolneux moisan morel edge detection helmholtz principle journal mathematical imaging vision desolneux moisan morel gestalt theory image analysis probabilistic approach springer publishing company incorporated akinlar topal edpf parameterfree edge segment detector false detection control int journal pattern recognition artificial intelligence jia kitchen image similarity computation using inductive learning relations ieee transactions image processing curvature weighted gradient based shape orientation pattern recognition taubin estimation planar curves surfaces nonplanar space curves defined implicit equations applications edge range image segmentation ieee trans pattern anal mach intell fitzgibbon pilu fisher direct least square fitting ellipses ieee trans pattern anal mach intell pami rosin assessing error fit functions ellipses graphical models image processing distance point ellipse accessed september ramanujan collected papers srinivasa ramanujan chelsea publishing new york cakir topal akinlar ellipse detection method joining coelliptic arcs european conference computer vision eccv topal benligiray akinlar robust css corner detector based turning angle curvature image gradients icassp pupil detector supplementary webpage http accessed september starburst source codes https accessed september swirski source codes https accessed september else source codes ftp accessed september prasad leung cho edge curvature convexity based ellipse detection method pattern recognition chia rahardja rajan leung split merge based ellipse detector selfcorrecting capability ieee transactions image processing fornaciari prati cucchiara fast effective ellipse detector embedded vision applications pattern recognition
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received month revised month accepted month doi adaptive control theory applications adaptive optimal output containment control problem linear heterogeneous systems relative output measurements mar majid mohammad bagher seyed kamal hosseini farzaneh hamidreza department electrical engineering ferdowsi university mashhad mashhad iran department electrical engineering university semnan semnan iran missouri university science technology rolla usa summary paper develops optimal relative based solution containment control problem linear heterogeneous systems distributed optimal control protocol presented followers assure outputs fall convex hull leaders output desired safe region also optimizes transient performance proposed optimal control solution composed feedback part depending followers state part depending convex hull leaders state comply applications feedback states assumed unavailable estimated using two distributed observers since followers directly sense absolute states distributed observer designed uses relative output measurements respect neighbors measured example using range sensors robotic information broadcasted neighbors estimate states moreover another adaptive distributed observer designed uses exchange information followers communication network estimate convex hull leaders state proposed observer relaxes restrictive requirement knowing complete knowledge leaders dynamics followers reinforcement learning algorithm structure next developed solve optimal containment control problem online using relative output measurements without requirement knowing leaders dynamics followers finally theoretical results verified numerical simulations keywords adaptive distributed observer cooperative output regulation output containment control optimal control reinforcement learning introduction distributed control systems attracted surge interest variety disciplines due broad applications including cooperation multiple robot systems satellite formation flying vehicles formation control transportation systems cooperative surveillance distributed sensor networks forth distributed cooperative control offers many advantages less communication requirement flexibility enhanced reliability scalability compared centralized counterpart fundamental problem distributed cooperative control systems consensus synchronization goal design distributed control policies agents ensure reach agreement certain quantities interest states outputs using local state output information available agent comprehensive review consensus synchronization problems provided based number leaders consensus synchronization problems categorized three classes namely leaderless containment control latest problem problem interest paper exist multiple leaders objective drive followers convex geometric space spanned leaders containment control problem extensively investigated recent years due numerous potential applications practical engineering example stellar observation satellite formation removing hazardous materials autonomous robots forth practical situations full state information agents unavailable measurement expensive measure instance group mobile agents navigating environments global navigation satellite systems signals rather attenuated forests urban canyons even building interiors position measurement might possible using ordinary global positioning system gps receivers situation one obvious solution might attained installing precise powerful gps receivers agents however precise powerful gps receiver costly also uses electrical power due fact requires amplifying attenuated signal possibly burden weight therefore real world scenarios solution may unfeasible costly motivating concept anchor agents localization problem context wireless sensor networks another solution aforementioned situation scenario equip small fraction agents leaders precise powerful gps receivers measure absolute states however rest agents followers equipped ordinary gps receivers access absolute position measurements access relative output measurements respect neighbors information broadcasted communication network neighbors existing containment control protocols focus case homogeneous agents agents identical dynamics recent results containment control problem considered case heterogeneous followers dynamics assumed dimensions agents however many applications different types agents performing different tasks agents dynamics also dimensions different requires designing distributed control protocols drive followers output convex hull spanned leaders output nevertheless existing results based relative state measurements zheng haghshenas used relative state make sense anymore followers different dimensions although design distributed relative based control protocols considered wen results still limited homogeneous systems output containment control heterogeneous systems considered zuo however method followers require absolute state output well complete knowledge leaders dynamics may available followers many applications moreover approach requires restrictive assumption requiring strongly connected communication graph besides mentioned shortcomings existing results another shortcoming take account transient behavior followers give importance response followers assure followers states outputs eventually converge convex combination leaders states outputs however desired find optimal solutions guarantee convergence also minimize transient containment error time another important issue considered existing results containment control designing online solutions require complete knowledge leaders reinforcement learning successfully used design adaptive optimal controllers systems systems online real time however knowledge solution optimal containment control problem overcome aforementioned shortcomings existing work paper presents adaptive optimal solution output containment control problem linear heterogeneous systems two distributed observers used distributed adaptive observer designed estimate followers state another distributed adaptive observer developed estimate convex hull leaders state proposed distributed adaptive observer relaxes restrictive requirement knowing complete knowledge leaders dynamics followers reinforcement learning algorithm structure developed solve optimal output containment control problem online real time proposed algorithm require knowledge leaders dynamics uses relative output measured data followers information broadcasted communication network neighbors main contributions paper follows novel distributed dynamic relative output feedback control protocol developed based cooperative output regulation framework solve output containment control problem linear fully heterogeneous systems adaptive distributed observer presented estimate leaders dynamics well outputs convex combination leaders states follower contrast existing work observer relaxes restrictive requirement knowing complete knowledge leaders dynamics followers optimal solution distributed containment control problem presented optimize transient output containment error followers well control efforts assuring state containment error algorithm developed solve formulated optimal output containment control problem online real time using relative output measurements followers respect neighbors information broadcasted neighbors without requirement knowing complete knowledge leaders dynamics followers subsequent sections organized follows basic concepts graph theory definitions notations presented section section states output containment control problem output regulation framework moreover analysis provided find containment control problem offline solution distributed adaptive observer designed section optimality explicitly imposed solving containment control problem section enables use techniques learn solution online real time numerical simulation given validate effectiveness theoretical results section finally section conclusions drawn preliminaries notations following notations used throughout paper let represent dimensional real vector space real matrix space respectively denotes matrix zeros let column vector entries equal represents identity matrix represents matrix matrices diagonal denotes euclidean norm matrix column denotes row matrix symbol represents kronecker product distance set denoted inf graph theory subsection basic concepts algebraic graph theory briefly reviewed let communication topology among agents presented weighted directed acyclic graph set nodes set edges weighted adjacency matrix adjacency elements edge node called parent node node child node neighbor assume directed graph acyclic graph directed cycle set node neighbors denoted directed path node node sequence edges distinct nodes directed graph directed graph strongly connected directed path every ordered pair nodes directed graph said spanning forest exists least one node directed path node nodes agent called leader receive information others neighbor otherwise called follower assume agents followers agents leaders notational convenience used denote set followers set leaders respectively laplacian matrix associated defined laplacian matrix associated partitioned note since last agents leaders last rows equal zero sequel assume communication graph satisfies following assumption assumption directed graph acyclic follower exists least one leader directed path lemma meng assumption invertible eigenvalues positive real parts entry row sum equal one proposition qin directed acyclic graph relabeled laplacian matrix lower triangular matrix definition rockafellar set said convex whenever convex hull finite set points minimal convex set containing points output containment control offline solution section output containment control problem first introduced standard assumptions listed distributed dynamic output feedback control protocol introduced follower uses relative output measured data followers information broadcasted communication network neighbors output containment control problem formulated linear cooperative output regulation problem offline solution provided consider set agents heterogeneous followers whose models described set homogeneous leaders set followers set leaders state agent control input output state leader output following standard assumptions made dynamics agents assumption pairs stabilizable assumption pairs detectable full row rank assumption leaders dynamics marginally stable full row rank assumption linear matrix equations unique solutions assumption follower access relative output measured data neighbors information gotten neighbors absolute output measurements followers available assuming full row rank assumption standard assumption output feedback see gadewadikar made avoid redundant measurements furthermore transmission zeros condition huang assumption standard assumption satisfied required full row rank therefore assumption standard assumption regarding assumption worth mention containment control problem leaders dynamic unstable due fact convex hull bounded reasonable unbounded convex hull assumption made relax need full relative state measurements existing literature might feasible scenarios output containment control problem defined follows definition system achieves output containment followers outputs converge convex hull spanned leaders outputs define containment error follower compact form yields stack column vectors respectively defined denotes connection weight leader diagonal matrix diagonal elements remark defined implies followers outputs converge convex hull spanned leaders outputs defined note expressed one stack column vector solve containment control problem definition following distributed dynamic measurement relative output feedback control protocol introduced paper denotes estimate state agent denotes estimate output denotes estimate convex combination leaders states moreover coupling gains gain matrices respectively designed lemma coupling gain finally design feedback gain matrices respectively note dynamic compensator compact form regarded observer convex combination leaders states distributed observer state follower note also absolute output measurements followers used control protocol remark since leaders act command generators usually equipped powerful sensors reasonably assume know states broadcast neighbors assumption standard assumption cooperative control relative measurements literature instant see zhang section eqn however followers assumed access absolute output measurements remark note leaders autonomous agents sense receive information agents acts autonomously guide followers leaders influenced followers therefore observer design well control design designed followers assure converge convex hull leaders output trajectory ultimate goal output containment control problem consider distributed state observer let global state estimation error stack column vectors respectively using dynamics global state estimation error yields proceeding following technical results required lemma let assumptions satisfied let eigenvalue design observer gain unique positive definite solution observer positive definite design matrices global state estimation error dynamic asymptotically stable coupling gain satisfies min min proof one show using proposition procedure theorem fax matrices hurwitz global state estimation error dynamic asymptotically stable assumption based lemma eigenvalues positive real parts rest proof similar theorem zhang omitted remark worth noting requirement acycle assumption lemma obviated agents identical dynamics see zhang details composition containment error control laws result following system state stack column vectors stack column vectors using ready describe output containment control problem defined definition cooperative output regulation problem follows problem cooperative output regulation problem given system digraph find control protocol form system following properties property origin system set zero asymptotically stable matrix hurwitz property initial conditions lim remark property indicates solutions system forget initial conditions converge zero exosystems leaders disconnected moreover properties together indicate solutions system forget initial conditions converge solutions depending exosignals see haghshenas problem huang problem details remark note error given actually output containment error therefore based remark solving problem output containment control problem described definition also solved order solve problem present following lemma system lemma suppose system satisfies property distributed control laws lim exists matrix satisfies following linear matrix equations proof let using one since hurwitz lim using manipulation rewritten follows lim lim following theorem shows problem solved using distributed control laws theorem consider system let assumptions satisfied control protocol designed lemma design hurwitz using solutions problem solved using distributed control laws positive constant proof nonsingular matrix one verify structure assumption exists hurwitz based lemma hurwitz positive constants chosen respectively based lemma assumption eigenvalues positive real parts therefore assumption one see hurwitz positive constant thus distributed control laws property problem satisfied next verify property problem let given assumption one let using yields note moreover hence satisfies linear matrix equations follows lemma property also satisfied lim completes proof remark seen proof theorem assumption guarantees feasibility solution lemma remark graph condition haghshenas zuo unnecessarily strong paper defining output containment error restrictive required assumption strongly connected communication graph haghshenas assumption zuo assumption relaxed milder assumption communication graph spanning forest remark solutions output regulator equations well distributed observer need complete knowledge leaders dynamics knowledge however available followers many applications order obviate requirement leaders dynamics distributed adaptive observer designed convex combination leaders states section optimality next implicitly incorporated design containment control problem section optimize transient containment errors agents algorithm developed solve optimal output containment control problem online real time without requiring knowledge leaders dynamics distributed adaptive observer leaders convex hull section distributed adaptive observer developed estimate convex combination leaders states outputs follower well leaders dynamics simultaneously estimate convex combination leaders states outputs follower consider following distributed adaptive observer estimation leaders dynamics respectively estimation convex combination leaders states outputs follower respectively following lemma used proof theorem observer design lemma cai consider following system asymptotically stable positive constant bounded continuous decay zero exponentially time infinity decays zero exponentially time infinity theorem consider leader dynamics adaptive observer let leaders dynamics estimation errors follower state estimation error convex combination leaders states outputs respectively initial conditions one obtains positive constant exponentially positive constant exponentially positive constant exponentially positive constant exponentially proof dynamics global leaders dynamics estimation error written using lemma easy show equivalently according assumption lemma eigenvalues positive real parts hence exponentially yields similar part one see exponentially positive constant next remains prove part part end using manipulations yields global form written according assumption matrix hurwitz decays zero exponentially hence applying lemma conclude exponentially adding subtracting right hand side therefore completes proof optimal output containment control problem section first optimal formulation output containment control problem introduced reinforcement learning algorithm proposed drive followers optimally convex hull spanned leaders outputs first assumed followers access states leaders states dynamics restrictive assumption relaxed combining based optimal control presented section distributed adaptive observer designed section distributed observer problem formulation offline solution aim optimal output containment control problem design distributed control laws followers converge convex hull spanned leaders outputs time transient output containment error captured well followers control efforts minimized optimal output containment control problem defined follows problem optimal output containment control problem consider control protocol let distributed state observer designed lemma moreover let distributed observers designed theorem design gain matrices control protocol properties problem satisfied also following discounted performance function minimized symmetric positive definite weight matrices discount factor containment error remark note rewritten follows therefore according lemma one see minimizing results minimizing output containment error shown state observer gains designed lemma moreover shown distributed observers designed theorem therefore control protocol becomes shown following control protocol designed minimize output containment control problem solved one write following form denotes augmented state augmented dynamics given value function written following quadratic form hamiltonian defined follows using stationary condition optimal control gain derived follows satisfies discounted algebraic riccati equation substituting yields discounted follows remark note consequently control gain requires complete knowledge leaders dynamics learn gains online fashion without requiring complete knowledge leaders dynamics algorithm designed subsection remark shown modares discount factor satisfies value function bounded control policy agents dynamics stable equivalent property problem following results show solving problem actually solves problem optimal manner lemma consider system control policy gain given dis count factors satisfies consequently output containment error goes zero asymptotically proof first let written compact form follows multiplying right left sides respectively one using observation observing one see null space subspace null space seen consequently yields consequently containment error defined therefore one conclude null space subspace containment error zero remains show providing discount factors satisfies null space attractive end choose following lyapunov function taking derivative gives mentioned remark hurwitz providing discount factors satisfies therefore assumption one conclude consequently marginally stable therefore exist positive based lasalle invariance principle semi definite matrix mentioned null space subspace converges largest invariance subspace consequently output output containment error equal zero therefore based containment error equal zero completes proof note depending absolute state follower requiring knowledge leaders dynamics graph topology available follower therefore used place implement optimal control without requiring knowledge leaders dynamics graph topology absolute state follower optimal control becomes obtained theorem consider system distributed adaptive observer along adaptation laws assumptions problem consequently problem solved using optimal control policy given long designed lemma discount factors satisfy positive constant proof using obtain nonsingular matrix one verify structure according separation principle distributed observers control gains designed separately shown theorem positive constant thus full row rank matrix asymptotically besides shown lemma asymptotically finally lemma shows asymptotically therefore asymptotically completes proof remark theorem shows one design observers gains appear control gains independently stated theorem chosen sufficiently large designed according lemma positive constants seen larger faster estimation errors distributed observer decays also seen larger faster estimation errors decays observer choosing sufficiently large makes convergence observers sufficiently fast therefore effects control performance negligible weight matrices design parameters chosen symmetric positive definite matrices discount factors used guarantee performance functions bounded given control policies chosen satisfy condition remark distributed optimal output containment control online optimal solution subsection method combined algorithm solve discounted learn optimal gain online real time using relative output measurements followers respect neighbors information broadcasted neighbors without requiring complete knowledge leaders dynamics since value function form quadratic polynomial quadratic polynomial basis vector critic neural network follower chosen optimal value function perfectly approximated optimal critic weight follower number neurons critic follower token optimal policy perfectly approximated actor follower form optimal actor weight follower number neurons actor follower ideal weights critic nns ideal weights actor nns unknown must estimated let value function corresponding written follows estimation iteration satisfies discounted let estimation iteration denoted value function gradient approximated correspondingly let denote estimation iteration estimation iteration one write following augmented form behavior policy admissible policy applied follower estimation target policy optimal policy iteration using manipulation discounted becomes consider control policy update law follows derivative respect time along system dynamics derived multiplying sides integrating sides time interval one obtain bellman equation follows remark note calculate first integrand knowledge leaders dynamic required due dependency obviate requirement term first integrand replaced estimation convex combination leaders outputs defined output estimation follower define exploiting critic actor weighted matrices remark put following form bellman approximation error minimized order drive critic weights actor weights toward ideal values rearranging bellman equation reformulated follows assume collected points time interval using least square method average sense one remark note calculate information follower knowledge graph topology absolute states followers required due dependency obviate requirements algorithm term used place end using place rewritten follows given simultaneously solve find optimal policy online solve problem algorithm given algorithm note algorithm solve problem without requiring knowledge graph topology absolute states outputs followers moreover algorithm solve problem without restrictive assumption followers knowledge leaders dynamics algorithm online algorithm start admissible control policy exploration noise collect required information solve least square problem obtain simultaneously let repeat step small predefined positive constant convergence set optimal control policy remark note due terms information required algorithm collected used beginning learning process however distributed observers converge information collected used algorithm note also mentioned remark observers gains appear control gains designed independently choosing sufficiently large makes convergence observers sufficiently fast therefore effect issue algorithm negligible simulation results section output containment control example given validate proposed approach consider system consists four heterogeneous followers three leaders leaders dynamics given initial leaders state vectors chosen dynamics heterogeneous followers given communication graph among agents given fig nodes represent leaders nodes represent four heterogeneous agents moreover communication weights fig chosen one verified assumptions satisfied distributed observer implemented observer matrices gains set figure systems communication graph time sec figure error state observer followers states error state observer followers states given fig observed fig error state observers followers states converges zero distributed adaptive observer along adaptation laws also implemented observer adaptive laws gains set seen fig error observers convex combination leaders states converges zero adaptive observers converge convex combination leaders states time sec figure error adaptive observer convex combination leaders states solving output regulator equation one obtain weight matrices discount factors chosen using lqr method optimal dynamic output feedback gains use algorithm find optimal dynamic output feedback control online time interval set sec figs show evaluation learned optimal dynamic output feedback controls along state observer adaptive observer adaptation laws heterogeneous systems fig confirms followers move convex hull formed leaders outputs note exist directed paths leaders followers directed path leader followers leader disconnected leaders follower follower follower follower sec sec sec figure outputs agents using algorithm along adaptive observer state observer leaders envelope followers leaders envelope followers time sec figure outputs four heterogeneous agents followers case convex hull line segment spanned outputs two leaders therefore two followers converge line segment spanned outputs leaders note also exists directed path leaders followers therefore two followers converge triangular area spanned outputs leaders followers leaders outputs along envelope leaders outputs shown fig seen fig followers outputs move envelopes formed leaders outputs stay time history output containment errors followers shown fig observed containment errors decay zero therefore seen followers outputs converge convex hull formed leaders outputs containment control problem solved results show introduced approach solves problem without requiring followers knowledge leaders dynamics states based relative output measurements followers respect neighbors information broadcasted communication network time sec figure time history output containment errors conclusions paper online optimal relative based solution output containment control problem linear systems presented followers assumed heterogeneous dynamics dimensions first distributed dynamic relative control protocol developed based cooperative output regulation framework provided offline solution output containment control problem hand proposed control protocol composed feedback part part however feedback states assumed unavailable followers estimated using two distributed observers distributed adaptive observer designed estimate followers states using relative output measurements followers respect neighbors information broadcasted communication network another distributed observer developed estimate convex hull leaders states relax restrictive assumption existent work follower knowledge leader dynamics adaptive distributed observer next designed estimate leaders dynamics convex combination leaders states optimality explicitly imposed finding feedback control gains assure convergence followers outputs convex combination leaders outputs also optimize transient output containment errors augmented ares employed solve optimal output containment control problem hand algorithm structure next developed solve ares online real time based using relative output measurements followers respect neighbors information broadcasted communication network without requirement knowing complete knowledge leaders dynamics followers finally simulation example verified effectiveness proposed algorithm references jadbabaie jie lin morse coordination groups mobile autonomous agents using nearest neighbor rules ieee transactions automatic control egerstedt buffa containment control mobile networks ieee transactions automatic control russell carpenter decentralized control 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adaptive control signal processing song lewis wei zhang structure optimal control unknown systems disturbances ieee transactions cybernetics luo liu huang wang optimal tracking control via ieee transactions neural networks learning systems yang liu optimal design reinforcement learning unknown nonlinear systems ieee transactions systems man cybernetics systems luo liu adaptive constrained optimal control design nonlinear systems structure ieee transactions neural networks learning systems modares nageshrao lopes gad babuska lewis optimal output synchronization heterogeneous systems using reinforcement learning automatica adib yaghmaie lewis output regulation heterogeneous linear systems differential graphical game international journal robust nonlinear control tatari vamvoudakis distributed learning algorithm differential graphical games transactions institute measurement control tatari vamvoudakis distributed optimal synchronization control linear networked systems unknown dynamics american control conference acc ieee mazouchi sani skh novel distributed optimal adaptive control algorithm nonlinear multiagent differential graphical games journal automatica sinica qin cluster consensus control generic linear systems directed topology acyclic partition automatica rockafellar convex analysis new jersey princeton university press huang nonlinear output regulation theory applications siam philadelphia huang chapter fourteen certainty equivalence separation principle cooperative output regulation systems distributed observer approach control complex systems elsevier gadewadikar lewis xie kucera parameterization stabilizing hinf static gains application design automatica hongwei zhang lewis das optimal design synchronization cooperative systems state feedback observer output feedback ieee transactions automatic control ugrinovskii allgower synchronization relative measurements unknown neighbour models australian control conference aucc ieee cai lewis huang adaptive distributed observer approach cooperative output regulation linear systems automatica lewis syrmos optimal control john wiley sons modares lewis jiang tracking control completely unknown systems via reinforcement learning ieee transactions neural networks learning systems
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apr decomposing cubic graphs connected subgraphs size three laurent guillaume anthony romeo irena ligm umr cnrs enpc esiee paris upem france laboratoire informatique umr cnrs nantes rue nantes cedex france department computer science university verona italy abstract let set connected graphs size study problem partitioning edge set graph graphs taken problem known possible choice general graphs paper assume input graph cubic study computational complexity problem partitioning edge set choice identify polynomial problems setting give characterisations cubic graphs cases introduction general context given connected graph set graphs sdecomposition problem asks whether represented union subgraphs isomorphic graph problem long history traced back kirkman intensively studied ever since pure mathematical algorithmic point views one notable results area proof dor tarsi holyer conjecture stated problem contains single graph least three edges many variants problem studied attempting prove holyer conjecture obtain algorithms restricted cases applications arise diverse fields traffic grooming graph drawing particular dyer frieze studied variant set connected graphs edges natural proved problem even assumption input graph planar bipartite see theorem claimed problem remains npcomplete additional constraint vertices input graph degree either interestingly one looks special case bipartite cubic graph vertex degree clearly decomposed polynomial time using selecting either part bipartition making vertex set center shows focusing case cubic graphs lead tractable results opposed general graphs non empty problems turn paper study problem cubic graphs case settle computational complexity problem showing problem cases table summarises state knowledge regarding complexity decomposing cubic arbitrary graphs using connected subgraphs size three puts results perspective allowed subgraphs complexity according graph class cubic arbitrary proposition theorem impossible theorem proposition theorem theorem theorem proposition theorem theorem theorem table known complexity results decomposing graphs using subsets terminology follow notation terminology graphs consider simple connected nontrivial given set graphs graph admits partitioned subgraphs isomorphic graph throughout paper denotes set connected graphs size study following problem input cubic graph set question admit let denote subgraph induced given graph removing subgraph consists removing edges well possibly resulting isolated vertices finally let two graphs subdividing edge consists inserting new vertex edge becomes replaced attaching vertex means building new graph identifying attaching edge consists subdividing using new vertex attaching figure illustrates process attaching edge edge cube graph shows small graphs occasionally use paper fig attaching new edge diamond graph graph net graph decompositions without section study decompositions cubic graphs use note cubic graph since vertices odd degree according bouchet fouquet kotzig proved cubic graph admits iff perfect matching however proof forward direction presented incomplete requires use proposition missing paper therefore provide following proposition completeness together another result also useful case proposition let cubic graph admits used three incident vertex proof partition three sets resp set vertices incident exactly one resp two three note exactly set vertices involved let goal show note vertex extremity three different vertex simultaneously extremity one inner vertex another vertex extremity one since two extremities two inner vertices number counting extremities counting inner vertices putting together two equalities yields completes proof since used cubic graphs proposition directly obtain following result implies proposition cubic graph admits iff perfect matching decompositions without section study decompositions cubic graphs use proposition cubic graph admits iff bipartite proof reverse direction select either set bipartition make vertex set center forward direction let let sets vertices containing respectively centers leaves show bipartition first since covers edges therefore vertices second since vertex would degree least finally edge connects center leaf another belong respectively therefore bipartite prove computed polynomial time recall graph contain induced subgraph isomorphic given graph since bipartite graphs admit decomposition proposition restrict attention graphs contain indeed would allowed proposition would imply admit decomposition strategy consists iteratively removing subgraphs adding initially empty empty case actual decomposition removal operations possible case decomposition exists analysis relies following notion induced vertices graph isolated contains vertex induces lemma cubic graph admits every isolated belongs proof contradiction isolated part decomposition exactly one vertex would center leaving remaining edge uncovered uncoverable minimal example cubic graph admits graph must belong decomposition lemma removal yields perfect matching observation let connected cubic graph sequence least one edge vertex removal yields cubic graph proof contradiction applying least one removal obtain cubic graph graph precedes removal sequence must vertex degree least four since connected proposition cubic graph whose isolated one decide polynomial time whether proof build iteratively removing add initially empty set lemma isolated must belong start adding removing therefore admits iff resulting subcubic graph admits observe contains vertices degree note vertex degree must leaf vertex degree must meeting point two ambiguity arises vertices degree may either center meeting point three however always exist least one vertex degree graph empty observation therefore safely remove graph add following rules stated order succeed deleting whole graph way otherwise decomposition exists conclude case graph may contain proposition cubic graph contains diamond one decide polynomial time whether proof cubic graph vertices assume let diamond induced vertices shown figure connected two vertices respectively adjacent two ways use edges shown figure regardless decomposition choose neighbourhood induce graph obtained removing parts added covered exists therefore assume figure show either must form thereby forcing either form cases removing yields graph contains vertices degree proof proposition observation allows make following helpful observations every leaf must leaf every vertex degree two must either belong leaf two distinct decided follows belongs must also belong otherwise would leaf graph obtained removing would contain cover otherwise must leaf two therefore iteratively remove subgraphs graph add according rules follow stated order succeed deleting whole graph way using either decomposition figure starting point otherwise decomposition exists fig diamond cubic graph two ways decompose arguments developed section lead following result proposition problem cubic graphs decompositions use section show problems hardness proof relies two intermediate problems define structured follows cubic planar monotone satisfiability marked edges marked edges theorem page lemma page lemma page start introducing following intermediate problem marked edges input cubic graph subset edges question admit edge middle edge every either one two edges drawings illustrate proofs section show marked edges dotted edges proof lemma uses following result lemma let bridge cubic graph admits decomposition must middle edge proof contradiction first note belong suppose part situation shown without loss generality bank bank remove contains summing terms degree sequence yields means mod mod therefore admits decomposition components size three argument shows belongs must middle edge completes proof lemma let instance marked edges graph obtained attaching every edge decomposed iff admits proof prove direction separately show transform decomposition decomposition subgraphs edge modified subgraphs distinguish four cases edge belongs attaching prevent adapting decomposition edge belongs extremity attaching prevent adapting part decomposition one edge adapt partition follows two edges adapt partition follows show transform decomposition parts need adapting connected inserted transforming since leaf inserted neighbour bridge middle edge lemma may therefore assume without loss generality starting point follows since simple therefore belong two cases consider belongs mapped onto replacing otherwise extremal edge since either edge remain removing replacing end either one two marked edges either way part added show restrict attention following variant marked edges say graph vertices degree marked edges input graph subset edges question admit edge middle edge every either one two edges following observation help observation let graph vertices mod fig adding net graph vertices dotted edges belong possible decomposition symmetry proof admits mod let subsets vertices degree mod prove allowing vertices make problem substantially difficult adding following gadgets vertices degree let instance marked edges least three vertices adding net mean attaching net leaves adding edges incident net leaves see figure proposition let instance marked edges least three vertices let instance obtained adding net three vertices less decomposed iff decomposed proof construction fewer vertices since degree instead vertices unchanged new vertices degree prove equivalence given decomposition need add induced induced cover edges added net order obtain decomposition see figure show valid decompositions must include choice made proof forward direction indeed marked edges middle edges induced appear decomposition moreover marked edge extremity two edges lying since would force another marked edge middle edge therefore possible decomposition net one defined symmetry safely remove preserving rest decomposition lemma marked edges marked edges proof given instance marked edges create instance successively adding net triple vertices triple remains proposition decomposable iff decomposable moreover either cubic hence instance marked edges trivially observation finally show marked edges reduction relies cubic planar monotone satisfiability problem cubic planar monotone satisfiability boolean formula without negations set exactly three distinct variables per clause literal appears exactly three clauses moreover graph clauses variables vertices edges joining clauses variables contain planar question exist assignment truth values true false exactly one literal true every clause input theorem marked edges proof first show transform instance cubic planar monotone satisfiability instance marked edges transformation proceeds mapping variable onto denoted whose edges belong mapping clause onto cycle five vertices way leaf coincides vertex cycle exactly two leaves adjacent cycle figure illustrates construction yields graph show satisfiable iff admits decomposition apply following rules transforming satisfying assignment decomposition variable set false corresponding added otherwise three edges meeting points three different decomposition one two edges current clause gadget two cases distinguished based whether leaf adjacent leaf cases rest clause gadget yields add decomposition see figure show convert decomposition satisfying truth assignment first observe must satisfy following crucial structural property clause exactly two subgraphs appear indeed construction appear remaining five edges clause gadget decomposed appear without loss generality must leaf either center clause gadget two edges clause gadget connect otherwise rest gadget decomposed cases remaining three edges clause gadget must form thereby causing appear contradiction similar argument allows handle finally none appear must leaf either center clause gadget two edges clause gadget cases remaining three edges clause gadget must form turn makes impossible decompose rest graph therefore yields satisfying assignment following simple way appears set false otherwise set true theorem proof immediate lemmas theorem fig connecting clause variable gadgets proof theorem dotted edges belong converting truth assignments decompositions proof theorem variable set true mapped onto decomposition shows case variable set true namely leaf adjacent leaf shows case set true leaves made adjacent clause gadget allows prove hardness decomposition theorem even graphs conclusions future work provided paper complete complexity landscape decomposition cubic graphs natural generalisation already studied authors study decompositions graphs connected components edges would like determine whether positive results generalise way setting would also interesting identify tractable classes graphs cases decomposition problems hard refine characterisation hard instances instance exist reduction theorem finally note applications relax size constraint allowing use graphs edges decomposition would like know impacts complexity problems study paper references bouchet fouquet trois types graphe combinatorial mathematics proceedings international colloquium graph theory combinatorics berge bresson camion maurras sterboul vol mathematics studies spinrad graph classes survey siam monographs discrete mathematics applications society industrial mathematics dor tarsi graph decomposition complete proof holyer conjecture siam dyer frieze complexity partitioning graphs connected subgraphs discrete appl fusy transversal structures triangulations combinatorial study drawings discrete holyer problems siam kirkman problem combinatorics cambridge dublin mathematical journal kotzig teorie grafov tretieho pro matematiky moore robson hard tiling problems simple tiles discrete comput see appendix details sau regular graphs ring traffic grooming priori placement adms siam discrete schaefer complexity satisfiability problems proc stoc san diego california usa may acm yuster combinatorial computational aspects graph packing graph decomposition computer science review appendix omitted proofs hardness proof uses ideas similar used based slightly different intermediate problem structure follows monotone marked edges theorem page lemma page use following intermediate problem marked edges input cubic graph subset edges question admit edge middle edge lemma let instance marked edges graph obtained attaching every edge decomposed iff admits proof proof forward direction exactly forward direction lemma reverse direction let belong inserted need show removing prevent adapting decomposition order obtain proof similar reverse direction lemma following modification since extremal edge map onto become extremal edge opposed possibly proof lemma give reduction following variant sat monotone boolean formula without negations set exactly three distinct variables per clause question exist assignment truth values true false exactly one two literals true every clause input theorem marked edges proof given instance monotone build instance marked edges follows variable occurrences assume create tree leaves called variable tree whose edges marked see figure edges incident leaf called border edges others called internal edges clause create called clause path marked edges join three border edges variable tree follows see figure one leaf joined another another one leaf joined another another one leaf joined another another note resulting graph indeed cubic inner vertex variable tree degree leaf also part clause path furthermore inner vertices clause path adjacent two vertices path one vertex variable tree endpoint clause path adjacent another vertex path two vertices different variable trees observe first cycle included variable trees length least since must included least tree tree path pair leaves length least therefore would use edges clause path uses one edge must joined leaves variable tree impossible since clause consists three different variables two edges used last edge would joining two leaves variable tree also impossible therefore prove satisfiable iff admits decomposition satisfying truth assignment create described figures specifically variable occurrences true cover edges figure false cover internal edges figure clause least variables among set false two variables corresponding variable trees border edges still uncovered variables assigned true border edges covered rest variable tree tree border edges coming clause path either edges cover clause path border variable trees shown figures always exist decomposition edges constraint marked edge middle overall edges variable trees clause paths covered therefore admits leaves fig variable tree variable occurrences edges marked leaves internal vertices partitioned two sets decomposition corresponding true decomposition internal edges corresponding false first consider variable tree show decompositions used true false assignments fact two possible decompositions internal edges tree indeed consider internal vertices partition see figure marked edges edges two internal vertices must part linking leaf center since internal vertices form path must alternate along path leaves centers either vertices centers vertices centers case yields one possible decomposition adjacent edges described respectively figures naturally assign true variable whose tree decomposed first case false variables make following observation false case consider leaf tree parent due marked edges center middle node endpoint consider clause show variables neither set true set false aiming contradiction assume first variables set true border edges trees already covered covering clause path may use edges path possibility decompose path way use two however would marked true true false true false true false true false false false true fig clause path connections variable trees truth assignment symmetry decomposition path edges neighboring uncovered edges variable trees false variables edges middle edges forbidden assume variables clause set false remains cover clause path border edges variable trees thanks observation made leaves tree false case vertices either centers middle nodes centers due marked edges must middle nodes however noted parents trees endpoints impossible finally variable assigned truth value clause must least one true one false variable therefore satisfying assignment instance monotone
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route formation truck platoons apr sebastian van hoef karl johansson fellow ieee dimos dimarogonas member ieee problem coordinate large fleet trucks given itinerary enable platooning considered platooning promising technology enables trucks save significant amounts fuel driving close together thus reducing air drag setting considered truck fleet provided start location destination departure time arrival deadline higher planning level plans computed plans consist routes speed profiles allow trucks arrive arrival deadlines hereby trucks meet common parts routes form platoons resulting decreased fuel consumption formulate combinatorial optimization problem combines plans involving two vehicles show problem hard solve large problem instances hence heuristic algorithm proposed resulting plans optimized using convex optimization techniques method evaluated monte carlo simulations realistic setting demonstrate proposed algorithm compute plans thousands trucks significant fuel savings achieved ntroduction latooning foreseen become common element intelligent transportation systems term refers group vehicles forming road train without physical coupling short distance maintained automatic control communication platooning received lot attention due potential increase road throughput reducing intervehicle gaps also help facilitate automatic operation vehicles paper focuses potential platooning reducing fuel consumption similar racing cyclists exploit follower vehicles lesser degree lead vehicle experience reduction air drag translates reduced fuel consumption advances wireless communication position advanced driver support systems made wide deployment platooning systems feasible attracted attention major truck manufacturers increased fuel costs environmental awareness make implementation systems likely near future using platooning reduce fuel consumption large number trucks leads challenging coordination problem consider two trucks travel two regions different locations approximately time trucks adjust speeds slightly beginning journeys form platoon start access linnaeus center school electrical engineering kth royal institute technology stockholm sweden shvdh kallej dimos common part route thus save fuel part trips approach might involve one trucks drive slightly faster two merge increases consequently fuel consumption initial phase one truck might instead slow let truck catch travel increased speed later arrive destination time many trucks involved straightforward compute optimal plan trucks main contribution paper derive efficient scalable method coordinate platooning large number trucks way explicitly considering effect speed platooning fuel consumption core novelty computation platoon plans done three computationally mathematically tractable stages first stage involves computation platoon plans taking account two vehicles time second stage selects one plan vehicle since problem solved second stage shown iterative algorithm compute heuristic solutions proposed third stage resulting plans optimized using convex optimization techniques potential method demonstrated using monte carlo simulations simulations demonstrate method able significantly reduce fuel consumption fleet vehicles well handle realistic number transport assignments example sweden efficient operation transport systems widely studied field due large economic environmental impact planning conducted transportation operators ranges way strategic tactical operational planning planning latter stage typically happens level departure arrival times input problem considered paper research aims reduce fuel consumption appropriate choice route travel speeds individual vehicles furthermore operators road infrastructure use variable speed limits ramp metering variable route recommendations etc improve safety efficiency road transportation system various aspects platoon coordination considered literature authors formulate mixed integer linear programming problem without considering speed dependency fuel consumption prove problem authors consider simple coordination scheme evaluate real fleet data local controllers coordinating formation platoons proposed authors use identify economic platoons based various criteria unlike paper method presented allows trucks wait trucks form platoon preliminary material used paper presented one key elements make problem tractable realistic numbers trucks select subset vehicles called coordination leaders vehicles adapt way select subset vehicles inspired clustering algorithm called partitioning around medoids clustering present variety different contexts large body research focuses clustering methods analysis tool structure understand classify large data sets examples include clustering graphs scope community detection closely related problem choosing coordination leaders leader election group agents jointly determine leader approach developed paper seen local leader election pairwise fuel savings interpreted preferences trucks coordination leaders paper organized follows start section formulating coordination problem introduce structure proposed solution section iii section problem broken purely combinatorial problem selecting coordination leaders problem shown section iterative algorithm developed find heuristic solutions combinatorial problem section discusses jointly optimize plans trucks constrained platoons proposed algorithm section section vii method demonstrated monte carlo simulations realistic scenario trucks swedish road network roblem ormulation section formulate problem introduce notation index set finitely many transport assignments tied specific truck transport assignment consists start position destination start time arrival deadline model road network directed graph nodes edges nodes correspond intersections endpoints road network edges correspond road segments connecting intersections function maps edge length corresponding road segment vehicle position pair indicates current road segment far vehicle traveled along segment goal compute plans trucks ensure arrival trucks individual deadline plan includes route road network start destination encodes piecewise constant speed trajectory speed constrained range feasible speeds vmin vmax supposed vehicles road sake planning reasonable assume trucks change speed instantaneously approach developed paper generalized order relax assumption definition vehicle plan vehicle plan consists route speed sequence time sequence route sequence edges road network speed sequence sequence speeds speeds within feasible speed range vmin vmax time sequence defines speed changes speed selected note changes speed happen principle everywhere beginning route segments symbols introduced notational convenience value may different different vehicle plans want compute vehicle plan truck valid vehicle plan brings truck start position time destination deadline vehicle plans constrained two conditions first condition requires trip start start time ends deadline second condition ensures truck arrives destination trip ends distance traveled introduce notion trajectories functions continuous time vehicle trajectory consists edge trajectory linear position trajectory edge trajectory given depends largest integer satisfies speed trajectory linear position second element position time given trucks platoon positions coincide model hence neglect physical dimension trucks platoon consists platoon leader number platoon followers introduce platoon trajectory tsn truck platoon trajectory equals truck platoon follower platoon leader traveling alone thus implies another truck model fuel consumption per distance traveled function speed whether truck platoon follower platoon leader assumed fuel consumption truck travels alone platoon follower reduced fuel consumption hereby neglect relatively small reduction fuel consumption truck platoon leader compared traveling denote fuel consumption per distance traveled vmin vmax function models fuel consumption truck platoon leader travels solo fuel consumption truck platoon follower functions either derived analytical model fitted data purposely omit fuel consumption depends road vehicle parameters order keep presentation concise presented results augmented handle additional parameters problem want solve find vehicle plan vehicle want minimize combined fuel consumption plans total fuel consumption associated vehicle plan given integrating fuel consumption according duration trip platoon coordinator assignments pns pnd vehicle plans fig schematic platoon coordinator trucks provide assignment data platoon coordinator computes fuel efficient vehicle plans ztn speed trajectory platoon trajectory tsn start time arrival time truck combined fuel consumption given primary goal compute vehicle plans minimize iii latoon oordinator consider centralized platoon coordinator fig trucks connect coordinator via communication share assignment data coordinator computes vehicle plans trucks plans sent trucks executed process repeated whenever updated information deviations plans new assignments current vehicle position new start position assignment already executed computation vehicle plans happens four stages computation routes routes calculated using algorithm route calculation road networks computation pairwise vehicle plans many plans involving two vehicles computed fuel savings plans recorded coordination graph introduced following selection pairwise plans consistent subset plans computed previous stage combined selecting subset called coordination leaders joint optimization selected pairwise plans jointly optimized lower fuel consumption platoon partners well merge split locations kept computed step stage computes routes stages compute speed sequences time sequences making use ability trucks form platoons order achieve low fuel consumption algorithms route calculation road networks well developed discussed paper discuss stages following sections electing pairwise ehicle lans section formulate combinatorial optimization problem corresponding second third computation stage introduced section iii problem proven motivates heuristic algorithm developed section begin need able compute call default plan valid vehicle plan according definition either lowest possible fuel optimal constant speed definition default plan default plan vehicle plan speed sequence vcd time sequence fuel optimal speed without platooning vcd computed vcd argmin vcm vmax vcm lowest constant speed arrive deadline vcm max vmin adapted plan introduced next speed sequence time sequence follower truck adapted way allows follower platoon part journey leader leader sticks default plan important order able compose plans plan computed way minimizes fuel consumption definition adapted plan adapted plan vehicle plan adapted vehicle plan denote merge time split time tsp truck becomes platoon follower truck time stays platoon follower tsp two trucks separate sequence events occurs tsp tsp fig overview relevant time instances adapted plan solid line illustrates route adapted plan dashed line one plan adapted index parallel sections line indicate trucks share route section lines top indicates trucks platoon fig illustrates adapted plan denote speed trajectory corresponding speed sequence time sequence adapted vehicle plan truck adapted truck fuel consumption truck plan adapted truck modeled denote platoon trajectory adapted plan tsp tsp fuel consumption altered fact platoon since speed trajectory change since takes role platoon leader reduction fuel consumption results implementing adapted plan default plan positive adapting saves fuel plan adapted exists define might exist adapted plan routes overlap constraint maximum speed conjunction arrival deadline makes impossible trucks form platoon compute interested adapted plans save fuel positive conveniently collect information weighted graph call coordination graph definition coordination graph coordination graph weighted directed graph recall elements represent trucks set edges edge weights edge adapted plan saves fuel compared default plan furthermore introduce set node nni set outneighbors nno define maximum empty set zero max definitions ready formulate problem finding fuel optimal set coordination leaders problem given input coordination graph find subset nodes maximizes fce max coordination leaders select default plans remaining assignments called coordination followers select input output select else end end fig clustering algorithm iterative algorithm compute set coordination leaders plans adapted coordination leader yields largest fuel savings since selection adapted plans alter speed trajectories coordination leaders several coordination followers select coordination leader without affecting fuel savings result adaptation potentially resulting platoons two vehicles objective function fce equals sum fuel savings arg max say coordination follower coordination leader nno problem combinatorial optimization problem compute optimal solution finite time using exhaustive search branch bound technique however computational complexity exact computation might high fact show following problem problem strong indicator searching algorithm computes solutions every coordination graph scales well size coordination graph futile proposition problem size input measured proof found appendix one disadvantage approach presented section truck join one platoon however somewhat mitigated frequent instance later point time might turn beneficial truck leave current platoon join another one terative election oordination eaders section present algorithm computes heuristic solutions problem motivated result problem apply iterative strategy converges local maximum consider algorithm fig call clustering algorithm input coordination graph output set coordination leaders initially empty set iteration node selected objective function fce increased added removed updated accordingly difference fce adding removing node set coordination leaders given function algorithm iterates increase fce possible function measures much gained switching whether belongs defined follows fce fce fce fce otherwise get fce fce max max max sum covers nodes select new coordination leader last summand accounts possibly coordination follower longer get fce fce max max max sum covers nodes coordination leader change last summand accounts possibly becoming coordination follower paper consider two methods select set first method select greedy manner according arg max second method choose randomly equal probability set clustering algorithm guaranteed converge finite time due number possible subsets finite thus number possible assignments finite every iteration fce strictly increases means changes every iteration assignment never reoccurs worst case clustering algorithm iterates subsets termination clustering algorithm efficient note instance function computed based induced neighbors means average complexity computing function average node degree number nodes coordination graph furthermore node added removed neighbors needs recomputed simulations suggest selecting greedy random manner makes little difference quality computed solution however greedy node selection tends lead less iterations algorithm thus better suited serial implementation random node selection might preferable parallel implementation due reduced need synchronization computed set coordination leaders immediately vehicle plan truck plans jointly optimized discussed following section oint ehicle lan ptimization section derive jointly optimize vehicle plans selected clustering algorithm formulating convex optimization problem linear constraints group consisting coordination leader coordination followers hereby timing platoons assembled broken apart adjusted locations happens changed trucks matched coordination leader coordination leaders follow default plans considered section consider coordination leader followers nfl arg max group agents denoted nfl construct ordered set time instances set contains start time arrival deadline coordination leader merge times split times followers divide distance traveled leader start destination according time instances get distances wnl points vcd speed leader according default plan distances points coordination followers join leave platoon similarly coordination follower nfl wnl wnl variables denote start time merge time split time arrival time follower according adapted plan first element distance along route start merge point part route follower platoons coordination leader entries coordination leader indices defined accordingly last element distance split point destination follower fig illustrates definition introduce sequences indicate segments journey coordination follower platoon follower truck platoon follower segment corresponds otherwise coordination leader pnl coordination follower nfl express speed time sequence truck nfl traversal times segments speed segment remains constant computed traversal times segments trucks routes optimization variables working traversal times rather sequence speeds allows state optimization problem linear constraints times speed changes computed tsn definitions ready state following optimization problem wnl wnl wnl wnl wnl wnl wnl wnl fig illustration sequences defined red dotted line represents route coordination leader black solid lines arrows represent routes coordination followers thin lines indicate distances elements wnl correspond problem min vmax vmin tsn nfl tsn tsnl convex objective sum convex functions hence convex instance polynomials arbitrary constant part coefficients fulfill requirement furthermore constraints linear thus problem convex optimization problem well developed numerical solvers readily available optimization initialized pairwise plans tnl tnl objective function equals combined fuel consumption assignments fig population density map start goal locations sampled brighter pixel larger population density area areas belonging mainland sweden shown blue part sum defines combined fuel consumption assignment defined composed fuel consumption coordination leader coordination followers coordination leader considered travel alone take role platoon leader throughout journey coordination followers travel alone first last segment journey become platoon followers segments two sets constraints first set applies trucks ensures sequences correspond valid vehicle plans particular constraints express trajectories stay within allowed range speed constraints express trucks arrive deadline second set constraints ensures platooning happens specified original pairwise plans constraints ensure coordination leader followers arrive time respective merge point constraints ensure speed leader speed follower supposed platoon vii imulations section evaluate coordination method outlined previous sections monte carlo simulations show coordination truck platooning lead significant reductions fuel consumption compared current situation trucks platoon well compared spontaneous platooning trucks form platoons happen vicinity another generate transport assignments randomly start goal locations sampled within mainland sweden probability assignment starting ending particular location proportional population density see fig resolution degrees longitude latitude road network node closest sampled coordinate chosen calculate routes open source routing machine assignments route found disregarded route longer kilometers kilometers long subsection route randomly selected take account merge points far current position considered coordination since uncertainty becomes large due traffic new assignments rest periods driver start locations along route considered since believe platoon coordination systems frequently replan assignments already route suspended driver rest kind coordination effort would present time gaps one minute order assess quality solution computed clustering algorithm establish upper bound solution problem upper bound based intuition assign every truck best coordination leader ignore coordination leaders contribute objective fce max fig routes platoon coordinator four coordination followers route coordination leader shown black routes coordination followers dashed beginning route marked star merge point follower indicated triangle split point triangle fuel model affine approximation around analytical fuel model fuel per distance traveled kilograms diesel per meter according model relative reduction fuel consumption platoon follower percent speed consider default speed assume speed freely chosen vmin vmax throughout entire journey sample start time assignments uniformly interval hours compute arrival deadlines according default speed pairwise plans trucks platoon long possible coordination follower splits coordination leader drives fast enough arrive time destination least default speed split points arriving time feasible thus trucks guaranteed meet deadlines initial value joint vehicle plan optimization fulfills constraints fig shows example routes coordination leader coordination followers coordination followers join leave platoon compare proposed platoon coordinator fuel savings arise spontaneous platooning trucks happen get others vicinity spontaneously form platoons end collect link arrival times according default plans link scenario sort times collect ascending order groups one minute difference edge arrival time assume groups forms platoon driving default speed default trajectory altered platooning generous estimate since neglects max max second inequality holds since bound tight optimal solution coordination leader otherwise coordination leaders contribute sum nevertheless bound helps assess far heuristic solution away optimum implemented platoon coordination python used cvxopt convex optimization execution clustering algorithm takes less second transport assignments even faster computation times could achieved optimizing implementation simulation consists following steps random generation transport assignments computation routes default plans computation coordination graph computation coordination leaders according section joint vehicle plan optimization according section evaluate different numbers assignments affect amount platooning fuel savings relative default plans comparison compute fuel savings spontaneous platooning run clustering algorithm greedy random node selection compute upper bound objective function fce results averaged simulation runs fig visualizes example coordination graph addition shows assignments selected step see small fraction assignment pairs save fuel forming platoon number assignments grows opportunities available assignment translate larger fuel savings fig shows effect fuel savings numbers transport assignments coordinated varied possible make number observations based data first fuel savings increase rapidly number transport assignments absolute number assignments small assignments added trend stagnates relative fuel savings increase slowly ideally approach asymptotically maximum fuel savings number transport assignments goes infinity since virtually every truck upper bound greedy random greedy random spontaneous relative fuel savings fig plots visualizes adjacency matrix coordination graphs assignments nonzero entries indicated black red dot corresponding edge coordination graph edges whose corresponding plans selected clustering algorithm correspond red dots platoon follower entire journey small difference greedy random node selection however greedy node selection outperforming random node selection consistently parallel even distributed implementation clustering algorithm random node selection would preferable due reduced need synchronization whereas greedy node selection faster centralized setting furthermore results selecting coordination leaders joint convex optimization less upper bound worse since upper bound tight indicates clustering algorithm performs well see clear improvement fuel savings joint optimization vehicle plans spontaneous platooning gives fuel savings less half achieved coordination also bear mind generous estimate fuel savings spontaneous platooning real difference would probably even larger conclude coordinated platooning yield significant fuel savings coordination crucial leveraging savings transport assignments starting course two hours get reduction fuel consumption number trucks starting time interval area like sweden realistic number total distance traveled simulated scenario order magnitude total distance traveled domestic road freight transport sweden within two hours assuming traffic volume equally spread year density road freight traffic simulated fraction total road freight traffic countries high population density small fuel savings platoon leaders neglected would increase platooning benefit fig shows distribution platoon sizes changes number transport assignments see larger number transport assignments distance traveled large platoons assignments half distance traveled platoon distance number transport assignments fig relative fuel savings due platooning compared default plans varying numbers assignments greedy indicates greedy node selection used clustering algorithm whereas random indicates random node selection keywords refer relative fuel savings joint optimization vehicle plans spontaneous relative fuel savings based estimate fuel savings due spontaneous platooning upper bound refers upper bound fce stated traveled platoons ten less vehicles promising since large platoons might difficult control thus platoon coordinator would prevent planning larger platoons since large platoons account small fraction distance traveled would large impact total fuel savings largest platoon formed vehicles noticeable effect occurs number transport assignments distance traveled relatively large platoons compared distribution number transport assignments seems kind phase transition occurs points enough assignments system one coordination leader many followers several coordination leaders better suited followers understand phenomenon subject future work simulations show computing plans large number vehicles form platoons feasible methods outlined paper motivates platoon coordination enables significant reductions fuel consumption might key leveraging full potential truck platooning viii onclusion centralized truck platoon coordinator proposed system provides trucks vehicle plans lead reduced fuel consumption making use platooning time evolves plans updated account deviations new greedy platoon size distribution number transport assignments random platoon size distribution network relate fuel savings achieved method furthermore want study receding horizon implementation platoon coordinator presence disturbances another direction study system setting practical details speed limits traffic driver rest times different vehicle types etc taken account finally similar coordination strategies might relevant types systems work carried scope companion project proposed platoon coordinator implemented demonstrator featuring simulated real trucks number transport assignments fig figure shows distribution platoon sizes per distance traveled number assignments percent upper plot shows results greedy node selection whereas lower plot shows random node selection clustering algorithm right size platoon indicated platoon size ten difference first second boundary instance means distance traveled member platoon size assignments order handle complexity coordination problem formulated way truck number plans adapted default plans vehicles computed adapted plan involves platooning distance platoon follower thus saving fuel derived plans systematically combined order maximize total fuel savings problem motivates proposed heuristic solution method furthermore derived jointly optimize vehicle plans resulting combination default plans adapted plans effectiveness method demonstrated realistic simulation study simulations motivate systems deployed trucks ability platoon commercially available various directions future work one direction understand transport assignments road ppendix roof roposition show result reduction optimization version set covering problem problem optimization version set covering problem well known reduction known hard problem common proof technique kind result constructing coordination graph correspondence coordination leaders selected sets cover show minimum number leaders corresponds set cover gives maximum value fce consider following set covering problem finite set furthermore let family subsets problem find smallest number subsets whose union construct coordination graph one shown fig introduce node element denote set nodes let bijective mapping elements nodes introduce node element denote set nodes let bijective mapping elements nodes consider node corresponds element nni weight corresponding edges introduce additional node edge node weight clearly reduction linear size input since membership increase fce since nodes inneighbors adding node decrease fce thus problem finding optimal reduces finding nodes belong optimal solution node least one otherwise could add node increase fce least therefore set cover otherwise would existed would node furthermore let set cover property set cover property nodes least one fig illustration graph used prove problem node contributes objective therefore optimal contains minimum number nodes every node least one since fulfills property maps set cover vice versa since solution set covering problem thus problem reduced problem shows problem acknowledgments work supported companion project knut alice wallenberg foundation swedish strategic research foundation swedish research council eferences horowitz varaiya control design automated highway system proc ieee vol jul bonnet fritz fuel consumption reduction platoon experimental results two electronically coupled 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optimal dynamic sensor subset selection tracking stochastic process nov arpan chattopadhyay urbashi mitra sensor networks cyberphysical systems problem dynamic sensor activation tracking process examined tradeoff energy efficiency decreases number active sensors fidelity increases number active sensors problem minimizing error infinite horizon examined constraint mean number active sensors proposed methods artfully combine three key ingredients gibbs sampling stochastic approximation learning modifications consensus algorithms create high performance energy efficient tracking mechanisms active sensor selection following progression scenarios considered centralized tracking process distributed tracking process finally distributed tracking markov chain challenge case process distribution parameterized known unknown parameter must learned key theoretical results prove proposed algorithms converge local optima two process cases numerical results suggest global optimality fact achieved proposed distributed tracking algorithm markov chain based filtering stochastic approximation seen offer error performance comparable competetive centralized kalman filter index sensor networks active sensing sensor subset selection distributed tracking data estimation gibbs sampling stochastic approximation filter ntroduction controlling monitoring physical processes via sensed data integral parts iot cyberphysical systems defense applications industrial process monitoring control localization tracking mobile objects environmental monitoring system identification disaster management applications sensors simultaneously resource constrained power bandwdith tasked achieve high performance sensing control communication tracking wireless sensor networks must contend interference fading one strategy balancing resource use performance activate subset total possible number sensors limit computation well bandwidth use parts paper published previous conferences see arpan chattopadhyay urbashi mitra ming hsieh department electrical engineering university southern california los angeles usa email achattop ubli work funded following grants onr nsf nsf afosr nsf herein address fundamental problem optimal dynamic sensor subset selection tracking stochastic process first examine centralized tracking iid process unknown parametric distribution serves benchmark first extension decentralized tracking process proposed algorithms almost sure convergence local optima proven next distributed algorithm tracking markov process known probability transition matrix developed algorithms numerically validated related literature optimal sensor subset selection problems broadly classified two categories optimal sensor subset selection static data known prior distribution unknown realization dynamic sensor subset selection track stochastic process several recent attempts solve first problem see problem poses two major challenges computing estimation error given observations subset sensors finding optimal sensor subset exponentially many number subsets tractable lower bound performance addressed first challenge greedy algorithm addressed second second challenge addressed via gibbs sampling approach dynamic sensor subset selection stochastic process considered markov process tracking centralized controllers single multiple sensor selection energy constraints sequential decision making previously studied optimal policy structural properties special case dynamic sensor selection infinite horizon examined single sensor selection broadcast sensor data studied knowledge combination gibbs sampling stochastic approximation previously applied iid process tracking herein paper using thompson sampling solved problem centralized tracking linear gaussian process unknown noise statistics via active sensing given consideration dynamic sensor subset selection distributed tracking markov process traditional kalman filtering applicable however much prior recent art developmemt kalman filtering adapt work gibbs mation approach iid case distributed tracking problem appear heavily studied perfect information sharing sensors assumed impractical assumes estimation done centralized fashion sensors make decentralized decisions whether sense communicate fusion center sparsity markov process also exploited iid process parametric distribution unknown parameters distributed tracking iid process discussed section distributed tracking markov chain described section numerical results presented section vii followed conclusion section viii contribution consider connected single wireless sensor network sensor nodes denoted set might also possible connectivity network maintained via relay nodes ignore possibility ease analysis fusion center connected sensors via single hop wireless links responsible control estimation operations network alternatively sensors operate autonomously multihop mesh network physical process measurement denoted discrete time index consider two models evolution paper make following contributions centralized tracking learning algorithm iid process unknown parametric distribution developed particular gibbs sampling minimizes computational complexity stochastic approximation achieve mean number activated sensors constraint furthermore simultaneous perturbation stochastic approximation spsa employed parameter estimation obviating need algorithm challenge overcome analysis handling updates different time scales precursor algorithm unknown parameters algorithms developed simpler version problem parameter distribution known centralized algorithm serves benchmark adapted distributed case exploiting partial consensus partial consensus novel spsa employed learn optimal consensus gains adaptively trick ensuring sensors employ similar sampling strategies sensor use seed random number generator centralized distributed algorithms prove almost sure convergence local optima furthermore prove resources needed communication learning made arbitrarily small exploiting properties updates final generalization develop algorithm sensor subset selection decentralized tracking markov chain known probability transition matrix adapt methods kalman consensus filtering framework gibbs sampling stochastic approximation numerical results show decentralized scheme markov chain performs close centralized scheme numerical results also show tradeoff performance message exchange learning furthermore numerical results show global local optima achieved tracking iid process organization rest paper organized follows system model described section centralized tracking iid process known distribution described section iii section deals centralized tracking ystem odel network data sensing model iid model process parametric distribution unknown parameter needs learnt via measurements lies inside interior compact subset markov model ergodic markov chain known transition probability matrix time sensor used sense process observation sensor provided column vector observation matrix appropriate dimension gaussian vector observation noise assumed independent across across let vector indicator sensor activated time sensor active time else decision activate sensor sensing communicating observation taken either fusion center sensor absence fusion center denote set possible configurations network generic configuration clearly configuration represents set activated sensors used represent configuration entry removed observation made sensor time define centralized estimation problem estimate fusion center connected sensors via direct wireless links denoted denote information available time fusion center history observations activations estimates time fusion center determines hand define current estimate denotes information used fusion center time estimate time varying process sufficient estimate obtain available deciding activation vector collecting observations order optimally decide fusion center needs knowledge performance configurations past hence two different information structures however see gibbs sampling algorithm determines using sufficient statistic calculated iteratively slot information structure used track markov chain different see section define policy pair mappings first goal solve following centralized problem minimizing mean squared error mse subject constraint mean number sensors active per unit time lim sup absence fusion center network estimate sensor denoted denote information available time sensor history sensor determines information available sensor time estimated define policy seek solve following distributed problem arg min lim sup arg min lim sup multiplier viewed cost incurred activating sensor time instant since unconstrained problem across exists one optimizer necessarily unique problem configuration chosen minimum cost achieved law large numbers hence known problem equivalently written distributed estimation problem relax using lagrance multiplier obtain following unconstrained problem arg min arg min lim sup lim sup relaxed version constrained problem iii entralized racking iid proces known section provide algorithm solving centralized problem done relaxing lagrange multiplier though final goal track process unknown discuss algorithms known section precursor algorithms developed subsequent sections unknown also tracking markov chain also extension distributed tracking discussed section unknown hence omitted section following result tells choose optimal solve theorem consider problem relaxed version exists lagrange multiplier optimal configuration constraint satisfied equality pair optimal configuration case exist multiple configurations multiplier probability mass function optimal problem optimal solution choose one configuration probability mass function proof see appendix remark theorem allows obtain solution solution choosing appropriate elaborated upon section basics gibbs sampling known finding optimal solution requires search possible configurations compute mmse configurations hence propose gibbs sampling based algorithms avoid computation let define probability distribution following terminology statistical physics call inverse temperature partition function viewed energy configuration hence figuration selected time probability distribution sufficiently large belong set minimizers high probability however computing requires addition operations hence use sequential subset selection algorithm based gibbs sampling see chapter order avoid explicit computation picking algorithm start initial configuration time pick random sensor uniformly set sensors choose bjt probability choose bjt probability choose activate sensors according theorem algorithm reversible ergodic markov chain stationary distribution proof follows theory chapter remark theorem tells fusion center runs algorithm reaches steady state distribution markov chain configuration chosen algorithm distribution large one runs sufficiently long finite time terminal state belong arg high probability also ergodicity occurence rates configurations match distribution almost surely exact solution algorithm operated fixed optimal soultion unconstrained problem obtained done updating slower iterates algorithm algorithm algorithm algorithm except time use log compute update probabilities theorem algorithm markov chain strongly ergodic limiting probability distribution satisfies proof see appendix used notion weak strong ergodicity markov chains chapter section provided appendix proof similar proof one theorem given completeness remark theorem shows solve exactly run algorithm infinite time contrast algorithm provides approximate solution remark time varying known joint distribution either find optimal configuration using algorithm use ever run algorithm timescale use running configuration sensor activation schemes minimize cost strong ergodicity optimal cost achieved algorithm convergence rate algorithm let denote probability distribution algorithm let consider transition probability matrix markov chain algorithm let recall definition dobrushin ergodic coefficient chapter section matrix using method similar proof theorem show chapter theorem say algorithm prove similar bounds unfortunately aware bound algorithm remark clearly algorithm convergence rate decreases increases hence convergence rate accuracy solution case also rate convergence decreases algorithm convergence rate expected decrease time gibbs sampling stochastic approximation based approach solve constrained problem section section presented gibbs sampling based algorithms provide algorithm updates time order meet constraint equality thereby solves via theorem lemma unconstrained problem optimal mean number active sensors decreases similarly optimal error increases proof see appendix remark optimal mean number active sensors unconstrained problem decreasing staircase function point discontinuity associated change optimizer lemma provides intuition update algorithm algorithm order solve seek provide one algorithm updates time instant based number active sensors previous time instant order maintain necessary timescale difference process update process use stochastic approximation based update rules remark tells optimal solution constrained problem requires randomize two values case optimal theorem belongs set discontinuities however randomization require update randomization probability another timescale stochastic approximations running multiple timescales leads slow convergence hence instead using varying use fixed large update iterative fashion using stochastic approximation proposed algorithm updates iteratively order solve algorithm choose initial discrete time instant pick random sensor independently uniformly sensor choose bjt probability choose bjt probability choose operation decision instant update node follows stepsize sequence constitutes positive nonnegative projection boundaries iterates defined assumption update algorithm inspired following result crucial convergence proof lemma algorithm lipschitz continuous decreasing function proof see appendix discussion algorithm increased hope reduce number active sensors subsequent iterations suggested lemma processes run two different timescales runs faster timescale whereas runs slower timescale understood fact stepsize update process decreases time faster timescale iterate view slower timescale iterate slower timescale iterate view faster timescale almost equilibriated reminiscent stochastic approximation see chapter make following feasibility assumption chosen assumption exists constraint algorithm met equality remark lemma continuously decreases hence feasible must exist intermediate value theorem let define let denote theorem algorithm assumption almost surely limiting distribution proof see appendix theorem says algorithm produces configuration distribution steady state hard constraint number activated sensors let consider following modified constrained problem min mcp easy see mcp easily solved using similar gibbs sampling algorithms section iii gibbs sampling algorithm runs set configurations activate number sensors thus also proposed methodology problem though framework general remark constraint weaker mcp remark choose large number sensors activated gibbslearning small variance allows solve mcp high probability entralized racking iid process nknown section iii described algorithms centralized tracking process known section deal centralized tracking process unknown case learnt time observations creates many nontrivial issues need addressed using gibbs sampling sensor subset selection proposed algorithm unknown since unknown estimate updated time using sensor observations hand solve constrained problem need update time attain optimal theorem iteratively mse configuration unknown since unknown learnt time using sensor observations hence combine gibbs sampling algorithm update schemes using stochastic approximation see algorithm also requires sufficiently large positive number large integer input let denote indicator time integer multiple define first describe key features steps algorithm provide brief summary algorithm step size stochastic approximation updates algorithm uses step sizes nonnegative sequences iii gibbs sampling step algorithm also maintains running estimate time selects random sensor uniformly independently sets bjt probability bjt probability similar algorithm sets operation even repeated multiple times sensors activated according observations collected algorithm declares occasional reading sensors fusion center reads sensors obtains required primarily seek update iteratively reach local maximum function log update since seek reach local maximum log gradient ascent scheme needs used gradient along coordinate computed perturbing two opposite directions along coordinate evaluating difference two perturbed values however estimating gradient along coordinate computationally intensive moreover evaluating requires compute expectation might also expensive hence perform noisy gradient estimation simultaneous perturbation stochastic approximation spsa algorithm generates uniformly sequences perturbs current estimate random vector two opposite directions estimates component gradient difference log log estimate noisy random component updated follows log log iterates projected onto compact set ensure boundedness diminishing sequence ensures gradient estimate becomes accurate time update updated follows intuition sensor activation cost needs increased prohibit activating large number sensors future motivated lemma goal converge defined theorem update since known initially true value known hence algorithm updates estimate using sensor observations fusion center obtains reading sensors goal obtain estimate mse configuration using observations update using however since unknown available alternative mse configuration fusion center uses trace conditional covariance matrix given assuming hence define random variable mmse estimate declared configuration observation made active sensors determined clearly random variable randomness coming two sources randomness randomness distribution since original process yields distribution computation simple gaussian mmse estimator since closed form expressions available compute using following update made iterates projected onto ensure boundedness goal converge ezb equal later argue occasional computation avoided convergence slow algorithm summary steps scheme provided algorithm show theorem algorithm almost surely converges set locally optimum solutions algorithm initialize iterates arbitrarily time perform gibbs sampling step section obtain observations estimate update according compute read sensors obtain update using using multiple timescales algorithm algorithm multiple iterations running multiple timescales see chapter process runs fastest timescale whereas update scheme runs slowest timescale basic idea faster timescale iterate views slower timescale iterate whereas slower timescale iterate views faster timescale iterate almost equilibriated example since iterates vary slowly compared iterates result iterates view complexity algorithm sampling communication complexity since sensors activated mean number additional active sensors per unit time observations need communicated fusion center made arbitrarily small choosing large enough computational complexity computation requires computations whenever however one chooses large additional computation per unit time small however one wants avoid computation also one simply compute update instead configurations however stepsize sequence used instead stepsize used updated using number times configuration chosen till time case convergence result theorem algorithm still hold however proof require technical condition lim inf almost surely satisfied gibbs sampler using finite bounded however discuss update paper sake simplicity convergence proof since technical details asynchrounous stochastic approximation required variant main theme paper one avoid computation step algorithm instead fusion center update time since iterates required gibbs sampling convergence algorithm list assumptions assumption distribution mapping distributed case defined lipschitz continuous assumption known fusion center centralized case known sensors distributed remark assumption allows focus sensor subset selection problem rather problem estimating process given sensor observations optimal mmse estimators computation depend exact functional form done using bayes theorem assumption let consider fixed algorithm suppose one uses algorithm solve given mse replaced objective function finds theorem meet constraint assume given exists optimal lagrange multiplier relax new unconstrained problem theorem also lipschitz continuous remark assumption makes sure iteration converges constraint met equality let define function assumption consider function log expected conditional function conditioned given assume ordinary differential equation globally asymptotically stable solution interior also lipschitz continuous remark one show iteration asymptotically tracks ordinary differential equation inside interior fact lies inside interior condition required make sure iteration converge unwanted point boundary due forced projection assumption makes sure iteration converges main result present convergence result algorithm result tells iterates algorithm almost surely converge set local optima theorem assumptions algorithm almost surely correspondingly almost surely also ezb almost surely process reaches distribution obtained replacing proof see appendix remark constraint satisfied policy constraint becomes redundant algorithm reaches global optimum remark sensors read one update based observations collected sensors determined case converge local maximum log different theorem general however numerical example section vii observe istributed tracking process next seek solve constrained problem problem brings additional challenges compared sensor access local measurement sharing measurements across network consume large amount energy bandwidth iii ideally iterates known sensors resolve issues propose algorithm combines algorithm consensus among sensor nodes see however approach different traditional consensus schemes following aspects traditional consensus schemes run many steps consensus iteration thus requiring many rounds message exchange among neighbouring nodes traditional consensus schemes care correctness data particular sensor node contrast proposed algorithm allows sensor broadcast local information neighbours time slot also since many sensors may use outdated estimates propose learning scheme based stochastic approximation order optimize coefficients linear combination used consensus following updates done links proposed algorithm note gibbs sampling step algorithm run sensors make individual activation decisions sensors supplied initial seed randomization sensors sample time exploit fact next algorithm however depending current configuration node uses linear combination estimate estimates made neighbours let denote initial estimate made node time estimation done node based actual estimate obtained method motivated kalman consensus filter proposed weight matrix used time configuration matrix entry zero nodes connected wireless network induce consensus matrices updated broadcast sensors update use spsa find optimal order minimize error first algorithm define ptdescribe special steps pof number times configuration sampled till time update sensors read obtain either supplied sensors sent specific node centralized computation done broadcasts results sensors estimates computed using denotes estimate node time quantity computed following update done remark algorithm similar algorithm except consensus used deciding estimates additional spsa algorithm used optimize consensus gains however scheme achieve perfect consensus optimal one round message exchange among neighbouring nodes allowed per slot remark since iteration depend run timescale iteration performance algorithm complexity distributed nature mean number additional sensors activated per slotis made small taking large enough argument applies computation moreover one deb cide compute update see section computation avoided without sacrificing convergence update algorithm requires another condition apart gradient descent scheme goal converge exists minimized estimated spsa outline proposed algorithm entire scheme described algorithm algorithm seed supplied sensors gibbs sampling iterates initialized arbitrarily time sensor run gibbs sampling section multiple times sensors activated according make observations compute locally update using read sensors obtain compute estimates using denotes estimate node time update using using update broadcast section ensure convergence iterates updated via following spsa algorithm whenever integer multiple random matrix generated neighbours otherwise independently equal probability discussed section algorithm converge slowly computation avoided gibbs sampling run nodes yield since sensors seed sensors update consensus gains updates need sent sensors however bounded delay broadcast affect convergence since nodes use local consensus periodic broadcast gibbs sampling step distributed algorithm distributed convergence algorithm assumption given function ykb section lipschitz continuous set ordinary differential equations ykb vectorized fixed globally asymptotically stable equilibrium lipschitz continuous present main result related algorithm shows iterates algorithm almost surely converge set locally optimal solutions theorem assumptions algorithm almost surely correspondingly almost surely result almost surely also ezb combine estimates made neighbouring sensors kalman filtering operation required since dynamical system expressed linear stochastic system kalman filter best linear mmse estimator since known conditional covariance matrix given known sensors almost surely process reaches distribution obtained replacing proof proof similar theorem global optimum reached istributed tracking arkov chain section seek track markov chain transition probability matrix finite state space order meaningful problem enumerate state denote state column vector location thus state space becomes also consider measurement process given configuration active sensors sensor makes observation mean covariance matrix depend state model centralized finite horizon version dynamic sensor subset selection problem solved shown sufficient statistic decision making belief vector state space conditioned history authors formulated partially observable markov decision process pomdp belief vector working proxy hidden state also proposed estimator make update belief vector using new observations made chosen sensors hence section skip centralized problem directly solve distributed problem centralized problem leads intractability sequential subset selection problem pomdp formulation provide structural result optimal policy hence solving distributed problem restrict class myopic policies seek minimize cost current time instant estimation scheme yields node belief vector state space generated use filter kcf consensus required since nodes access complete observation set consensus requires sensor kalman consensus filtering kcf use adapted additional consideration sensor maintains estimate time making observation observations made sensor computes using kcf estimates evolve follows called consensus gain kalman gain matrices sensor nbr set neighbours sensor projection probability simple done ensure valid probability belief vector theorem kalman gains nodes arep optimized given consensus gains mse current time step imized computational communication complexity per node implementation grows rapidly hence section also provides alternative suboptimal kcf algorithm compute sensor low complexity easily implementable hence adapt suboptimal algorithm section problem kcf gain update scheme section maintains two matrices viewed proxies covariance matrices two errors also requires system noise covariance matrix since system noise dependent whose pnexact value unknown sensors use node estimate covariance matrix similarly used alternative covariance matrix kcf filter also maintains abstract iterate overall kcf gain update equations section follows small enough making arbitrarily large finite broadcast even done slots avoid network congestion particular slot interestingly updated using local iterates need communication computing involves simple matrix operations polynomial complexity hkt hkt fkt kkt amk vii umerical esults proposed algorithm sensor subset selection done via gibbs sampling run nodes supplied seed nodes generate configuration time quantity updated via stochastic approximation converge mse configuration since mse pnunder computed directly sensor uses past slot update varied slower timescale proposed scheme provided algorithm algorithm input stepsize sequences section consensus gain matrices seed randomization sensors initial covariance matrix iterates initialized arbitrarily define time sensor select sensor running gibbs sampling step section activate sensors autonomously according common selected make observations accordingly sensor perform state estimation gain update update sensors using compute either computation bottleneck update broadcast sensors step performed time remark algorithm suboptimal greedily chooses via gibbs sampling without caring future cost kcf update suboptimal low complexity performs well see section vii complexity algorithm time sensor needs obtain neighbours nbr consensus also needs broadcast nodes update done node per slot communication broadcast made demonstrate performance algorithm centralized algorithm distributed consider following parameter values gibbs sampling run times per slot performance algorithm illustration purpose assume scalar zero mean gaussian noise independent across standard deviation chosen uniformly independently interval initial estimate consider three possible algorithms algorithm basic form call gibbs variation algorithm called lowcomplexgibbs sensors read relatively expensive updates done every slots iii algorithm greedy sensors picked arbitrarily used ever wrong estimate without update mse per slot mean number active sensors per slot plotted figure mse three algorithms much smaller mmse without observation notice gibbs lowcomplexgibbs significantly outperform greedy terms mse shows power gibbs sampling learning time plotted one sample path since gibbs lowcomplexgibbs converge almost surely global optimum also observe gibbs converges faster lowcomplexgibbs since uses computational communication resources observe almost surely gibbs lowcomplexgibbs verified simulating multiple sample paths interesting note numerical example recall theorem remark gibbs lowcomplexgibbs converge true parameter convergence rate vary stepsize parameters performance algorithm consider states assume sensors form line topology transition probability matrix chosen randomly set algorithm call gibbskcf values chosen uniformly scalar values case set compared mse performance three algorithms gibbskcf algorithm centralkalman gibbs greedy lowcomplexgibbs mse gibbs mean number active sensors per slot gibbs lowcomplexgibbs gibbs lowcomplexgibbs fig performance algorithm centralized tracking iid process centralized kalman filter tracks using observations two arbitrary sensors iii perfectblind time state known perfectly sensors observation allowed compute mse perfectblind figure plot mse three algorithms along one sample path gibbskcf observe converges given sample path gibbskcf provides better mse perfectblind mse seen slightly worse along sample paths also observe mse gibbskcf slightly worse centralkalman given instance close many problem instances verified numerically establishes efficacy gibbskcf power gibbs sampling based sensor subset selection despite using one round consensus per slot basically dynamic subset selection compensates performance loss due lack fusion center viii onclusions proposed centralized distributed learning algorithms dynamic sensor subset selection tracking well markovian processes first provided algorithms based gibbs sampling stochastic approximation data unknown parametric distribution proved almost sure convergence next provided algorthm based kalman consensus filtering gibbs sampling stochastic approximation distributed tracking markov chain numerical results demonstrate efficacy algorithms simple algorithms without learning eferences chattopadhyay mitra optimal sensing data estimation large sensor network technical report available https shorter version accepted ieee globecom chattopadhyay mitra optimal dynamic sensor subset selection tracking stochastic process submitted ieee infocom https wang fisher iii liu efficient observation selection probabilistic graphical models using bayesian lower bounds mse perfectblind centralkalman gibbskcf mean number active sensors per slot gibbskcf das moura distributed kalman filter optimized gains ieee transactions signal processing kar moura gossip distributed kalman filtering weak consensus weak detectability ieee transactions signal processing filter optimality stability performance conference decision control pages ieee xiao boyd lall scheme robust distributed sensor fusion based average consensus international symposium information processing sensor networks ipsn pages ieee tzoumas jadbabaie pappas sensor scheduling batch state estimation complexity algorithms limits conference decision control pages ieee michelusi mitra estimation control cognitive radio exploiting sparse network dynamics ieee transactions cognitive communications networking chattopadhyay baszczyszyn gibbsian distributed content caching strategy cellular networks https spall multivariate stochastic approximation using simultaneous perturbation gradient approximation ieee transactions automatic control chattopadhyay coupechoux kumar sequential decision algorithms impromptu deployment wireless relay network along line transactions networking longer version available http fig chain performance algorithm distributed tracking markov proceedings conference uncertainty artificial intelligence uai pages acm zois levorato mitra active classification pomdps state estimator ieee transactions signal processing zois levorato mitra heterogeneous sensor selection physical activity detection wireless body area networks ieee transactions signal processing krishnamurthy djonin structured threshold policies dynamic sensor schedulinga partially observed markov decision process approach ieee transactions signal processing arapostathis optimal sensor querying general markovian lqg models controlled observations ieee transactions automatic control gupta chung hassibi murray stochastic sensor selection algorithm applications sensor scheduling sensor coverage automatica bertrand moonen efficient sensor subset selection link failure response linear mmse signal estimation wireless sensor networks european signal processing conference eusipco pages eurasip bremaud markov chains gibbs fields monte carlo simulation queues springer vivek borkar stochastic approximation dynamical systems viewpoint cambridge university press schnitzler mannor sensor selection crowdsensing dynamical systems international conference artificial intelligence statistics aistats pages arpan chattopadhyay obtained electronics telecommunication engineering jadavpur university kolkata india year telecommunication engineering indian institute science bangalore india year respectively currently working ming hsieh department electrical engineering university southern california los angeles postdoctoral researcher previously worked postdoc paris research interests include wireless networks systems machine learning control urbashi mitra deans professor departments electrical engineering computer science university southern california los angeles usa previous appointments inplace clude bellcore ohio state university photo honors include fulbright leverhulme trust visiting professorship ieee transactions molecular biological multiscale communications ieee communications society distinguished lecturer nae galbreth lectureship okawa foundation award nsf career award research wireless communications ppendix roof heorem prove first part theorem exists one second part theorem proved similarly let denote optimizer possibly different definition since feasible solution constrained problem assumption hence completes proof hence ppendix eak trong rgodicity consider markov chain possibly timehomogeneous transition probability matrix denote collection possible probasbility distributions state space let denote total variation distance two distributions called weakly ergodic markov chain called strongly ergodic exists ppendix roof heorem first show markov chain weakly ergodic let define consider transition probability matrix inhomogeneous markov chain dobrushin ergodic coefficient given see chapter section definition inf min sufficient condition markov chain weakly ergodic chapter theorem positive probability activation states nodes updated period slots hence also node chosen modifiedgibbs algorithm sampling probability hence activation state slot greater independent sampling slots pairs min log log first inequality uses fact cardinality second inequality follows replacing numerator thirdp inequality follows lowerln bounding last equality follows fact diverges hence markov chain weakly ergodic order prove strong ergodicity invoke chapter theorem denote specific time given specific matrix evolves infinite time fixed reach stationary distribution hence claim condition chapter theorem satisfied next check condition chapter rem arg minb argue increases sufficiently large verified considering derivative arg minb probability decreases large using fact bounded sequence converges write monotone hence chapter theorem markov chain strongly ergodic straightforward verify claim regarding limiting distribution ppendix roof emma let corresponding optimal error mean number active sensors multiplier values respectively definition adding two inequalities obtain since obtain completes first part proof second part proof follows using similar arguments convergence fastest timescale let denote probability distribution algorithm column vector indexed cofigurations ppendix corresponding transition probability matrix tpm roof emma let denote form similar standard stochastic straightforward see continuously approximation scheme chapter except differentiable let denote simplicity step size sequence iteration constant sequence derivative given also constant time also constant time stationary distribution tpm exist lipschitz continuous constant slower timescale iterates hence using similar argument chapter lemma straightforward verify hence one show following equivalent noting dividing numerator denominator condition reduced true since hence decreasing also easy verify hence lipschitz continuous ppendix roof heorem let distribution algorithm since follows total variation distance rewrite update equation follows martingale difference noise sequence easy see derivative bouned hence lipschitz continuous function also easy see sequence bounded hence theory presented chapter chapter section converges unique zero almost surely hence almost surely since continuous limiting distribution becomes ppendix roof heorem proof involves several steps brief outline steps provided one one lim convergence iteration note depends iteration depends estimation function updated faster timescale compared let consider iterations constitute twotimescale stochastic approximation note given iteration remains bounded inside compact set independent hence using chapter theorem additional modification suggested chapter section projected stochastic approximation claim almost surely kept fixed value also since lipschitz continuous claim lipschitz continuous also hence using analysis similar appendix section uses chapter lemma one claim lim proves desired convergence iteration convergence iteration iteration view iterations equilibriated let assume kept fixed work timescale situation asymptotically tracks iteration martingale differenece sequence lipschitz continuous using assumption assumption little algebra expression large enough theory chapter theorem chapter section one claim almost surely lipschitz continuous assumption hence using similar analysis appendix section uses chapter lemma say iteration lim convergence iteration note slowest timescale iteration hence view iterations three different timescales equilibriated however iteration affected iterations hence iteration example simultaneous perturbation stochastic approximation since lies inside interior projection operation applied iterates hence combining proposition discussion chapter section say almost surely completing proof seen almost surely hence almost surely almost surely almost surely hence theorem proved
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priv private sample efficient identity testing bryan cai eecs mit bcai constantinos daskalakis eecs csail mit costis gautam kamath eecs csail mit jun june abstract develop differentially private hypothesis testing methods small sample regime given sample categorical distribution domain explicitly described distribution privacy parameter accuracy parameter requirements type type errors test goal distinguish dtv provide theoretical bounds sample size method satisfies differential privacy guarantees type type errors show differential privacy may come free regimes parameters always beat sample complexity resulting running noisy counts standard approaches repetition endowing statistics differential privacy guarantees experimentally compare sample complexity method recently proposed methods private hypothesis testing introduction hypothesis testing problem deciding whether observations unknown phenomenon conform model often viewed distribution alphabet goal determine using samples whether equal model distribution type test lifeblood scientific method received tremendous study statistics since beginnings naturally focus minimizing number observations unknown distribution needed determine confidence whether several fields research application however samples may contain sensitive information individuals consider example individuals participating clinical study disease carries social stigma may thus crucial guarantee operating samples needed test statistical hypothesis protects sensitive information samples odds goal hypothesis testing since latter verifying property population samples drawn samples without care however sensitive information sample might actually divulged statistical processing improperly designed recently exhibited example may possible determine whether individuals participated study data would typically published association studies motivated part realization increased recent interest developing data sharing techniques private protecting privacy computing data extensively studied several fields ranging statistics diverse branches computer science including algorithms cryptography database theory machine learning see references notion privacy proposed theoretical computer scientists found lot traction differential privacy roughly speaking requires output algorithm two neighboring datasets differ value one element statistically close formal definition see section goal paper develop tools privately performing statistical hypothesis testing particular interested studying tradeoffs statistical accuracy power significance privacy sample size precise given samples categorical distribution domain explicitly described distribution privacy parameter accuracy parameter requirements type type errors test goal distinguish dtv want output test private probability make type type error respectively treating hard constraints want minimize number samples draw notice correctness constraint test pertains whether draw right conclusion compares privacy constraint pertains whether respect privacy samples draw pertinent question much privacy constraint increases number samples needed guarantee correctness main result privacy may come free certain regimes parameters mild cost regimes parameters precise without privacy constraints well known identity testing performed log samples size min tight main theoretical result privacy constraints number samples needed max log statistical test provided section upper bound number samples requires proven theorem notice privacy comes free privacy requirement example required statistical accuracy precise constants sitting notation given proof theorem experimentally verify sample efficiency tests comparing recently proposed private statistical tests discussed detail shortly fixing differential privacy type type error constraints compare many samples required methods distinguish hypotheses apart total variation distance find different algorithms efficient depending regime properties desired analyst experiments discussion tradeoffs presented section approach standard approach turn algorithm differentially private use repetition already mentioned absent differential privacy constraints statistical tests provided use optimal log number samples trivial way get privacy using test create datasets comprising samples run test one datasets chosen randomly clear changing value single element combined dataset may affect output test probability thus output private see proof issue approach total number samples draws log higher target see corollary different approach towards private hypothesis testing look deeper tests try privatize tests variations classical compute number times element appears sample aggregate counts using statistic equals close empirical distribution defined counts hypothesis distribution accept statistic low reject high using threshold reasonable approach privatize test add noise laplace noise count running test well known adding laplace noise set counts makes differentially private see theorem however also increases variance statistic noticed empirically recent work show variance optimal style test statistic significantly increases add laplace noise counts section thus increasing sample complexity route seems problematic last approach towards designing differentially private tests exploit distance beween null alternative hypotheses correct test accept null probability close reject alternative null probability close requirements correctness alternative close null could thus try interpolate smoothly datasets expect see sampling null datasets expect see sampling alternative far null rather outputting accept reject merely thresholding statistic would like tune probability output reject based value statistic make reject probability function dataset moreover probability close datasets expect see null close datasets expect see show section statistics high sensitivity requiring samples made appropriately lipschitz approach adding noise counts turning output test lipschitz fail isolation test actually goes intricately combining two approaches two steps filtering step whose goal reject blatantly far step performed comparing counts expectations added laplace noise counts noisy counts deviate expectation taking account extra variance introduced noise safely moreover noise added step differentially private filtering step fails reject perform statistical step step computes statistic without adding noise counts crucial observation filtering step reject statistic actually respect counts thus value statistic still differentially private use value statistic determine bias coin outputs details test given section related work identity testing one classical problems statistics traditionally called hypothesis testing see classical contemporary references field focus often asymptotic analysis number samples goes infinity wish get grasp asymptotic distributions error exponents past twenty years problem enjoyed significant interest theoretical computer science community see survey focus instead finite sample regime rather asymptotics specifically goal minimize number samples required still remaining computationally tractable number recent works simultaneous work focused independence testing investigate differential privacy former set goals particular algorithms focus fixing desired significance type error privacy requirement study asymptotic distribution test statistics hand first work apply differential privacy latter line inquiry goal minimize number samples required ensure desired significance power privacy point comparison two worlds provide empirical evaluation method versus methods problem distribution estimation rather testing also recently studied lens differential privacy another classical statistics problem recently piqued interest theoretical computer science community note techniques required setting quite different must deal issues arise sparsely sampled data preliminaries paper focus discrete probability distributions distribution use notation denote mass places symbol definition total variation distance defined dtv definition randomized algorithm domain private range pairs inputs guarantee called pure differential privacy context distribution testing neighboring dataset definition corresponds two datasets one dataset generated removing one sample factor equivalent alternative definition one dataset generated arbitrarily changing one sample definition algorithm testing problem respect known distribution takes samples unknown distribution following guarantees probability least outputs dtv probability least outputs particular type type errors test parameter radius distinguishing accuracy notice satisfies neither cases algorithm output may arbitrary note algorithm satisfy definitions latter condition correctness property need satisfied falls one two cases former condition privacy property must satisfied realizations samples particular fall two cases recall classical laplace mechanism states applying independent laplace noise set counts differentially private theorem theorem given set counts noised counts private random variables drawn laplace finally recall definition differential privacy relationship differential privacy definition randomized algorithm domain differentially private pairs inputs divergence distribution proposition propositions mechanism satisfies privacy satisfies mechanism satisfies satisfies log differential privacy simple upper bound section provide upper bound differentially private identity testing problem generally show algorithm requires dataset size decision problem made private multiplicative cost sample size folklore result include prove completeness theorem suppose exists algorithm decision problem succeeds probability least requires dataset size exists private algorithm succeeds probability least requires dataset size proof first probability flip coin output yes equal probability guarantees probability least either outcome allow satisfy multiplicative guarantee differential privacy draw datasets size solve decision problem finally select random one computations output outcome correctness follows since randomly choose right answer probability probability solve problem correctly probability privacy note remove single element dataset may change outcome one computations since pick random computation selected probability thus probability outcome additively shifted since know minimum probability output gives desired multiplicative guarantee required privacy obtain following corollary noting tester among others requires samples identity testing corollary exists private testing algorithm testing problem distribution requires log samples min roadblocks differentially private testing section describe roadblocks prevent two natural approaches differentially private testing working section show one simply adds laplace noise empirical counts dataset runs laplace mechanism theorem attempts run optimal identity tester variance statistic increases dramatically thus results much larger sample complexity even case uniformity testing intuition behind phenomenon follows performing uniformity testing small sample regime number samples square root domain size see elements times elements time elements times add laplace noise guarantee privacy obliterates signal provided collision statistics thus many samples required signal prevails section demonstrate statistics high sensitivity thus naturally differentially private words consider statistic two datasets differ one record may quite large implies methods rescaling statistic interpreting probability applying noise statistic differentially private taken large number samples laplaced large variance proposition applying laplace mechanism dataset applying identity tester results significant increase variance even considering case uniformity precisely consider statistic number occurrences symbol dataset size oisson laplace uniform var particular distribution total variation distance uniform variance statistic compared unnoised statistic upper bounded see noised statistic larger variance proof first compute mean note since oisson independently distributed oisson mpi see additional discussion words mean rescaling distance shifted constant amount expectation second term focus case even consider even otherwise uniform total variation distance furthermore direct calculation thus expectation case next examine variance let mpi mqi similar computation var since four summands expression var wish use chebyshev inequality separate two cases require var square mean separation words require high sensitivity consider primary statistic use algorithm mqi mqi shown section dtv variance two cases separated constant probability natural approach truncate statistic range interpret probability output result bernoulli result likely dtv result likely one might hope statistic naturally private specifically would like statistic low sensitivity change much remove single individual unfortunately case consider datasets identical one fewer occurrence symbol shown difference mqi letting uniform distribution requiring sake privacy constraint roughly form particular achieve desired sample complexity one may observe large looking symbol alone sufficient conclude uniform even count laplace noise added indeed main algorithm section works part due formalization quantification intuition priv differentially private identity tester section prove main testing upper bound theorem exists private testing algorithm testing problem distribution requires max log samples min pseudocode algorithm provided algorithm fix constants overview algorithm approach refer reader approach paragraph section proof theorem prove theorem case general case follows cost multiplicative log sample complexity standard amplification argument precise consider splitting dataset log run test one independently return majority result since test correct probability correctness overall test follows chernoff bound remains argue privacy note neighboring dataset result single changed since take majority result conditioning result result either irrelvant equal overall output former case test private latter case know individual test private overall privacy follows applying law total probability require following two claims give bounds random variables note due fact draw oisson samples oisson mpi independently claim log simultaneously probability exactly proof survival function folded laplace distribution exp probability sample exceeding value log equal probability probability exceed value since independent probability none exceeds value desired algorithm priv differentially private identity tester input explicit distribution sample access distribution define sample laplace exists log return either equal probability end draw multiset oisson samples let number occurrences ith domain element mqi log max mqi log log return end end mqi let closest value contained interval sample bernoulli return else return end claim mpi max mpi log log simultaneously probability least proof consider two cases let oisson random variable first assume log bennett inequality following tail bound exp log consider log point log thus log exp log focus case log appeal proposition implies following via stirling approximation exp log set log giving upper bound conclude taking union bound argument depending whether mpi large small proceed proving two desiderata algorithm correctness privacy correctness use following two properties statistic rely condition proofs properties identical proofs lemma omitted claim dtv claim var dtv var first note claim probability return line exactly consider case note claim probability output line thus negligible chebyshev inequality get probability least output probability least note subtract since conditioning event probability union bound similarly dtv chebyshev inequality gives probability least therefore output probability least privacy prove privacy claim probability return line exactly thus minimum probability output algorithm least therefore privacy implies privacy first consider possibility rejecting line consider two neighboring datasets differ frequency symbol coupling randomness two datasets case output differs value mqi lies opposite sides threshold two datasets since differs two datasets probability mass assigned pdf interval length probability outputs differ therefore step private next consider value two neighboring datasets one fewer occurrence symbol consider case already returned line otherwise value irrelevant determining output algorithm mqi mqi mqi mqi mqi mqi mqi mqi mqi mqi since return line log max log log mqi log max mqi log log implies mqi log mqi log enforce terms log log log log since terms step private combining previous step gives desired privacy thus argued beginning privacy section proof differential privacy experiments performed empirical evaluation algorithm priv synthetic datasets experiments performed laptop computer ghz intel core cpu ram significant discussion required provide full comparison prior work area since performance algorithms varies depending regime compared algorithm two recent algorithms differentially private hypothesis testing monte carlo goodness fit test laplace noise mcgof projected goodness fit test note implemented modified version priv differs algorithm lines particular instead consider statistic mqi mqi add laplace noise scale parameter sensitivity guarantees privacy similar algorithms choose threshold noised statistic desired type error algorithm analyzed provide identical theoretical guarantees algorithm practical advantage fewer parameters tune begin experimental evaluation started uniformity testing experimental setup follows algorithms provided uniform distribution algorithms also provided samples distribution unknown case distribution call paninski construction case dtv paninski construction distribution half elements support mass half mass use name construction showed example one hardest distinguish uniform one requires samples distinguish random permutation construction uniform distribution fixed parameters addition recall proposition implies pure differential privacy privacy guaranteed priv stronger zcdp privacy guaranteed particular guarantee differential privacy implies result ran privacy parameter equivalent amount zcdp algorithm provides experiments conducted number different support sizes ranging ran testing algorithms increasing sample sizes order discover minimum sample size type type errors empirically determine empirical error rates ran algorithms times recorded fraction time algorithm correct algorithms take parameter target type error input parameter results first test provided figure indicates support size indicates minimum number samples required plot three lines demonstrate empirical number samples required obtain type type error different algorithms see case statistically efficient followed mcgof priv explain difference statistical efficiency note theoretical guarantees priv imply performs well even data sparsely sampled precisely one benefits tester reduce variance induced elements whose expected number occurrences less since none testers reach regime even expects see element times reap benefits priv ideally would run algorithms uniform distribution sufficiently large support sizes however since prohibitively expensive uniformity testing priv mcgof sample complexity support size figure sample complexities priv mcgof uniformity testing thousands repetitions methods instead demonstrate advantages tester different distribution second test conducted vanishing fraction probability mass concentrated small constant fraction serves proxy large support since elements expected number occurrences algorithms provided samples distribution either similar paninski construction total variation distance placed support elements containing mass ran test support sizes ranging parameters previous test results second test provided figure case compare priv note test slightly better support sizes though difference pronounced diminished depending construction distribution found mcgof incredibly inefficient construction even required samples factor worse support size explain phenomenon inspect contribution single domain element statistic mqi mqi case mqi approximately equal mqi standard deviation term order made arbitrarily large mqi may naively seem susceptible pitfall projection method appears elegantly avoid final test note guarantees zcdp priv guarantees vanilla tial privacy previous tests guarantee privacy proposition guarantees imply third test revisit uniformity testing guarantees imply specifically ran guarantee zcdp priv guarantee log various note often thought theory cryptographically small compare wide range large small test conducted support sizes ranging results third test provided figure found tested priv required fewer samples unsurprising large small since differential privacy guarantees become easy satisfy found true even moderate values distribution domain partitioned intervals distribution uniform interval particular figure support elements contained probability mass similar trends hold modifications parameters identity testing priv sample complexity support size figure sample complexities priv identity testing uniformity testing revisited priv priv priv priv sample complexity priv support size figure sample complexities priv uniformity testing approximate differential privacy implies analyst satisfied approximate differential privacy might better using priv rather algorithm guarantees zcdp main focus evaluation statistical nature note priv efficient runtime implementation mcgof efficient memory usage implementation former point observed noting amount time priv able reach trial corresponding support size mcgof able reach latter point observed noting ran memory support size likely requires matrix computations matrix size plausible implementations could made time memory efficient found implementations sufficient sake comparison acknowledgments authors would like thank jon ullman helpful discussions early stages work authors supported nsf onr references dakshi agrawal charu aggarwal design quantification privacy preserving data mining algorithms proceedings acm symposium principles database systems pods pages new york usa acm jayadev acharya constantinos daskalakis gautam kamath optimal testing properties distributions advances neural information processing systems nips pages curran associates alan agresti categorical data analysis wiley nabil adam john worthmann methods statistical databases comparative study acm computing surveys csur tugkan batu eldar fischer lance fortnow ravi kumar ronitt rubinfeld patrick white testing random variables independence identity proceedings annual ieee symposium foundations computer science focs pages washington usa ieee computer society mark bun thomas steinke concentrated differential privacy 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random sampling philosophical magazine series david pollard good inequalities http jon rao alastair scott analysis categorical data complex sample surveys tests goodness fit independence tables journal americal statistical association sean simmons cenk sahinalp bonnie berger enabling gwass heterogeneous human populations cell systems vincent tan animashree anandkumar alan willsky error exponents composite hypothesis testing markov forest distributions proceedings ieee international symposium information theory isit pages washington usa ieee computer society caroline uhler aleksandra stephen fienberg data sharing association studies journal privacy confidentiality gregory valiant paul valiant automatic inequality prover instance optimal identity testing proceedings annual ieee symposium foundations computer science focs pages washington usa ieee computer society yue wang jaewoo lee daniel kifer differentially private hypothesis testing revisited arxiv preprint fei stephen fienberg aleksandra caroline uhler scalable privacypreserving data sharing methodology association studies journal biomedical informatics
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review conference paper iclr dentifying earning bsorption ump eep nov min sudeep gaddam xiaolin department electrical computer engineering university florida gainesville usa minli sudeepgaddam andyli yinan zhao jingzhe jian department astronomy university florida gainesville usa yinanzhao jingzhema jge bstract pervasive interstellar dust grains provide significant insights understand formation evolution stars planetary systems galaxies may harbor building blocks life one effective way analyze dust via interaction light background sources observed extinction curves spectral features carry size composition information dust broad absorption bump prominent feature extinction curves traditionally statistical methods applied detect existence absorption bump methods require heavy preprocessing reference features alleviate influence noises paper apply deep learning techniques detect broad absorption bump demonstrate key steps training selected models results success deep learning based method inspires generalize common methodology broader science discovery problems present work build deepdis system kind applications ntroduction pervasive interstellar dust contains footprints cosmos evolving dust grains come condensation heavy elements produced stars considered primary reaction sites molecules form essentially source interstellar medium gould salpeter hollenbach salpeter dust grains also play important role controlling metal abundances thermodynamic evolution interstellar medium besides coagulation interstellar dust grains protostellar disk along catalyzed complex organic molecules eventually leads planets dust thought harbor secrets planetary formation even life research dust gas conducted analyzing reddening extinction effects spectra background light sources emission local universe feasible compare stellar spectra explore extinction curves investigate dust grains however luminous background light sources quasars necessary probe dust gas ingredients galaxies larger distances among research topics cosmic dust grains broad absorption bump stands significant values although precise characteristics yet established broad absorption bump believed tightly bounded types aromatic carbonaceous materials draine greenberg promising carriers bump use broad absorption bump absorption bump short rest paper review conference paper iclr polycyclic aromatic hydrocarbon pah molecules pahs recognized abundant organic molecules milk way neighboring galaxies peeters believed building blocks organic life bernstein caro traditional way discovering absorption bump mainly depends statistic techniques jiang generally involves three steps first composite quasar spectrum constructed using median combining jiang composite spectrum reddened used fit every spectrum candidate absorption bump spectra parameterized absorption profile finally applying constrains peak position bump width bump height large portion spectra without bump features filtered order exclude false positives caused noises simulation technique jiang applied determine detection significance among steps curve fitting occupies considerable amount time order convolve absorption bump composite spectrum extra information needed determine absorption redshift significantly constrains available observations due lack extra information cases composite quasar spectrum could potentially reused related computations fitting procedure evolves iterative error minimization get best bump profile using curve fitting method required every new observation fitting restricted relatively smooth continuum quasar spectrum however complicated types emission absorption candidate spectra could potentially disturb fitting process reducing effectiveness method issues tightly coupled method easily resolved paper propose apply deep learning based method detect absorption bump try alleviate aforementioned issues deep learning recently recognized ability automatically extract high level features accurately recognize classify target objects hinton salakhutdinov lecun deep learning effective model trained sufficient data features complex absorption bump application normal observed spectrum consists raw features flux value certain wavelength due complex cosmos environment relation space among features could complicated also observe traditional method curving fitting process actually generating various spectra bumps two facts make deep learning model natural fit application rest paper present details deep learning based method absorption bump detection specifically first give background information problem traditional method show generate raw training testing data sets next present transform raw data different formats models including fully connected neural networks two convolutional neural networks details results also reported finally present generalization methodology simplify improve efficiency similar astronomy problems also resembling problems science fields discuss workflow design details ongoing work deepdis system aims provide services raditional method discovering absorption bump previous research fitzpatrick massa showed extinction curve variation intrinsic spectral slopes quasar represented parameterized linear component drude component added approximate possible absorption bump overall representation extinction spectrum absorption bump expressed wavelength component drude profile definition peak position full width half maximum fwhm drude profile respectively strength measured area bump abump review conference paper iclr rest wavelength absorber frame flux erg sdss model bump reddened linear component bump mgii observed wavelength figure absorption bump example popular method discovering absorption bump filtering fitting main idea get best fitted parameters observed quasar spectrum following equation equation whole procedure consists three major steps first obtaining median quasar composite spectrum basic assumption quasars similar condition similar observed spectrum median quasar composite spectrum regarded base spectrum created combining spectra set observed quasars rest frame fitzpatrick massa schneider second curve fitting based method testing median quasar spectrum first reddened quasar emission redshift absorption bump profile convolved absorption redshift absorption redshift obtained referring unique absorption lines absorption lines parameters equation equation selected minimize fitting error parameters last step conducted applying set rules filter low confidence candidates one fitting example shown figure black curve observed spectrum green curve reddened normal extinction spectrum continuum red line continuum absorption bump couple issues method first hard dependency absorption lines absorption redshift obtained referring absorption lines absorption lines identified using extra step research topic quider zhu however observations contain absorption feature due different emission absorption redshift ranges second error minimizing curve fitting method effective cases broader emission sometimes requires human identification third fitting procedure observation time consuming multiple iterations error minimizing steps required produce best profile dentifying bsorption ump eep earning several aforementioned drawbacks traditional method propose deep learning based method could alleviate issues provide flexible usage specifically proposed method exhibits following merits large data set empowers accurate representation learning curve fitting step traditional method repeatedly generating training samples carefully designing generation able train accurate model absorption bump detection hard dependency absorption lines generating samples control selecting emission absorption redshifts model trained extra information needed new observations easy use share trained model prediction newly added observations one feed forward process trained models could also easily shared among researchers beyond trained models simulation parameters simulated data sets review conference paper iclr easy extend extensions already trained models happen two levels first additional training samples could included cover cases second multiple targets could also added based feature set extensions could achieved conveniently fine tuning trained model order validate feasibility present work applying deep learning based method absorption bump detection following subsections specifically show details training data generation experience model selection data transformation data eneration applied two methods generate data set first method follows corresponding part curve fitting procedure refer berk obtain sdss data release composite quasar spectrum intrinsic spectrum change emission absorption redshifts add absorption bump profile according equation choose typical bump profile abump order increase diversity closely simulate observations apply second method selected subset spectra sdss data release according catalog zhu follow fitting method described jiang convolve absorption bump profile one generated samples extremely close cases note refer existing catalog generate data however dependency required predicting new observations total samples generated two methods described half contains absorption bump half generated sample stored file following wavelength flux pattern around pairs file divide data two parts training data set testing data set ase ully onnected eural etwork table configurations test results fully connected neural network neurons accuracy neurons accuracy neurons accuracy neurons accuracy neurons accuracy first straightforward choice model fully connected neural network effectiveness capturing relationships among input features tested past decades order feed training data fully connected neural network extracted flux value vector use vector train model perform data transformation based observation absorption bump related absolute wavelength value due redshifts rather neighboring flux value relationship addition wavelength interval two consecutive pair relative stable data simulated realworld due data collection mechanism number pairs one sample could slightly different padded samples zeros round total flux values changed configurations fully connected neural network vary number hidden layers number neurons per layer output layer softmax two classes without absorption bump initial learning rate set step decreasing police applied testing results converged training shown table performance one hidden review conference paper iclr layer significantly worse neural networks due incapability capture best performance observed network neurons per layer total number trainable parameters around million ase mage onvolutional eural etwork quasar spectrum plotted black curve figure inspected researchers inspires transform raw data images first plot training testing samples coverage span wavelength around flux raw image drawn large hundreds images scaled gray images used training testing convolutional neural network cnn lecun bengio recently proved effectiveness image related recognition classification due capability parameter sharing capturing localized patterns adapted two popular cnn based models alexnet krizhevsky googlenet szegedy results get modified alexnet googlenet figure visualization alexnet shows filters first convolutional layer plot two test images absorption bump marked red dashed circle last convolutional layer feature maps two test images maps effective activation marked red dashed rectangles better viewed electronic version order understand learned representation select trained alexnet model visualize first convolutional layer filters also last convolutional layer feature maps shown figure first convolutional layer filters typical edge detection filters effective activation area feature maps figure marked red dash rectangles corresponds absorption bump locations test images proves representation sensitive absorption bump also reconstructed input images randomly generated image respect given classes without bump using gradient ascend method reconstructed images shown image bump clearly shows model sensitive absorption bump input spectrum ase hree atrix onvolutional eural etwork image plus cnn model improves accuracy however input image sparse considering large white background order condense input well empower cnn model perform another data transformation transformation also based assumption used fully connected neural network different previous one input vector pad fold vector matrix folded matrices fed cnn models transformation makes possible filters convolutional layer capture localized information neighboring flux values use different configurations cnn review conference paper iclr without bump bump figure reconstructed input image without bump post best results find table fix cnn models two fully connected layers convolutional layers alter number convolutional layers number filters convolutional layer kernel size best model get achieves test accuracy model convolutional layers filters filter size convolutional layers respectively trainable parameters around instead several tens millions previous two cases table configurations results cnn matrix configurations conv layers filters kernel size best results iscussion previous sections present three kinds models corresponding data transformations select trained models plot roc curves shown figure corresponding aucs given table roc curves selected models consistent previously reported testing accuracy absorption bump detection application could improve sensitivity trained models slightly decreasing decision threshold could result slightly false positive signals confident capturing potential absorption bump events interested true positive rate table auc roc curves alexnet googlenet false positive rate figure roc curves various models type alexnet googlenet auc review conference paper iclr based experience absorption bump detection application couple things worth notice data preparation model selection generating raw data order eliminate bias trained model carefully designed criteria followed cover cases much possible another necessary step feeding data training shuffling raw data generated systematical manner using data localized characteristics training batches could impede convergence model could also cause diverged performance training testing data set experienced problems early stage accuracy best model get data low presented details believe deep learning based method absorption bump detection effective trained model information required filter new observations candidates specifications data generation model configuration trained model easy share reuse observe experience data generation could also modeled programming model significantly relieving users heavy programming still benefiting power distributed computing following section present generalization current work providing deep learning based framework similar problems eneralization oosting cience iscovery eep earning cience iscovery roblem similar absorption bump detection category problems science field involves detection certain phenomenon among background events call science discovery problem discoveries building blocks scientific research scientists rely propose ideas validate hypotheses prove theories typical science discovery problem consists two main components feature set relation among features science discovery problem expressed rtarget rtarget targeted discoveries parameters constants one event rtarget defines relations among features depict targeted phenomenon rtarget complementary set rctarget compose rtarget scientists aim pick discoveries observations mainstream method perform detection filtering filtering rules constructed upon raw features extracted high level features high level feature extraction done numerically combining raw features fitting method similar absorption bump application scientists concerned accuracy approximation theories continue improving effectiveness issues methods feature engineering feature engineering procedure also depends lot experience numerous experiments complex mysterious nature science discovery problems also increase difficulty extracting meaningful high level features effectively capture target event characteristics even carefully developed feature extraction technique excellent theoretical explanation data could also offset expected effectiveness considering existence various noise uncertainty data collection phase extra dependency science discovery cases order get better approximated representation targeted event extra effort required obtain additional information example approximated absorption bump representation requires emission absorption redshifts emission redshift included public data set absorption redshift calculated based absorption lines similar requirements occur science fields two potential drawbacks scenarios impede science exploration first extra dependencies incur considerable amount time efforts second may target discovery thus restricting exploration scope due incomplete data sets review conference paper iclr sharing collaboration information exchange collaboration two significant factors speed science discoveries theories findings well shared researchers heavy data processing required generate results existing methods new observation even though distributed computing technologies pervasive crossdiscipline knowledge requirement impedes wide adoption eep earning based ethod inspired advance practical effectiveness deep learning propose method science discovery problems proposed method rooted three observations first deep learning proven practical success classification recognition problems second deep learning powerful relations among features complicated often case science discovery problems finally deep learning requires large amount training data effective despite fact data might adequate sometimes biased targeted discoveries normally rare compared background events observe relatively easy simulate enough eligible training samples desired information embedded control training data sets ensures accurately capture targeted phenomena solves biased training issue satisfies deep learning requirement training samples existing knowledge raw data set training samples train model trained model additional knowledge raw data set fine tuning trained model real data decision training samples model figure workflow deepdis mark top left corner means workflow proposed method shown figure three main steps data preparation model training model service data preparation two operations performed data generation data transformation approximation theory rtarget rctarget constant variables set range parameters used generate enough labeled raw data raw data represented series pairs data transformation phase maps raw data sets desired formats training images matrix transformed data sets used model training step train model converge criteria satisfied trained model published ready use model service step trained model used give decisions data data set similar transformations simulated raw data sets science discovery problems various phenomena could detected upon feature set inspires introduce fine tuning transfer learning techniques workflow specifically additional data could generated tune already trained model substituting layers previous model apply fine tuning using extra data sets new model could trained identify new phenomenon architecture proposed deepdis system shown figure four layers deepdis namely user interface layer control layer runtime layer compute storage resource layer beginning design several main considerations planted gene deepdis first one flexibility second one harnessing power various computing storage resources third one sharing show emphasize aspects following descriptions user interface layer two kinds information required users first part data generation function parameters users requested implement generation functions function take dictionary parameters including constants range parameters output possible pairs constrained ranges resembles flat map function many programming languages data specifications also contains instructions review conference paper iclr generation function parameter specs data specs model specs user interface deepdis controller data coordinator model coordinator generation training service coordinator control detection transformation sharing deepdis caffe runtime compute storage resources figure architecture overview deepdis transform raw data sets data format second part model configuration specifications passed deepdis configurations required users makes deepdis even scientists less knowledge distributed computing deep learning control layer layer responsible understanding users requests translate computing tasks specifically deepdis controller three coordinators data coordinator model coordinator service coordinator fulfill controller divides requests several tasks use dag schedule coordinators responsible designated tasks preparing dividing distributing underlying runtime layer also opens pluggable extensible interfaces advanced users provide new data format transformation functions etc runtime layer compute storage resources layer deepdis integrates distributed computing framework spark zaharia caffe jia deep learning toolbox efficiently execute data generation model training tasks spark extended incorporate adaptations control layer collaborate caffe distributed model training compute storage resources layer different types resources utilized achieve efficient data processing model training sharing onclusion paper presented deep learning based method detecting absorption bump discussed detail generated transformed training data according selected models different models presented results order get sense models learned chose cnn based alexnet model provided visualizations filters feature maps reconstructed maximum activation input images resulted images prove model identifying absorption bump rather background noise success applying deep learning based method absorption bump application generalized methodology broad science discovery problems deep learning models effectively trained upon sufficient amount data also showed ongoing work building specialized system deepdis support proposed method distributed data processing techniques used automatically handle data generation data transformation model training sharing deepdis designed provide science discovery service researchers without much knowledge distributed computing deep learning deepdis hope boost science discovery process eferences berk daniel vanden richards gordon bauer amanda strauss michael schneider donald heckman timothy york donald hall patrick fan xiaohui knapp review conference paper iclr composite quasar spectra sloan digital sky survey astronomical journal bernstein max elsila jamie dworkin jason sandford scott allamandola louis zare richard side group addition polycyclic aromatic hydrocarbon coronene ultraviolet photolysis cosmic ice analogs astrophysical journal caro munoz meierhenrich schutte barbier segovia arcones rosenbauer thiemann brack greenberg amino acids ultraviolet irradiation interstellar ice analogues nature draine interstellar dust grains annu rev astron astrophys fitzpatrick edward massa derck analysis shapes ultraviolet extinction curves atlas ultraviolet extinction curves astrophysical journal supplement series gould robert salpeter edwin interstellar abundance hydrogen molecule basic processes astrophysical journal hinton geoffrey salakhutdinov ruslan reducing dimensionality data neural networks science hollenbach david salpeter surface recombination hydrogen molecules astrophysical journal jia yangqing shelhamer evan donahue jeff karayev sergey long jonathan girshick ross guadarrama sergio darrell trevor caffe convolutional architecture fast feature embedding proceedings acm international conference multimedia acm jiang prochaska kulkarni zhou dusty absorber associated quasar sdss astrophysical journal jiang peng jian prochaska xavier wang junfeng zhou hongyan wang tinggui high dust depletion two intervening quasar absorption line systems extinction bump astrophysical journal jiang peng jian zhou hongyan wang junxian wang tinggui toward detecting dust feature associated strong absorption lines astrophysical journal krizhevsky alex sutskever ilya hinton geoffrey imagenet classification deep convolutional neural networks advances neural information processing systems lecun yann bengio yoshua convolutional networks images speech time series handbook brain theory neural networks lecun yann bengio yoshua hinton geoffrey deep learning nature aigen greenberg mayo dust trust overview observations theories interstellar dust solid state astrochemistry springer peeters els allamandola hudgins hony tielens aggm unidentified infrared features iso arxiv preprint quider anna nestor daniel turnshek david rao sandhya monier eric weyant anja busche joseph pittsburgh sloan digital sky survey quasar absorptionline survey catalog astronomical journal review conference paper iclr schneider donald richards gordon hall patrick strauss michael anderson scott boroson todd ross nicholas shen yue brandt fan xiaohui sloan digital sky survey quasar catalog seventh data release astronomical journal szegedy christian liu wei jia yangqing sermanet pierre reed scott anguelov dragomir erhan dumitru vanhoucke vincent rabinovich andrew going deeper convolutions arxiv preprint zaharia matei chowdhury mosharaf das tathagata dave ankur justin mccauley murphy franklin michael shenker scott stoica ion resilient distributed datasets faulttolerant abstraction cluster computing proceedings usenix conference networked systems design implementation usenix association zhu guangtun brice metal absorption line catalog redshift evolution properties absorbers astrophysical journal
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control class uncertain nonlinear systems utilizing input position feedback feb spandan roy indra narayan kar senior member ieee abstract paper tracking control problem class systems subjected unknown uncertainties addressed control strategy christened adaptive robust control tarc presented proposed control strategy approximates unknown dynamics logic switching logic provides robustness approximation error novel adaptation law switching gain contrast conventional control methodologies require either nominal modelling predefined bounds uncertainties also proposed adaptive law circumvents problem switching gain state derivatives proposed control law estimated past data state alleviate measurement error state derivatives available directly moreover new stability notion control proposed turn provides selection criterion controller gain sampling interval experimental result proposed methodology using nonholonomic wheeled mobile robot wmr presented improved tracking accuracy proposed control law noted compared control adaptive sliding mode control index terms control system control state derivative estimation wheeled mobile robot ntroduction background motivation esign efficient controller nonlinear systems subjected parametric nonparametric uncertainties always challenging task among many approaches adaptive control robust control two popular control strategies researchers extensively employed dealing uncertain nonlinear systems general adaptive control uses predefined parameter adaptation laws equivalence principle based control law adjusts parameters controller fly according pertaining uncertainties however approach poor transient performance online calculation unknown system parameters controller gains complex systems computationally intensive whereas robust control aims tackling uncertainties system within uncertainty bound defined priori reduces computation complexity great extent complex systems compared adaptive control exclusive online estimation uncertain parameters required however nominal modelling uncertainties necessary decide upon bounds always possible increase operating region controller often higher uncertainty bounds assumed turn leads problems like higher controller gain consequent possibility chattering switching law based robust controller like sliding mode control smc effect reduces controller accuracy higher order sliding mode alleviate chattering problem prerequisite uncertainty bound still exists control tdc utilized implement state derivative feedback enhancing stability margin siso linear time invariant lti systems tdc used provide robustness uncertainties process uncertain terms represented single function approximated using control input state information immediate past time instant advantage robust control approach uncertain systems reduces burden tedious modelling complex system great extent spite unattended approximation error commonly termed error tde causes detrimental effect performance closed system stability front work carried tackle tde includes internal model gradient estimator ideal velocity feedback nonlinear damping sliding mode based approach stability closed loop system depends boundedness tde shown method approximates continuous time closed loop system discrete form without considering effect discretization error stability criterion mentioned restricts allowable range perturbation thus limits controller working range stability system established frequency domain makes approach inapplicable nonlinear systems moreover controllers designed require nominal modelling upper bound tde respectively always possible practical circumstances also best knowledge authors controller design issues selection controller gains sampling interval achieve efficient performance still open problem contrast tdc works reported use low pass filter approximate unknown uncertainties disturbances however frequency range system dynamics external disturbances required determine time constant filter furthermore order low pass filter needs adjusted according order disturbance maintain stability controller considering individual limitations adaptive robust control recently global research reoriented towards adaptiverobust control arc switching gain controller adjusted online series publications regarding arc estimates uncertain terms online based predefined projection function predefined bound uncertainties still requirement work reported attempts estimate maximum uncertainty bound integral adaptive law makes controller susceptible high switching gain consequent chattering adaptive sliding mode control asmc presented proposed two laws switching gain adapt online according incurred error first adaptive law switching gain decreases increases depending predefined threshold value however threshold value achieved switching gain may still increasing resp decreasing even tracking error decreases resp increases thus creates overestimation resp underestimation problem switching gain moreover decide threshold value maximum bound uncertainty required second adaptive law threshold value changes online according switching gain yet nominal model uncertainties needed defining control law limits adaptive nature control law applicability controller problem definitions contributions paper three specific related problems tdc dealt corresponding solutions also contributions paper summarized problem stability analysis tdc provided approximates continuous time system discrete time domain without considering effects discretization error choice delay time relation controller gains still open problem paper new stability analysis tdc based method provided continuous time domain furthermore proposed stability approach relation sampling interval controller gain established problem tdc reported velocity acceleration feedback necessary compute control law velocity feedback required acceleration term approximated numerically using time delay however many applications velocity acceleration feedback available explicitly numerical approximation terms invokes measurement error second contribution paper filtered control control law formulated position feedback sufficient velocity acceleration terms estimated using past present position information curb effect measurement error stability analysis proposed provided also maintains relation controller gains sampling interval problem robustness property tde essential achieve good tracking accuracy robust controllers reported literature either requires nominal model uncertainties predefined bound required devise control law would avoid prior knowledge uncertainties providing robustness tde towards last contribution article control strategy adaptive robust control tarc formulated class uncertain systems proposed control law approximates uncertainties logic provides robustness tde arising logic based estimation switching control novel adaptive law presented aims overcoming problem switching gain without prior knowledge uncertainties proposed adaptive law provides flexibility control designer select suitable error function according application requirement maintaining similar system stability notion proof concept experimental validation proposed control methodology provided using nonholonomic wmr comparison tdc asmc organization article organized follows new stability analysis tdc along design issues first discussed section followed proposed control methodology detail analysis section iii presents experimental results proposed controller comparison tdc asmc section concludes entire work notations following notations assumed entirety paper variable delayed amount denoted represent minimum eigen value euclidean norm argument respectively represents identity matrix ontroller esign control revisited general system second order dynamics devoid delay written system state control input matrix denotes combination system dynamics terms based system properties practice assumed unmodelled dynamics disturbances subsumed control input defined auxiliary control input nominal values respectively reduce modelling effort complex systems approximated data previous instances using logic system definition fixed small delay time substituting system dynamics converted input well state delayed dynamics let desired trajectory tracked tracking error auxiliary control input defined following way two positive definite matrices appropriate dimensions putting following error dynamics obtained treated overall uncertainty written state space form noting derivative inside integral respect error dynamics modified assumed choice controller gains makes hurwitz always possible also assumed unknown uncertainties bounded paper new stability criterion based method presented theorem addresses issues defined problem theorem system employing control input auxiliary control input uub controller gains delay time selected following condition holds scalar solution lyapunov equation proof let consider following lyapunov function using time derivative yields using two non zero vectors exists scalar matrix following inequality holds using jensen inequality following inequality holds applying following inequalities obtained since write assuming uncertainties square integrable within delay let exists scalar following inequality holds deh eth deh substituting adding yields eth let controller gains delay time selected make one find positive scalar would established thus would uub ultimate bound let denote smallest level surface containing ball radius centred initial time solution remains decreases long time required reach zero otherwise finite time reach given remark since depends controller gains provides selection criterion choice delay given controller gains design issue previously unaddressed literature moreover approximation error would reduce small values however selected smaller sampling interval input output data available sampling intervals lowest possible selection sampling interval choice sampling interval governed corresponding hardware response time computation time etc hence proposed stability approach provides necessary step selection sampling interval given controller gains filtered control noticed state derivatives necessary compute control law tdc however many circumstances available amongst scenario new control strategy proposed estimates state derivatives state information past instances proposing control structure following two lemmas stated instrumental formulation well stability analysis lemma time order time derivative degree polynomial computed following way prespecified scalar lemma non zero vector constant matrix following relation holds structure similar except auxiliary control input selected following way evaluated stability system employing derived sense uniformly ultimately bounded uub notion stated theorem theorem system employing control input auxiliary input uub selected following condition holds proof proof provided appendix control related work observed tde degrades tracking performance tdc face uncertainties control methods attempt counter uncertainties reported requires predefined bound uncertainties always possible practical circumstances circumvent situation adaptive sliding mode control asmc proposed control input asmc given nominal values switching control input choice sliding surface defined follows switching control calculated switching gain scalar adaptive gain threshold value small scalar always keep positive evaluation done two ways scalar sampling interval choice requires predefined bound uncertainties noted even decreases resp increases unless falls resp goes switching gain decrease resp increase causes overestimation resp underestimation switching gain controller accuracy compromised improper low choice may lead high switching gain consequent chattering hand method assumes nominal value uncertainties always greater perturbations assumption may hold due effect unmodelled dynamics thus necessitates rigorous nominal modelling uncertainties design control law either two situations bound estimation uncertainty modelling always feasible practical circumstances consequently compromises adaptive nature controller adaptive robust control considering limitations existing controllers aim negotiating uncertainties discussed earlier novel control law named adaptive robust control tarc proposed endeavour neither requires nominal model predefined bound uncertainties well eliminates overestimationunderestimation problem switching gain structure control input tarc similar also evaluated according however auxiliary control input selected nominal control input selected similar switching control law responsible negotiating tde defined ksk small scalar following novel adaptive control law evaluation proposed scalar adaptive gain represents small scalar suitable function error defined designer selected according adaptive law present choice increases resp decreases whenever error trajectories move away resp close advantages proposed tarc summarized follows tarc reduces complex system modelling effort knowledge suffices controller design since along uncertainties approximated using logic turn reduces tedious modelling effort complex nonlinear systems evaluation switching gain require either nominal model predefined bound uncertainties also removes problem state derivatives required compute control law explicitly evaluated past state information using stability system employing tarc analysed sense uub stated theorem assumption let unknown scalar quantity knowledge however required stability analysis compute control law theorem system employing adaptive law uub provided selection holds condition proof let define lyapunov functional defined putting error dynamics becomes also following similar steps proving theorem provided appendix defined appendix positive scalar let define following evaluating structure one find two positive scalars using stability analysis employing tarc carried following various cases case utilizing would established thus using relation system would uub following ultimate bound case ksk utilizing case would achieved system would uub following ultimate bound case iii ksk since using adaptive law case iii similarly argued earlier system would uub following ultimate bound remark performance tarc characterized various error bounds various conditions noticed low value high value would result better accuracy however large may result high control input also one may choose different values moreover noticed stability notion tarc invariant choice thus provides designer flexibility select suitable according application requirement iii onclusion selection controller gain sampling interval crucial performance tdc design issue addressed paper new stability approach bound delay derived select suitable sampling interval new control approach devised state derivatives estimated previous state information moreover novel control law tarc proposed class uncertain nonlinear systems subjected unknown uncertainties proposed controller approximates unknown dynamics law negotiates approximation error surfaces due approximation uncertainties state derivatives switching logic adaptive law eliminates problem online evaluation switching gain without prior knowledge uncertainties experimentation wmr shows improved path tracking performance tarc compared tdc conventional asmc proposed framework also extended systems autonomous underwater vehicle unmanned aerial vehicle robotic manipulator etc ppendix roof heorem let define lyapunov functional eferences kristic kanellakopoulos kokotovic nonlinear adaptive control design wiley new york liu yao adaptive robust control class uncertain nonlinear systems unknown sinusoidal disturbances ieee conference decision control corless leitmann continuous state feedback guaranteeing uniform ultimate boundness uncertain dynamic system ieee transactions automatic control vol lee utkin chattering suppression methods sliding mode control systems annual reviews control vol levant sliding modes differentiation control international journal control vol ulsoy control siso systems improved stability margins asme journal dynamic systems measurement control vol hsia gao robot manipulator control using decentralized linear joint controllers ieee international conference robotics automation ito time delay controller systems unknown dynamics asme journal dynamic systems measurement control vol cho chang park jin robust tracking nonlinear friction using time delay control internal model ieee transactions control system technology vol chang robust tracking robot manipulator nonlinear friction using time delay control gradient estimator journal mechanical science technology vol jin kang chang robust compliant motion control robot nonlinear friction using estimation ieee transactions industrial electronics vol jin chang jin gweon stability guaranteed control manipulators using nonlinear damping terminal sliding mode ieee transactions industrial electronics vol chang park improving control certain hard nonlinearities mechatronics vol roy nandy ray shome time delay sliding mode control nonholonomic wheeled mobile robot experimental validation ieee international conference robotics automation zhu tao yao cao adaptive robust posture control parallel manipulator driven pneumatic muscles redundancy transactions mechatronics vol zhu tao yao cao integrated adaptive robust posture control parallel manipulator driven pneumatic muscles ieee transactions control system technology vol sun zhao gao saturated adaptive robust control active suspension systems ieee transactions industrial electronics vol islam liu saddik robust control four rotor unmanned aerial vehicle disturbance uncertainty ieee transactions industrial electronics vol chen yao wang based adaptive robust posture control linear motor driven stages high frequency dynamics case study transactions mechatronics vol liu pan new adaptive sliding mode control uncertain nonlinear dynamics asian journal control vol chen yeh chang design implementation adaptive dynamic controller wheeled mobile robots mechatronics vol plestan shtessel bregeault poznyak new methodologies adaptive sliding mode control international journal control vol plestan shtessel bregeault poznyak sliding mode control gain adaptation application electropneumatic actuator state control engineering practice vol bandyopadhayay janardhanan spurgeon advances sliding mode control new york leitmann efficiency nonlinear control uncertain linear systems asme journal dynamic systems measurement control vol chang lee model reference observer control application robot trajectory control ieee transactions control system technology vol shin kim performance enhancement pneumatic vibration isolation tables low frequency range time delay control journal sound vibration vol kuperman zhong robust control uncertain nonlinear systems based uncertainty disturbance estimator international journal robust nonlinear control vol talole phadke robust linearization using uncertainty disturbance estimator international journal control vol suryawanshi shengde phadke robust sliding mode control class nonlinear systems using inertial delay control nonlinear dynamics vol reger jouffroy algebraic estimation deadbeat state reconstruction joint ieee conference decision control chinese control conference khartionov chen stability systems boston nasiri nguang swain adaptive sliding mode control class mimo nonlinear systems uncertainty journal franklin institute vol roy nandy ray shome robust path tracking control nonholonomic wheeled mobile robot international journal control automation systems vol kim joe lee kim time delay controller design position control autonomous underwater vehicle disturbances ieee transactions industrial electronics doi
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arxiv aug computing fixpoint set boolean equations viktor kuncak rustan leino mit vkuncak microsoft research leino december technical report paper presents method computing least fixpoint system equations booleans resulting computation significantly shorter result iteratively evaluating entire system fixpoint reached microsoft research microsoft corporation one microsoft way redmond http krml introduction many problems computer science particular arising context program analysis involve computation least dually greatest fixpoint system equations paper consider way compute least fixpoint equations involved booleans important cases resulting computation significantly shorter computation iteratively evaluates entire system fixpoint reached let begin overview result restrict attention finite lattice finite lattice complete lattice infinite ascending chains monotonic function lattice also continuous hence kleene fixpoint theorem states least fixpoint monotonic function lattice join sequence elements exponentiation denotes successive function applications denotes bottom element lattice sequence ascending lattice finite exists natural number least fixpoint call least fixpoint depth able evaluate function able determine whether two given lattice elements equal compute least fixpoint starting value repeatedly apply application leaves value unchanged existence fixpoint depth guarantees process terminates paper consider problem computing expression least fixpoint without computing value expression first computing small expression least fixpoint relegate computation value expression external tool sat solver sequel therefore assume able compute value expression particular lattice element fixpoint depth function lattice bounded height lattice therefore lattice booleans height least fixpoint given lattice cartesian product space booleans height least fixpoint given krml function represented isomorphically functions write tuple functions defined function follows booleans example let let least fixpoint equals argued terms functions expands refer closed form fixpoint expanded closed form different way write expanded closed form shares common subexpressions let let let let representation cubic means computing may take time space cubic allow write functions arguments functions obtain quadratic representation example let let let let krml let consider another closed form call pruned closed form pruned closed form application function replaced occurs another application function example pruned closed form interpretation functions words symbolic name uninterpreted cubicsized expanded closed form may reasonably small representation fixpoint pruned closed form generally much larger cubic every subset function appears expanded context set enclosing functions smaller pruned closed form obtained taking advantage common subexpressions however cases pruned closed form significantly smaller expanded closed form example fixpoint computation dominated computation local fixpoints meaning fixpoints involve small number functions important situation program analysis case applies function represents control point given program function defined terms functions corresponding successor predecessor control points given program contains many local loops example suppose functions expanded closed form let let let let krml contrast pruned closed form yields much shorter expression generally even suppose odd even expanded closed form still cubic whereas pruned closed form expression rest paper define pruned closed form precisely prove yields value expanded closed form using theorem section sketch obtain pruned closed form applications theorem write lattice meet values satisfy predicate monotonic function write denote least fixpoint tarski fixpoint theorem says meet fixpoints fixpoint using function isomorphic representation functions write equivalently krml state theorem monotonic functions possibly different lattices note side equality expresses fixpoint lattice functions types respectively consequence theorem kleene fixpoint theorem known fixpoint depth following lemma lemma lattice domain monotonic functions proof theorem function therefore fixpoint depth therefore using lemma show pruned closed form indeed least fixpoint monotonic boolean functions lemma substitute equals equals lemma krml calculation shows expression least solution equation symmetric argument expression least solution equation expression pruned closed form using result show pruned closed form also least fixpoint monotonic boolean functions lemma isomorphic representation functions substitute equals equals lemma substitute equals equals first steps calculation reverse order lemma substitute equals equals lemma krml substitute equals equals lemma calculation shows expression least solution similarly main result pruned closed form least fixpoint next section prove result directly using lemma theorem given monotonic functions boolean domain ordered represent indexed things like list booleans write fact given functions monotonic written follows index tuples booleans infix dot highest operator precedence denotes function cation order ordering tuples interested viewing functions specifying system equations namely krml variables left colon show unknowns take tuple functions also write function one produces tuple results applying given argument functions example functions given argument thus write system equations interested least sense ordering solution satisfies equation interested least fixpoint function lattice boolean height least fixpoint reached applying times starting bottom element lattice least fixpoint given exponentiation denotes successive function applications tuple precisely specify pruned closed form introduce notation keeps track functions applied enclosing context particular use set contains indices functions already applied formally define following family functions index set indices taking advantage previous notation using denote function always returns boolean extended pointwise boolean function write definition follows goal prove following theorem krml proof start proving lemmas use proof theorem lemma index proof induction let denote consider three cases ase definition since definition bottom element ase definition since distribute index induction hypothesis since monotonicity distribute distribute exponentiation krml ase see first steps previous case index identity function following corollary lemma proves one direction theorem corollary proof index lemma distribute exponentiation support remaining lemmas define one family functions index set indices lemma index monotonic function proof prove term quantification follows krml monotonicity since antecedent lemma index set indices proof consequent follows trivially prove term quantification induction ase gives identity function ase exponentiation since distribute index see distribute induction hypothesis exponentiation proof third step calculation definition since krml lemma using antecedent lemma fulfill antecedent lemma definition definition since need one lemma lemma index set indices satisfying proof induction consider three cases ase definition since left zero element definition since ase exponentiation since definition since monotonicity since definition since krml ase suffices prove side whenever side therefore assume latter prove former definition since exponentiation since distribute see monotonicity definition since assumption calculation used following fact every index prove divide proof two case formula follows immediately case first derive consequences assumption induction hypothesis definition since lemma krml calculating assumption made induction hypothesis concludes proof lemma finally proof theorem proof theorem proof argument corollary exponentiation since distribute definition index thus also lemma distribute related work acknowledgments theorem already found use namely translation boolean programs satisfiability formulas krml knew theorem one kuncak proved theorem detailed section tony hoare proposed way prove theorem way would eliminate recursive uses variables one one hoare also proved essentially amounts theorem appealing tarski fixpoint theorem elaborated format section whose formulation carroll morgan also contributed learnt theorem patrick cousot theorem often called simply theorem bakker traces independent proof thereof finally grateful feedback eindhoven tuesday afternoon club participants ifip meeting biarritz france march references bakker mathematical theory program correctness hans definable operation general algebras theory automata flowcharts cliff jones editor programming languages volume lecture notes computer science pages springer stephen cole kleene introduction metamathematics van nostrand new york rustan leino sat characterization correctness thomas ball sriram rajamani editors model checking software volume lecture notes computer science pages springer may jacek theorem resolving equations space languages bull acad polon alfred tarski fixpoint theorem applications pacific journal mathematics
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modular structural operational semantics delimited continuations neil sculthorpe paolo torrini peter mosses omp project department computer science swansea university grac ful project department computer science leuven belgium omp project department computer science swansea university open question whether modular structural operational semantics framework express dynamic semantics paper shows furthermore demonstrates express general delimited control operators control shift introduction modular structural operational semantics msos variant structural operational semantics sos framework principal innovation msos relative sos allows semantics programming construct specified independently semantic entities directly interact example function application specified msos rules without mentioning stores exception propagation known msos specify semantics programming constructs exception handling unclear whether msos specify complex operators indeed perceived difficulty handling control operators regarded one main limitations msos relative modular semantic frameworks section paper demonstrates dynamic semantics specified msos extensions msos framework required approach first specifying general delimited control operators control shift specifying terms control contrast operational specifications control operators given direct style based labelled transitions rather evaluation contexts begin giving brief overview delimited continuations section msos section material two sections novel skipped familiar reader present msos specification dynamic semantics delimited control operators section ensure msos specification indeed define control operators described literature provide proof equivalence specification one based evaluation contexts section delimited continuations point execution program current continuation represents rest computation sense continuation understood context program term evaluated control operators allow current continuation treated object language reifying abstraction applied manipulated classic example control operator olivier danvy ugo liguoro eds woc postproceedings eptcs neil sculthorpe paolo torrini peter mosses work licensed creative commons attribution license modular structural operational semantics delimited continuations delimited continuations generalise notion continuation allow representations partial contexts relying distinction inner outer context control operators manipulate delimited continuations always associated control delimiters delimited control operators control associated prompt delimiter shift associated reset delimiter used simulate general idea control shift capture current continuation innermost enclosing delimiter representing inner context give informal description control section formal msos specification control given section also specify shift terms control control unary operator takes function argument expects reified continuation argument executed control reifies current continuation innermost enclosing prompt function inner context discarded replaced application interaction control prompt simply unary operator evaluates argument returns resulting value let consider examples following expression continuation bound function result prompt application expression evaluates prompt control reified continuation applied multiple times example prompt control furthermore continuation need applied example following expression multiplication two discarded prompt control preceding examples continuation could computed statically however general current continuation context point program execution control executed time computation source program may already performed example following program print abb prompt print control print abb command print executed control operator form part continuation reified control case bound print printed every application examples control found online test suite accompanying paper literature modular sos rules paper presented using implicitly modular sos variant msos notational style similar conventional sos viewed syntactic sugar msos assume reader familiar sos basics msos key notational convenience semantic entities stores environments mentioned rule implicitly propagated premise conclusion allowing entities interact programming construct specified omitted neil sculthorpe paolo torrini peter mosses rule two types semantic entities relevant paper inherited entities environments unmentioned implicitly propagated conclusion premises observable entities emitted signals exceptions unmentioned implicitly propagated sole premise conclusion observable entities required default value implicitly used conclusion rules lack premise mention entity note premise refer specifically transition relation side conditions rule notational convenience also write line demonstrate specification control operators using rules paper use funcon framework framework contains open collection modular fundamental constructs funcons semantics specified independently rules funcons facilitate formal specification programming languages serving target language specification given inductive translation style denotational semantics however paper concerned translation control operators specific language aim give msos specifications control operators funcon framework convenient environment specifying prototypical control operators examples translations funcons found present examples funcons specifications rules familiarity funcon framework required purposes understanding paper funcons may simply regarded abstract syntax typeset funcon names bold capitalised italic names semantic entities come funcons control operators continue use italic referring control operator general bold referring funcon specifically figure presents rules funcons throw catch idea throw emits exception signal catch detects handles signal first argument catch expression evaluated second argument function exception handler exception signals use observable entity named exc written label transition arrow exc entity either value none denoting absence exception denoting occurrence exception value side condition val requires term value thereby controlling order rules applied case throw first argument evaluated value rule exception carrying value emitted rule case catch first argument evaluated exception occurs rule exception occur handler applied exception value computation abandoned rule evaluates value discarded returned rule observe rules mention exc entity rule implicitly propagated premise conclusion rule implicitly default value none also observe none rules figure mention entities environments stores entities also implicitly propagated exc none throw throw val exc throw stuck exc none catch catch exc exc none catch apply val catch figure rules exception handling modular structural operational semantics delimited continuations env env lambda closure val closure apply apply val apply apply val val apply closure val env env apply closure apply closure figure rules lambda calculus figure presents rules identifier lookup abstraction lambda application apply note closure funcon value constructor specified rule thus transition rules present rules completeness funcons used defining semantics control operators section observe rules mention environment env propagated implicitly furthermore consider none rules figure mention environment env none rules figure mention exc signal however modular nature specifications allows two sets rules combined without modification implicit propagation handling unmentioned entities comparison figure present conventional sos specification lambda calculus combined exception handling semantic entities mentioned explicitly every rule exc none env exc env exc env throw throw exc none env lambda closure val exc env exc env throw stuck exc env apply apply exc none env exc none env catch catch exc env exc none env catch apply val exc none env catch exc env val exc env apply apply val exc env exc env apply closure apply closure val val exc none env apply closure figure sos rules lambda calculus exception handling neil sculthorpe paolo torrini peter mosses specifications control operators present dynamic semantics control operators msos framework specify control prompt directly specify shift reset terms control prompt approach similar manner specifications exceptions figure control operator emits signal executed delimiter catches signal handles note implicit delimiter around funcon translation funcons language implicit delimiter insert explicit delimiter overview approach whether semantics control operators specified using msos considered open problem section suspect explicit representation term context msos given rule access current subterm contents semantic immediately obvious capture context abstraction approach construct current continuation control operator rule enclosing delimiter achieve exploiting way semantics step computation builds derivation tree root program term current operation thus step control operator executed rule control operator part derivation rule enclosing delimiter step current continuation corresponds abstraction control operator argument subterm enclosing delimiter thus constructed subterm represent reified continuations abstractions using lambda funcon section constructing abstraction achieved two stages rule control replaces occurrence control argument fresh identifier rule prompt constructs abstraction updated subterm first approximation suggests following rules ctrl control ctrl ctrl none lambda prompt prompt apply side condition requires identifier introduced rule already occur program rule replaces term control emits signal ctrl containing function identifier signal caught handled prompt rule abstraction representing continuation executed control operator constructed combining updated subterm contain place control auxiliary environment one problem approach outlined identifier introduced dynamically control operator executes time closures may already formed particular control occurs inside body lambda enclosing prompt outside lambda funcon would introduced inside closure already formed hence modular structural operational semantics delimited continuations contain binding example consider evaluation following term prompt lambda control lambda prompt closure control lambda prompt apply lambda lambda closure occurrence inside closure containing empty environment closure applied say subterm larger program body closure would get stuck rule would provide environment containing rule could match problem arises consequence choice specify semantics lambda calculus using environments closures instead given semantics using substitution problem would arisen however prefer use environments enable modular specification semantics requires substitution defined every construct language moreover environments allow straightforward semantics dynamic scope solution introduce auxiliary environment captured closures figure specifies looks identifier auxiliary environment binds identifier value auxiliary environment scopes binding expression make use funcons next subsection give complete specification control prompt val val val figure rules bindings auxiliary environment dynamic semantics control prompt specify control follows control control val ctrl control neil sculthorpe paolo torrini peter mosses rule combination val premise rule ensures argument function evaluated closure rule applied notice rule uses contrast preliminary rule used specify prompt follows val prompt ctrl none ctrl none prompt prompt ctrl lambda ctrl none prompt prompt apply rule case argument value prompt discarded rule evaluates argument expression ctrl signal emitted evaluation rule handles case ctrl signal detected reifying current continuation passing argument function notice unlike preliminary rule rebound using rules complete specification dynamic semantics control prompt relying existence funcons figures rules modular valid independently whether control operators coexist mutable store exceptions signals semantic entities except use auxiliary environment rules correspond closely specifications control prompt based evaluation contexts however rules communicate control prompt emitting signals thus require evaluation contexts section present proof equivalence specification conventional one based evaluation contexts dynamic semantics shift reset shift operator differs control every application reified continuation implicitly wrapped delimiter effect separating context application inner context difference control shift analogous dynamic static scoping insofar shift application reified continuation access context way statically scoped function access environment applied shift funcon specified terms control follows shift shift val shift control lambda apply lambda reset apply key point insertion reset delimiter rest merely exposes application continuation delimiter inserted following given definition shift reset delimiter coincides exactly prompt reset prompt modular structural operational semantics delimited continuations alternatively insertion extra delimiter could handled semantics reset rather shift val reset ctrl none ctrl none reset reset ctrl lambda reset ctrl none reset reset apply difference rules rules funcon names definition rule delimiter wrapped around body continuation given definition reset shift operator coincides exactly control shift control specification rules similar based specification shift reset section dynamic semantics abort operator traditionally undelimited considers current continuation entirety rest program setting delimited continuations simulated requiring single delimiter appear program otherwise two distinguishing features relative control shift first applied continuation never returns second body invoke continuation current continuation applied result application returns specify follow sitaram felleisen section first introduce auxiliary operator abort specify terms control prompt abort purpose abort terminate computation innermost enclosing prompt given value abort abort val abort control lambda achieve first distinguishing feature placing abort around application continuation preventing returning value achieve second applying continuation result application resumes current continuation returns value callcc callcc val callcc control lambda apply apply lambda abort apply neil sculthorpe paolo torrini peter mosses control effects section presented direct specification exception handling using dedicated semantic entity throw catch figure used program together control operators section would give rise two sets independent control effects independent delimiters alternative would specify exception handling indirectly terms control operators following sitaram felleisen case delimiters semantic entity would shared msos specify either approach required language specified beyond control operators discussed section general operators manipulating delimited continuations exist cps hierarchy beyond scope paper remain avenue future work adequacy sos model lambda calculus extended delimited control presented using rules provably equivalent one based reduction semantics lambda terms evaluation strategy specified using evaluation contexts reduction models delimited control based evaluation contexts originally introduced refined adequacy proof section prop carried respect version models formalism call sos model differs reduction models framework relies particular sos model uses environments signals whereas uses substitution evaluation contexts moreover difference notion value sos model function application computed using closures whereas uses substitution order focus operational content models convenient get differences achieve embedding sos explicit congruence rules embedding call lifting formalism called define notion adequacy two systems relation parametric translation show adequacy two systems proving derivationally equivalent sense relation reasoning induction structure derivations adequacy proof sos split three main parts equivalence sos challenging approach would involve giving formal derivation model sos along lines goes beyond scope paper intend focus equivalence respect delimited control given equivalence sos respect lambda calculus show sos equivalent respect extension delimited control specifically define syntactic representation environments standard auxiliary ones using contexts lambda terms use representation define encoding sos adequacy sos provable respect simple translation relation define version obtained consider two distinct extensions model delimited control first one call lrdc uses lifted control rules original model thus equivalence model straightforward second one call uses version sos control rules difference model provably equivalent boils sos transitions based congruence rules also using closures transitions based evaluation contexts using lambda expressions adequacy proved respect identity translation prop gives result primary interest modular structural operational semantics delimited continuations reduction semantics presentation reduction semantics evaluation contexts follows main lines model assumption evaluate closed expressions values terms defined follows lambda apply control prompt general notion context term hole defined following grammar lambda apply apply prompt control cbv evaluation strategy specified using restrictive notion cbv context apply apply prompt control order represent delimited continuations even restrictive notion needed pure context include control delimiters apply apply control notation used represent term factored context subterm fills think form term annotation factorisation unique cases considering reduction rules needed specify context propagation presented follows giving model using notation captureavoiding uniform substitution apply lambda reduction rules prompt control formulated follows giving model making use prompt val lambda prompt control prompt apply system based reduction semantics observational equivalence defined smallest congruence relation terms extends reduction equivalence functional extensionality apply apply presenting models based typeset construct names refer sos values valsos ones valrc needed subscript accordingly define derivational equivalence adequacy respect translation relation mentioned case identity follows neil sculthorpe paolo torrini peter mosses def given two systems respectively defined languages values vala valb relation say adequate respect whenever following hold vala exist valb valb exist vala derivationally equivalent respect whenever following hold whenever exist whenever exist define relational composition representing sos section define encoding sos lambda terms unlike reduction semantics sos models rely internally linguistic extension account closures auxiliary environment notation reason need extended internal language including following additional constructs closure closures auxiliary identifier lookup auxiliary let bindings constructs meant included source language definition expression require used order represent environments introduce notion apply lambda tacitly assume bound variables distinct sos transition specified env embedded apply lambda apply lambda silently assume permutation introduce represent auxiliary environment similar manner though using order represent signals extend notion one introducing new ternary value constructor ctrl part expression definition representation sos ctrl sos transitions specified ctrl none env ctrl env represented respectively ctrl modular structural operational semantics delimited continuations assume variables distinct ones silently assume permutation way define translation relation sos configurations expressions applying translation sos rules gives models resp particular rule uses environment expressed follows apply lambda apply lambda rule uses auxiliary environment takes following form model extends three rules sos rule control val control ctrl additional congruence rule match rule ctrl ctrl encoding sos rule rule ctrl lambda prompt prompt apply since correspondence sos transitions treating mpermutations silent transitions taking simplicity identity relation modulo reordering environments observational equivalence sos following straightforward prop model corresponding sos one derivationally equivalent adequate respect translation similarly sos models proof first prove derivational equivalence straightforward control rules correspond sos ones given representation auxiliary environments signals adequacy follows definition value systems lifting facilitate comparison define version using represent environments also extend language closure models reduction rules specified relying evaluation contexts way closure used change needed definition context however since reduction steps lifted replace single context propagation rule sufficed four following rules lifting lifted congruence neil sculthorpe paolo torrini peter mosses apply lambda apply lambda apply lambda apply lambda assume models include well rules rule notice evaluation also apply open terms however need change definition value substitution free variables dealt rule also need following rule deal closures closure apply lambda apply lambda lambda gives model extend model rules delimited control two ways simulate simulate sos rules prompt control based one lifted version rules involve use auxiliary notation extension rules hand model obtained extending control rules rules rely auxiliary notation following proved systems considering respect prop valsos whenever exists valrc apply lambda valsos whenever exist valrc apply closure following provable equivalences correspond respectively rule elimination apply lambda apply lambda apply lambda following proved prop apply lambda apply lambda proof apply lambda equiv equiv observing occur free must used apply lambda equiv following immediate consequence prop applying functional extensionality lambda lambda modular structural operational semantics delimited continuations adequacy sos first show models ones equivalent prop adequate proof language included hence take identity denoted idrc translation models obtained refactoring models gives equivalent systems change evaluation strategy affects top level prove derivational equivalence induction structure derivations relying equiv rule eliminate additional syntax observing closure inessential eliminated without leading new values adequacy follows immediately values defined way two systems consider relationship different representations prop model corresponding one adequate proof two models use language hence take identity translation differ congruence rules reduction function application congruence rules expressed using unlike equivalent specifications cbv equivalence respect values function application rely prop extend result delimited control prop adequate proof first prove derivational equivalence respect identity using prop fact two extensions obtained adding rules finally compare delimited control prop derivationally equivalent adequate proof prove derivational equivalence induction structure derivations respect identity translation two systems equivalent prop interesting case delimited control respect stepwise behaves lifted version prompt rule rule systems rules added without expanding set derivable values thus possible difference two systems natural control rule lifted version rule rule rule show two rules interderivable given system one rule one derivable first observe specification continuation either rule equivalent therefore interchangeable order derive rule one observe lifted expression control value reduced ctrl using control rule rule applicable congruence rule rule gives premise application rule prompt control way simulates control rule transition ctrl possible provided control possibly relying conversion closures function applications therefore control rule applied prompt simulate rule neil sculthorpe paolo torrini peter mosses diagramatically overall proof presented follows vertical arrows denote model inclusion prop derivationally equivalent adequate respect idrc proof based props using fact adequacy transitive composition translation relations transitivity observational equivalence related work direct way specify control operators giving operational semantics based transition rules continuations taken direct approach though contrast direct specifications control operators approach based emitting signals via labelled transitions rather evaluation contexts control operators also given denotational semantics transformation style cps operational specification translation code higher level algebraic characterisations control operators given terms equational theories denotationally function rewritten cps taking continuation represented function additional argument applying continuation value function would returned straightforward extension transformation suffices express shift reset however sophisticated cps transformations needed express control prompt felleisen initial specification control prompt used operational semantics without evaluation contexts however specification otherwise differs quite significantly based exchange rules push control outwards term encounters prompt exchange rule defined every construct language approach inherently modular later specifications control prompt used evaluation contexts algebraic characterisations based notion abstract continuations continuations represented evaluation contexts exchange rules needed felleisen also gave operational specification based cek abstract machine continuations treated frame stacks shift reset operators originally specified denotationally terms cps semantics continuations treated functions relying approach distinguishes outer inner continuations correspondingly transformation produces abstractions take two continuation parameters translated standard cps style operational semantics shift given danvy yang specification based evaluation contexts given kameyama hasegawa together algebraic characterisation giving cps semantics control significantly complex shift continuations reified shift always delimited applied treated functions case control different approaches problem developed including abstract continuations monadic framework operational framework relying introduction recursive continuations provides alternative approach based refined cps transform conversely difference control shift manifest modular structural operational semantics delimited continuations quite intuitively direct specification specifications section specifications using evaluation contexts shown filinski shift implemented terms mutable state point view expressiveness monad functionally expressible represented lambda calculus shift reset moreover control shift equally expressive untyped lambda calculus direct implementation control shift given gasbichler sperber implementation control operators monadic framework given dyvbig semantics based efficient implementation evaluation contexts provided framework conclusion presented dynamic semantics control operators msos framework settling question whether msos expressive enough control operators definitions concise modular require use evaluation contexts definitions based evaluation contexts often even concise corresponding msos definitions since single alternative contextfree grammar evaluation contexts subsumes entire msos rule allowing evaluation particular subexpression however grammars significantly less modular msos rules adding new control operator specified language may require duplication potentially large grammar pages inherent lack modularity evaluation context grammars addressed plt redex tools use ellipsis initially validated specifications suite test programs accumulated examples literature control operators language used testing caml light pedagogical sublanguage precursor ocaml existing translation funcons previous case study extended caml light control operators specified semantics operators direct translations corresponding funcons presented paper generated funcon programs tested prototype funcon interpreter directly interprets specifications suite test programs accompanying translator interpreter available online test programs demonstrated successfully specified control operator behaves similarly operator control described literature prove specified exactly operator addressed section proved msos specification equivalent conventional specification using reduction semantics based evaluation contexts acknowledgments thank casper bach poulsen ferdinand vesely anonymous reviewers feedback earlier versions paper also thank martin churchill exploratory notes adding evaluation contexts msos olivier danvy suggesting additional test programs reported work supported epsrc grant swansea university omp project funding horizon grant leuven grac ful project neil sculthorpe paolo torrini peter mosses references abelson dybvig haynes rozas adams friedman kohlbecker steele bartley halstead oxley sussman brooks hanson pitman wand report algorithmic language scheme symbolic computation kenichi asai yukiyoshi kameyama polymorphic delimited continuations asian symposium programming languages systems lecture notes computer science springer egidio astesiano inductive operational semantics formal description programming concepts ifip reports springer isbn malgorzata biernacka dariusz biernacki olivier danvy operational foundation delimited continuations cps hierarchy logical methods computer science dariusz biernacki olivier danvy simple proof folklore theorem delimited control journal functional programming dariusz biernacki olivier danvy static dynamic extents delimited continuations science computer programming martin churchill peter mosses modular bisimulation theory computations values international conference foundations software science computation structures lecture notes computer science springer martin churchill peter mosses neil sculthorpe paolo torrini reusable components semantic specifications transactions software development xii lecture notes computer science springer william clinger scheme environment continuations sigplan lisp pointers olivier danvy analytical approach programs data objects dsc thesis department computer science aarhus university available http olivier danvy defunctionalized interpreters programming languages international conference functional programming acm olivier danvy andrzej filinski functional abstraction texts technical report diku university copenhagen http typed conavailable olivier danvy andrzej filinski abstracting control conference lisp functional programming acm olivier danvy andrzej filinski representing control study cps transformation mathematical structures computer science olivier danvy zhe yang operational investigation cps hierarchy european symposium programming languages systems lecture notes computer science springer kent dyvbig simon peyton jones amr sabry monadic framework delimited continuations journal functional programming matthias felleisen robert bruce findler matthew flatt semantics engineering plt redex mit press isbn matthias felleisen mitch wand daniel friedman bruce duba abstract continuations mathematical semantics handling full jumps conference lisp functional programming acm modular structural operational semantics delimited continuations mattias felleisen theory practice prompts symposium principles programming languages acm andrzej filinski representing monads symposium principles programming languages acm martin gasbichler michael sperber final shift direct implementation shift reset international conference functional programming acm carl gunter didier jon riecke generalization exceptions control languages international conference functional programming languages computer architecture acm yukiyoshi kameyama masahito hasegawa sound complete axiomatization delimited continuations international conference functional programming acm yukiyoshi kameyama takuo yonezawa typed dynamic control operators delimited continuations international symposium functional logic programming lecture notes computer science springer peter mosses pragmatics modular sos international conference algebraic methodology software technology lecture notes computer science springer peter mosses modular structural operational semantics journal logic algebraic programming peter mosses mark new implicit propagation structural operational semantics workshop structural operational semantics electronic notes theoretical computer science elsevier peter mosses ferdinand vesely funkons semantics international workshop rewriting logic applications lecture notes computer science springer gordon plotkin structural approach operational semantics journal logic algebraic programming reprint technical report daimi aarhus university grigore traian florin overview semantic framework journal logic algebraic programming amr sabry matthias felleisen reasoning programs style lisp symbolic computation neil sculthorpe paolo torrini peter mosses modular structural operational semantics delimited continuations additional material available http static simulation dynamic delimited control symbolic computation dorai sitaram matthias felleisen control delimiters hierarchies lisp symbolic computation
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may configuration equivalence equivalent isomorphism ali rejali meisam soleimani malekan abstract giving condition amenability groups rosenblatt willis first introduced concept configuration beginning theory question whether concept configuration equivalence coincides concept group isomorphism posed negatively answer question introducing two solvable hence amenable groups configuration equivalent also prove conjecture due rosenblatt willis configuration equivalent groups include free group rank show equivalent groups class numbers introduction definitions paper groups assumed finitely generated discrete let group denote identity group given finite ordered subset cayley graph denoted vertices elements directed edges let group ordered generating set corresponded cayley graph assume finite partition considered coloring colors configuration colors one colors color case may say configuration date february mathematics subject classification key words phrases configuration configuration group isomorphism conjugacy class class number paradoxical decomposition first authors would like express gratitude toward banach algebra center excellence mathematics university isfahan also thank yves cornulier ideas proof theorem configuration equivalence equivalent isomorphism simultaneously considering right multiplication concept twosided configuration reached configuration tuple satisfying exists eci xgi first concept configuration introduced rosenblatt willis give characterization amenability groups characterize normal sets concept configuration suggested called configuration pair set configurations configurations resp corresponding configuration pair denoted con cont set configuration configuration sets denoted con cont respectively origin theory configuration conjecture raised combinatorial properties configurations used characterize various kinds behavior groups like group abelian group containing free subgroup conjecture leads notion configuration equivalence group configuration contained group written con con two groups configuration equivalent written con con concepts configuration contained configuration equivalent similarly defined notations used denoting concepts first steps theory taken shown finiteness periodicity properties characterized configuration paper authors proved two configuration equivalent groups isomorphism classes finite quotients finite index property extended abelian quotient property see also shown two configuration equivalent groups satisfy laws result generalized proving group laws established configuration equivalent groups hence particular abelian group property nilpotent class particular another properties characterized configuration see shown torsion free nilpotent group hirsch length interesting know answer question whether conserved equivalence configuration question answered assumption question affirmatively answered without extra hypothesis configuration equivalence equivalent isomorphism addition shown paper solubility group recovered con recent paper showed notion normality obtained configuration equivalence also presence normality showed equivalent normal subgroup quotient finitely presented recognizable configuration pair contains normal subgroup shown class polynomial type groups involving finite abelian free polycyclic groups satisfied recognizability condition also interested investigating question subclasses class groups configuration equivalence coincide isomorphism question answered positively class finite free abelian groups shown groups form group integers arbitrary finite group determined isomorphism configurations proved infinite dihedral group pointed existence golden configuration pairs implied isomorphism indeed showed class finitely presented hopfian groups golden configuration pair configuration equivalence coincided isomorphism light configuration proved polynomial type groups groups finite commutator subgroup equivalent specially polycyclic two notation equivalent prove two configuration equivalent groups contains free group rank also contains present paper define configuration sets finite sigma algebras group help show twosided equivalence groups number normal subgroups finite index also cardinality normal subgroups quotient polycyclic class number different conjugacy classeswill shown equal equivalent groups class equivalent groups finite class number study type subgroups show class set finite polycyclic subgroups configuration equivalence equivalent isomorphism question whether configuration equivalence implies isomorphism seen open since beginning theory configuration negatively answer open question presenting two solvable hence amenable groups configuration sets like two recent papers use following notation notation let group ordered subset let denote product call pair representative pair word corresponding speak representative pair assume number components number components denotes configuration finite sigma algebras really important configuration image subsets group left translations finite subsets seems sets con cont group replace sigma algebra indeed involve sigma algebras theory configuration follows let group correspondence finite sigma algebras finite partitions indeed finite sigma algebra set atoms partition denote atomic sets sigma algebra atom also finite collection subsets use denote sigma algebra generated following always consider sigma algebras finite try rewrite symbols configuration sigma algebras sigma algebra define con con atom similarly define cont remember following efficient symbol notation let two groups generating set respectively coloring cayley graph colors get partitions respectively two sets write esf show two sets color particular con con cont cont configuration pairs groups respectively implies corresponded cayley graphs colored colors configuration equivalence equivalent isomorphism also use sigma algebras let partitions resp sfi say asb words asb following technical lemma used following lemma let two groups sigma algebras resp con con suppose sbi corresponded proof set atom atom without loss generality let match also set assumptions con proves proving note con cont cont one easily get analog lemma left multiplication replaced right multiplication let two groups consider partitions respectively refinements assume may say two pairs similar may write configuration equivalence equivalent isomorphism also sigma algebras sigma subalgebras resp say similar written atom atom atom atom rewrite lemma sigma algebras lemma let sigma algebras groups resp suppose generating sets con con sigma con con sigma algebraic version lemma similarly obtained use sigma algebras equality tarski numbers configuration equivalent groups easily obtained recall following definition paradoxical decomposition tarski number definition group admits paradoxical decomposition exist positive integers disjoint subsets elements minimal possible value paradoxical decomposition tarski number denoted group paradoxical decomposition means amenable case define theorem let groups configuration equivalent proof amenable done hence without loss generality suppose finite fix generating set say let elements definition consider sigma algebra generated sets set since exists generating set sigma algebra con con corresponded resp suppose spi sqj therefore lemma leads configuration equivalence equivalent isomorphism sxr sys satisfy conditions definition hence symetry also completes proof say group admits decomposition exists disjoint sets along elements paradoxical decomposition group decomposition group admits decomposition denote minimal amount using method proof theorem easily show theorem let two configuration equivalent group admits decomposition also admits decomposition subgroups configuration equivalent groups studied yet however use tarski number pieces information free subgroups obtained theorem dekker see example theorem contains free subgroup theorem implies groups configuration equivalent free subgroups fact stating result reminders needed power function polynomial type group function following properties finite set associate finite partition power function called consisting disjoint subsets generating set find representative pair con con considered color configuration equivalence equivalent isomorphism use power function speak subgroups configuration equivalent groups theorem let two groups configuration equivalent suppose contains free subgroup rank contains free subgroup rank proof fix generating set say let generating set subgroup free rank assume power function assume sigma algebra generated set con con denote set one easily check arbitrary representative pair say hence means subgroup free rank following conjugacy classes involved element group denote conjugacy class obvious normal subset see definition nothing disjoint union conjugacy classes show following theorem two groups configuration equivalent class numbers theorem let finitely generated groups configuration equivalent class numbers proof let generating set suppose class number least elements pairwise disjoint consider finite sigma algebra containing following sets assume sigma algebra along generating set cont cont elements scl get lemma normal sets class number least completes proof obvious central element suppose class number finite implies finite subgroup stands center lemma using finiteness class number present configuration equivalence equivalent isomorphism certain type configuration pair make possible study subgroups efficiently lemma let finitely generated group finite class number fix generating set partition containing cont cont configuration pair group singleton without loss generality assume proof suppose let elements conjugacy classes consider sigma algebra generated following subsets cont cont sigma algebra generating set group scl sets normal see lemma theorem elements since sets written union least two atoms central without loss generality assume therefore atom works well power function polynomial type groups say configuration equivalent groups finite class number polynomial type subgroups isomorphic theorem let two groups configuration equivalent assume groups finite class number polynomial type subgroup contains subgroup isomorphic furthermore isomorphic centers proof fix generating set let properties page page established applying lemma get partition cont cont configuration equivalence equivalent lemma property one easily see since becomes finite subgroup finite class number adding repeating proof one easily obtain interest study finitely generated groups finite class numbers large prime number exists infinite group exponent exactly conjugacy classes theorem osin recently constructed finitely generated example major breakthrough since problem open years groups infinitely presented still open problem question whether infinite finitely presented group finite class number group exists following theorem make sense theorem let finitely presented group finite class number suppose finitely generated group quotient proof fix generating set say apply lemma get partition consider set defining relators let sigma algebra generated sets max sigma algebra cont cont lemma lemma imply assume max particular hence map ranges arbitrary representative pairs introduced epimorphism configuration quotients results quotients configuration equivalent groups obtained section study number quotients configuration equivalence equivalent let normal subgroup denote quotient map cases ambiguity may drop recall partition becomes partition refer partition say one easily check intersection two refinement let ordered subset generating set lemma ordered subset becomes generating set called pair mean configuration pair generating set notions preserving presentation recognizability configuration pair defined lets first consider concept recognizable configuration pair definition let group normal subgroup pair may say recognizable whenever cont cont configuration pair groups every representative pair color element contains consequence lemma one easily show see lemma lemma let group normal subgroup assume recognizable configuration pair refinement configuration pair recognizable really makes working recognizable configuration pair useful accompaniment notion following one definition let regarded say configuration pair preserves presentation following held configuration equivalence equivalent cont cont configuration pair groups nsf normal function bellow defined cosets considered arbitrary representative pair remark theorem know finitely presented pair exists preserves presentation let two partitions group define partition follows atom regarding notation lemma well see matter work one one partition precisely lemma let group generating set assume collection partitions group generating set collection partitions cont cont proof set let configuration pair cont cont partition lemma implies cont cont concept configuration study number finite index subgroups end provide following definition definition let group finite collection normal subgroups may say recognizable following held closed intersection finitely presented exists generating set collection partition recognizable configuration pair configuration equivalence equivalent case collection properties definition next theorem worthy attention theorem suppose group recognizable collection normal subgroups let group configuration equivalent collection normal subgroups proof let regarded definition lemma generating set partitions cont cont hence lemma obtain normal subgroups along following isomorphisms isomorphisms equation easily proved know theorem number subgroups finite index finitely generated group finite since intersection finite index subgroups finite index subgroup theorem obtain corollary let finitely generated groups configuration sets let contain exactly number normal subgroups index moreover proof suppose collection normal subgroups index let collection normal subgroups obtained intersection assume power function yields configuration pair ordered set partition let gnk generating set hence one see conditions definition satisfied therefore theorem shows existence least normal subgroups index symmetry concept configuration completes proof configuration equivalence equivalent denote collection normal subgroups quotient polycyclic since every polycyclic group recognizable configuration pair since closed intersection argument like one proof previous corollary leads corollary let finitely generated groups cardinality furthermore finite bijection every concept configuration equivalence equivalent isomorphism question whether concepts configuration equivalence isomorphism groups open since beginning configuration theory indeed answer groups configuration sets worth noting groups example solvable thus natural conjecture two concepts may equivalent amenable groups rejected provide example may need provide following two technical lemmas lemma let epimorphism groups suppose generating set partition cont cont proof assume cont exists configuration means fck therefore fck hence configuration cont conversely let tuple configuration cont means fck choose configuration belonging cont configuration equivalence equivalent may call homomorphism every ordered generating set image ordered generating set clear becomes epimorphism call two groups following lemma understood groups configuration sets lemma let two finitely generated groups suppose cont cont proof consider configuration pair exists generating set thus previous lemma cont cont cont completes proof obvious isomorphic groups converse always true following theorem show theorem exist finitely generating groups configuration sets proof put ring laurent polynomials let group matrices belongs hti group easily checked finitely generated indeed denote matrix becomes generating set following equations integer center group consists unipotent matrices single possibly element upper right corner clearly isomorphic rewrite product group bellow configuration equivalence equivalent equality one see map introduced automorphism set automorphism implies let groups isomorphic torsion free note ordered set image natural quotient map forms generating set also image form generating set thus lemma complete proof remark groups theorem solvable shows two concepts configuration equivalence isomorphism equivalent solvable hence amenable groups groups finitely presented residually finite hopfian references abdollahi rejali willis group properties characterized configurations illinois math abdollahi rejali yousofzadeh configuration nilpotent groups isomorphism algebra appl denis osin small cancellations relatively hyperbolic groups embedding theorems ann math rejali soleimani malekan strong configuration equivalence isomorphism http rejali soleimani malekan configuration polycyclic groups isomorphism http rejali yousofzadeh group properties characterized configurations algebra colloq tavakoli rejali yousofzadeh abdollahi note configuration group matematika volume number rosenblatt willis weak convergence strong convergence amenable groups canad math bull walker cancellation direct sums groups proc amer math vol bogopolski introduction group theory european mathematical society magnus karrass solitar combinatorial group theory presentations groups terms generators relations dover publications inc new york geometry defining relations groups kluwer configuration equivalence equivalent robinson course theory groups edition springer new york segal polycyclic groups cambridge university press london sims computation finitely presented groups cambridge cambridge university press sapir combinatorial algebra syntax semantics http tomkinsin edition pitman publishing limited london university isfahan current address department mathematics faculty sciences university isfahan isfahan iran address rejali department mathematics student university isfahan isfahan iran address
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dec nearly commuting matrices zhibek kadyrsizova abstract prove algebraic set pairs matrices diagonal commutator field positive prime characteristic irreducible components intersection size matrices equal furthermore show algebraic set reduced intersection irreducible components irreducible characteristic pairs matrices size addition discuss various conjectures singularities algebraic sets system parameters corresponding coordinate rings keywords frobenius singularities commuting matrices introduction preliminaries paper study algebraic sets pairs matrices commutator either nonzero diagonal zero also consider related algebraic sets first let define relevant notions let xij yij matrices indeterminates field let polynomial ring xij yij let denote ideal generated entries commutator matrix denote ideal generated entries ideal defines algebraic set pairs matrices diagonal commutator called algebraic set nearly commuting matrices ideal defines algebraic set pairs commuting matrices zhibek kadyrsizova let uij denote entry matrix uij uij theorem algebraic set commuting matrices irreducible variety equivalently rad prime following results due knutson characteristic field characteristics theorem algebraic set nearly commuting matrices complete intersection variety commuting matrices one irreducible components particular set uij regular sequence dimension theorem characteristic zero radical ideal knutson paper conjectured two irreducible components proved characteristics thesis theorem algebraic set nearly commuting matrices two irreducible components one variety commuting matrices skew component two minimal primes one rad let rad let denote minimal prime rad following conjecture made artin hochster nearly commuting matrices conjecture reduced answered positively mary thompson thesis case matrices theorem domain let back algebraic sets nearly commuting matrices irreducible components first take look everything trivial precisely without loss generality may replace respectively identity matrix size denote generators minors diagonal entries ideal generated size minors moreover therefore prime also radical prime zhibek kadyrsizova therefore monomial term coefficient since see fedder criterion lemma furthermore determinantal rings regular see therefore rest paper shall use following notations notation let integer let xij yij matrices indeterminates field let polynomial ring xij yij let denote ideal generated entries commutator matrix denote ideal generated entries let denote radical minimal prime rad prove following results paper theorem let ring notation assume also field positive prime characteristic pure rings words algebraic set nearly commuting matrices size irreducible components intersection particular skew component reduced case theorem let ring notation reduced words algebraic set nearly commuting matrices reduced matrices sizes characteristics theorem intersection variety commuting matrices skew component irreducible rad prime nearly commuting matrices section show coordinate ring algebraic set pairs matrices diagonal commutator case matrices moreover also show implies corresponding fact irreducible components variety commuting matrices intersection first state two lemmas due fedder include criterion finitely generated particularly convenient form complete intersections lemma fedder let regular local ring polynomial ring field characteristic unmixed proper ideal homogeneous polynomial case primary decomposition lemma fedder criterion let regular local ring polynomial ring field homogeneous maximal ideal characteristic proper ideal homogeneous polynomial case next result straightforward consequence two lemmas prove quite useful lemma let regular local ring polynomial ring field suppose characteristic ideal homogeneous polynomial case suppose also primary decomposition aim zhibek kadyrsizova proof observe first aim aim aij aij rest immediate lemma lemma lemma closely related results compatibly split ideals immediately get corresponding result algebraic set corollary suppose coordinate ring algebraic set nearly commuting matrices next use fedder criterion show case theorem let field characteristic let let ring notation proof recall generated regular sequence uij therefore thus fedder criterion uij sufficient prove homogeneous maximal ideal show proving following claim claim monomial term nonzero coefficient modulo proof compute coefficient obtained choosing monomial every uij following way nearly commuting matrices exponents ast bst xst yst respectively addition denote goal find nonnegative integer tuples ast bst cij notice linear system nonzero determinant sum first equations twice sum rest equations therefore unique solution linear system solved using standard methods linear algebra following solution zhibek kadyrsizova column vector represents matrix solutions arbitrary integers since look integer solutions must hence therefore coefficient sum expressions form nearly commuting matrices run solutions linear system modulo equivalent also written following lemma shows expression equal values fact purpose prime lemma let proof prove stronger statement claim let proof first observe hence may assume zhibek kadyrsizova let consider difference using pascal identity get thus therefore nearly commuting matrices thus case finally use induction conclude thus finally complete proof theorem corollary let ring notation rings gorenstein proof ring since ideals linked via also see moreover theory linkage also implies isomorphic canonical modules respectively hence gorenstein dimension corollary let ring notation radical zhibek kadyrsizova remark prove next section radical prime implies prime particular domain irreducibility section prove intersection variety commuting matrices irreducible first define notions definition let matrix indeterminates matrix whose ith column defined diagonal entries numbered upper left corner lower right corner let denote determinant theorem irreducible polynomial remark identity matrix next two lemmas due young give connection variety defined algebraic set nearly commuting matrices lemma given matrix exists matrix diagonal matrix lemma dense open set variety defined every point exists matrix nonzero diagonal matrix nearly commuting matrices following notion discriminant significant importance matrix theory use section order reduce study case commuting matrices particularly simple characterization definition let discriminant discriminant characteristic polynomial contains eigenvalues fact let matrix equivalently distinct eigenvalues matrix commutes polynomial degree see theorem remark irreducible polynomial degree polynomial degree moreover divide proved showing exists matrix property example purpose one use following matrices otherwise characteristic polynomials outline prove main result section rad prime dim equidimensional zhibek kadyrsizova dim minimal primes rad prime ideal observe minimal primes height larger one first need following theorem due hartshorne theorem proposition let noetherian local ring maximal ideal spec disconnected depth lemma let ideals notation every minimal prime height proof suppose exists minimal prime ideal height least localize moreover disjoint punctured spectrum spec however theorem shows possible let define need state prove next result let positive integer fix partition choose positive integers let upper triangular jordan form nilpotent matrix size finitely many choices let let denote identity matrix size distinct elements let matrix blocks main diagonal direct sum matrices nearly commuting matrices let open subset therefore irreducible dimension let similar distinct let denote dimension set matrices commute number independent choice since commutes matrix direct sum matrices size commutes moreover dimension set invertible matrices commute lemma dimension proof define surjective map algebraic sets gln fix gln dimension set gln set invertible matrices commuting zhibek kadyrsizova let let hence therefore dimension since dimension dimension gln minus dimension generic fiber dimension moreover set pairs matrices commute dimension claim let injective map proof let lemma therefore since closure hence claim dimension set proof let distinct eigenvalues dim nearly commuting matrices therefore dim notice since similarly let distinct eigenvalues value therefore dimension moreover closed subset defined vanishing prove claim need show dim showing contain component dimension words show pairs matrices either let matrix distinct eigenvalues similar jordan matrix case similar two possible forms take diag diagonal matrix distinct entries diagonal case similar zhibek kadyrsizova write take matrix identity matrix goal show first prove case matrices observe denote nearly commuting matrices moreover particular diagonal diag det finally det final expression determinant nonzero hence dim thus minimal primes prove one minimal prime zhibek kadyrsizova theorem let notation irreducible rad prime proof let dense open subset algebraic set defined lemma let suppose lemma exists matrix algebraic set nearly commuting matrices let polynomial ring one independent variable fix since defines closed set must well since dense subset recall arbitrary element assume also every matrix commutes polynomial degree thus polynomial degree moreover since every element form polynomial degree identify polynomials degree consider map moreover map bijective morphism therefore irreducible irreducible nontrivial irreducible nearly commuting matrices decomposition give nontrivial irreducible decomposition since minimal primes thus result corollary let notation prime ideal nearly commuting matrices radical ideal section prove radical ideal characteristics know rad unmixed heights equal prove result sufficient show becomes prime radical localize theorem defining ideal algebraic set nearly commuting matrices radical proof simplicity notation let denote since every must vanish set therefore disjoint hence localize injective homomorphism ideal generated linear equations entries coefficients always choose least variables yij write rest combinations chosen ones thus yij prime zhibek kadyrsizova next observe clearly prove direction let nonzero lemma words exists matrix property nonzero diagonal therefore injective homomorphism discrete valuation domain generators become linear polynomials entries coefficients let matrix coefficients linear system rows indexed columns indexed entry xih spot entry spot xii xjj spot zero everywhere else let denote entries yij generated entries matrix elementary row operations transform diagonal matrix gives new generators prove radical sufficient show diagonal entries order one end reduces show rank ideal generated minors size contained nearly commuting matrices sufficient prove original matrix hence suffices show claim submatrix obtained first columns nonzero determinant determinant proof sufficient prove first part claim frac invert case since nearly commute must commute see lemma moreover generic matrix hence discriminant nonzero divisible thus distinct eigenvalues polynomial degree write equations become notice invertible every choice values unique solution equation furthermore bottom rows linearly independent generic matrix true even holds permutation zhibek kadyrsizova matrix bottom rows standard basis vectors thus given bottom row exist equals bottom row uniquely determined entries bottom row therefore invertible first let show det matrix open dense subset closed set defined exists matrix commutator nonzero diagonal matrix see lemma hence polynomials degree therefore space solutions dimension showed must therefore minors must vanish whenever vanishes nearly commuting matrices notice degree polynomial degree det therefore prove part sufficient show det multiple let put grading entries let deg xij deg yij products sums property well deg deg therefore commutator matrix fact polynomial property notice diagonal entries degree thus degree however case determinant matrix nonzero entry corresponding degree therefore product entries nonzero term determinant degree hence det multiple factor minors remaining expression divisible ready finish discussion let matrix obtained elementary row transformations diag cii diagonal matrix proved exists property cii unit ckk denote ckk unit zhibek kadyrsizova ideal generated following equations ykk reduced show consider two cases last factor ring isomorphic unit factor ring isomorphic ykk either case reduced therefore since injective map irq radical conjectures section state conjectures made research many appeared result computations performed computer algebra program conjecture let notation remark case conjecture true following lemma allows reduce conjecture regularity nearly commuting matrices lemma let noetherian local ring prime characteristic let ideal homogeneous graded case generated regular sequence let ideals height linked via let suppose equivalently proof canonical module isomorphic similarly canonical module gorenstein hence regular recall graded ring localization homogeneous maximal ideal applying corollary implies thus want prove variety commuting matrices skew component sufficient prove statement intersection course need know whether conjecture conjecture solved proving following one conjecture let monomial term modulo xij yij xii upij coefficient equal zhibek kadyrsizova remark monomial obtained taking product variables dividing variables according following pattern denote variable divided conjecture let matrix indeterminates size field let irreducible polynomial definition conjecture following regular sequence hence part system parameters xst xnn nearly commuting matrices mod remark conjecture verified using software small prime characteristics case equivalent following identifications variables matrices conjecture let uij subset cardinality let ideal generated elements particular prime ideal acknowledgement paper part thesis written author doctorate study university michigan author would like express enormous gratitude mel hochster support guidance received working paper work supported part nsf grant barbour scholarship university michigan zhibek kadyrsizova references enescu applications pseudocanonical covers tight closure problems pure appl algebra fedder rational singularity graded complete intersection rings transactions american mathematical society gerstenhaber dominance varieties commuting matrices ann math grayson stillman tem algebraic geometry macaulay commutative computer algebra sysalgebra available http hartshorne complete intersections connectedness american journal mathematics hochster huneke tight closure parameter ideals splitting extensions algebraic geom horn johnson matrix analysis cambridge univ press cambridge knutson schemes related commuting variety journal algebraic geometry lyubeznik smith strong weak equivalent graded rings american journal mathematics peskine szpiro liaison des invent math schwede tucker survey test ideals progress commutative algebra closures finiteness factorization walter gruyter gmbh berlin thompson topics ideal theory commutative noetherian rings thesis university michigan young components algebraic sets commuting nearly commuting matrices thesis university michigan nearly commuting matrices email address department mathematics school science technology nazarbayev university kabanbay batyr ave astana kazakhstan
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learning classification digits spiking neural networks shruti kulkarni john alexiades bipin rajendran nov department electrical computer engineering new jersey institute technology newark usa email bipin describe novel spiking neural network snn automated handwritten digit classification implementation platform information processing within network feature extraction classification implemented mimicking basic aspects neuronal spike initiation propagation brain feature extraction layer snn uses fixed synaptic weight maps extract key features image classifier layer uses recently developed normad approximate gradient descent based supervised learning algorithm spiking neural networks adjust synaptic weights standard mnist database images handwritten digits network achieves accuracy training set test set nearly fewer parameters compared spiking networks use network gpu based system demonstrating snn simulation infer digits written different users test set images platform achieves accuracy exceeding making prediction within snn emulation time less index neural networks classification supervised learning gpu based acceleration processing ntroduction human brain computational marvel compared systems ability learn execute highly complex cognitive tasks well energy efficiency computational efficiency brain stems use sparsely issued binary signals spikes encode process information inspired spiking neural networks snns proposed computational framework learning inference general purpose graphical processing units become ideal platform accelerated implementation large scale machine learning algorithms multiple gpu based implementations simulating large snns targeting forward communication spikes large networks spiking neurons local weight update based spike timing difference contrast demonstrate highly optimized real time implementation scheme spike based supervised learning gpu platforms use framework real time inference digits captured different users interface previous efforts develop deep convolutional spiking networks started using second generation artificial neural research supported part grants national science foundation award cisco mcnair fellowship program networks anns errors train network thereafter converting spiking versions several supervised learning algorithms proposed train snns explicitly using time spikes neurons encode information derive appropriate weight update rules minimize distance desired spike times observed spike times network use normalized approximate descent normad algorithm design system identify handwritten digits normad algorithm shown superior convergence speed compared methods remote supervised method resume snn trained mnist database consisting training images test images highest accuracy snn mnist reported convolution neural network achieved accuracy test set network contrast three layers learning synapses fewer parameters compared achieves accuracy mnist test dataset paper organized follows computational units snn network architecture described section section iii details network simulation divided among different cuda kernels system image steps explained section present results network simulation speed related optimizations section section concludes gpu based system implementation study piking eural etwork basic units snn spiking neurons synapses interconnecting computational tractability use leaky integrate fire lif model neurons evolution membrane potential described dvm total input current resting potential model membrane capacitance leak conductance respectively membrane potential crosses threshold reset resting value remains value till neuron comes refractory period tref synapse weight connecting input neuron output neuron transforms incoming spikes arriving ieee personal use material permitted permission ieee must obtained uses current future media including material advertising promotional purposes creating new collective works resale redistribution servers lists reuse copyrighted component work works times current based following transformation tik summed function represents incoming spike train double decaying exponentials represent synaptic kernel values closely match biological time constants output layer generates spike train frequency close output neurons issue spikes presentation duration set baseline experiments also network effect network classification ability discussed section layer also lateral inhibitory connections helps prevent neurons spiking given input output neuron highest number spikes declared winner classification learning layer network architecture spike encoding use network hidden layer performs feature extraction output layer performs classification see fig network designed take input pixel mnist digit image translate pixel value set spike streams passing pixels currents layer neurons first layer current applied neuron corresponding pixel value range obtained following linear relation scaling factor minimum current lif neuron generate spike parameters chosen equation spike streams weighted twelve synaptic weight maps filters priori chosen values generate equivalent current streams using equations spatial filter maps chosen detect various edges corners image convolution layer input currents neurons pixel input image input layer spike trains twelve spatial filters size learning synapses fully connected layer neurons neurons output layer lateral inhibitory connections synapses connecting hidden layer neurons output layer neurons modified course training using normad rule strength weights adjusted based error observed desired spike streams term denoting effect incoming spike kernels neuron membrane potential according relation exp represents neuron impulse response learning rate iii cuda implementation snn implemented gpu platform using programming framework gpu divided streaming multiprocessors consists stream processors optimized execute math operations programming framework exploits hardware parallelism gpus launches jobs gpu grid blocks mapped blocks divided multiple threads scheduled run also called cuda core since memory transfer cpu gpu local memory one main bottlenecks network variables neuron membrane potentials synaptic currents declared global gpu memory implementation simulation equations evaluated numerically iterative manner time step next epoch next time step output layer consists neurons one ten digits train network correct neuron read images mnist generate spike trains gpu device memory lif layer neurons image fig pixel images mnist database converted spike trains presented duration weighted twelve synaptic weight maps resulting twelve current streams feed corresponding feature map neurons output neurons corresponding digit weights fullyconnected synapses output layer neurons adjusted using normad learning rule additionally output layer neurons also lateral inhibitory connections lif neurons synaptic current current summation currents spike trains synaptic currents lif convolution kernels neurons synapses per map synapses spike errors weight update synapses netowork output compute compute synapses spike trains fig diagram showing different variables network computed time step signals flow across different layers dimensions within brackets sizes variables respective cuda kernels fig shows forward pass backward pass weight update training phase image pixels read gpu memory passed currents layer one neurons grid size presentation duration filtering process involves convolution incoming spike kernels weight matrix computation parallelized across cuda kernels grid size threads thread computes current hidden layer neurons indexed timestep based following spatial convolution relation iin wconv represents synaptic kernel equation calculated spike trains pixels wconv represents weights filter matrix membrane potential array lif neurons applied current described equation evaluated using second order method thread independently checks membrane potential exceeded threshold artificially reset vmk vmk eal time inference user data used cuda based snn described previous section design user interface capture identify images digits written users touchscreen interface drawing application capture digit drawn user built using opencv image processing library captured image touch screen using standard methods similar used generate mnist dataset images convert user drawn images required format grayscale image size pixels network implemented nvidia gtx gpu cuda cores preprocessing phase takes image passed trained snn inference cuda process takes initialize network gpu memory network simulation time depends presentation time time step interval refractory period implemented storing latest spike issue time nlast neuron vector membrane potential neuron updated current time step nlast tref synaptic current neuron hidden layer neuron output layer given equation evaluated iterative manner thereby avoiding evaluation expensive exponential difference current time previous spike times nik synaptic current computation time step synapse spawned cuda across kernels exp nik nik exp update synaptic weights parallel evaluation total synaptic current norm performed using parallel reduction cuda inference testing phase calculate synaptic currents membrane potentials neurons layers determine spike times evaluate term weight update represent rising falling regions double exponential synaptic kernel strength synapses hidden output layers initialized zero training every time step error function output neuron calculated based difference observed desired spikes next equation spikes originating neuron computed evaluated compute norm across neurons determine instantaneous synapses parallel spike error end presentation accumulated used drawn image inverted image bounding boxed image normalized image fig outline preprocessing steps used convert user input image fed network examples user input left pixel images fed snn right image preprocessing fig shows preprocessing steps used create input signal snn captured image fig shows sample images image captured user first binarized thresholding cropped remove excess background image resized pixels along longer dimension maintaining aspect ratio thereafter resized image placed bounding box image center mass coincides center bounding box finally image passed blurring filter create images similar ones mnist dataset esults trained network mnist training consisting images epochs network achieves error training set test set time step network simulated table lists networks ann snn mnist classification problem seen though networks classification accuracies exceeding use number parameters compared network designed simplify computational load developing system table comparison snn network learning algorithm deep learning ann converted snn convolution snn snn normad work learning synapses accuracy test accuracy integration time step interval used inference approximating neuronal integration instead mnist test error increases see fig reduction processing time hence touch screen based interface system simulate snn infer users digits digit presented network simulated average wall clock time making processing possible fig tested image presentation time image capture preprocessing complete output spikes snn prediction result display time fig mnist accuracy function presentation time integration time step various stages classifying user input image takes snn emulation completed network accuracy set handwritten digits collected various users system measure accuracy set captured images mnist slight loss performance compared mnist dataset attributed deviations statistical characteristics captured images compared mnist dataset onclusion developed simple spiking neural network performs spike encoding feature extraction classification information processing learning within network performed entirely spike domain approximately times lesser number synaptic weight parameters compared state art spiking networks show approach achieves classification accuracy exceeding training set mnist database test set trained network implemented cuda parallel computing platform also able successfully identify digits written users demonstrating true generalization capability also demonstrated general framework implementing spike based neural networks supervised learning weight update rules gpu platform time step neuronal spike transmission synaptic current computation weight update calculation network executed parallel framework using gpu implementation demonstrated based platform classification images eferences maass networks spiking neurons third generation neural network models neural networks vol coates deep learning cots hpc systems international conference machine learning fidjeland shanahan accelerated simulation spiking neural networks using gpus international joint conference neural networks ijcnn ieee nageswaran configurable simulation environment efficient simulation spiking neural networks graphics processors neural networks vol yavuz turner nowotny genn code generation framework accelerated brain simulations nature scientific reports vol yudanov simulation spiking neural networks performance high accuracy international joint conference neural networks july naveros techniques using parallel cpugpu spiking neural networks front neuroinformatics vol krichmar coussy dutt spiking neural networks using neuromorphic hardware compatible models acm journal emerging technologies computing systems jetc vol diehl spiking deep networks weight threshold balancing international joint conference neural networks ijcnn july cao chen khosla spiking deep convolutional neural networks object recognition international journal computer vision vol rueckauer theory tools conversion analog spiking convolutional neural networks arxiv preprint hunsberger eliasmith training spiking deep networks neuromorphic hardware arxiv preprint anwani rajendran normad approximate descent based supervised learning rule spiking neurons international joint conference neural networks july ponulak kasinski supervised learning spiking neural networks resume sequence learning classification spike shifting neural computation vol mohemmed span spike pattern association neuron learning spike patterns international journal neural systems vol lee delbruck pfeiffer training deep spiking neural networks using backpropagation frontiers neuroscience vol lee kukreja thakor cone efficacies temporally precise spike mapping ieee transactions neural networks learning systems vol april mnist database handwritten digits available http dayan abbott theoretical neuroscience computational mathematical modeling neural systems journal cognitive neuroscience vol harris optimizing parallel reduction cuda available http culjak abram pribanic dzapo cifrek brief introduction opencv proceedings international convention mipro may wan regularization neural networks using dropconnect proceedings international conference machine learning
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nov hypercollecting semantics application static analysis information flow mounir assaf david naumann julien signoles stevens institute technology hoboken stevens institute technology hoboken software reliability security lab cea list saclay totel tronel cidre rennes cidre rennes abstract show static analysis secure information flow expressed proved correct entirely within framework abstract interpretation key idea define galois connection directly approximates hyperproperty interest enable use galois connections introduce fixpoint characterisation hypercollecting semantics set sets transformer makes possible systematically derive static analyses hyperproperties entirely within calculational framework abstract interpretation evaluate technique deriving example static analyses qualitative information flow derive dependence analysis similar logic amtoft banerjee sas type system hunt sands popl quantitative information flow derive novel cardinality analysis bounds leakage conveyed program instead simply deciding whether exists encompasses problems hypersafety put framework use introduce variations achieve precision rivalling recent precise static analyses information flow categories subject descriptors software engineering checkers programming languages logics meanings programs semantics programming language keywords static analysis abstract interpretation information flow hyperproperties introduction static analyses tell something executions program needed example validate compiler optimizations functional correctness also formulated terms predicate observable behaviours less abstract execution traces 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 acm must honored abstracting credit permitted copy otherwise republish post servers redistribute lists contact request permissions permissions publications acm fax popl january paris france copyright held publication rights licensed acm acm doi http program correct traces satisfy predicate contrast trace properties extensional definitions dependences involve one trace express final value variable may depend initial value variable noninterference security literature sabelfeld myers two traces initial value result final value sophisticated information flow policies allow dependences subject quantitative formalisations involve two traces sometimes unboundedly many secure information flow formulated decision problems theory hyperproperties classifies simplest form noninterference quantitative flow properties hypersafety properties clarkson schneider number approaches explored analysis dependences including type systems program logics dependence graphs several works used abstract interpretation way one approach forming product program encodes execution pairs barthe terauchi aiken darvas thereby reducing problem ordinary safety checked abstract interpretation means alternatively property checked dedicated analyses may rely part ordinary abstract interpretations trace properties amtoft theory abstract interpretation serves specify guide design static analyses well known effective application theory requires choosing appropriate notion observable behaviour property interest cousot bertrane notion trace chosen one program semantics executions formalized terms collecting semantics used define trace property interest thus specify abstract interpretation cousot cousot cousot foundation abstract interpretation quite general based galois connections semantic domains collecting semantics defined clarkson schneider formalize notion hyperproperty general way set sets traces remarkably prior works using abstract interpretation secure information flow directly address dimension instead involve various hoc formulations paper presents new approach deriving information flow static analyses within calculational framework abstract interpretation first contribution lift collecting semantics sets trace sets dubbed hypercollecting semantics fixpoint formulation simply lifted direct image composed galois connections specify hyperproperties beyond without recourse hoc additional notions basis foundational advance becomes possible derive static analyses entirely within calculational framework abstract interpretation cousot cousot cousot second contribution use hypercollecting semantics derive analysis ordinary dependences seen rational reconstruction type system hunt sands logic amtoft banerjee determine variable conservative approximation variables whose initial values influence final value third contribution derive novel analysis quantitative information flow shows benefit taking hyperproperties seriously means abstract interpretation noninterference variables depends fixed values one final value quantitative information flow one interested measuring extent variables influence given range variation high inputs range variation final values directly address question hyperproperty given set traces agree low inputs cardinality possible final values using hypercollecting semantics derive novel cardinality abstraction show used analysis quantitative information problems including bounding problem calculational approach disentangles key design decisions enabled identify opportunities improving precision assess precision analyses provide formal characterisation precision quantitative information flow analysis vis vis qualitative versions analyses rival state art analyses qualitative quantitative information flow technical development uses simplest programming language semantic model ideas exposed one benefit working entirely within framework abstract interpretation wide range semantics analyses already available rich programming languages outline following background section introduce domains galois connections hyperproperties section hypercollecting semantics section hyperproperties information flow defined section use framework derive static analyses section section section uses examples evaluate precision analyses shows existing analyses leveraged improve precision discuss related work section conclude appendices provide detailed proofs results well table symbols background collecting semantics galois connections formal development uses deterministic imperative programs integer variables let range literal integers variables resp cmp arithmetic resp comparison operators skip else cmp different program analyses may consider different semantic domains needed express given class program properties imperative programs usual domains based states states map variable value winskel trc states figure fragment hierarchy semantic domains abstraction program properties require use traces include intermediate states others use abstract domains information flow properties involving intermediate outputs restricted explicit data flow schoepe details intermediate steps needed contrast bounding range variables expressed terms final states another example consider determining variables left unchanged express need initial final states paper use succinct term trace elements trc defined trc states states interpreting trc initial final state literature known relational traces contrast maximal trace semantics using set finite sequences uniform framework describes relationships correspondences many semantic domains using galois connections cousot three domains depicted figure given partially ordered sets monotone functions comprise galois connection provided satisfy proposition write iff example specify analysis determines variables never changed let sets variables define trc vars trc hierarchy usual domains depicted figure connections defined abstraction define elt trc elt lifts abstraction trc lemma abstraction let elt function sets let elt elt elt domain states suffices describe final reachable states program abstraction relational domain trc elt paper focus domain trc simplest express dependences program semantics define denotational semantics jck commands denotational semantics jek trc val expressions val adds bottom element using flat ordering jck standard semantics commands jek jskipkt jif else jbkt jbkt jwhile ckt jckt jbkt otherwise let trace denotation jekt evaluates current state sect also use jekpre evaluates initial state denotation jckt execution leads denotation case diverges boolean expressions evaluate either assume programs wrong denote lifting approximation order terminating computations written image initial traces jckt initrc jckt initrc states specify properties hold executions use collecting semantics lifts denotational semantics arbitrary sets trc traces idea direct image jck precise paper focus terminationinsensitive properties thus set traces jckt later also use collecting semantics expressions jekt importantly collecting semantics trc trc defined compositionally using fixpoints cousot sec conditional guard write grdb filter defined grdb jbkt trc trc collecting semantics ekt else grdb else clause loops uses denotation constructed conditional command definitional denotation compositional given galois connection trc one unmodified variables desired analysis specified since computable general require approximation sound sense denotes lifting partial order explain significance specification suppose one wishes prove program satisfies trace property trc prove initrc given suffices find abstract value approximates initrc initrc show equivalent property galois connections implies monotonicity implies initrc beauty specification obtained abstract interpretation derived systematically calculating left side shown cousot domains galois connections hyperproperties express hyperproperties need galois connections domains involve sets sets observable behaviours section spells powerset domains form hierarchy illustrated along top figure describe dependences trc states trc states abstraction figure extended hierarchy semantic domains cardinalities quantitative information flow formulated galois connections spell methodology whereby standard notions techniques abstract interpretation applied specify form equation static analyses hyperproperties first example consider condition final value depends initial value expression needs least two traces two traces denoted agree initial value agree final value implies must hold two traces program equivalent following sets traces traces agree initial value agree final value later extend example analysis infers dependences hold consider problem quantifying information flow mincapacity smith program two integer variables problem infer much information conveyed via considering traces agree initial value many final values possible example program mod two final values initial though many possible initial values cardinality problem generalizes prior work quantitative flow analysis typically low inputs considered whereas simple dependence problem formulated terms traces cardinality problem involves trace sets unbounded size terminology hyperproperties hyperproperty yasuoka terauchi sec although hypersafety clarkson schneider fixed problem variable final values means formulated terms sets traces turns using galois connections sets sets develop general theory encompasses many hyperproperties enables derivation interesting abstract interpreters applications use relational traces notion observable behavior thus trc approach works well notions hierarchy domains shown top figure parallel ordinary hierarchy shown along bottom abstractions hierarchy obtained lifting abstraction two standard collecting semantics cousot hypercollecting versions abstraction lemma instance lemma justifies abstraction trc states lifting abstraction trc states cousot sec additionally diagonal lines figure represent abstractions hypercollecting semantics defined form observations corresponding collecting semantics defined observations lemma let set define form galois connection hpp noted clarkson schneider trace property lifted unique hyperproperty lifting exactly concretisation lemma although model clarkson schneider quite general focus infinite traces hyperproperties formulated terms notions observation illustrated figure cardinality abstraction lay groundwork quantitative information flow analysis consider abstracting set values cardinality cardinality one ingredient many quantitative information flow analyses estimating amount sensitive information program may leak smith backes braun rybalchenko mardziel doychev lattice abstract representations consider set denotes infinite cardinal number use natural order max join consider abstraction operator crdval val computing cardinality given crdval operator crdval additive preserve joins crdval max crdval crdval thus exists associated concretisation crdval lower adjoint galois connection yet lift abstraction operator crdval galois connection val called supremus abstraction cousot lemma supremus abstraction let elt function set codomain forming complete lattice let elt elt elt example define maxv crdval crdval thus obtain galois connection val another example let consider simplified form ingredient dependency noninterference analysis program variable agreex states determines whether set states contains states agree value agreex function agreex additive part galois connection states problem arises agreements multiple variables concrete domains like finite maximal trace semantics lift operator agreex galois connection states supremus abstraction yields agreex agreex abstract security requirement yields set sets traces namely hyperproperty hints intuition appear literature mclean volpano rushby zakinthinos lerner security policies predicates sets traces higher order rushby however recently comprehensive framework proposed sharp characterisation security policies hyperproperties clarkson schneider abstract interpretation hyperproperties basic methodology verification hyperproperty may described follows step design approximate representations forming complete lattice choose collecting semantics among extended hierarchy set sets domains trc define galois connection step compute approximation semantics program interest step prove inferred approximation implies satisfies concretisation set trace sets program trace set contrast approximations trace properties infer single trace set program trace set subset suffices prove step guided need describes hyperproperty implies calculational design cousot abstract domains greatly systematises step relying galois connection defined step collecting semantics adapted additional structure sets show section hypercollecting semantics following introduce hypercollecting semantics defined sets trc sets traces used subsequent sections derive static analyses step methodology spelled detail given built galois connection trc supremus abstraction approximation initial traces initrc find approximation analysed program initrc prove program satisfies hyperproperty interest order compute define hypercollecting semantics lcm trc trc serve manner equation static analysis correct construction hypercollecting semantics states examples consistent many formulations noninterference goguen meseguer volpano smith giacobazzi mastroeni amtoft banerjee hunt sands motivated characterisation security requirements hyperproperties clarkson schneider concretising abstract value seen defining denotation type expression instance benton sec hunt sands defining set objects satisfy description thus concretising interpreted satisfies property requirement naturally yields set traces concretising interpreted satisfies security requirement yields set sets traces intuitively abstract property requirement defined terms set traces trc trc emt lskipmt lif else grdb lwhile lif else skipm lgrdb grdb recall section standard collecting semantics formulation captures direct image sets underlying program semantics proved example cachera pichardie assaf naumann fixpoint formulation level use simply direct image standard collecting semantics direct image standard collecting semantics would yield set inner fixpoints sets traces whereas outer fixpoint sets sets traces enables straightforward application fixpoint transfer theorem theorem trc singleton set lcm trc necessarily singleton set containing element loop lcm yields set sets traces set traces contains traces exit loop less iterations prove theorem corollary following trc lcmt proved structural induction commands loops secondary induction iterations loop body summary suppose one wishes prove program satisfies hyperproperty trc one wishes prove initrc suppose approximation hypercollecting semantics similarly lcm given suffices find abstract value approximates initrc initrc show property equation equivalent lcm galois connections initrc lcm using initrc theorem information flow section gives number technical definitions build definition galois connections specify information flow policies explicitly hyperproperties fixed main program considered refer variables varp analyses parametrised program analyse initial typing context varp mapping variable security level initial value assume finite lattice concrete case may defined universal flow lattice powerset variables varp information flow types inferred suitable abstraction hunt sands sec initial typing context defined initial variety key notion information flow two states iff agree values variables security level introduce notion set traces requiring initial states let first denote jekpre trc val evaluation expression initial state trace jek trc val evaluates expression final state denote judgement traces set trc initially initially agree value variables security level example case universal flow lattice means jxkpre jxkpre jykpre jykpre initial iff varp jxkpre jxkpre notion variety cohen underlies definitions qualitative quantitative information flow security information transmitted execution program varying initial value exploring variety resulting value execution also vary showing variety conveyed cohen define expression set sets values may take considering initially traces variety defined first function trc val trace sets obtain function lem trc val sets trace sets intuitively expression variety conveyed varying input values variables security level lem trc val lem trc val lemt set values results initially lequivalent traces thus expression leak sensitive information attackers security clearance set singleton sets indeed sensitive data attackers security clearance data security level attackers access denning denning thus set singleton sets means matter sensitive information varies variety conveyed expression besides pedagogical purpose define resp lem instead simply lifting denotational semantics jek expressions sets traces resp sets sets traces since want build modular abstractions traces relying underlying abstractions values thus enables pass information initially traces underlying domain values keeping disjoint values originate traces initially specifying information flow ingredients needed describe information flow command respect typing context varp quantitative security metric introduced smith relies mincapacity order estimate leakage program let assume program characterized set trc traces initrc simplicity assume attackers observe value single variable varp generalization multiple variables straightforward leakage measured attackers security clearance defined mll definition follows lemma purposes suffices know quantity aims measure bits remaining uncertainty sensitive data attackers security clearance refer original work smith details leaving aside logarithm definition mll quantitative security requirement may enforce limit amount information leaked attackers security clearance requiring variable less equal integer denote hyperproperty characterises security requirement set program denotations satisfying trc note implicitly depends choice initial typing noninterference policy final value depends initial values variables labelled corresponds hyperproperty therefore program satisfies let lpm initrc since theorem satisfies lxmt monotony lxmt definition dependences rely abstract interpretation derive static analysis similar existing ones inferring dependences amtoft banerjee hunt sands amtoft hunt sands recall analyses parametrised security lattice program denote atomic dependence constraint varp read agreement security level leads agreement atomic contract expressing final value must depend initial values security level said otherwise states noninterference variable data sensitive attackers security clearance inputs security level dependences similar information flow types hunt sands dual independences assertions amtoft banerjee interpretations equivalent hunt sands sec dep dep lattice dependence constraints given lattice program define dep varp agree agree deptr deptr val val agree val val agree trc dep varp trc dep deptr dep trc deptr dep trc deptr note deptr set dependences holds instance initial typing context varp determines initial dependences program initrc varp initrc varp derive approximation lem approximation lem dep called expression determines whether set dependence constraints guarantees variety conveyed expression inputs security level fixed notice use symbol subscript contrast similar notation using subscript later sections dep rest section fixed together typing context varp semantic characterisation dependences tightly linked variety atomic constraint holds variety conveyed inputs security level fixed use intuition define galois connections linking hypercollecting semantics lattice dep instantiating supremus abstraction lemma agreement abstraction approximates set val determining whether contains variety agree deptr dependence abstraction expressions agreements abstraction level holds one final value cmp deriving clauses defining amounts constructive proof following lemma sound lem dependence abstract semantics derive dependence abstract semantics approximating hypercollecting semantics lcm abstract semantics dep dep dependence constraints hold execution command inputs satisfying initial dependence constraints assume static analysis approximating variables command modifies mod com modifiable variables val exists trc jckt jxkpre mod note val agree val also agree iff dependence abstraction approximates set trc dependence constraint dep recall set final values variable traces agree inputs abstract semantics assignments discards atomic constraints related variable input set constraints adds atomic constraints guarantees expression conditionals security level input set guarantees conditional guard agree abstract semantics computes join dependences conditional branches projecting atomic constraints related notation guarantee conditional guard atomic constraints related variables possibly modified discarded intuitively guarantees conditional guard variable branches guarantees conditional command otherwise guaranteed conditional hold conditional variables modified dep dep dependence abstract semantics comparison previous analyses dependence analysis similar logic amtoft banerjee well flowsensitive type system hunt sands relationship sets dep dependence constraints type environments varp hunt sands formalised abstraction lif else let let let mod else otherwise dep varp varp dep varp fact isomorphism way interpret dependences indeed holds also corollary appendix observation suggests reformulating sets dep dependence constraints contain elements minimal level refrain simplicity presentation dependence analysis least precise type system hunt sands state result denote bottom element lattice also assume modified variables precise enough simulate effect program counter used type system mod subset variables targets assignments theorem dep varp holds lwhile lfpv lif else varp theorem dependence semantics sound lcm denote lifting partial order derive abstract semantics directly approximating relational hypercollecting semantics lcm dependence galois connection derivation structural induction commands leverages mathematical properties galois connections start specification best abstract transformer lcm dep dep successively approximate finally obtain definition dependence abstract semantics form command derivation proof obtained definition abstract semantics correct construction let showcase simplest derivation sequence commands order illustrate process hby definition hypercollecting semanticsi hby extensive hby induction hypothesis lattice cardinality constraints card card program lattice say valid set constraints iff varp let card set valid sets constraints complete lattice iff max lcm htake last approximation cardinality abstraction dependence analysis concerned whether variety conveyed refine analysis deriving cardinality abstraction enumerates variety denote atomic cardinality constraint varp read agreement security level leads variety values variable alternatively leverage galois connections give analysis approximation cardinality analysis work lemmas introduced section rest section fixed together typing context varp valid constraint set essentially function essentially pointwise order functions ensure antisymmetric cardinality abstraction relies abstraction introduced section order approximate variable cardinality crdtr cardinality abstraction crdtr crdtr trc card varp trc card crdtr card trc crdtr theorem cardinality abstract semantics sound card trc crdtr cardinality abstraction enables derive approximation lem lem approximation lem card called expression enumerates conveyed expression assuming set card cardinality constraints holds note infinite cardinal absorbing lem card expressions lnm lxm cmp min lem sound lemma lem lem derive cardinality abstract semantics approximating relational hypercollecting semantics section uses definitions follow cardinality abstract semantics lskipm abstract semantics conditionals also similar dependences conditional guard convey initially traces follow execution path join operator defined max cardinality conditional branches conditional otherwise conditional branches variables may modified conditional soundly approximate conditional lcm card card varp lem lif else let let let mod else lbm add otherwise lwhile lfpv lif else varp add abstract semantics assignments similar spirit one dependences discard atomic constraints related add new ones computing expression lcm lcm lattice card complete although finite may define widening operator card card card ensure convergence analysis cousot cousot nielson cortesi zanioli sec else occurrence widening depends iteration strategy employed static analyser widening accelerates forces convergence fixpoint computations simplest setting analyser passes arguments widening operator old set cardinality well new set computed atomic cardinality constraint widening operator compares old cardinality new cardinality cardinality still strictly increasing widening forces convergence setting cardinality decreasing widening operator sets maximum cardinality order force convergence ensure sequence computed cardinalities stationary leakage far showed one derive static analyses abstract representations interpreted approximating hypercollecting semantics let recall security requirement introduced section order illustrate analyses may prove program satisfies hyperproperty step methodology section see also equation consider program characterised set trc traces initrc prove satisfies hyperproperty use cardinality analysis prove variable indeed approximates lxm thus inferred program guaranteed satisfy hyperproperty since approximates assumption lxm hence hyperproperty hyperproperty clarkson schneider requires exhibiting traces order prove program satisfy example noninterference security level corresponds hyperproperty hyperproperty program reduced safety property product program barthe terauchi aiken darvas clarkson schneider various quantitative information flow properties example bounding problem cardinality analysis targets namely leakage hyperproperty yasuoka terauchi sec instead bounding problem hypersafety clarkson schneider cardinalities dependences quantitative security metrics natural generalisations qualitative metrics noninterference cardinality abstraction natural generalisation dependence analysis instead deciding variety conveyed cardinality analysis enumerates variety words dependences abstractions cardinalities factor galois connections suitable improve precision simplicity consider two point lattice initial typing context variables low variables usual low may flow high consider following program lemma composition two galois connections listing leaking bit secret crdval val crdval lqone cardinality abstraction determines values execution program listing initially traces fixed low inputs one value branch one value else branch cardinalities get summed conditional since conditional guard may evaluate different values thus cardinality abstraction proves example program satisfies hyperproperty otherwise otherwise stronger trace properties another way proving hyperproperty proving stronger trace property program proven satisfy trace property trc proving stronger hyperproperty trc sense program satisfies hyperproperty instance proving program output variable ranges interval integer values whose size prove program satisfies however approximating hyperproperty trace property may coarse programs illustrate interval analysis cousot cousot example program listing interval analysis loses much precision initial state program since maps low input variables conditional determines belongs interval coarse overapproximation also polyhedron cousot halbwachs capture disjunction needed example program abstract domains many existing ones suitable task inferring cardinalities dependences convex using basis extract counting information delivers leakage coarse one especially presence low inputs disjunction two polyhedra powerset domains disjunctive postconditions partitioning bourdoncle precise cardinality analysis example however disjunctions tractable general soon one fixes maximum number disjunctive elements quantitative information flow analysis mardziel defines widening operator guarantee convergence one loses relative precision wrt classical dependence analyses amtoft banerjee hunt sands cardinality analysis guarantees corollary future work investigate relying cardinality analysis strategy guiding trace partitioning rival mauborgne combining analyses existing domains also deliver better precision consider following program lemma composition two galois connections crdtr card dep trc crdtr lqonecc use lemmas abstract cardinality abstract semantics derive correct construction dependence analysis section derivation found appendix proves lemma theorem stated earlier corollary theorem also proves precision cardinality analysis relative amtoft banerjee logic amtoft banerjee well hunt sands type system hunt sands corollary leakage programs card varp lcm holds varp lxm cardinality analysis determines leakage programs type system hunt sands mean final typing environment computed type system allows attackers security clearance observe variable varp best knowledge cardinality abstraction first analysis quantitative information flow provides formal precision guarantee wrt traditional analyses qualitative information flow advantage makes cardinality analysis appealing even interested proving qualitative security policy since cardinality abstraction provides quantitative information may assist making better informed decisions declassification necessary nonetheless need experimentation compare quantitative analyses section towards precision section introduces examples evaluate precision analyses shows existing analyses leveraged secret else secret else listing leaking cardinal abstraction determines variable leaks two possible values fixed low inputs two possible values whereas one possible value relational abstract domains polyhedra cousot halbwachs octogons support expressions therefore unable compute precise bound leakage variable consider analysis disjunction polyhedra linearisation intervals linearisation expressions compute following constraints variable linearisation happens right side expressions constraint linearisation happens left side expressions two combinations constraints possible none deduce variable values underlying domain intervals lacks required precision linearisation intervals cardinalities delivers better precision also improve dependence abstraction therefore deduce min min comparison operators use lgrdb comparison operators also use lgrdb new definitions update abstract semantics conditionals loops dependences cardinalities leverage transfer functions improved dependences abstract semantics dep dep lif else let lgrdb let let mod else otherwise cardinality abstraction determines initially memories lead variety pointer conditional whereas variety assuming aliasing analysis determines may point cardinality analysis determines variable variety initially memories listing leaking bit secret secret else min min scaling richer languages rely existing abstract domains support richer language constructs pointers aliasing consider following variation listing improving precision improve precision cardinality abstraction augment existing abstract domains one shortcoming cardinality analysis fact relational assuming attackers security clearance observe variables execution program listing cardinality abstraction leads compute leakage two bits four different possible values instead possible values initially memories relying relational domain linearisation cardinalities captures required constraints compute leakage one bit constraints interpreted initially memories result equal one fixed integer times values leave extensions cardinality analysis abstraction dependence future work following focus one particular improvement previous analyses order gain precision uncovered case deriving analyses relying calculational framework abstract interpretation indeed notice following holds lwhile lfpv lif else improved cardinality abs semantics lcm card card lif else let lgrdb let let mod else lbm add otherwise lwhile lfpv lif else illustrate benefits improvement consider following example secret secret secret secret listing improved precision cardinality analysis determines initially memories result infinity values grows widened contrast cardinalities also determine variables secret value assuming lequivalent memories reduction concerns variable secret loop specifically similarly improved dependence analysis also determines variables secret low sound precision gains noninterference askarov discusses guarantees provided security requirement remarkably overlooked many previous analyses fact simple improvement makes dependence analysis strictly precise amtoft banerjee hunt sands analyses incomparable recent dependence analysis combination intervals consider following example inspired secret else listing example program analysis determines low whereas cardinality abstraction determines memories result values variable track actual values variables combine cardinality interval analysis precise cases reduced product cousot cousot granger cortesi assume set stint interval environments provided int assume also usual partial order denote int int galois connection enabling derivation interval analysis approximation standard collecting semantics defined trc lift galois connection trc obtain galois connection compositing obtain int int hpp int trc stint trc granger reduced product granger cardinality abstraction interval analysis may defined pair functions toint card stint stint tocard card stint card verifying following conditions soundness toint tocard reduction toint tocard int let denote size function returns size interval one granger reduced product defined tocard tocard toint toint card stint card min size card stint card enhanced reduced product cardinality analysis determines program listing memories result one possible value variable dependence analysis improved similarly reduction function defined follows todep dep stint dep todep size extended reduced product intervals dependence analysis also able determine variable low program listing skip else listing leakage variable reduced products final example let consider listing inspired besson program annotate result improved cardinality abstraction best knowledge existing automated static analysis determines variable low end program also prior monitor one recently presented besson accepts executions program assuming attackers clearance observe variable initially memories cardinality abstraction determines variables two values result precise precise challenge let see required gain precision determine variables possible value low tackle challenge need consider cardinality combined interval analysis simple relational domain tracking equalities equality exit loop reduced singleton interval conditional still deduce different values thanks cardinality abstraction using intervals deduce variable one value singleton interval finally last assignment cardinalities abstraction determines variable one possible value similarly combination analyses put use let dependence analysis reach desired precision related work although noninterference important applications many security requirements strong one motivation research quantitative information flow analysis addition number works investigate weakenings noninterference downgrading policies conditioned events data values askarov sabelfeld banerjee sabelfeld sands mastroeni banerjee assaf chapter proposes take guarantees provided noninterference askarov explicit definition security relative secrecy requirement inspired volpano smith propose preventing programs leaking secrets polynomial time giacobazzi mastroeni introduce abstract noninterference generalizes noninterference means abstract interpretations specify example limits attacker power extent partial releases declassification survey mastroeni generalizes notion highlights among things applicability range underlying semantics galois connections work level trace sets sets sets abstract noninterference retains explicit formulation volpano sabelfeld myers two related initial states two executions lead related final states relations defined terms abstract interpretations individual mastroeni banerjee show infer indistinguishability attackers find best abstract noninterference policy holds inference algorithm iteratively refines relation using abstract domain completion cousot cousot structures occur work abstraction nondeterministic programs works one level sets powerdomains nondeterminacy properties considered trace properties schmidt hunt sands develop binding time analysis strictness analysis hunt based partial equivalence relations concretisations sets equivalence classes cousot cousot point analysis could achieved collecting semantics defined simply direct image best knowledge explored literature except unpublished work paper builds assaf assaf clarkson finkbeiner extend temporal logic means quantify multiple traces order express hyperproperties provide model checking algorithms finite space systems agrawal bonakdarpour introduce technique runtime verification properties dependences analysis derive similar information flow logic amtoft banerjee equivalent type system hunt sands amtoft banerjee use domain trc basis relational logic validate forward analysis effect interpretation independences galois connection sets sets analysis formulated proved correct abstract interpretation deal dynamically allocated state amtoft augment relational assertions information flow logic region assertions computed abstract interpretation used express agreement relations two executions approximate modifiable locations approach generalized banerjee relational hoare logic programs encompasses information flow properties conditional downgrading banerjee give backwards analysis infers dependencies proved strictly precise hunt sands amtoft banerjee achieved product construction facilitates inferring relations variables executions follow different control paths correctness analysis proved way relational hoare logic variations proposed analyses section rivals terms incomparable dependence analysis relies approximation modifiable variables soundly track implicit flows due control flow instead labelling program counter variable account implicit flows sabelfeld myers zanioli cortesi also derive similar analysis syntactic galois syntactic assignment abstracted propositional formula denoting information flow variables variable soundness analysis wrt semantic property noninterference requires justification though remarkable concretisation propositional formula yields roughly speaking set program texts zanotti also provides abstract interpretation account type system volpano enforcing noninterference guaranteeing stronger safety property namely sensitive locations influence public locations boudol explicitly formulate noninterference abstract interpretation namely merge twin computations makes explicit aspect need analysis relate aligned intermediate states analysis like many others based reducing problem safety property product programs sousa dillig implement algorithm automates reasoning hoare logic implicitly constructing product programs performance compares favorably explicit construction product programs program dependency graphs another approach dependency shown correct noninterference wasserrab using slicing simulation argument denning chap proposes first quantitative measure program leakage terms shannon entropy shannon quantitative metrics emerge literature braun clarkson smith dwork smith alvim quantitative security metrics model different scenarios suitable different policies existing static analyses quantitative information flow leverage existing model checking tools abstract domains safety prove program satisfies quantitative security requirement proving stronger safety property contrast cardinal abstraction proves hyperproperty inferring stronger hyperproperty satisfied analysed program key target quantitative information flow mutlilevel security lattices beyond lattice backes synthesize equivalence classes induced outputs low equivalent memories relying software model checkers order bound various quantitative metrics heusser malacaria also rely similar technique quantify information flow database queries rybalchenko note exact computation characteristics prohibitively hard propose rely analyses among randomisation techniques abstract interpretation ones also propose rely product program model scenario attackers may refine knowledge influencing low inputs klebanov relies similar techniques handle programs low inputs uses polyhedra synthesize linear constraints cousot halbwachs variables mardziel decide whether answering query sensitive data augments attackers knowledge beyond certain threshold using probabilistic polyhedra conclusion galois semantic characterisations program analyses provide new perspectives insights lead improved techniques extended framework fully encompass hyperproperties remarkable form hypercollecting semantics enables calculational derivation analyses new foundation raises questions numerous list one promising direction combine dependence cardinality analysis existing abstract domains advanced symbolic methods partitioning handjieva tzolovski rival mauborgne static analysis secure information flow yet catch recent advances dynamic information flow monitoring besson bello hedin assaf naumann besson discussed section existing static analyses may use statically secure information flow seems likely hypercollecting semantics also use dynamic analyses acknowledgments thanks anindya banerjee anonymous reviewers thoughtful comments helpful 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collecting semantics hypercollecting semantics atomic dependence agreement security level leads agreement set atomic dependency constraints atomic cardinality agreement security level leads values valid set atomic cardinality constraints dep card appendix background collecting semantics galois connections lemma abstraction let elt function sets let elt elt elt proof let elt elt elt appendix domains galois connections hyperproperties lemma let set define form galois connection hpp proof special case supremus abstraction cousot defined lemma indeed instantiate supremus abstraction taking hpp thus obtain galois connection hpp notice powerset set provided set inclusion partial order complete lattice required supremus abstraction lemma supremus abstraction let elt function set codomain forming complete lattice let elt elt elt elt proof notice assumption lattice complete guarantees set elt supremum let proof goes definitions elt elt elt appendix hypercollecting semantics proving main result section theorem first prove lemma proofs lemma theorem structural induction cases follow definitions important cases loops proof technique classical one using denotational semantics order prove equality two denotations characterised fixpoint suffices introduce two sequences converge towards fixpoint characterisations prove equality sequences ensures limits denotations characterised fixpoint equal let prove lemma lemma used later proof case loops theorem lemma commands sets traces trc standard collecting semantics section expressed direct image denotational semantics jckt trc proof proof proceeds structural induction commands important case case loops case skip jskipkt case ekt case hby induction trc hby induction trc trc case else else grdb hby induction hypothesis trc grdb trc jif else trc case let first prove following intermediate result trc grd jwhile ckt trc indeed let sequence xtn defined defined xtn trc jckt jbkt otherwise notice sequence converges equal evaluation loop state jwhile ckt definition denotational semantics loops thus sequence xtn converges jwhile ckt trc let also sequences ynt gnt defined ynt grdb notice trc sequence gnt converges grd written otherwise lfpt grd also means sequence converges grd grd thus suffices prove trc xtn ynt proof proceeds induction let trc xtn ynt trc jckt trc grdb trc grdb grdb hby induction hypothesisi grdb hby definition grd grd grd grd hbecause gnt grd grd grdb let prove grd else indeed let sequence fnt defined else therefore induction holds let fnt gnt grdb hsince gnt grdb else hby induction hypothesisi else concludes induction thus passing limit sequences obtain desired result finally conclude else intermediate result grd intermediate result trc conclude proof structural induction cases theorem trc proof prove theorem corollary general result trc lcmt proof proceeds structural induction commands important case one loops ones follow definition case skip case lskipmt case emt hby structural induction monotonicity hypercollecting semantics lcmi hby structural induction case else lif else else else case let xtn sequence defined xtn trc jckt jbkt otherwise notice limit sequence xtn trc ordinary collecting semantics loop proved lemma thus sequence xtn converges let also ytn gtn sequences defined lif else skipmgtn ytn mgtn hypercollecting semantics loop lwhile cmt notice limit thus suffices prove sequences xtn ytn verify following result trc passing limit inequality leads required result trc lwhile cmt prove following precise characterisation sequences xtn ytn implies ytn trc ytn remaining proof proceeds induction case trc hsince definition let ytn lif else lif else lif else lif else skipm lif else skipm lif else skipm lif else skipm lif else skipm lif else skipm lif else skipm lif else skipm else hthe set else set traces exiting loop body less iterations equal definition concludes induction conclude proof structural induction cases appendix dependences dep lemma yields galois connection trc deptr proof lattice dep finite therefore complete thus galois connection since instance supremus abstraction presented lemma reasoning applies val agree proofs lemma theorem deferred appendix explain lemmas derive dependence abstract semantics approximation cardinality semantics appendix cardinality abstraction crdtr card lemma yields galois connection trc crdtr proof lattice card complete since subsets card infimum supremum wrt partial order notably closed interval complete wrt partial order thus instance supremus abstraction lemma lemma lem sound lem lem proof derivation proof structural induction expressions case start left side derive definition right side interesting case binary arithmetic operations case integer literal let card lnm lnm crdtr hnb precision loss simplicity presentation bottomi max crdval lnm lnm defined use indicate case variable let card lidm preserves joinsi max reductive case let card lidm hby induction hypothesisi case cmp derivation similar case difference booleans evaluate different values cmp min cmp case conclusion conclude structural induction expressions cases theorem cardinality abstract semantics sound lcm lcm proof derivation proof structural induction commands interesting case conditionals case skip let card lskipm reductive lskipm case extensive monotonei hby induction hypothesisi case first proceed towards intermediate derivation lid crdtr max consider two cases variables modified assignment variable case notice varp thus max max preserves joinsi hby definition lxmi lxm lxm hby soundness lxm lemma lxm case max max lem hby soundness lxm lemma lem final derivation lid hrecall intermediate derivation case max hby cases lem hnb set constraints remains valid owing exclusion lefti lem lid case else intermediate derivation lif else crdtr else else else max else case lbm let assume lbm let varp lif else exists else since lcm lifting set sets semantics loops general lcmt existence also rely fact merely convenient shortcut avoid lengthy details possible use fact perform derivation let lif else else since lbm lbm traces evaluate exclusively sets partitioned sets evaluating evaluating therefore exists thus else max monotonei max lxm lcj lxm monotone extensive max lxm lcj hby induction hypothesisi max lxm lcj hby soundness abstract variety lemma max lxm lcj max lcj case lbm mod else let assume lbm let varp let lif else else notice first mod else monotonei case lbm mod else let assume lbm let varp lxm lxm let lif else else else grdb grdb hby monotonicity theorem lxm lxm lxm lxm first approximation simply use lgrdb refine section lxm lxm final derivation lif else hby intermediate derivation case max else add else lbm otherwise varp add com com mod com case lwhile lgrd lfp lif else skipm monotone extensivei lif else skipm hby assuming lgrdb soundi lif else skipm hby fixpoint transfer theoremi lfpv lif else hprecision loss simplicity first approximation idi lfpv lif else lwhile case conclusion conclude structural induction commands cases appendix appendix dependencies reloaded soundness proof dependences semantics noted text derive dependency analysis calculation specification derivation looks similar one appendix cardinality abstraction choose different way proving soundness dependency analysis formulate abstraction cardinality abstraction another illustration benefit gained working hyperproperties entirely within framework abstract interpretation proof soundness also implies cardinality abstraction least precise type system hunt sands hunt sands logic amtoft banerjee amtoft banerjee corollary theorem lemma composition two galois connections crdval lqone val otherwise otherwise proof notice agree crdval also agree crdval max crdval otherwise val agree val crdval otherwise notice lqone thus obtain iff well crdval lqone val lemma composition two galois connections crdtr lqonecc card dep trc proof first deptr hby decomposition case hwith crdtr preserves unionsi crdtr also deptr crdtr crdtr crdtr therefore crdtr lqonecc card dep trc lemma sound lem proof derivation agreements abstraction cardinalities security level lnm henceforth derive case let dep lnm lem abstraction cardinalities lem derivation goes structural induction expressions lnm case let dep lidm case let dep case cmp case similar case case conclusion conclude structural induction expressions theorem dependence semantics sound proof recall lcm crdtr lqonecc dep card trc since lcm lcm lcm continue derivation dependences abstract semantics abstraction lcm make explicit derivations assignments conditionals cases similar derivation cardinalities abstract semantics case lid lem case else lif else lbm otherwise let let let mod else otherwise conclude structural induction commands appendix let let let mod else add precision proof lemma trc proof assume thus therefore holds corollary trc lemt lemt proof direct result lemma definition lem corollary trc proof let assume corollary lidmt lidmt monotonicity lidmt lidmt thus lidmt also lidmt thus corollary dep proof note thus therefore monotony also let deptr definition thus also corollary also means deptr finally concludes proof case henceforth assume dep well formed meaning dep conjecture proven dependence analysis derived given well formed initial set dependence constraints analysis always yields well formed set dependence constraints simplicity use corollary argue still augment set dependence constraints ensure well formed adding appropriate atomic constraints alternative approach would reduce set dependence constraints change slightly abstract semantics order leverage corollary guarantee precision refrain simplicity consider constructive version hunt sands flow sensitive type system proposed hunt sands lemma dep varp holds proof proof proceeds structural induction expressions case definition case definition type system thus hsince assumed case definition type system thus induction assuming thus since well formed holds therefore also case cmp case similar case conclude structural induction cases let denote bottom element lattice theorem dep varp holds proof proof goes structural induction commands conditional case explicitly assumes modified variables analysis precise enough enable simulation program counter achieved collecting variable names language case skip case stems premice case assume case also case otherwise lemma since thus finally cases case case proceeds induction remarking type system types command case else assume let also let mod mod intuitively program counter simulated modified variables analysis precise enough language achieved simply collecting variable names let dep assuming else lbm otherwise induction induction assume prove since therefore implies thus since atomic constraints related variables explicitly written discarded likewise explicitly written thus meaning assume amd prove since explicitly written one branches least also meaning notice well formed thus exists since also using lemma well formed thus thus case assume output type environment defined lfp written differently given lfp else skip let sequence defined let else skip also let sequence defined lif else prove induction case case holds assumption case assume prove let else skip assumption thus using proof case lif else therefore lif else therefore proves least fixpoints equal finally conclude cases structural induction commands
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dynamic topology adaptation based adaptive link selection algorithms distributed estimation oct songcen rodrigo lamare senior member ieee vincent poor fellow ieee paper presents adaptive link selection algorithms distributed estimation considers application wireless sensor networks smart grids particular exhaustive lms least squares rls link selection algorithms link selection algorithms exploit topology networks links considered proposed link selection algorithms analyzed terms stability tracking performance computational complexity comparison existing centralized distributed estimation strategies key features proposed algorithms accurate estimates faster convergence speed obtained network equipped ability link selection circumvent link failures improve estimation performance performance proposed algorithms distributed estimation illustrated via simulations applications wireless sensor networks smart grids index link selection distributed estimation wireless sensor networks smart grids ntroduction istributed signal processing algorithms become key approach statistical inference wireless networks applications wireless sensor networks smart grids well known distributed processing techniques deal extraction information data collected nodes distributed geographic area context specific node set neighbor nodes collect local information transmit estimates specific node specific node combines collected information together local estimate generate improved estimate prior related work several works literature proposed strategies distributed processing include incremental diffusion strategies incremental strategy processing follows hamiltonian cycle information flows nodes one direction means node passes information adjacent node uniform direction however order determine cyclic path covers nodes method needs solve problem addition nodes fails communications research group department electronics university york york lamare cetuc brazil department electronics university york poor department electrical engineering princeton university princeton usa poor part work presented ieee international conference acoustics speech signal processing vancouver canada ieee international workshop computational advances multisensor adaptive processing saint martin edics data communication cycle interrupted distributed processing breaks distributed diffusion strategies neighbors node fixed combining coefficients calculated network topology deployed starts operation one disadvantage approach estimation procedure may affected poorly performing links specifically fixed neighbors calculated combining coefficients may provide optimized estimation performance specified node links severely affected noise fading moreover number neighbor nodes large node requires large bandwidth transmit power prior work topology design adjustment techniques includes studies dynamic sense track changes network mitigate effects poor links contributions paper proposes studies adaptive link selection algorithms distributed estimation problems specifically develop adaptive link selection algorithms exploit knowledge poor links selecting subset data neighbor nodes first approach consists exhaustive lms least squares rls link selection algorithms whereas second technique based link selection algorithms approaches distributed processing divided two steps first step called adaptation step node employs lms rls perform adaptation local information following adaptation step node combine collected estimates neighbors local estimate proposed adaptive link selection algorithms proposed algorithms result improved estimation performance terms error mse associated estimates contrast previously reported techniques key feature proposed algorithms combination step involves subset data associated best performing links algorithms consider possible combinations node neighbors choose combination associated smallest mse value algorithms incorporate reweighted zero attraction rza strategy adaptive link selection algorithms rza approach often employed applications dealing sparse systems way shrinks small values parameter vector zero results better convergence performance unlike prior work aware algorithms proposed algorithms exploit possible sparsity mse values associated links different way contrast existing methods shrink signal samples zero shrink zero links poor performance high mse values using algorithms data associated unsatisfactory performance discarded means effective network topology used estimation procedure change well although physical topology changed proposed algorithms choice data coming neighbor nodes node dynamic leads change combination weights results improved performance also remark topology could altered aid proposed algorithms feedback channel could inform nodes whether switched proposed algorithms considered wireless sensor networks also tool distributed state estimation could used smart grids summary main contributions paper present adaptive link selection algorithms distributed estimation able achieve significantly better performance existing algorithms devise distributed lms rls algorithms link selection capabilities perform distributed estimation analyze mse convergence tracking performance proposed algorithms computational complexities derive analytical formulas predict mse performance simulation study proposed existing distributed estimation algorithms conducted along applications wireless sensor networks smart grids paper organized follows section describes system model problem statement section iii proposed link selection algorithms introduced analyze proposed algorithms terms stability tracking performance computational complexity section numerical simulation results provided section finally conclude paper section notation use boldface upper case letters denote matrices boldface lower case letters denote vectors use denote transpose inverse operators respectively conjugate transposition complex conjugate ystem odel roblem tatement fig network topology nodes consider set nodes limited processing capabilities distributed given geographical area depicted fig nodes connected form network assumed partially connected nodes exchange information neighbors determined connectivity topology call network property partially connected network whereas fully connected network means data broadcast node captured nodes network one hop think network wireless network analysis also applies wired networks power grids work order perform link selection strategies assume node least two neighbors aim network estimate unknown parameter vector length every time instant node takes scalar measurement according random regression input signal vector denotes gaussian noise node zero mean variance linear model able capture approximate well many relations estimation purposes assume compute estimate distributed fashion need node minimize mse cost function denotes expectation estimated vector generated node time instant equation also definition mse solve problem diffusion strategies proposed strategies estimate node generated fixed combination strategy given ckl denotes set neighbors node including node ckl combining coefficient local estimate generated node local information many ways calculate combining coefficient ckl include hastings metropolis laplacian nearest neighbor rules work due simplicity good performance adopt metropolis rule given max linked ckl ckl denotes cardinality set coefficients ckl satisfy ckl combination strategy mentioned choice neighbors node fixed results problems limitations namely nodes may face high levels noise interference may lead inaccurate estimates number neighbors node high large communication bandwidth high transmit power required nodes may shut collapse due network problems result local estimates neighbors may affected circumstances performance degradation likely occur network discard contribution poorly performing links associated data estimation procedure next section proposed adaptive link selection algorithms presented equip network ability improve estimation procedure proposed scheme node able dynamically select data coming neighbors order optimize performance distributed estimation techniques table lgorithm iii roposed daptive ink election lgorithms section present proposed adaptive link selection algorithms goal proposed algorithms optimize distributed estimation improve performance network dynamically changing topology algorithmic strategies give nodes ability choose neighbors based mse performance develop two categories adaptive link selection algorithms first one based exhaustive search second based relaxation details illustrated following subsections exhaustive link selection proposed algorithms employ exhaustive search select links yield best performance terms mse first describe define adaptation step two strategies lms algorithm employ adaptation strategy given step size node algorithm employ following steps adaptation forgetting factor let following adaptation step introduce combination step algorithms based exhaustive search strategy first introduce tentative set using combinatorial approach described set nonempty set elements tentative set defined write cost function node initialize time instant node end node find possible sets ckl arg min ckl end end ckl local estimator calculated depending algorithm different choices set combining coefficients ckl ensure condition satisfied introduce error pattern node defined ckl node strategy finds best set must solve following optimization problem arg min steps completed combination step performed described ckl stage main steps algorithms completed proposed minimizes error algorithms find set pattern use set nodes obtain algorithms briefly summarized follows step node performs adaptation local information based lms rls algorithm step node finds best set satisfies step node combines information obtained best set neighbors details proposed algorithms shown tables algorithms implemented networks large number small sensors computational complexity cost may become high algorithm requires exhaustive search needs computations examine possible sets time instant table lgorithm initialize small positive constant time instant node end node find possible sets ofp ckl end end arg min ckl link selection algorithms previously outlined need examine possible sets find solution time instant might result high computational complexity large networks operating scenarios solve combinatorial problem reduced complexity propose based algorithms simple standard diffusion lms rls algorithms suitable adaptive implementations scenarios parameters estimated slowly strategy reweighted strategy rza reported sparsity aware techniques approaches usually employed applications dealing sparse systems scenarios shrink small values parameter vector zero results better convergence rate performance unlike existing methods shrink signal samples zero proposed algorithms shrink zero links poor performance high mse values detail novelty proposed inspired link selection algorithms illustrate processing mse value mse value sparsity aware technique nodes sparsity aware technique mse value nodes mse value algorithms nodes sills sirls algorithms fig fig following show proposed algorithms employed realize link selection strategy automatically adaptation step follow procedure algorithms algorithms respectively reformulate combination step first introduce sum penalty combination step different penalty terms considered task adopted heuristic approach known reweighted attracting strategy combination step strategy shown excellent performance simple implement regularization function penalty defined log nodes sparsity aware signal processing strategies fig shows standard type processing see processed aware algorithm nodes small mse values shrunk zero contrast proposed algorithms keep nodes lower mse values shrink nodes large mse values zero illustrated error pattern ekl stands neighbor node node including node defined ekl shrinkage magnitude introduce vector matrix quantities required describe combination step first define vector contains combining coefficients neighbor node including node described ckl define matrix includes estimated vectors generated adaptation step lms neighbor node including node given note adaptation steps identical respectively error vector contains error values calculated neighbor node including node expressed ekl use hat distinguish defined original error devise approach modified vector following way element largest absolute value employed element smallest absolute value set process ensure node smallest error pattern reward combining coefficient remaining entries set zero point combination step defined stand jth element parameter used control algorithm shrinkage intensity calculate partial derivative log ekl sign ekl sign ensure replace denominator parameter stands minimum absolute value ekl rewritten table iii lgorithms initialize small positive constant time instant node adaptation step computing exactly algorithms respectively end node ekl ckl ekl find maximum minimum absolute terms modified max min min sign sign stage mse cost function governs adaptation step combination step employs combining coefficients derivative penalty performs shrinkage selects set estimates neighbor nodes best performance function sign defined sign inserting proposed combination step given sign note condition sign enforced sign means corresponding node discarded combination step following time instant node still largest mse changes combining coefficients set nodes guarantee stability parameter assumed sufficiently small penalty takes effect element magnitude comparable moreover little shrinkage exerted element whose rls algorithms perform link selection adjustment combining coefficients point emphasized process satisfies condition penalty reward amounts nodes maximum minimum error pattern respectively computing multiplications therefore additional complexity introduced described jth element used calculation neighbor node largest mse value modifications ekl value positive number lead term sign positive means combining coefficient min end end node shrunk weight node build shrunk words node encounters high noise interference levels corresponding mse value might large result need reduce contribution group nodes contrast neighbor node smallest mse ekl value negative number term sign negative result weight node associated smallest mse build increased remaining neighbor nodes entry ekl zero means term sign zero change weights build main steps proposed algorithms listed follows step node carries adaptation local information based lms rls algorithm step node calculates error pattern step node modifies error vector step node combines information obtained selected neighbors algorithms detailed table iii algorithms design different combination steps employ adaptation procedure means proposed algorithms ability equip wireless networks operating lms rls algorithms includes example diffusion conjugate gradient strategy nalysis proposed algorithms section statistical analysis proposed algorithms developed including stability analysis mse analysis tracking performance addition computational complexity proposed algorithms also detailed start analysis make assumptions common literature assumption vector input signal vector statistically independent vector node defined denotes optimum wiener solution actual parameter vector estimated estimate produced proposed algorithms time instant assumption input signal vector drawn stochastic process ergodic autocorrelation function assumption iii vector represents stationary sequence independent vectors positive definite autocorrelation matrix independent assumption independence regressor input signals spatially temporally independent stability analysis general ensure network converge global network performance information propagated across network work shows central performance node able reach nodes one multiple hops section discuss stability analysis proposed algorithms first substitute obtain ckl also define matrix combining coefficients ckl correspond entries matrix matrix kronecker structure denotes kronecker product inserting global version written estimation error produced wiener filter node described diag define take expectation arrive proceed let define introduce lemma lemma let denote arbitrary mab trices real entries columns ckl adding one matrix stable choice stable proof assume stable true every ckl square matrix every exists submultib plicative matrix norm satisfies ckl submultiplicative matrix norm holds spectral radius since stable choose subject ckl obtain ckl since entries columns add one bounded unity means ckl frobenius norm bounded well equivalence norms norm particular result ckl lim converges zero matrix large employing assumption start establishes stability lms algorithm define global vectors matrices square matrix stable satisfies view lemma need matrix stable result requires stable diag holds following condition satisfied diag vector largest eigenvalue correlation matrix difference lms algorithms strategy calculate matrix lemma indicates choice needs stable result convergence condition given convergence conditions proposed algorithms adapt according network connectivity choosing group nodes best available performance construct estimates comparing results existing algorithms seen proposed link selection techniques change set combining coefficients indicated matrix employs chosen set assumption mse expression derived denotes trace matrix minimum error mmse node correlation matrix inputs node vector inputs measurement correlation matrix emse defined difference error time instant minimum square error write proposed adaptive link selection algorithms derive emse formulas separately based assume input signal modeled stationary process update estimate employ ckl ckl subtracting sides arrive ckl ckl mse analysis part analysis devise formulas predict excess mse emse proposed algorithms error signal node expressed ckl ckl ckl ckl ckl let introduce random variables otherwise time instant node generate data associated network covariance matrices size reflect network topology according exhaustive search strategy network covariance matrices value equal means nodes linked value means nodes linked example suppose network nodes node two neighbor nodes namely nodes possible configurations set evaluating possible sets relevant covariance matrices size time instant described table coefficients obtained according covariance matrices example three sets respectively shown table parameters ckl calculated different choices matrices coefficients ckl calculated error pattern possible calculated set smallest error selected according newly defined rewritten ckl table ovariance matrices different sets simplify analysis assume samples input signal uncorrelated variance using diagonal matrices write ckl ckq table oefficients different sets starting focus term determined network topology subset term ckl calculated metropolis rule assume ckl statistically independent terms upon convergence parameters ckl vary steady state choice subset node fixed assumptions substituting arrive ckl ckq due structure equations approximations quantities involved decouple kln kkn ckl ckq ckl ckq ckl ckq ckl ckq kkn nth element main diagonal assumption ckl statistically independent terms rewrite kkn kln ckl ckq table ovariance matrix upon convergence employ relation sign ekl sign according eqs proposed algorithm converges node time instant goes infinity assume error equal noise variance node asymptotic value hkl divided three situations due rule algorithm sign node largest mse sign node smallest mse qkl remaining nodes taking obtain noticed situation time instant goes infinity assumption upon convergence choice parameters hkl neighbor node node obcovariance matrix node fixed means tained quantity hkl deterministic deterministic vary example taken expectation assume choice fixed show table last removing random variables inserting coefficients also fixed given asymptotic values kkn algorithm obtained point theoretical results deterministic mse algorithm given kkn well parameters ckl computed using metropolis combining rule result coefficients proposed algorithm start coefficients ckl deterministic taken inserting expectation ckl mse given kkn algorithm need consider possible combinations algorithm simply adjusts combining coefficients node neighbors order select neighbor nodes yield smallest mse values thus redefine combining coefficients ckl sign node time instant received estimates neighbors calculates error pattern ekl every estimate received find nodes largest smallest error error vector defined contains error pattern ekl node procedure detailed carried modifies error vector example suppose node three neighbor nodes nodes error vector form described modification error vector edited quantity hkl defined sign hkl term error pattern ekl modified error vector ckl ckl subtracting sides arrive ckl ckl ckl ckl random variables rewritten ckl kkn kkn ckl hkl ckl hkl ckl ckq ckl ckq ckl hkl ckq hkq ckl hkl ckq hkq ckl ckq since modify ckl nth elements main diagonals assumption upon convergence ckl vary steady state choice node fixed rewrite subset mse given point compare find difference replaced also result arrive kkn basis tends infinity mse approaches mmse theory proposed algorithm insert remove random variables following procedure algorithm obtain hkl hkq satisfy rule mse given kkn ckl ckq conclusion according help modified combining coefficients proposed neighbor node lowest mse contributes combination neighbor node ckl ckq highest mse contributes least therefore proposed algorithms perform better standard diffusion algorithms fixed combining coefficients tracking analysis section assess proposed algorithms environment due structure equations approximations algorithms track minimum point surface quantities involved decouple scenarios interest optimum estimate assumed vary according model kln kkn denotes random perturbation order facilitate analysis typical context tracking analysis adaptive algorithms tracking analysis employ assumption iii start ckl ckq subtracting sides obtain ckl ckl hkl ckl ckq ckl hkl ckq hkq ckl hkl ckq hkq instant adaptation method lms rls multiplications additions divisions table viii omputational complexity combination step per node per time ckl instant ckl algorithms ckl ckl ckq table vii ckl ckl computational complexity adaptation step per node per time ckl hkl kkn kkn multiplications additions divisions start similar procedure algorithm kkn using assumption iii arrive conclusion scenarios one additional term mse expression algorithms additional term value algorithms result proposed algorithms still perform better standard diffusion algorithms fixed combining coefficients according conclusion obtained previous subsection first part right side already obtained mse analysis part section second part decomposed computational complexity analysis computational cost algorithms studied assume data first analyze adaptation step mse obtained algorithms adaptation cost depends type rem cursions lms rls strategy employs kkn details shown table vii combination step analyze computational complexity table viii overall complexity algorithm summarized table recursions follow three tables total number nodes procedure algorithm obtain linked node including node number nodes chosen proposed algorithms require extra kkn computations compared existing distributed lms rls algorithms extra cost ranges small additional number operations algorithms algorithm follow significant extra cost depends esprocedure algorithm arrive algorithms kkn imulations section investigate performance proposed link selection strategies distributed estimation table omputational algorithm complexity per node per time instant multiplications additions divisions diffusion lms strategy diffusion rls mmse bound mse fig time diffusion wireless sensor networks topology nodes fig network mse curves static scenario two scenarios wireless sensor networks smart grids applications simulate proposed link selection strategies static scenarios also show analytical results mse tracking performances obtained section diffusion lms strategy diffusion rls mmse bound mse diffusion wireless sensor networks subsection compare proposed rls algorithms diffusion lms algorithm diffusion rls algorithm link strategy terms mse performance network topology illustrated fig employ nodes simulations length unknown parameter vector generated randomly input signal generated correlation coefficient white noise process variance ensure variance noise samples modeled circular gaussian noise zero mean variance step size diffusion lms algorithms diffusion rls algorithm forgetting factor set equal static scenario sparsity parameters algorithms set metropolis rule used calculate combining coefficients ckl mse mmse defined respectively results averaged independent runs fig see best performance mse convergence rate obtains gain standard diffusion rls algorithm bit worse still significantly better standard diffusion rls algorithm regarding complexity processing time simple standard diffusion rls algorithm time fig network mse curves scenario complex proposed algorithms superior standard diffusion lms algorithm scenario sparsity parameters algorithms set unknown parameter vector varies according markov vector process independent gaussian vector process variance fig shows similarly static scenario best performance obtains gain standard diffusion rls algorithm slightly worse still better standard diffusion rls algorithm proposed lms algorithms advantage standard diffusion lms algorithm scenario notice scenario large proposed algorithms still better performance compared standard techniques illustrate link selection algorithms provide figs two figures see upon convergence proposed link selection state node algorithm static scenarios nodes simulation theory nodes simulation theory mse mse mse fig time set selected links node static scenario link selection state node algorithm scenarios snr snr fig mse mse performance snr nodes simulation theory nodes simulation theory time fig link selection state node scenario algorithms converge fixed selected set links mse analytical results aim section validate analytical results obtained section first verify mse performance specifically compare analytical results results obtained simulations different snr values snr indicates signal variance noise variance ratio assess mse snr show figs rls algorithms use compute mse convergence results show analytical curves coincide obtained simulations within indicates validity analysis assessed proposed algorithms snr equal respectively nodes network parameters follow definitions used obtain network mse curves static scenario details shown top sub figure figs theoretical curves close simulation curves remain within one another tracking analysis proposed algorithms varying scenario discussed follows verify results subsection tracking analysis provide means estimating mse mse fig snr snr mse performance snr mse consider model next examples employ nodes network parameters used obtain network mse curves scenario comparison curves obtained simulations analytical formulas shown figs curves verify gap simulation analysis results within different snr values details parameters shown top sub figure figs smart grids proposed algorithms provide tool could used distributed state estimation smart grid applications order test proposed algorithms possible smart grid scenario consider ieee system number substations every nodes simulation theory nodes simulation theory measurement jacobian vector bus aim distributed estimation algorithm compute estimate minimize cost function given compare proposed algorithms algorithm single link strategy standard diffusion rls algorithm standard diffusion lms algorithm terms mse performance phase angle gap mse comparison used determine accuracy algorithms phase angle gap used compare rate convergence scenario phase angle gap refers phase angle difference actual parameter vector target estimate buses define bus system fig mse mse snr snr fig mse performance snr scenario nodes simulation theory nodes simulation theory fig buses corrupted additive white gaussian noise equal variance step size standard diffusion lms proposed lms algorithms parameter vector set vector diffusion rls rls algorithms forgetting factor set equal sparsity parameters algorithms set results averaged independent runs simulate proposed algorithms smart grids static scenario mse mse snr ieee system simulation snr diffusion rls diffusion lms strategy mmse bound fig mse performance snr scenario time instant bus takes scalar measurement according state vector entire interconnected system nonlinear measurement function bus quantity measurement error mean equal zero corresponds bus initially focus linearized state estimation problem system built per unit voltage magnitudes buses branch impedance state vector taken voltage phase angle vector buses therefore nonlinear measurement model state estimation approximated mse time fig mse performance curves smart grids fig seen best performance significantly outperforms standard diffusion lms algorithms slightly worse outperforms remaining techniques worse still better remains better diffusion rls lms algorithms single link strategy phase angle gap comparision bus phase angle gap diffusion rls diffusion lms single link strategy time fig phase angle gap comparison bus order compare convergence rate employ phase angle gap describe results choose bus first iterations example illustrate results fig fastest convergence rate lms second fastest followed standard diffusion rls standard diffusion lms algorithms algorithm single link strategy worst performance estimates obtained proposed algorithms quickly reach target means phase angle gap converge zero onclusions paper proposed algorithms distributed estimation applications wireless sensor networks smart grids compared proposed algorithms existing methods also devised analytical expressions predict mse performance tracking behavior simulation experiments conducted verify analytical results illustrate proposed algorithms significantly outperform existing strategies static varying scenarios examples wireless sensor networks smart grids vii acknowledgements authors wish thank anonymous reviewers whose comments suggestions greatly improved presentation results eferences lopes sayed incremental adaptive strategies distributed networks ieee trans signal vol aug diffusion squares adaptive networks formulation performance analysis ieee trans signal vol july chen hero sparse lms system identification proc ieee icassp taipei taiwan may xie choi kar poor fully distributed state estimation monitoring 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haardt blind adaptive constrained parameter estimation based constant modulus design cdma interference suppression ieee transactions signal processing vol june song lamare haardt wolf adaptive widely linear interference suppression based wiener filter ieee transactions signal processing vol august ruan lamare robust adaptive beamforming using mismatch estimation algorithm signal processing letters ieee vol jan chen hero regularized algorithms technical report afosr december lamare distributed conjugate gradient strategies distributed estimation sensor networks proc sensor signal processing defence london cattivelli sayed diffusion strategies distributed kalman filtering smoothing ieee trans autom control vol september sayed fundamentals adaptive filtering hoboken usa john wiley sons kailath sayed hassibi linear estimation englewood cliffs usa lamare diniz blind adaptive interference suppression based constrained algorithms dynamic bounds ieee trans signal vol march cai lamare variable 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dec binomial edge ideals cactus graphs giancarlo rinaldo abstract classify binomial edge ideals cactus bicyclic graphs introduction binomial edge ideals introduced appeared independently also let polynomial ring variables coefficients field let graph vertex set edges associate binomial fij ideal generated fij called binomial edge ideal ideal generated set indeterminates may viewed binomial edge ideal graph algebraic properties binomial edge ideals terms properties underlying graph studied authors considered property graphs recently nice results bipartite graphs blocks obtained see however classification binomial edge ideals terms underlying graphs still widely open case monomial edge ideals introduced seems rather hopeless give full classification aim paper extend results classify unmixed binomial edge ideals deviation namely difference minimum number generators height less equal invariant interesting combinatorial interpretation unmixed deviation number connected components see remark hence deviation represents minimum number edges must removed graph break cycles making forest see chapter section give classification unmixed binomial edge ideals cactus graph graph whose blocks cycles natural extension result obtained useful study binomial edge ideals deviation invariant giancarlo rinaldo section application results obtained section classify unmixed binomial edge ideals bicyclic graph case deviation preliminaries section recall concepts notation graphs simplicial complexes use article let simple graph vertex set edge set subset called clique belonging one vertex graph called cutpoint removal vertex increases number connected components connected subgraph cutpoint maximal respect property block block graph graph whose vertices blocks two vertices adjacent whenever corresponding blocks contain common cutpoint connected graph cactus blocks cycles edges set simplicial complex vertex set collection subsets imply element called face maximal face respect inclusion called facet vertex called free vertex belongs exactly one facet clique complex simplicial complex whose faces cliques hence vertex graph called free vertex belongs one clique need notation results section recall sake completeness let let let connected components induced subgraph namely complete graph vertex set set denote prime ideal radical ideal primary decomposition see corollary theorem possible confusion write simply instead moreover height see lemma denote set minimal prime ideals cutpoint graph say cutset denote set cutsets lemma lemma let connected graph unmixed binomial edge ideals cactus graphs following observation gives motivation consider block graphs context proposition let unmixed block graph tree proof theorem block block graph complete graph recall two vertices adjacent corresponding blocks contain common cutpoint moreover cutpoint lemma assertion follows observe following see also remark proposition let vertex neighbor set let graph proof free vertex fact definition one clique containing vertices adjacent adjacent hence assume free vertex definition graph free vertex implies proposition cutsets containing let let connected components graph either intersects hence edge since connected way let let connected components graph either intersects edge implies moreover particular true cutsets cutpoint graph number connected components decreases happens hence argument works direction corollary let vertex free vertex let graph giancarlo rinaldo proof let applying proposition cutsets exactly cutsets containing moreover connected components induced set vertices stated proof proposition see using notation introduced connected component mentioned proof example sometime ideal corollary natural interpretation example let vertices edges obtain known diamond graph ring related isolated vertex similar argument obtain complete graph vertices interpretation fails let edges consider graph obtained adding edge cycle expected corollary passing block complete complete one see example figure useful aim thanks following nice result see theorem state using notation theorem let graph whose blocks complete graphs following conditions equivalent unmixed tree classification cactus graphs section provide classification binomial edge ideal cactus graphs since binomial edge ideal resp unmixed resp unmixed connected component assume graph connected unless otherwise stated start following binomial edge ideals cactus graphs proposition let unmixed binomial edge ideal cycle block proof begin observing unmixed binomial edge ideals containing blocks cycles length exist assume exists cycle unmixed since block represent subgraphs since block cutpoint paths pass see theorem let edge edge observe moreover least two connected components induced remaining vertices assuming unmixed exactly connected components implies one vertices cutpoint cutpoint thanks observation easily obtain contradiction give proof sake completeness observe cutset may assume without loss generality cutpoint cutpoint argument focusing cutpoint consider obtain cutpoints let since connected components obtain contradiction proposition let unmixed binomial edge ideal cycle block satisfies following exactly two cutpoints adjacent proof proposition know graph satisfying thesis exists prove cases unmixed let let assume representation given suppose either cutpoint hence unmixed let assume cutpoints hence unmixed definition graph decomposable resp indecomposable exists resp exist decomposition free vertex recursive decomposition applied finite giancarlo rinaldo figure path number steps obtain indecomposable either free vertex decomposition unique ordering say decomposable indecomposable graphs lemma see let decomposable graph decomposition resp unmixed resp unmixed aim prove every cactus graph decomposable indecomposable graphs block graph path hence necessary first step classify indecomposable cactus graphs whose block graph path reach goal use following notation let graph path defined following sets see figure vertices edges cutpoints lemma let graph path use notation power set subset proof use induction cardinality since cutpoint claim follows let wit let assume without loss generality observe graph connected component obtained removing vertex block since hypothesis two vertices binomial edge ideals cactus graphs contains least one vertex cutpoint hence two connected components one containing vertex one containing vertex cutpoint proposition example theorem useful aim compute primary decompositions whose blocks complete graphs cycles diamond graphs see also figure proposition let graph path use notation let kml kmi following labelling vertices blocks satisfying following conditions proof sake completeness give equivalent conditions suppose cutset free vertex hence follows fact vertex diamond graph free vertex clique holds vertices cutpoints block complete graph moreover satisfies condition trivial way hence assume either consider case since cases follow similar argument since cutset graph cutpoint hence two vertices adjacent must belong reason since cutpoint since adjacent condition satisfied suppose satisfying conditions let prove cutset induction cutset lemma giancarlo rinaldo let let see figure assume since case similar conditions neither belongs graph contains connected components isolated vertex subgraph obtained removing vertex claim cutset moreover connected component containing vertex adjacent claim easily follows cutset contains following connected components isolated vertex connected components complete graph complete graph hence satisfies hypothesis proposition cutset induction hypothesis complete graph moreover condition applied hence set contains cutpoint also case graph satisfies hypothesis proposition cutset induction hypothesis path length whose edges note also case condition contains one vertices fact free vertex path one cutpoints appear moreover apply induction hypothesis subgraph obtained removing similar argument previous case obtain assertion assumption case contains following connected components isolated vertex obtained removing vertex connected components corollary let graph satisfies hypothesis proposition unmixed proof lemma sufficient show thanks lemma wit cutset note bit bij block obtained removing block obtained removing easily observe connected hence let wit binomial edge ideals cactus graphs figure cases theorem implies either focus representation holds case without loss generality let since cutpoint induces connected component condition proposition hence last summand induced isolated vertex argument holds theorem let graph satisfies hypothesis proposition proof start observing dim follows corollary formula see dim max hence sufficient prove depth using induction number blocks strategy focus block particular see figure consider vertex following exact sequence corollary binomial edge ideal graph satisfies hypothesis theorem second block less cutsets see figure moreover respectively tensor product respectively quotient rings whose definining ideals binomial edge ideals using induction obtain goal three cases study start induction case theorem case set proposition second block complete graph vertices corollary obtained removing vertex complete graph depth giancarlo rinaldo lemma applied sequence obtain assertion fact case depth let obtain quotient rings associated complete graphs isolated vertex using formula ring adding results thanks obtain depth argument depth fact edge hence theorem depth lemma assertion follows case set proposition second block diamond hence case moreover depth depth fact complete graph obtained removing vertex graph using representation similar easily obtain assertion moreover edge also case obtain assertion using representation equivalent one used let case lemma applied graphs since complete graph induction hypothesis obtain cohenmacaulay case case setting second block namely complete graph vertices hence therefore using induction hypothesis tand lemma applied corollary thanks condition proposition hence claim obtained removing vertex claim defining binomial edge ideals cactus graphs obtain representation tensor product similar using induction hypothesis obtain assertion claim follows proving condition proof case let vertices satisfies proposition let prove enough check conditions vertices satisfies condition satisfies condition respect since satisfies conditions respect end implication observing either exists block either satisfied exists cases satisfy respect cases follow similar argument point facts path defined vertex set edges case useful consider edge obtained removing vertex satisfies hypothesis proposition induction lemma applied obtain moreover lemma observe graph studied case decomposable respect vertex hence also graphs case let case second block case claim complete graph obtained removing moreover using notation introduced proposition claim follows defining following representation giancarlo rinaldo figure tree unicyclic indecomposable graphs focus last factor since ones exactly equivalent ones already studied block either complete graph diamond graph free vertex cases induction hypothesis claim follows condition prove use proposition similar arguments ones used prove lemma let indecomposable cactus graph path use notation following conditions equivalent unmixed one following cases occurs subgraphs satisfy iii proof known fact proposition admissible blocks since every vertex complete graph free vertex observe every cactus graph whose blocks decomposable single blocks suppose block graph path containing one blocks proposition block two cutpoints thus neither since hypothesis indecomposable two complete graphs adjacent end observing two cycles unmixed see also remark implication follows applying theorem example thanks lemma obtain indecomposable graph trees unicyclic graphs shown figure bicyclic properties figure underline free vertices circle around binomial edge ideals cactus graphs figure bicyclic indecomposable cactus graphs figure non unmixed cactus graphs tree theorem let cactus graph use notation following conditions equivalent unmixed tree decomposable indecomposable graphs path satisfies one equivalent conditions lemma proof known fact follows lemma proposition block graph tree let number vertices degree greater two make induction block graph path assertion follows lemma lemma let let vertex whose degree greater proposition proposition suppose vertex free vertex since cactus number blocks degree greater less induction hypothesis done suppose contradiction vertices graph decomposable adjacent blocks figure intended cactus subgraphs observe set containing vertices indicated filled dots figure cutset components hence unmixed contradiction giancarlo rinaldo figure cactus graphs example figure union indecomposable cactus graphs joined free vertices surrounded circle one containing one containing one classification bicyclic graphs section thanks classification binomial edge ideals cactus graphs classify ones bicyclic namely ideals deviation fact cactus graph cycles blocks particular bicyclic graph hence focus attention bicyclic graphs cactus case exists one block paths block edge remark set defined cutset assume ideal unmixed let length path cutpoint cutpoint fact lemma exactly two connected components induced paths hence exists another connected component subgraph assertion follows similar argument neither cutpoint section use notation defined remark call graphs bicyclic graphs lemma let bicyclic graph unmixed use notation binomial edge ideals cactus graphs path length less one path length proof suppose contradiction exists path length let distinct vertices let suppose neither cutpoint observe cutset hence two connected components subgraphs one containing vertex one containing vertex hence exists another block different contains vertex let argument exists block set cutset induces components first containing second containing third containing fourth containing contradiction similar argument left reader check cutpoint cutpoint cutset cutpoint hence contradiction focusing cutset suppose contradiction exist paths length remark assume vertex cutpoint focus cutset two connected components subgraphs one containing vertex one containing isolated vertex hence exists another block different contains vertex let argument exists block set cutset components first containing second respectively third containing respectively isolated vertices contradiction lemma let bicyclic graph cohenmacaulay respectively unmixed tree decomposable indecomposable graphs one graphs figure respectively figure proof let length paths remark lemma study following cases case since obtain graph left figure fact consider cone vertex two connected components given isolated vertex path whose edges theorem assertion giancarlo rinaldo figure bicyclic graphs case since obtain graph right figure removing vertex using similar argument one used proposition obtain candidate exactly graph figure focus cutpoint corollary obtain moreover binomial edge ideal path ring dimension two components path vertices isolated vertex also observe cone vertex vertex graph obtained attaching edge complete graph whose vertices hence dimension moreover binomial edge ideal equal previous cone removing vertex hence depth depth lemma applied following exact sequence assertion follows cases two cases unmixed add edges figure moreover already found example paper symbolic computation observed argument non bipartite one one right figure ready give main result section corollary let bicyclic graph tree decomposable indecomposable graphs one following cases occurs set unicyclic graphs figure one bicyclic cactus graphs figure one bicyclic graphs figure binomial edge ideals cactus graphs figure bicyclic unmixed non noncactus graphs acknowledgements author partially supported gnsaga indam italy references banerjee graph connectivity binomial edge ideals proc amer math vol bolognini macchia strazzanti binomial edge ideals bipartite graphs cocoateam cocoa system computations commutative algebra available http ene herzog hibi binomial edge ideals nagoya math vol harary graph theory series mathematics herzog hibi hreinsdottir kahle rauh binomial edge ideals conditional independence statements adv appl vol kiani saeedi madani unmixed binomial edge ideals comm algebra vol ohtani graphs ideals generated comm algebra vol rauf rinaldo construction binomial edge ideals comm vol rinaldo binomial edge ideals small deviation bull math soc sci math roumanie tome villarreal graphs manuscripta vol department mathematics university trento via sommarive povo trento italy
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nov institut national recherche informatique automatique building tangent adjoint codes ocean general circulation model opa automatic differentiation tool tapenade tber vidard dauvergne novembre num apport recherche building tangent adjoint odes ean general cir ulation model opa automati tool tapenade tber vidard dauvergne num projet tropi rapport abstra novembre pages ean general ulation model opa developed lodyc team paris university opa ently undergone major rewriting migrating adjoint ode needs built earlier versions adjoint opa written hand high development ost use automati tool tapenade build hani aly tangent adjoint odes odes opa validate omparison divided also identi twin experiment apply methods improve performan adjoint walther binomial ode parti ular implement griewank kpointing algorithm whi gives optimal time memory onsumption apply spe egy iterative linear solver omes impli time stepping heme opa general ulation model tapenade automati ferentiation reverse mode adjoint code che kpointing inria sophia antipolis fran inria grenoble fran recherche inria sophia antipolis constru tion des odes tangent adjoint ulation opa par automatique tapenade ulation opa est par lodyc paris nouvelle version onstitue une majeure ave les parti ulier une migration vers odes tangent adjoint qui auparavant rits main doivent nous utilisons automatique tapenade pour gent adjoint raison ave onstruire les odes nous validons les obtenues par les sur deux appli ations test luant des jumelles nous utilisons kpointing ursif mial griewank walther pour les performan ode adjoint nous utilisons une que pour solveur provenant impli ite ement temps montrent raisonnable tant nos onsommation que pour temps ution opa cir ulation tapenade tomatique mode inverse code adjoint che kpointing tangent adjoint opa introdu tion development tangent adjoint models important step addressing sensitivity analysis variational data assimilation problems eanography sensitivity analysis study model output varies hanges model inputs sensitivity information given adjoint model used dire tly gain understanding physi pro esses data assimilation one onsiders ost fun tion whi measure mis adjoint sensitivities used build gradient des ent algorithms similarly tangent model used remental algorithms linearize ontext ost fun tion around kground ontrol previous version ean general cir ulation model opa weaver developed numeri tangent adjoint hand using odes lassi hniques sin opa model dergone major update parti ularly new versions fully rewritten adjoint paper report development tangent odes opa using automati tool nade brief des ription opa model guration used work given next tion tion present prin iples ted fun tionalities tool tapenade tion interesting ulties ountered appli ation large ode tion shows experiments validate derivatives presents two illustrative appli ations using omputational aspe rather impli ations eanography outlook work given lusion ean general cir ulation model opa developed lodyc team paris university opa exible ean ulation model guration used either regional global ean opa ean model omponent nemo leus european modelling ean widely used ienti munity moreover oming major tor operational eanography mer ator ecmwf formulation based alled primitive equations temporal evolution ean velo ity urrents tber perature salinity three horizontal verti dimensions equations derived equations oupled state tion water density heat equation boussinesq hydrostati approximations let introdu following variables subs ript tial temperature velo ity tor denotes horizontal tor salinity pressure density tor invariant form primitive equations orthogonal set unit tors linked earth written follows generalized derivative tor operator time verti oordinate referen density coriolis eleration gravity eleration parametrization small ale physi momentum temperature salinity luding surfa ing terms full des ription model basi dis retization physi numeri details found paper opa used global free surfa guration guration model uses rotated grid poles north ameri asia order avoid singularity problem north pole spa resolution roughly equivalent geographi mesh meridional resolution near equator see gure verti domain spreading surfa depth meshed using levels levels top meters time step minutes time steps per day model heat freshwater momentum uxes atmosphere inria tangent adjoint opa figure orca mesh solar radiation penetrates upper layers ean zero uxes heat salt applied bottom lateral solid boundaries ondition also applied initialization model temperature salinity based levitus limatology null initial velo ity eld details spa ean physi refer page dedi ated guration ial website guration routinely used ompute eani omponent seasonal fore asting system size modules ning pro edures lines makes largest appli ation tapenade date omputational kernel whi tually pro edures http ounts tber prin iples tool tapenade tapenade tool developed tropi team inria given sour original program evaluates mathemati fun tion given sele tion input output variables tapenade produ new sour program omputes partial tives sele ted outputs respe sele ted inputs basi ally tapenade inserting additional statements opy original program like tools tapenade based fundamental observation original program whatever size run time omputes fun tion whi omposition elementary fun tions words omputed instru tion exe utes sequen elementary statements tually evaluates fun tion implemented hain rule derivative derivatives calling omponent variables therefore one apply partial ulus get obian matrix essive respe omponent essive values intermediate states memory throughout exe ution get derivatives elementary instru tion easily built must inserted program values dire tly available use pro ess yields analyti exa numeri derivatives ura pra two sorts derivatives parti ular importan ienti omputing tangent dire tional derivatives adjoint reverse derivatives parti ular tangent adjoint two sorts derivative programs required opa tapenade provides tangent derivative produ full obian times dire tion input spa equation http inria tangent adjoint opa whi heaply exe uted right left ause tor prodheaper produ also onvenient exe ution order ause uses intermediate values builds order program hand adjoint derivative produ transposed output spa resulting equation weight tor dot produ obian times gradient whi also heaply exe uted right left however uses intermediate values inverse building order regarding runtime adjoint ost small multiple original program slowdown tor less theory pra tangent whereas despite higher runs less ost adjoint ode still large heapest get gradient tangent mode would tangent independent rea adjoint reason dis ussed way obtain gradients require ost obtaining derivatives tangent ode one per dimension whereas ost adjoint mode ulty adjoint mode lies needs intermediate values reverse order end tapenade basi ally uses strategy alled rst sweep opy original program run together statements store intermediate values sta get overwritten ond sweep kward derivative statements ompute elementary derivatives using statements restore intermediate values required urs ost memory spa maximum sta size needed attained end forward sweep thus proportional length program also runtime penalty sta manipulations tapenade implements number strategies mitigate analysis program values need stored ontrol graph redu ing number however long programs opa involving unsteady simulations work alone ombines omputation ost based stati tapenade alled tber che kpointing redu maximum sta size exe utions consider pie original program ost dupli ated che kpointing illustrated gure means main forward sweep pushes value sta kward sweep rea hes pla intermediate values missing sta runs ond time time strategy pushing values sta kward sweep resume safely run twi requires enough input values stored size snapshot generally less sta size used adjoint program well hosen obviously also slows kpointing peak size sta tor two che kpoints whi divide nested ase sta peak size adjoint runtime slowdown grow little logarithm size applies kpointing pro edure default mode tapenade successive sweeps figure che kpointing applied program pie rightwards arrows represent forward sweeps thi store intermediate values sta thin otherwise leftwards arrows represent kward sweeps bla dots stores white dots retrieves small dots push pops big dots snapshots tapenade apa ity generate robust ient tangent adjoint odes demonstrated several test appli ations regarding appli ation language fortran taking handle programs written ount new programming onstru provided required important programming past years mostly handle modules stru tured data types array notation pointers dynami memory allo ation sin new opa written opa realisti test new tapenade inria tangent adjoint opa exist several tools restri ting tools whi like tapenade operate sour transformation provide tangent adjoint modes use global program analysis optimize demonstrated appli ability large industrial odes ode tion taf pioneer meteorology standard tool popular mit global cir ulation model unlike tapenade adjoint mode taf regenerates intermediate values given initial point strategy getting blurred nested joint omputation alled strategy comparison odes grow alike kpointing kpoints inserted openad essor adifor adic uses strategy periments also apply openad mit gcm tool although using operator overloading instead sour transformation popular applied adjoint mode essfully many industrial appli ations seen extension strategy intermediate values stored sta also omputation graph allows tool perform optimizations graph ost higher memory onsumption applying tapenade opa generated working tangent adjoint odes omputational kernel opa using tapenade depending nal appli ation tion tual fun tion well input output variables may hni ulties ountered essentially tion des ribes points onstru new opa uses extensively modular extend internal representation tapenade handle nesting modules pro edures ause module essentially nesting mirrored ode private ferentiated modules onstru omponents subroutines ess variables original module tber therefore module must ontain opy original module variables types pro edures model original ode must ode ontain hange tapenade annot use parts opies words ode need linked original interfa hanism way implement overloaded pro edures stati overloading whi resolved ompile time therefore extend tapenade king phase pletely solve alls interfa pro edures conversely tapenade able generate interfa pro edures general stru ture ode preserved array notation used systemati ally opa time requires many alls intrinsi fun tions split propagate derivatives fun tions used arrays elemental intrinsi tapenade must generate ode whi far trivial instan single statement opa zws sqrt abs psal generates adjoint mode abs psal mask psal elsewhere zwsb sqrt end psalb psalb elsewhere psalb psalb end without going detail adjoint model observe test needed prote ode inria tangent adjoint opa abs sqrt well test turned ontrols onstru keep runtime bene array notation temporary variables introdu automati ally store isions mask although tapenade still optimal way example opa uses pointers dynami deallocate memory allo ation alls allocate appli ation pointer analysis available tapenade nding whether variable derivative even variable essed pointer unfortunately dynami allo ation handled partly tangent mode tapenade adjoint mode general strategy memory allo ation tapenade sometimes annot produ working allo ate ode understand adjoint deallocate made hand hanges must ode make work che kpointing hidden variables opa reads writes several data les postpro essing stages also omputational kernel sour terms wind stress read intermediate time steps also modules pro edures private annot save variables whose value preserved essed outside examples although unrelated two points ommon problem reentrant alled pro edure modi internal sible outside alling ontext identi result similarly save make pro edure variable omes pro edure ond time alled pro edure reads previously opened moves read pointer omes impossible pro edure twi obtain values read non reentrant pro edures problem adjoint mode saw tion kpointing strategy kpointing relies alling kpointed pie twi way ond equivalent rst end ient subset exe ution ontext snapshot must saved restored hidden variables like internal read pointer inside opened save variable annot saved restored general tber kpointing would require hidden variables put snapshot kpointing forbidden similarly pro edure allo ates memory allo ation must done twi pro edure kpointed one must deallo ate memory restoring snapshot dupli ate tapenade yet able automati ally tapenade fun tionalities problem ope hidden variables ases intera tion user essary first nade issues warning message subroutine ause private save annot kpointed variable message issued able would part snapshot pro edure happened opa turned hand variables question publi variables original ode prin iple global ould also done ally however handful variables thus developing priority subroutine reentrant ause pointers ause isolated memory allo ation deallo ation tapenade lets user label subroutine must kpointed opa took another strategy modi main subroutines always rst make sure opened use dire read without using read pointer thus subroutines reentrant binomial che kpointing automati opa one ambitious appli ations tapenade far means building adjoint pie ode performs unsteady nonlinear simulation large number time steps time step omputes new state whose size ranges hundreds megabytes adjoint mode kpointing applied whi means intermediate values stored sta ould exe ute handful time steps run memory even largest workstation che kpointing ompulsory ompute adjoint several thousands time steps whi goal saw tion tapenade applies subroutine alls kpointing level kpointed easy strategy often inria tangent adjoint opa far optimal one hand several alls better tapenade option mark sele ted hand kpointed alls kpointing kpointing applied ations example top level simulation program loop many time steps nitely need ient kpointing heme applied level time iterations one lassi solution used taf mit gcm multi level ursive kpointing basi ally splits ode alled omplete time val small number equidistant intervals apply strategy instan time steps split large intervals small intervals time steps sket hed gure onsumes maximum simultaneous snapshots average number dupli ate exe utions time step time steps realisti situation split large intervals small intervals time steps one gure onsumes maximum simultaneous snapshots average number dupli ate exe utions time step figure ward kpointing time steps snapshots omputations right adjoint omputations left bla resent snapshot white snapshot les les represent reading available tber however shown strategy optimal reasonable assumptions time steps ost run time snapshot needed run time step step later step optimal distribution nested run griewank walther hara terized kpoints whi follows binomial law optimal strategy spatial temporal omplexity adjoint ode grow logarithmi ally respe number time steps original simulation words slowdown tor whi grows like number times time step exe uted memory whi grows like number simultaneous snapshots grow logarithmi ally total number time steps real appli ations memory spa behave symmetrially one always wait little longer result whereas memory spa bounded therefore maximum number snapshots stored simultaneously xed shows optimal strategy gives slowdown tor grows like root total number time steps whi still good figure shows optimal kpointing strategy problem gure time steps memory snapshots average number dupli ate exe utions time step realisti situation time steps memory snapshots average number dupli ate exe utions implemented optimal strategy adjoint ode opa made rst experiments hand modi ation adjoint ode produ tapenade still tapenade produ automati ally pro edures store retrieve snapshot therefore hand modi ation benign given number time steps general pro edure hedules optimal sequen tions store snapshot retrieve snapshot run time step run adjoint time step omplete simulation sions tapenade fully automate pro ess figure shows performan opa good agreement theory noti parti ular two small tion points iterations urve around iterations going optimality proof see optimal strategy parti ularly ient number time steps implementation heduling pro edure found inria tangent adjoint opa figure optimal binomial kpointing time steps snapshots slowdown factor total number time steps figure optimal binomial kpointing snapshots slowdown tor fun tion total length initial simulation slowdown tor ratio adjoint ode ompared original ode tber exa tly number snapshots number dupli ate exe utions allowed per time step target hine whi orresponds tion points gure previous version opa adjoint written hand nevertheless even adjoint must implement strategies retrieve intermediate states reverse order something lose looking adjoint rst observe kpointing kpointing strategy neither multi level optimal binomial like single level strategy one snapshot stored every xed number time steps reverse sweep states two stored snapshots rebuilt approximately using linear interpolation advantage time steps evaluated twi therefore slowdown tor remains well see least two drawba first hand manipulation requires deep knowledge original program underlying equations method blend easily automati yet automated tool therefore tedious ode manipulations would still essary ond introdu approximation errors omputed derivatives whose mathemati behavior lear gradient obtained end used omplex optimizations loops small errors may result poor onvergen ase large numbers time steps believe exa binomial kpointing approximate interpolation worth experimenting interpolation probably good enough many variables vary slowly whi ould designated variables would need stored iterative linear solver opa model solves ellipti equation end time step using iterative method generates sequen approximations exa solution hani appli ation kind methods gives sequen derivatives approximate solutions number inria tangent adjoint opa iterations original solver reason keeps ontrol original program program parti ular onvergen tests still based variables naturally one may ask whether derivatives reasonable approximations desired derivative exa solution issues derivative onvergen iterative solvers relation dis ussed opa provides two alternative algorithms solve ellipti pre onditioned conjugate gradient relaxation method algorithms give equation essive orre results original ode pcg generally preferred thanks ien torization properties however algorithms figure tives obtained divided sor algorithm remain reases ome ode gives results using two ompares approximate derivawe see derivatives obtained orre number time steps ontrary derivatives obtained pcg algorithm ompletely wrong time steps noti urs gent mode well adjoint mode derivatives obtained pcg although wrong remain identi tangent adjoint explanation iteration pcg involves omputation alar produ ables depend state tor thus making numeri algorithm nonlinear even though ellipti equation linear gilbert shown appli ation xed point iteration gives derivative xed point iteration onverges desired derivative parti ular ase large ontra tive iterate ant updating unfortunately ase iterative solvers pcg whi similar onvergen result knowledge solve problem opa exploit linearity ellipti system opa ploit self adjointness property ellipti operator thus use original pcg routine solve linear systems pra ally using alled feature provided tapenade figure shows tangent mode pcg gives ura sor solver tber figure evolution relative error tangent derivative divided three strategies sor straightforward pcg straightforward pcg bla box strategy inria tangent adjoint opa another experiment tried use straightforward pcg solver time xing number pcg iterations high value observed derivatives ome expensive oherent divided ould another way solve problem ertainly hoi iteration number deli ate lem nitely deserves study rms general ommendation solvers nonlinear kind use bla strategy instead validation experiments corre tness test lassi way gent adjoint orre tness automati ally generated odes follows choose arbitrary input arbitrary dire tion compute divided good enough small using tangent program using adjoint program nally test derivative ompute ompute performed omplete global simulation time steps odes results shown table values mat table dot produ test time steps tber mat well last digits whi shows tangent adjoint odes really ompute derivatives omputation order shown equations values mat well ause weakness divided approximation figure shows weakness small figure relative error divided respe derivatives value omputed various values step size dominant error due hine ura large value dominant error due ond derivatives minimizes errors annot eliminate best ompletely sensitivity analysis long simulation one main appli ation adjoint models sensitivity analysis study model output varies hanges model inputs using dire statisti methods would require many integration non linear model one adjoint model integration enough ompute sensitivity example gure shows output map sensitivity north atlanti meridional heat hanges initial sea surfa temperature one year integration period starting january inria tangent adjoint opa done omputing gradient respe dxdzdt ross tion north atlanti zonal temperature meridional urrent velo ity contours gure show variation initial sst would upon heat transport shows large ale patterns mainly ated north parallel caribbean sea strong spot moro results onsistent obtained marotzke map omputed adjoint opa global grid time steps year experiment done sor algorithm iterative linear solver tapenadegenerated adjoint omputed sensitivity map time times original simulation data assimilation validation automati ally generated derivatives data assimilation experiment arried done alled twin periment framework whereby dire model traje tory used generate syntheti observations initial sea surfa temperature perturbed white noise overed using variational data assimilation hniques syntheti observation given sea surfa height ssh sea surfa salinity sss generated model original outputs starting unperturbed sst ost fun tion minimised sst ssh ssh sss sss supers ript sss stands syntheti observation ssh model output omputing ost issues antar zoom ered minimisation done iterative gradient sear algorithm tber figure sensitivity map north atlanti line respe heat transport dotted hanges initial surfa temperature inria tangent adjoint opa gradient umputed using adjoint hniques figure illustrates performan optimization loop integration period month time steps ost fun tion reases two orders magnitude figure indi ates true solution top panel overed good approximation bottom panel randomly perturbed one middle panel showing quality derivatives obtained cost function iterations figure twin experiment convergen ost fun tion con lusion outlook build tangent adjoint odes previous version opa ean general cir ulation model ost several months opment experien resear new version opa written use tool tapenade signi antly redu rst numeri appli ations show quality derivatives obtained works validates hoi strategy obtain tangent adjoint opa versions ome time opa largest appli ation tapenade work pointed number limitations tber figure twin experiment true eld top initial perturbed eld middle identi optimal sea surfa temperatures bottom inria tangent adjoint opa tapenade lifted limitations remain nonreentrant pro edures whi need addressed future work opa nitely reases essful den tapenade works also additional illustration superiority binomial kpointing strategy ompared multi level kpointing standards appli ation elds cfd slowdown adjoint nonsteady simulation time steps would ode onsidered good standards weather simulation ean modeling however ientists expe yet faster adjoints tion even ost radi onsider approximations hange mathemati nature optimization pro ess understand essary shall study proposed option tool work underlined several dire tions resear tools already studied resear groups considering appli ation language two onstru need better next experiment made soon apply tapenade parallelized version opa essary generated tangent adjoint odes used produ tion ontext opa sour makes extensive use prepro essor dire tives ifdef tapenade deal dire tives ause respe synta stru ture ode handling dire tives tool opinion hopeless might done though generate odes possible prepro essed devise tool put dire tives made easier original ode ode odes losely follows stru ture ase tapenade considering spe ally adjoint hope obtain ient ode systemati ellipti operator exploitation also hope optimize present version tapenade applies using pro ling information believe whi kpointing useless kpointing strategy kpointing pro edure dete several pro edure alls tive tapenade already able use information produ better adjoint tber referen castaings dartus honnorat dimet loukili monnier automati tool variational data lation adjoint sensitivity analysis ood modeling ker editor automati appli ations theory tations lncse pages springer christianson reverse umulation attra tive xed points mization methods software courtier hollingsworth strategy operational implementation using remental approa meteorol giering tangent linear adjoint model compiler users manual http giering kaminski ipes adjoint ode acm onstru tion transa tions mathemati software gilbert automati iterative pro esses tion methods software giles ghate duta using automati adjoint cfd ode development uthup editor aerospa design optimization ent trends pages graw hill new delhi bangalore india griewank hieving logarithmi plexity reverse automati software growth temporal spatial optimization methods griewank bis hof corliss carle williamson derivative onvergen iterative equation solvers software optimization methods inria tangent adjoint opa griewank juedes srinivasan tyner kage automati algorithms written trans math software adjoint analyses malization properties appli ations appli ations theory tools acm automati ture notes computational ien engineering springer sele ted papers chi ago pas ual tapenade user guide hni report inria dervieux automati optimum design applied soni boom redu tion kumar editor pro eedings montreal canada pages lncs springer heimba hill giering parallel mit general ient exa adjoint ulation model generated via automati future generation computer systems kim hunke lips omb sensitivity analysis parameter tuning heme global modeling ean modeling journal lauvernet baret ledimet improved estimates vegetation biophysi variables meris toa images using spatial temporal beijing china onstraints pro eedings ispmsrs made dele luse imbard levy ean general ulation model referen manual hni report pole modelisation ipsl marotzke giering zhang stammer hill lee constru tion adjoint mit ean general cir ulation model tber appli ation atlanti heat transport sensitivity geophys talagrand use adjoint equations numeri modeling automati algorithms theory implementation appliation pages philadelphia penn siam atmospheri ulation griewank corliss editors utke naumann fagan tallent strout heimba hill wuns automati modular tool fortran odes hni report argonne national laboratory submitted acm toms walther griewank advantages binomial kpointing numeri mathemati advan appli ations pages springer berlin adjoint ulations feistauer editor pro eeding enumath weaver vialard anderson variational assimilation ean general tropi ulation model ean formulation internal diagnosti monthly weather review onsisten inria recherche inria sophia antipolis route des lucioles sophia antipolis cedex france recherche inria futurs parc club orsay zac des vignes rue jacques monod orsay cedex france recherche inria lorraine loria campus scientifique rue jardin botanique cedex france recherche inria rennes irisa campus universitaire beaulieu rennes cedex france recherche inria avenue europe montbonnot france recherche inria rocquencourt domaine voluceau rocquencourt chesnay cedex france inria domaine voluceau rocquencourt chesnay cedex france http issn
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jun entra energy transparency kerstin edera john gallagherb pedro henk mullerd zorana kyriakos georgioua manuel hermenegildoc bishoksan kafleb steve kerrisona maja kirkebyb maximiliano klemenc xueliang lib umer liqatc jeremy morsea morten rhigerb mads rosendahlb university bristol university denmark imdea software institute spain xmos bristol spanish council scientific research spain technical university madrid spain roskilde abstract promoting energy efficiency first class system design goal important research challenge although hardware designed software controls hardware given system potential energy savings likely much greater higher levels abstraction system stack thus greatest savings expected software development vision entra project article presents concept energy transparency foundation energyaware software development show energy modelling hardware combined static analysis allow programmer understand energy consumption program without executing thus enabling exploration design space taking energy consideration paper concludes summarising current future challenges identified entra project introduction energy efficiency major concern systems engineering future emerging technologies minecc programme aims lay foundations radically new technologies computation strive theoretical limits energy research objectives range physics software include among others new elementary devices well software models programming methods supporting strive energetic entra project addresses latter objective corresponding author email address jpg john gallagher preprint submitted micpro june dissemination software engineering tested energy optimization enables enables energy transparency case studies measured benchmarks relies relies energy modelling program analysis project management figure overview entra project work plan focus energy transparency regard key prerequisite new system development methods tools entra project ran october december months funded european commission framework programme consortium contained three research institutions one industrial partner specialising design advanced multicore microcontrollers xmos xcore overview project structure shown figure foundations central concept energy transparency developed two work packages energy modelling energy analysis respectively energy transparency enables energy optimisations studied concerned development tools techniques applicable software development finally work packages dealing benchmarking evaluation dissemination project management paper summarises mainly outcomes work packages public deliverables project available project website http introduction discuss two main areas research supporting energy transparency section presents approaches building models software energy consumption different levels abstraction section contains overview static resource analysis techniques showing energy model used analysis program energy consumption section summarises role energy transparency software development discusses achievements project far outlines current challenges directions future research computing computing research challenge requires investigating entire system stack application software algorithms via programming languages compilers instruction sets micro architectures design manufacture hardware energy consumed hardware performing computations control computation ultimately lies within applications running hardware hardware designed save modest amount energy potential savings far greater higher levels abstraction system stack estimate intel software realize savings factor three five beyond achieved energy efficient hardware roy johnson list five objectives help make software design decisions consistent objectives power minimization match algorithm hardware minimize memory size expensive memory accesses optimise performance making maximum use available parallelism take advantage hardware support power management select instructions sequence order operations way minimizes switching cpu datapath achieve objectives requires programmer tools understand relationship code energy usage energy transparency aims enable exactly energy transparency concept energy transparency odds trend modern software engineering desire abstract away details using languages abstract data types classes libraries layers interpretation compilation interests portability programmer productivity understandability software reuse contrast energy transparency requires making visible software impacts energy consumption executed hardware availability information enables system designers find optimal performance accuracy energy usage computation achieve energy transparency models energy consumed computation required discussed section models established different levels abstraction ranging models characterize individual functional hardware blocks via instruction set architecture isa characterization models models based intermediate representations used compiler final energy models provide information feeds static resource usage analysis algorithms represent energy usage elementary parts computation discussed section energy modelling energy models rely information several possible abstraction levels hardware description functional block instruction set architecture isa performance counter transaction based abstractions energy models higher levels tend faster use lower accuracy models lower levels abstraction entra aim provide accurate modelling exploited analysis applied order estimate energy consumed software defining constructing energy model isa practical level abstraction energy modelling expresses underlying hardware operations relationship intent software constructing model level gives following benefits energy costs attributed directly individual machine instructions output back end compiler instruction properties energy consumption strongly correlated energy consumption typically increases increasing numbers operands machine instructions traced back original source code statements written software developer well various intermediate representations however energy modelling isa level requires additional effort order produce useful models instruction costs must captured profiling suite measurement device power addition indirect statistical approaches required characterise instructions profiled direct measurements furthermore architectures properties cost running multiple threads cost idle periods must determined target architecture energy modelling analysis xmos xcore embedded microcontroller beyond offering instruction execution xcore hardware comes variety configurations xcore architecture simple design thus ideal investigate advanced energy modelling static analysis techniques required achieve energy transparency techniques developed readily transferable deeply embedded iottype processors arm cortex series atmel avr fact xcore offers made particularly interesting target entra project shown xcore number active threads impact upon energy consumption model must take account traditional models attribute energy costs simply instructions transitions instructions additional effects impact energy consumption cache hits misses although build principle parallelism considered yielding complex model equation energy consumed program nidl tclk tclk energy consumption split two parts capturing idle active processor behaviour respectively former consider base processor idle power present even device waiting external events multiplied number cycles active threads nidl clock period tclk latter individual instruction costs accounted based costs well aggregated overhead parallelism scaling factor determined number active threads calculated isa instruction multiplied number occurrences target program particular level parallelism well clock period tclk model parameters separated two groups values first group constants obtained profiling processor fixed clock period tclk yielding base power overheads costs parallelism scaling measured second group must determined analysis target program include number idle cycles nidl number threads program instruction counts instruction number active threads analysis produce values used estimate program energy demonstrated various static methods follow principle illustrate instruction profiling example heat map representing device power interleaving selected subset data manipulation instructions shown fig profiling framework executes tightly coupled threads xcore pipeline random valid operand values produce average power estimate instruction random input data shown cause higher power dissipation constrained data would found programs due data dependencies thus creating modest cases instruction profiling used determine costs instructions executed repeatedly succession cost instructions estimated generic average grouped operand count accurately regression tree approach identifies significant set features including instruction length whether memory accessed others find similar directly profiled instruction latter accurate three approaches adds significant modelling overhead form model used various analysis methods instruction set simulation iss produce instruction trace instruction counts thread activity determined iss achieve best model accuracy alternatively instruction execution statistics used instead extrapolate model terms faster power buf zext rus sext rus ldc ldapf ldapb zext sext andnot mkmsk rus mkmsk clz ldapf ldapb ldc neg byterev bitrev ldw ldw stw ldaw shl shr shl shr lsu lss add sub ldaw sub add ashr ldawb ldawf ashr ldawb xor ldawf mul maccu maccs ladd lsub lmul even threads instruction name encoding lmul lsub ladd maccs maccu mul ldawf xor ldawb ashr ldawf ldawb ashr add sub ldaw sub add lss lsu shr shl shr shl ldaw stw ldw ldw bitrev byterev neg ldc ldapb ldapf clz mkmsk mkmsk rus andnot sext zext ldapb ldapf ldc sext rus zext rus buf odd threads instruction name encoding figure power dissipation instruction interleaving xmos xcore processor dashed lines denote change operand count axis label colour indicates green red instruction encoding producing full trace increase estimation error shown yield acceptable margin benchmarks see section terms model calculated static analysis removing simulation step allowing rapid exploration well parameterised bounded energy estimations values nidl analysed functions input state allows derivation energy functions characterising energy consumption possible runs program rather specific run set runs experiments tracing two orders magnitude slower simulation latter typically taking less one minute however mitigated analysing single functions blocks ignoring parts trace terminating soon blocks interest executed making trace simulations take less minute cases observed static analysis course achieve faster results analysing code blocks without simulation key prerequisite achieving high accuracy energy modelling isa level predictable architecture instruction set gives accurate view processor behave xcore thread scheduling sram memory subsystem together absence performance enhancing complexity enabled achieve accurate model essential obtaining accurate energy predictions thus worst case execution time wcet analysis results energy modelling energy consumption analysis influenced processor architecture predictability determines accuracy achievable well complexity modelling analysis techniques energy modelling system iss must accurately simulate network behaviour order capture timing data allows accurate estimation communicating processes static analysis must provide similar characterisation therefore instruction execution network behaviour flow communication processes must predictable order enable energy transparency processor model achieves average error suite benchmarks standard deviation multicore model demonstrates average energy estimation error standard deviation models less error provide suitable accuracy energy estimations energy modelling higher levels software abstraction performing static analysis isa level benefit good accuracy due closeness hardware suffer loss useful information program structure types good compromise found modelling intermediate representation used compilers program information preserved since compiler natural place optimisations modelling predicting energy consumption level could therefore enable energy specific optimisations using mapping technique lifted energy model level compiler implemented within llvm tool chain isa level energy models thus propagated llvm level allowing energy consumption estimation programs level enables static analysis performed higher level isa thus making energy consumption information accessible directly compiler optimiser mapping technique determines energy characteristics llvm instructions provides energy characterization takes consideration compiler behavior control flow graph cfg structure types aspects instructions taking account information improved accuracy llvm characterization significantly experimental evaluation demonstrated mapping technique allowed energy consumption transparency llvm level accuracy keeping within estimations cases principle mapping technique may used map energy consumption programs even higher levels source code however fine grained characterization line source code using method impractical due numerous transformations optimisations introduced compiler loss accuracy resulting difficulty associating energy consumption costs obtained lower levels source code lines alternative approach building energy model investigated target language java android platform attempt map energy model source code would need deal explicitly java virtual machine well operating system layers highly impractical strategy instead basic operations source code identified correlations energy costs found measuring energy consumption large number execution cases analyzing results using techniques based regression analysis resulting energy model basic operations implicitly includes effect layers software stack hardware approach inherently approximate nevertheless approach may feasible one complex software stacks energy models needed instance give programmer energy profile code development static analysis energy consumption static analysis key component energy transparency infers information energy consumed programs without actually running energy models analysis performed program representations different levels software stack ranging source code different programming languages intermediate compiler representations llvm isa static analysis given level consists reasoning execution traces program level order infer information among things many times certain basic elements program executed role energy model provide information energy consumption basic elements used analysis infer information energy consumption entities procedures functions loops blocks whole program analysis also performed given software level using energy models lower level model needs mapped upwards higher level described section information inferred static analysis lower level also reflected upwards higher level using suitable mapping information entra project approach applied static analysis programs running xcore architectures however framework return section hypothesis choice level affects accuracy energy models precision analysis opposite ways energy models lower levels precise higher levels lower levels program structure data structure information lost often implies corresponding loss accuracy analysis hypothesis level potential choices illustrated figure entra explored different points space combinations analysis modelling experimental results confirm expected exists also suggest performing static analysis llvm level good compromise llvm close enough source code level preserve program information needed static analysis whilst close enough isa level allow propagation isa energy model llvm level without significant loss accuracy information inferred analysis information inferred analyzers guided final use program optimisation verification helping software developers make design decisions example infer safe approximations namely upper lower bounds energy consumed program parts approximations functions parametrised sizes input data hardware features clock frequency voltage analyzers infer concrete values parameters yield energy consumption program parts static energy profiling determines distribution energy usage parts code useful developer showing parts program functions blocks program perhaps particularly expensive energy internal representa transforma including asser ons energy model program including asser ons layer analysis informa loss energy modelling precision loss energy model transforma source code analysis compiler llvm energy model layer energy model miza ons transforma mized llvm analyser llvm code generator transforma isa energy consump ons layer hardware figure level potential choices called many times parts natural targets optimisation since small improvement yield considerable savings note safety bounds depends energy models giving safe bounds instruction challenging problem discussed section safe bounds vital applying energy analysis verifying certifying energy consumption generic resource analysis framework resource analysis framework developed parametric respect resources programming languages regarding resources common assertion language allows definition different resources basic components program affect use resources concretely allows encoding different energy models specific hardware architectures particular energy models developed xcore architecture llvm isa levels described section regarding programming languages differentiate input language entra currently source llvm isa intermediate semantic program representation resource analysis actually operates use horn clauses intermediate program representation referred transformation performed supported input language passed resource analysis explored different approaches transformation one approach perform direct transformation applied isa llvm code another approach consists producing applying partial evaluation techniques instrumented interpreters directly implement semantics language analysed cases resulting programs analysed ciaopp tool see section horn clauses offer several features make convenient analysis instance representation inherently supports static single assignment ssa recursive forms current trend favouring use horn clause programs intermediate representations analysis verification tools using generic representation assertion language ciaopp analysis tools instantiated general framework produce series concrete energy analyzers allowed study advantages limitations different techniques well implied different choices analysis energy modelling levels common assertion language defined common assertion language vehicle propagating information throughout system levels communication among different analysis verification optimisation tools actors involved software development assertion language allows expressing energy models different architectures writing energy consumption specifications describing energy consumption components available analysis time expressing analysis results ensuring interoperability refer reader references full description ciao assertion language basis assertions used internal common assertion language entra entra common assertion language also includes front end express energy specifications energy related information source code energy analysis using ciaopp input ciaopp parametric static resource usage analyzer along assertions common assertion language expressing energy model llvm blocks individual isa instructions possibly additional trusted information analyzer based approach recursive equations cost relations representing cost running program extracted program solved obtaining cost functions may polynomial exponential logarithmic terms program inputs output cost functions express energy consumption block mapped directly back language represented generic resource analysis engine fully based abstract interpretation defines resource analysis abstract domain integrated plai abstract interpretation framework ciaopp brings features multivariance efficient fixpoints assertionbased verification user interaction setting solving recurrence relations inferring functions representing bounds sizes output arguments resource usage predicates program integrated plai framework abstract operation direct energy analysis llvm mentioned llvm offers good analyzability accuracy addition using generic approach based ciaopp translation entra project experimented approach uses similar analysis techniques operates directly llvm representation advantage approach integrated directly llvm toolchain principle applicable languages targeting llvm energy model used exactly one applied described section energy consumption static analysis since underlying challenges analysing timing energy consumption behaviour program appear quite similar applied well known wcet analysis techniques retrieve energy consumption estimations one popular wcet techniques implicit path enumeration ipet retrieves worst case control flow path programs based cost model assigns timing cost cfg basic block replaced timing cost model isa energy model given equation absence architectural performance enhancing features caches technique provide safe upper bounds timing experimental evaluation demonstrated case energy energy consumption contrast time data sensitive see section order explore value limits applying ipet energy consumption estimations also extended analysis llvm level using llvm energy characterization given mapping technique referred section furthermore extended energy consumption analysis embedded programs two commonly used concurrency patterns task farms pipelines experimental evaluation set mainly industrial programs demonstrates although energy bounds retrieved considered safe still provide valuable information energy aware development delivering energy transparency software developers absence widely accessible software energy monitoring probabilistic resource analysis bounds energy consumption useful information distribution consumption within bounds even example may execution cases program result consumption close lower bound upper bound reached outlying cases vice versa distribution estimates average energy consumption derived one approach obtaining kind information perform probabilistic static analysis program respect energy consumption special case probabilistic output analysis whose aim derive probability distribution possible output values program probability distribution input output case energy consumption discussion software development needs energy transparency designers programmers understand energy consumption early stage development lifecycle order explore design space taking energy consumption consideration many decisions taken early process hardware platform degree parallelism fundamental algorithms data structures determine overall energy efficiency final application software development lifecycle includes activities providing energy specification energy budget making initial rough estimates energy consumption based models application allowing exploration design space respect energy consumption choosing configuring hardware platform suits application example reducing energy cost frequent communications memory accesses developing code constant reference energy consumption program parts allowing energy bugs identified early performing energy optimisation critical code sections using precise energy models taking account compiler generated machine instructions energy critical applications providing guarantees form tight upper lower bounds energy consumption important note development platform seldom final deployment platform emphasising importance energy modelling final target hardware alternative energy transparency wait application run final intended platform measure energy usage stage likely late much excessive energy consumption software energy modelling challenges verification energy consumption requires development energy model difficult since energy cost executing instruction depends operands obtain consumption instruction must therefore measure execution operands induce energy models built entra based averages obtained measuring energy consumed random valid data processed demonstrated variation due data range examine impact operand values instruction level energy consumption propose probabilistic approach developing worstcase energy models suitable safe energy consumption analysis energy modelling serious challenge determining maximum amount circuit switching instruction data tends take exponential amount time evaluate making difficult determine guarantee worst case data instruction sequence even model built energy consumed instruction given function operands would still challenges encoding functions suitable way exploited effectively static analysis algorithms explored technique models upper lower bounds energy branchless blocks instructions order take account switching costs within block uses evolutionary algorithm faster exploration search space also reduced fact algorithm deal program model fed static analysis takes account program infers energy information whole program procedures software energy analysis challenges static analysis always involves precision complexity analysis obtaining tight bounds energy usage depends several factors including accurate propagation data size measures extraction solution relations expressing energy consumption terms data sizes problems solvable large class useful programs program structure departs standard patterns precision may rapidly lost instance realisation general framework described section using ciaopp resource analyser described section uses deal recursive programs including multiple mutual recursion programs iterative programs nested loops programs programs operation complex data structures nested lists arrays analysis produces parametrised energy bounds depend input data sizes expressed large class functions polynomial factorial exponential logarithmic summations contrast approaches limited polynomial functions absolute bounds experiments reported realisation analysis framework perform analysis isa llvm levels compare energy values obtained evaluating inferred energy functions different input data sizes actual hardware measurements xcore platform results show llvm level analysis reasonably accurate less average deviation hardware measurements powerful analysis isa level sense deal larger class programs programs involving structured types average deviation smaller set benchmarks analysis produced results although tested prototype tools relatively small programs exhibit features also present bigger real programs could analysed bigger scale since designed analysis tools enable scalability thus interpret experimental results promising encourage continue research following incremental approach making prototype tools robust powerful scalable well evaluating bigger real programs implementation research challenge already said approach developing energy analysers architecture neutral claim supported experimental results performed direct energy analysis llvm described section arm xmos xcore platforms benchmarks also contain nested loops complex control flow predicates perform bitwise operations well operations arrays matrices overall final deviation hardware measurements typically less xmos arm platforms respectively showing general trend static analysis results relied upon give estimate energy consumption instantiating general approach platforms include hardware operating system assessing different application domains another challenge intend address future work static analysis code difficult since precision easily lost due thread interleaving accurate analysis timing synchronisation behaviour threads energy analysis using model given energy instruction depends many threads active simultaneously extending results less predictable architectures entra approach generic consists framework parametrised energy model generic static analysis tools translate code common analysable form llvm principle constructed programming language however much experimentation investigation remains done apply approach effectively architectures contain sources unpredictability caches complex pipelines interrupts present xcore architecture likely path research follow approach wcet analysis unpredictable architectures employing supporting analyses permit accurate energy analyses example approximation cache contents specific program points leading accurate models guaranteed memory access definitely hit definitely miss cache energy optimisations enabled energy transparency different types optimisations different levels software stack performed taking advantage information provided multilevel entra tools static dynamic energy optimisations enabled energy transparency investigated entra dynamic optimisations framework stochastic scheduling based evolutionary algorithms eas developed cases tasks independent dependent latter case dependence modelled using copula theory particular archimedean copulas eas also used improve allocation scheduling multicore environments example algorithms described able deal task migration preemption ones allow decreasing program accuracy performing loop perforation order save energy optimisations include use energy analysis choose software parameters order transform programs ensure energy target met minimizing loss quality service energy performance also improved optimisation techniques task placement xcore network identifying communication patterns among tasks using swallow experimental open platform many xcores source model data demonstrated incorporating energy consumption timing energy modelling process aid identifying impact task placement communicating applications energy optimisations already incorporated recently released compiler xmos ltd optimisations object code obtained aggressively applying global dataflow analyses inspired entra project research results case studies showing power reduction approximately using global optimiser found entra project report experiment energy optimisation android code carried using source code energy modelling mentioned section energy consumption mobile devices smartphones increasing concern developers software devices study concerned optimisation code interactive games energy model combined execution profiling enabled developer discover code blocks several hundred blocks considered code consumed overall energy aggressive source code optimisation refactoring blocks enabled energy savings various scenarios conclusions goal entra project provide techniques tools supporting energy transparency software level results obtained include energy models xcore processor different levels abstraction isa level llvm well preliminary energy models less predictable architectures model incorporating execution xcore also developed models incorporated static analyses corresponding level code experiments compared predictions energy consumption programs actual measured consumption encouraging results percentage error obtained addition accuracy analysability different levels explored leading preliminary conclusion analysis llvm level provides good compromise project identified challenging problems future research extending analyses code building energy models acknowledgements research received funding european union framework programme grant agreement entra whole systems energy transparency grant agreement coordination support action spanish mineco tin strongsoft project madrid program references lack software support marks low power scorecard dac electronics roy johnson software design low power nebel mermet eds low power design deep submicron electronics vol kluwer academic url http brooks tiwari martonosi wattch framework power analysis optimizations proceedings annual international symposium computer architecture isca acm new york usa doi http url http tiwari malik wolfe lee instruction level power analysis optimization software journal vlsi signal processing url http steinke knauer wehmeyer marwedel accurate fine grain energy model supporting software optimizations proceedings patmos kerrison eder energy modeling software hardware multithreaded embedded microprocessor acm trans embedded comput syst url http brandolese corbetta fornaciari software energy estimation based statistical characterization intermediate compilation code low power electronics design islped international symposium georgiou kerrison eder value limits multilevel energy consumption static analysis deeply embedded single programs tech url http wegbreit mechanical program analysis commun acm rosendahl automatic complexity analysis acm conference functional programming 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overview low power electronics digest technical ieee symposium ieee pallister kerrison morse eder data dependent energy modeling worst case energy consumption analysis tech may url http kerrison eder modeling visualizing networked embedded software energy consumption tech arxiv url http heckmann langenbach thesing wilhelm influence processor architecture design results wcet tools proceedings ieee liqat kerrison serrano georgiou grech hermenegildo eder energy consumption analysis programs based xmos models gupta pea eds logicbased program synthesis transformation international symposium lopstr revised selected papers vol lecture notes computer science springer url http lattner adve llvm compilation framework lifelong program analysis transformation proc international symposium code generation optimization cgo ieee computer society url http gallagher view energy analysis mobile application source code tech submitted conference october url http liqat georgiou kerrison hermenegildo gallagher eder inferring parametric energy consumption functions different software levels isa llvm eekelen lago eds foundational practical aspects resource analysis fourth international workshop fopara revised selected papers lecture notes computer science springer press url http liqat klemen gallagher hermenegildo transformational approach parametric static profiling kiselyov king eds international symposium functional logic programming flops vol lncs springer doi url http henriksen gallagher abstract interpretation pic programs logic programming sixth ieee international workshop source code analysis manipulation scam ieee computer society navas hermenegildo flexible approach analysis programs international symposium program synthesis transformation lopstr lncs grebenshchikov gupta lopes popeea rybalchenko hsf software verifier based horn clauses competition contribution flanagan eds tacas vol lncs springer hojjat garnier iosif kuncak verification toolkit numerical transition systems 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analysis embedded software using implicit path enumeration design integrated circuits systems ieee transactions doi rosendahl kirkeby probabilistic output analysis program manipulation bertrand tribastone eds proceedings thirteenth workshop quantitative aspects programming languages systems qapl london april vol eptcs qapl london url http kirkeby rosendahl probabilistic resource analysis program transformation proc foundational practical aspects resource analysis lncs springer press url http morse kerrison eder infeasibility analysing dynamic energy tech mar url http liqat hermenegildo inferring energy bounds statically evolutionary analysis basic blocks workshop high performance energy efficient embedded systems url http stochastic deterministic evolutionary allocation scheduling xmos chips neurocomputing doi http url http improved stochastic scheduling based evolutionary algorithms via modeling task dependences herrero sedano baruque corchado eds international conference soft 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shape inverse rendering different classes objects single input image shima zohreh nov computer science engineering information technology shiraz university shiraz iran november abstract paper deep framework proposed problem shape inverse rendering single input image main structure proposed framework consists unsupervised components significantly reduce need labeled data training whole framework using labeled data advantage achieving accurate results without need predefined assumptions image formation process three main components used proposed network encoder maps input image representation space decoder decodes representation structure mapping component order map representation part needs label training mapping part many parameters components network unsupervised using images data case way reconstructing shapes decoder component inspired model based methods reconstruction maps low dimensional representation shape space advantage extracting basis vectors shape space training data restricted small set examples used predefined models therefore proposed framework deals directly coordinate values point cloud representation leads achieve dense shapes output experimental results several benchmark datasets objects human faces comparing recent similar methods shows power proposed network recovering details single images introduction inverse rendering stands specific class machine learning techniques aim recovering properties scene like camera extrinsic parameters scene lighting shape scene existing measurements images attracted high interest recent academic research due wide applications computer vision inverse rendering well posed problem existing infinite number structure result image facing problems needs additional assumptions prior knowledge class structures recovered images instance statistical model used determine solution likely specific class objects aim inverse rendering problem actually inverting image formation process called rendering computer graphics rendering expressed equation formulated reflected radiance light surface intensity pixels image represents way object surface reflects lights various spatial angular positions describes lighting scene measurement showing light transferred existing surfaces scene light reflected surfaces scene affecting knowing quantities means properties scene known computed using quantities direct problem solve inverse rendering problem arises information available aim compute values side paper focus recovering shape geometry object single input image different poses implicitly embedded variable spatial properties surface light reflection inverse rendering problem machine learning task faced main steps solved selection model assuming specific class hypotheses defining score criterion defining search strategy selection model selection model one main necessities solving machine learning problems due free lunch theorem case recovering structures images step important problems deal problem based hadmard definition problems might one solutions give image solution depends continuously data means small errors measurements may cause large errors solutions solving kinds problems assumptions usually made properties solutions restrict solution space feasible regions selection suitable model defining target scene serves prior knowledge solution space search performed reliable like using assumptions method may costs like losing promising regions spanned selected model inverse rendering problems various types models prior knowledge target scene literature make existing methods different approaches use additional information like multiple images video sequence landmarks scene find unique solution work certain statistical model determine feasibility solutions perform regressing techniques relates output measurements structures paper focus pose invariant recovering structure object single image help model describing properties object main structure proposed framework based finding relation measurements single image scene structure scene constrained used model defining score criterion selection suitable model describing solution space optimum solution searched space aim first criterion defined guides search toward finding optimum solution satisfying constraints problem hand instance case inverse rendering human face image face recognition task first main objective recover distinctiveness characteristics face image good solution generic determined tradeoff two properties objective criterion paper since objective minimize reconstruction error used root mean square error rmse loss function training evaluation defining search strategy exist different types search mechanisms search solution based defined score criterion case convex criteria closed form solution exists search done linear time however nature inverse rendering problem highly nonlinear inference finding relation image corresponding structure intractable assumptions incorporated problem like deterministic models formulating behavior variables rendering equation defined accurately solution computed closed form however strong assumptions inverse rendering problems cases use learning based methods training models use test samples popular learning framework recent years deep networks show power solving nonlinear problems deep networks like learning based structures trained ways supervised unsupervised supervised methods use set labeled data guide training result accurate availability ground truth training procedure suffer lack enough labeled training data domains case inverse rendering applications providing enough realistic training data deep network may impractical one solution problem use generative model generating synthetic training data trained network figure main configuration proposed framework another solution found attempts recent literature unsupervised training network reconstruction structure generated part network rendering mechanism used transform generated structure something form input measurement network try reduce difference input rendered output network need labeled data training existing unsupervised methods literature used analytical form rendering structure assumptions image formation process characteristic although significant improvement reducing need training data may affects quality solutions due existing assumptions rendering mechanism camera properties scene illumination reflectance properties case inverse rendering problem well known models help generate enough data training deep networks paper propose mechanism parts network trained using unlabeled data whole network using labeled training data aggregation components believe properties reduces nedd deep network large set training data therefore framework inverse rendering problem paper organized follows first briefly describe framework section section review analyze recent related works proposed framework network structure define details section experimental results verify performance proposed structure several benchmarks demonstrated section finally discussions conclusions found sections overview proposed framework composed main components based fact inverse rendering problem look way transforming input image corresponding structure mapping input image corresponding structure suitable representations finding transformation obtained representations show experimentally nonlinear mapping suitable representations results accurate solutions finding suitable representations done using autoencoders unsupervised tools finding representations convolutional networks finding suitable representations either different types autoencoder existing well known networks literature case using autoencoders finding representations inspired morphable models network extracts basis vectors training data size uses basis functions reconstruct accurate shape object rather using figure shows overall structure proposed framework details proposed idea found section related works section review analyze recent attempts inverse rendering single images using deep frameworks related works concentrating search strategies richardson proposed deep iterative framework inverse rendering human faces iteration framework output network fed input additional information previous result training process method supervised using generating synthetic labeled training data final output using form shape shading sfs method achieve authors improve work sfs component network implemented another deep network output framework coefficients predefined computed using hundred samples framework directly deal shape structures supervised framework reconstruction human faces single image proposed also used generating training data aim fitted images via optimization method used model ground truth training data strategy finding ground truth may cause found solutions dependent power optimization method powers new work may ignored loss function also used training network human faces tuned make generic face generation overdetermined one suitable purpose face recognition unsupervised framework reconstruction objects single multiple input images volumes proposed authors suggested inference network encoding input data low dimensional representation fed generative network generating objects using encoded data obtained object rendered image using fixed differentiable rendering method objective training network minimize difference rendered input images using fixed rendering component may restrict accuracy generated shapes input image rendering mechanism differs network rendering authors also suggested using convolutional network rendering component using another deep network rendering component may overcome mentioned limitation autoencoder network proposed reconstruction human faces using unsupervised training mechanism network encoder part autoencoder convolutional network like alexnet networks decoder part analytical differentiable formula uses encoded data assumptions image formation process form face render image paper objective also minimize difference input image rendered shape using fix rendering method generative adversarial network gan trained generating shapes different objects method unsupervised training approach random vector distribution fed network shape object generated rendered form image input discriminator way discriminator need shape control process learning shapes rendering method fixed network used generate training data inverse rendering frameworks objects cnn designed recover geometry texture human faces single image framework directly works coordinates form proposed binary volumes maps images directly coordinates binary volume using binary volume coordinates representing human face may result blur shapes paper aim design deep framework shape inverse rendering different types shape representations point clouds volume representations single input image training process utilizes labeled unlabeled realistic data improving quality resulted shapes following section demonstrates proposed approach details proposed framework main idea work mapping representation single input image representation corresponding shape structure defining structure actually based main functionality deep networks using different layers nonlinear computations order feature extraction transformation proposed framework also made components feature extraction transformation stated section used three main components framework first component trained compute representations input images second component trained reconstructing shape representations third components used map representation computed first component one used second component representations component satisfy objective instance component computing representation image computes representation suitable mapped representation hand representation suitable reconstructing shape form point cloud binary volume observed experiments complex representations representation component achieve better performance decoder part linear autoencoder reconstruction component suitable reconstructing shape form point cloud third component structure mapping representation corresponding representation actually proposed component manifold images may different manifold shapes especially case complicated shapes like human faces therefore nonlinear mapping function found obtain accurate representation image representation note proposed frameworks deals directly dense shapes objects instead using predefined models representing face characteristic advantage restricted spanning obtained samples using larger set training data extracting suitable basis vectors shape spaces main contributions paper follows using deep structures different components interpretable framework mapping representations manifold another directly reconstructing results using deep structures component improves handling stage possibility unsupervised component using unlabeled data representations improved using realistic data therefore reducing need labeled data training suitable initialization training fact labeled data needed final aggregation components using training data extract bases representation space instead using predefined analytical models ability training framework inverse rendering different object classes single input image following subsection proposed idea paper demonstrated simplest case linear terms singular value decomposition svd describing proposed idea simple case linear terms singular value decomposition svd section analytically describe proposed idea paper simplest case linear case calculations gives mathematical insight motivation proposing framework section evaluate idea using different datasets compare possible method case solving shape inverse rendering problem aim assume linear case component linear autoencoder encoder decoder components linear feedforward network mapping component proven optimal representation space resulted linear autoencoder using loss function spanning resulted performing pca data therefore consider use linear autoencoder equivalent performing pca training data therefore inferred weights hidden output layer trained linear autoencoder fact bases resulted representation subspace found using training data autoencoder characteristic means using linear autoencoder obtain bases representation space data size without concern basis vector decomposition large data matrix let denotes matrix formed concatenating samples input images vectorized column vectors applying svd subtracting mean value bases eigen vectors pca subspace obtained using svd matrix columns representing eigenvectors diagonal matrix eigenvalues diagonal elements denotes matrix eigenvectors columns therefore use columns basis vectors resulted representation subspace images representation resulting spanned first eigenvectors largest eigen values expressed uxt similarly considering column matrix containing vectorized training samples shape structures represented point clouds binary volumes resulted representation using another autoencoder expressed uzt find transformation two obtained representations inverse rendering problem linear feedforward net used equivalent performing least square mechanism find linear relation argmint result used reconstructing shape say hand also solve direct optimization problem linearly tries find linear relation images corresponding shapes argminb two approaches look subspace best describes shape objects image first approach first tries find low dimensional representation input image corresponding structure set training data linear solution found transforming representation one via least squares note matrices used mapping functions test data examples used calculating necessarily correspond used obtain mechanism shows possibility unsupervised training different components method second closed form approach directly finds linear mapping input image corresponding structure believe first approach significantly improves process finding solution inverse rendering problem table section verified claim experimentally compared second approach loss function mentioned purpose proposed framework paper recover shape object deal regression problem therefore used root mean squared error rmse loss criterion training components proposed framework criterion standard deviation prediction error standard way regression analysis xgt erm xigt denote prediction ground truth ith training sample stands sample size sample size stands size mini batch different loss functions proposed used inverse rendering problems different researches different applications like reconstruction recognition binary voxel representation used representing faces sigmoid cross entropy loss function used training network framework deal mesh representation voxel representations choose standard rmse training networks experiments section perform experiments evaluate performance proposed framework different inverse rendering scenarios used types datasets objects human faces different shape representations like point clouds binary volumes order report compare reconstruction results also analyse structures different components framework datasets report results performance proposed framework two types datasets objects human faces case trained separate structure using type data representation also used unlabeled datasets order unsupervised components case human faces used lfw dataset unlabeled data unsupervised besel face model generating synthetic data labeled training data along bosphorus dataset realistic labeled data training respectively note aggregating components using supervised training used faces generated besel face model bfm natural expression rendered poses degrees axis therefore objective human face shape inverse rendering work pose invariant identity recovering faces single image faces framework represented point clouds vertices space normalized dimension generated training test samples natural expression rendered images size mentioned poses input images case objects used categories imagenet dataset unlabeled data pretraining shapenet datasets labeled training data best knowledge datasets object reconstructions used categories training networks form binary voxel grid rendered training sample gray scale image viewpoints axis case linear autoencoders vectorized voxel grid binary column vector size incorporate alexnet experiments evaluating use deep cnn computing representations face dataset case rendered used colored images faces existing dataset size standard input size alexnet framework parameter setting table includes configuration structure used type dataset type used structure since applied inverse rendering different data types use different structures network components case encoder mapping decoder object binary volume linear nonlinear convolutional human face point cloud nonlinear convolutional linear table network configuration components using different structures besel face dataset modelnet dataset shapenet dataset using low dim representation directly finding table comparison average test rmse obtained different datasets using closed form linear solutions section note case object datasets since classes objects need layers larger sizes representation units learning rate set configuration gives better performance cases uses epochs reduce another epochs case mesh representation learning slower needed smaller learning rate longer training time batch size set constant cases equal training trained networks scratch waterloo university servers machines servers several gpus suitable training deep networks evaluation mechanism used rmse evaluation criterion test validation sets different parts experiments case face datasets also showed rmse form heat maps showing rmse reconstructed face ground truth face suitable tool error visualization different regions reconstructed face start experiments first reporting evaluation results closed form linear solutions described section using different data types next step use equivalent deep structure changing mapping component nonlinear structure show results significantly improved nonlinear component encoder part framework alexnet using labeled human face dataset report results show reconstruction results realistic human face dataset compare results recent methods reconstruction results solving linear closed form formula inverse rendering regarding analytical description proposed idea section show finding linear mapping low dimensional representations data achieves significantly better results rather finding direct linear mapping data table figures include numerical visual results applying methods different datasets using low dim representation stands finding mapping low dimensional representations data using least squares directly finding stands finding least squares solution direct mapping input images data numerical results table indicate effectiveness proposed idea mapping low dimensional representations instead direct mapping terms rmse existing datasets looking figure first clear using linear methods powerful enough addressing shape inverse rendering problem single input image need input image ground truth using low dim representation directly finding figure visual results using linear methods finding mapping spaces bese face dataset use nonlinear tools fields especially case human face inverse rendering second see using low dimensional representation mapping works better direct mapping visually input images ground truth using low dim representation directly finding figure visual results comparing linear least squares finding mapping space dataset next step experiments set deep structure consist linear autoencoders finding representations nonlinear feedforward network mapping representations strat point using nonlinearity proposed framework using linear encoders decoders nonlinear feedforward network mapping component start point using nonlinear deep structure framework used nonlinear feedforward network mapping component trained network using samples besel face model dataset training set dataset epochs figures show results test data face object datasets respectively observed using nonlinearity mapping component framework results significant improvement visually numerically instance case face datasets reconstructions observation also shows relation subspaces representations nonlinear relation next subsection fixing nonlinear feedforward network mapping component consider use complicated representations data alexnet encoder input image transfer learning alexnet encoder figure shows proposed framework uses alexnet model encoder component model trained one million images categories images imagenet database like human chair pencil many categories model extracting rich features images classification object categories input images ground truth network reconstruction heat map rmse average rmse test error figure results using nonlinear feedforward network mapping component besel face dataset input images ground truth network reconstruction average rmse test error figure results using nonlinear feedforward network mapping component modelnet dataset using model encoder component first verify possibility using components framework second analyze use complicated representations solving inverse rendering problem proposed framework alexnet using set colored figure alexnet encoder component proposed framework input images samples generated besel face dataset samples results shown figures respectively case since texture associated object rendered color image represent input images size feed network figures observable using configuration computing complex representations input images accuracy reconstruction improved compared linear autoencoder actually objective encoder component compute representations suitable mapped representation results show complicated representations appropriate aim case decoder component aim recovering sharp shapes input image ground truth network reconstruction heat map rmse average rmse figure visual numerical results using alexnet encoder proposed framework besel face dataset input image ground truth network reconstruction average rmse figure visual numerical results using alexnet encoder proposed framework dataset tations therefore representation component shape inferred case point cloud shapes directly deal real coordinate values linear decoder gives promising result compared nonlinear convolutional structures blur results case binary volume shapes output digitized using threshold value nonlinear structures show promising performances figure shows digaram rmse epoch using different mentioned structures different types shapes pose invariant reconstruction figure shows power proposed framework reconstruction sample face input image captured different poses reconstruction results realistic datasets evaluated final configuration framework alexnet encoder nonlinear mapping linear decoder bosphorus face dataset realistic face dataset figure shows results heat map terms rmse bosphorus also compared visual reconstruction results proposed framework realistic images modelnet dataset besel face dataset figure validation error epoch different structures used components proposed frameworks modelnet dataset face dataset lin linear encoder decoder nonlinear feedforward network conv convolutional encoder decoder input image network reconstruction heat map rmse figure result proposed framework bosphorus face dataset case realistic images first manually cropped face input image aligned tha intensity image sample image training samples framework using procrustes analysis used use input image framework note order compute rmse heat map computed minimum rmse ground truth network reconstruction results figure show network could successfully recover main features face note using linear decoder case face datasets equivalent performing pca training samples causes achieving better results compared predefined using small set training data figure shows visual reconstruction results using realistic images obtained proposed framework compared two similar recent reconstruction methods described section see results obtained proposed framework comparable unsupervised raining methods detect shape details compared binary volume representation shapes believe dealing directly point clouds extracting basis vectors large set training data discussion observed experiments using convolutional network encoder mapping component framework significantly improves performance however case decoder using network linear results better reconstruction results believe convolutional autoencoder reconstruction representation figure result proposed framework bosphorus face dataset obtained data may inaccurate information loss resulted convolution pooling units another property proposed framework use autoencoder directly reconstruct shape structure learning large set examples structure uses training data extract bases shape space believe characteristic leads unbiased reconstructions rather using predefined model biased toward mean shape training samples hand working directly point cloud representation representing shape helps framework achieve detailed results disadvantage regressing process framework mention existing loss functions case situation output network exact values regression need define suitable loss function guide learning process toward finding desired solution turn problem regression form classification problem using losses like cross entropy achieve desirable solutions like results obtained generator adversarial netwok gan frameworks conclusions future works paper interpretable framework proposed pose invariant shape inverse rendering single input image using autoencoder based mapping suitable low dimensional representations computed data components extracting representations reconstruction final shapes proposed framework one hand makes possible use unlabeled data unsupervised therefore reducing need large sets labeled data training framework hand helps achieve promising detailed reconstructions extraction bases vector shape space large set training data compared predefined models proposed framework two types shape representations used reconstruction point cloud binary volumes obtained results show binary volumes suitable representations complicated deep networks however express blur shapes high dimensional representation express less details compared point cloud representations hand working exact coordinates point clouds easy deep networks linear structures preferred shape reconstructions future attempts one unify inverse rendering framework working point cloud volume shape structures different class objects input image tewari jackson figure comparing viual reconstruction results references aldrian smith inverse rendering faces morphable model ieee transactions pattern analysis machine intelligence baldi hornik neural networks principal component analysis learning examples without local minima neural networks blanz vetter morphable model synthesis faces proceedings annual conference computer graphics interactive techniques acm publishing chang funkhouser guibas hanrahan huang savarese savva song shapenet model repository arxiv preprint deng dong socher imagenet hierarchical image database computer vision pattern recognition cvpr ieee conference ieee dou shah kakadiaris face reconstruction deep neural networks arxiv preprint fouhey gupta zisserman understanding shape via shape attributes arxiv preprint gadelha maji wang shape induction views multiple objects arxiv preprint gao yuille exploiting symmetry manhattan properties object structure estimation single multiple images arxiv preprint garrido casas valgaerts varanasi theobalt reconstruction personalized face rigs monocular video acm transactions graphics tog huang ramesh berg labeled faces wild database studying face recognition unconstrained environments tech university massachusetts amherst october jackson bulat argyriou tzimiropoulos large pose face reconstruction single image via direct volumetric cnn regression arxiv preprint jiang zhang deng liu face reconstruction geometry details single image arxiv preprint kim tewari thies richardt theobalt inversefacenet deep inverse face rendering single image arxiv preprint kim torii okutomi inverse rendering arbitrary illumination albedo european conference computer vision springer krizhevsky sutskever hinton imagenet classification deep convolutional neural networks advances neural information processing systems liu funkhouser interactive modeling generative adversarial network arxiv preprint lun gadelha kalogerakis maji wang shape reconstruction sketches via convolutional networks arxiv preprint marschner greenberg inverse rendering computer graphics cornell university parkhi vedaldi zisserman deep face recognition bmvc vol patow pueyo survey inverse rendering problems computer graphics forum vol wiley online library paysan knothe amberg romdhani vetter face model pose illumination invariant face recognition advanced video signal based surveillance avss sixth ieee international conference ieee piotraschke blanz automated face reconstruction multiple images using quality measures proceedings ieee conference computer vision pattern recognition poggio koch problems early vision computational theory analogue networks proceedings royal society london biological sciences rezende eslami mohamed battaglia jaderberg heess unsupervised learning structure images advances neural information processing systems richardson sela kimmel face reconstruction learning synthetic data vision fourth international conference ieee richardson sela kimmel learning detailed face reconstruction single image arxiv preprint savran sankur akarun bosphorus database face analysis biometrics identity management skoglund face modelling analysis master thesis technical university denmark dtu kgs lyngby denmark tewari kim garrido bernard theobalt mofa deep convolutional face autoencoder unsupervised monocular reconstruction arxiv preprint thies zollhofer stamminger theobalt face capture reenactment rgb videos proceedings ieee conference computer vision pattern recognition tran hassner masi medioni regressing robust discriminative morphable models deep neural network arxiv preprint wang che galeotti horvath gorantla stetten ultrasound tracking using probesight camera pose estimation relative external anatomy inverse rendering prior surface map applications computer vision wacv ieee winter conference ieee wolpert macready free lunch theorems optimization ieee transactions evolutionary computation wood morency robinson bulling morphable eye region model gaze estimation european conference computer vision springer song khosla zhang tang xiao shapenets deep representation volumetric shapes proceedings ieee conference computer vision pattern recognition deng automatic speech recognition deep learning approach springer zhu lei liu shi face alignment across large poses solution proceedings ieee conference computer vision pattern recognition
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computation optimal control problems terminal constraint via variation evolution sheng zhang liao fei liao abstract enlightened inverse consideration stable dynamics evolution variation evolving method vem analogizes optimal solution equilibrium point dynamic system solves asymptotically evolving way paper compact version vem developed computation optimal control problems ocps terminal constraint corresponding evolution partial differential equation epde describes variation motion towards optimal solution derived optimality conditions established explicit analytic expressions costates lagrange multipliers adjoining terminal constraint related states control variables presented method field pde numerical calculation epde discretized problems ivps solved common ordinary differential equation ode numerical integration methods key words optimal control dynamics stability variation evolution problem optimality condition introduction optimal control theory aims determine inputs dynamic system optimize specified performance index satisfying constraints motion system closely related engineering widely studied complexity optimal control problems ocps usually solved numerical methods various numerical methods developed generally divided two classes namely direct methods indirect methods direct methods discretize control state variables obtain nonlinear programming nlp problem example direct shooting method classic collocation method methods easy apply whereas results obtained usually suboptimal optimal may infinitely approached indirect methods transform ocp problem bvp optimality conditions typical methods type include indirect shooting method novel symplectic method although precise indirect methods often suffer significant numerical difficulty due hamiltonian dynamics stability costates dynamics adverse states dynamics recent development representatively method blends two types methods unifies nlp bvp dualization view methods inherit advantages types blur difference theories control field often enlighten strategies optimal control computation example variable transformation reduce variables recently new variation evolving method vem enlightened states evolution within stable dynamic system proposed optimal control computation first author computational aerodynamics institution china aerodynamics research development center mianyang china zszhangshengzs vem also synthesizes direct indirect methods new standpoint evolution partial differential equation epde describes evolution variables towards optimal solution derived viewpoint variation motion optimality conditions asymptotically met frame refs besides states controls costates also employed developing epde increases complexity computation ref proposed compact version vem uses original variables handles class ocps free terminal states paper vem developed accommodate ocps terminal constraint corresponding evolution equations derived throughout paper work built upon assumption solution optimization problem exists describe existing conditions purpose brevity relevant researches theorem documented following first principle vem results regarding ocps without terminal constraint reviewed vem ocps terminal constraint developed course equivalent optimality conditions established explicit analytic solution costates lagrange multipliers classic treatment obtained later illustrative examples solved verify effectiveness method preliminaries principle vem vem newly developed method optimal solutions originates dynamics stability theory control field lemma small adaptation autonomous dynamic system like state time derivative vector function let contained within domain equilibrium point satisfies exists continuously differentiable function constant asymptotically stable point lemma aims dynamic system states may directly generalized case lemma dynamic system described presented equivalently pde form denotes variation operator denotes partial differential operator independent variable function vector vector function let contained within certain function set equilibrium function satisfies exists continuously differentiable functional constant asymptotically stable solution system dynamics theory stable dynamics may construct monotonously decreasing function functional achieve minimum equilibrium reached inspired consider inverse problem performance index function derive dynamics minimize performance index optimization problems right platform practice thought optimal solution analogized equilibrium dynamic system anticipated obtained asymptotically evolving way accordingly virtual dimension variation time introduced implement idea variable evolves optimal solution minimize performance index within dynamics governed variation dynamic evolution equations fig illustrates variation evolution process vem solving ocp variation motion initial guess variable evolve optimal solution fig illustration variable evolving along variation time vem vem bred idea demonstrated unconstrained problems ocps free terminal states variation dynamic evolution equations derived frame vem may reformulated epde evolution differential equation ede replacing variation operator partial differential operator differential operator since right function epde depends time suitable solved method field pde numerical calculation discretization along normal time dimension epde transformed problem ivp solved mature ordinary differential equation ode integration methods note resulting ivp defined respect variation time normal time results ocps free terminal states clear development first results ocps free terminal states obtained frame vem recalled problem performance index bolza form subject dynamic equation time state vector elements belong control vector elements belong function partial derivatives continuous respect function partial derivatives continuous respect vector function partial derivatives continuous lipschitz initial time fixed terminal time free initial boundary conditions prescribed terminal states free find optimal solution minimizes arg min class ocps presuming already feasible initial solution states control variables satisfies eqs variation dynamic evolution equations derived vem kpu dimensional matrix positive constant dimensional state transition matrix time point time point satisfies use partial differential operator differential operator variation dynamic evolution equations may reformulated get following epde ede kpu definite conditions including initial guess initial feasible solution solution determined eqs satisfy eqs optimality conditions problem without employment costates proved equivalent traditional ones costates iii vem ocps terminal constraint problem definition paper consider ocps terminal constraint defined problem performance index bolza form subject dynamic equation time state vector elements belong control vector elements belong function partial derivatives continuous respect function partial derivatives continuous respect vector function partial derivatives continuous lipschitz initial time fixed terminal time free initial terminal boundary conditions respectively prescribed dimensional vector function continuous partial derivatives find optimal solution minimizes arg min compared ocps free terminal states defined problem problem covers problem may considered special case derivation variation dynamic evolution equations instead circumventing problem constructing equivalent unconstrained functional problem extremum address problem way similar ref also consider problem within feasible solution domain solution satisfies eqs first transform bolza performance index equivalent lagrange type partial derivatives notated differentiating respect variation time gives partial derivatives form column vector matrix jacobi matrixes solutions related need satisfies following variation equation initial condition note matrixes linearized feasible solution linear equation zero initial value thus according linear system theory solution may explicitly expressed use follow derivation ref obtain different ocps free terminal states use variation dynamic evolution equations achieve way terminal constraint guaranteed find solution guarantees also satisfies variation equation theorem following variation dynamic evolution equations guarantees solution stays feasible domain performance index dimensional matrix positive constant defined parameter vector solution linear matrix equation dimensional matrix dimension vector gxf kpu moreover evolution equations occurs optimal value satisfies dimensional matrixes dimension vectors proof derive eqs though optimization theory reformulate constrained optimization problem subject note decision variable decision parameter since minimum optimization problem may negative infinity penalize large decision variable parameter introduce another performance index formulate optimization problem mop min gxf use weighting method solve pareto optimal solution mop resulting performance index get solution minimizes get solution minimize otherwise get compromising solution mop obviously case pareto optimal solution value performance indexes cases compromising solution guarantees set introduce lagrange multiplier adjoin constraint may get unconstrained optimization problem use optimality conditions may get eqs substitute eqs gxf deduction gives gxf gxf thus definition determines obtained furthermore may reformulated eqs holds means occurs eqs hold optimal value since may arbitrary matrix may arbitrary positive constant consider three case iii dimensional identity matrix comparing three cases substituting specific values may obtain irrelevant specific value regarding linear equation assuming control satisfies controllability requirement solution guaranteed invertible parameter may calculated equivalence classic optimality conditions actually eqs optimality conditions problem show equivalent traditional ones costates adjoining method may constructed functional costate variable vector lagrange multiplier parameter variation may derived hamiltonian transversality conditions theorem problem optimality conditions given eqs equivalent optimality conditions given proof define quantity obviously simplified differentiate respect process use leibniz rule property dimensional identity matrix means conforms dynamics costates furthermore use eqs hold feasible solution domain compare may arbitrary small achieve extremal condition eqs since generally hold arbitrary conclude implies sameness also conforms boundary conditions costates relation established therefore eqs identical investigating optimality condition found optimal control related future state thus optimal feedback control law analytic form exists ocps theorem actually got explicit analytic expression costates lagrange multipliers classic treatment formerly obtained numerically solving bvp proof theorem variables evolving direction using vem easy determine theorem solving ivp respect defined variation dynamic evolution equations feasible initial solution satisfy optimality conditions problem proof lemma lyapunov functional may claim minimum solution peoblem asymptotically stable solution within feasibility domain dynamics governed eqs feasible initial solution evolution dynamics maintains feasibility variables also guarantee functional decrease occurs due asymptotical approach determines optimal conditions namely eqs presume already feasible initial solution satisfy eqs theorem guarantees variation dynamic evolution equations may used obtain optimal solution minimizes formulation epde use partial differential operator differential operator reformulate variation dynamic evolution equations may get epde ede put perspective definite conditions initial feasible solution eqs realize anticipated variable evolving along variation time depicted fig initial conditions belong feasible solution domain value represents optimal solution ocp right part epde also vector function time thus may apply method discretize along normal time dimension use ode integration methods get numerical solution already pointed previously problem contains problem thus results developed paper general meaning see evolution equations terminal states free degraded case ref moreover paper considers ocps defined free terminal time yet results obtained also applicable simpler case fixed terminal time ocps equation regarding terminal time necessary anymore evolution equations regarding state variable control variable still similar simplified gxf gxf kpu illustrative examples first linear example taken xie considered example consider following dynamic system find solution minimizes performance index boundary conditions initial time terminal time fixed solving example using vem epde derived particular problem constant may calculated matrix set definite conditions epde initial guess states control obtained numerical integration achieve feasibility designed following control law damp parameter frequency parameter set using method time horizon discretized uniformly points thus dynamic system states obtained ocp transformed ivp ode integrator matlab default relative error tolerance default absolute error tolerance employed solve ivp comparison analytic solution solving bvp presented figs show evolving process solutions optimal respectively numerical solutions indistinguishable optimal shows effectiveness vem fig plots profile performance index value variation time declines rapidly first almost reaches minimum keeps approaching analytic minimum monotonously addition may compute exactly identical proved theorem analytic solution numerical solutions vem fig evolution numerical solutions optimal solution optimal solution numerical solutions vem fig evolution numerical solutions optimal solution optimal solution numerical solutions vem fig evolution numerical solutions optimal solution minimum funcitonal value numerical solution fig approach minimum performance index consider nonlinear example free terminal time brachistochrone problem describes motion curve fastest descending example consider following dynamic system sin cos gravity constant find solution minimizes performance index cos boundary conditions example fixed terminal position boundary conditions free terminal velocity thus gxf specific form epde ede parameters set respectively definite conditions obtained physical motion along straight line connects initial position terminal position also discretized time horizon uniformly points thus large ivp states including terminal time obtained still employed matlab numerical integration integrator setting default relative error tolerance absolute error tolerance respectively comparison computed optimal solution radau method based ocp solver fig gives states curve coordinate plane showing numerical results starting straight line approach optimal solution time control solutions plotted fig asymptotical approach numerical results demonstrated fig terminal time profile variation time plotted result declines rapidly first gradually approaches minimum decline time changes slightly compute vem close result fig presents evolution profiles lagrange multipliers also approach optimal value rapidly optimal solution numerical solutions vem fig evolution numerical solutions coordinate plane optimal solution optimal solution numerical solutions vem fig evolution numerical solutions optimal solution minimum decline time evolving profile fig evolution profile minimum decline time corresponding corresponding fig evolution profiles lagrange multipliers conclusion paper developes compact variation evolving method vem address computation optimal control problems ocps terminal constraint set general evolution equations derived optimality conditions established ocps prescribed terminal boundary conditions especially equivalence proof optimality conditions even costates completely considered derivation analytic expressions related states control variables obtained analytic relations lagrange multipliers adjoin terminal constraints original variables also explicitly got results may help deepen understanding towards optimal control theory however currently proposed method requires initial solution feasible inflexible especially problem terminal constraint studies carried address issue references pesch plail maximum principle optimal control history ingenious ideas missed opportunities control cybernetics vol betts survey numerical methods trajectory optimization guid control vol lin loxton teo control parameterization method nonlinear optimal control survey journal industrial management optimization vol hargraves paris direct trajectory optimization using nonlinear programming collocation guid control vol stryk bulirsch direct indirect methods trajectory optimization ann oper vol peng gao zhong symplectic approaches solving problems guid control vol rao survey numerical methods optimal control proc astrodynam specialist pittsburgh aas paper garg patterson hager rao unified framework numerical solution optimal control problems using pseudospectral methods automatica vol ross fahroo perspective methods trajectory optimization proc astrodynam monterey aiaa paper ross fahroo pseudospectral methods optimal motion planning differentially flat systems ieee trans autom control vol zhang yong qian variation evolving method optimal control arxiv arxiv zhang qian computation control problem variation evolution principle zhang variation evolving optimal control computation compact way hartl sethi vickson survey maximum principles optimal control problems state constraint siam vol khalil nonlinear systems new jersey usa prentice hall zhang shen computational fluid dynamics fundamentals applications finite difference methods beijing china national defense industry press bryson applied optimal control optimization estimation control washington usa hemisphere zhen linear system theory beijing china tsinghua university press hull optimal control theory applications new york usa springer wang luo zhu advanced mathematics changsha china national university defense technology press xie optimal control theory application beijing china tsinghua univ press sussmann willems brachistochrone problem modern control theory bonnard gauthier eds contemporary trends nonlinear geometric control theory applications singapore world scientific patterson rao amatlab software solving optimal control problems using gaussian quadrature collocation methods sparse nonlinear programming acm trans math software vol
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type system theory intensional logic support variable bindings hybrid programs gipsy joey paquet concordia university montreal canada paquet jun serguei mokhov concordia university montreal canada mokhov abstract describe type system platform called general intensional programming system gipsy designed support intensional programming languages built upon intensional logic imperative intensional execution model gipsy type system glues static dynamic typing intensional imperative languages compiler runtime environments support intensional evaluation expressions written various dialects intensional programming language lucid intensionality makes expressions explicitly take account multidimensional context evaluation context value serves number applications need notion context proceed describe discuss properties type system related type theory well particularities semantics design implementation gipsy type system keywords intensional programming imperative programming type system type theory general intensional programming system gipsy context introduction gipsy ongoing effort development flexible adaptable programming language development framework aimed investigation lucid family intensional programming languages using platform programs written many flavors lucid compiled executed framework approach adopted aimed providing possibility easily developing compiler components languages intensional nature execute system lucid functional language programs executed distributed processing environment however standard lucid algebra types operators extremely hardly benefit parallel evaluation operands adding granularity data elements manipulated lucid inevitably comes addition data types corresponding operators lucid semantics defined typeless solution problem consists adding hybrid counterpart lucid allow external language define algebra types operators turn solution raises need elaborated type system bridge implicit types lucid semantics explicit types operators functions defined hybrid counterpart language paper presents type system used gipsy compile time static type checking well dynamic type checking problem statement short presentation proposed solution section presents brief introduction gipsy project beginnings gipsy type system gipsy support hybrid intensional programming lucx context type extension known context calculus contexts act firstclass values present complete gipsy type system used compiler general intensional programming compiler gipc system general eduction engine gee producing executing binary gipsy program called general eduction engine resources geer respectively type system hoil support gipsy mokhov paquet problem statement proposed solution problem statement data types implicit lucid well dialects many functional languages type declarations never appear lucid programs syntactical level data type value inferred result evaluation expression imperative languages like java like types explicit programmers must declare types variables function parameters return values used evaluation gipsy want allow lucid dialect able uniformly invoke written imperative languages way around perform semantic analysis form type assignment checking statically compile time dynamically run time perform type conversion needed evaluate hybrid expressions time need allow programmer specify declare types variables parameters return values intensional imperative functions binding contract invocations despite fact lucid explicitly typed thus need general type system well designed implemented support scenarios proposed solution uniqueness type system presented specific programming language model either lucid family languages imperative languages designed bridge programming language paradigms two general would intensional imperative paradigms gipsy framework designed support common environment intensional imperative languages thus type system generic gipsy program include code segments written theoretically arbitrary number intensional imperative dialects supported system type system specific language follows details proposed solution specification type system introduction gipsy type system introduction jlucid objective lucid general imperative compiler framework gicf prompted development gipsy type system implicitly understood lucid language incarnation within gipsy handle types general manner glue imperative intensional languages within system evolution different lucid variants lucx introducing contexts values jooip java intensional programming highlighted need development type system accommodate general properties intensional hybrid languages matching lucid java data types present case interaction lucid java allowing lucid call java methods brings set issues related data types especially comes type checks lucid java parts hybrid program pertinent lucid variables expressions used parameters java methods java method returns result assigned lucid variable used intensional expression sets types cases exactly basic set lucid data types defined grogono int bool double string dimension lucid int size java long gipsy java double boolean string equivalent lucid string java string simply mapped internally stringbuffer thus one think lucid string reference evaluated intensional program based fact lengths lucid string java string java string also object java however point lucid program direct access string properties though internally may expose later programmers also distinguish float data type system hoil support gipsy mokhov paquet table matching data types lucid java return types java methods types lucid expressions internal gipsy types int byte long float double boolean char string method method object object void int dimension float double bool char string dimension function operator class url bool gipsyinteger gipsyfloat gipsydouble gipsyboolean gipsycharacter gipsystring gipsyfunction gipsyoperator gipsyarray gipsyobject gipsyembed gipsyvoid parameter types used lucid corresponding java types internal gipsy types string float double int dimension bool class url operator function string float double int int string boolean object object method method gipsystring gipsyfloat gipsydouble gipsyinteger dimension gipsyboolean gipsyobject gipsyembed gipsyarray gipsyoperator gipsyfunction type floating point operations dimension index type said integer string far dimension tag values concerned might types eventually discussed therefore perform data type matching presented table additionally allow void java return type always matched boolean expression true lucid expression always evaluate something types mapping restrictions per table mapping table type adapter table would exist mapping language back type adapter simple theory gipsy types simple theory gipsy types stgt based simple theory types stt mendelson theoretical practical considerations described sections follows stt partitions qualification domain ascending hierarchy types every individual value assigned type type assignment dynamic intensional dialects resulting type value intensional expression may known compile time assignment types constant literals done however hybrid system mostly statically typed boundaries type assignment also occurs boundary intensional programming languages ipls imperative languages prior type system hoil support gipsy mokhov paquet imperative procedure invoked type assignment procedure parameters ipl expression computed dynamically matched type mapping table similar table subsequently procedure call returns back ipl type imperative expression matched back table assigned intensional expression expects table used call made procedure ipl back reverse order stt quantified variables range one type making logic applicable underlying logic theory also means elements domain type stt states atomic type member elements members basic atomic type type next higher type similarly succ peano arithmetic next operator lucid also consistent describe composite types arrays objects recursively decomposed flattened see primitive types stt applies symbolically stt uses primed unprimed variables infix set notation formulas rely fact unprimed variables type similar notion lucid stream elements stream type primed variables stt range next higher type two atomic formulas stt form identity set membership stt defines four basic axioms variables types range identity extensionality comprehension infinity add intensionality fifth axiom variables definition identity relationship extensionality comprehension axioms typically range elements one two nearby types set membership unprimed variables range lower type hierarchy appear left conversely primed ones range higher types appear right axioms defined identity extensionality comprehension covers objects arrays collection elements may form object next higher type stt states comprehension axiom schematic respect types infinity xry yrx exists binary relation elements atomic type transitive irreflexive strongly connected intensionality intensional types operators based intensional logic context calculus extensively described works related present type system accommodates two common hybrid execution environment gipsy context finite subset relation dim dim set possible dimensions set possible tags types types types types generally referred kinds kinds provide categorization types similar nature type systems provide kinds first class entities available programmers gipsy expose functionality type system point however implementation level provisions may later decide expose use programmers internally define several broad kinds types presented sections follow type system hoil support gipsy mokhov paquet numeric kind primitive types category numerical values represented gipsyinteger gipsyfloat gipsydouble provide implementation common arithmetic operators addition multiplication well logical comparison operators ordering equality thus numerical type following common operators provided resulting type arithmetic operator largest two operands terms length range double length say covers range int length say appear arguments operator resulting value type without loss information largest length double result logical comparison operators always boolean regardless length numerical types tmax tmultiply tmax tdivide tmax tadd tmax tsubtract tmax tmod tmax tpow tmax generalized implementation arithmetic operators done realization interface called iarithmeticoperatorsprovider concrete implementation developed general delegate class genericarithmeticoperatorsdelegate design implementation allow exposure values later several iterations refinement also allow operator type overloading replacement type handling implementation altogether researchers wish implementation engine gee thus changed refer interface type implementation dealing operators equivalently logic comparison operators ilogiccomparisonoperatorsprovider class corresponding genericlogiccomparisonoperatorsdelegate class latter relies comparator implemented numerical kind numericcomparator using comparators classes implement standard comparator interface allows java use optimize buildin sorting searching algorithms collections types case class called genericlogiccomparisonoperatorsdelegate implementation delegate class also relies example numeric types described design figure important mention grouping numeric kind integers numbers violate ieee standard kinds wrap corresponding java types also grouped numeric kind semantics including implementation ieee java accordance java language specification type system hoil support gipsy mokhov paquet figure example provider interfaces comparators logic kind similarly numeric types primitive type gipsyboolean fits category types used boolean expressions operations type provides expect arguments type boolean following set operators logic type provide gipsy type system band bor bnot bxor bnand bnor note logical xor operator denoted different corresponding bitwise operator section similarly bitwise logical similarly generalized implementation arithmetic operators logic operator providers implement ilogicoperatorsprovider interface general implementation delegate class genericlogicoperatorsdelegate bitwise kind bitwise kind types covers types support bitwise operations entire bit length particular type types category include numerical logic kinds described earlier type system hoil support gipsy mokhov paquet section section parameters sides operators resulting type always implicit compatible type casts performed unlike numeric kind generalized implementation kind operators done interface called ibitwiseoperatorsprovider generically implemented corresponding delegate class genericbitwiseoperatorsdelegate composite kind name suggests composite kind types consist compositions types possibly basic types examples kind arrays structs abstract data types realization objects collections studied type system gipsyobject gipsyarray gipsyembed kind characterized constructors dot operator decide membership well invoke member methods define equality design types like entire type system adheres composite design pattern prominent examples kind figure including gipsycontext composed dimensions indirectly tagsets figure composite types intensional kind intentional types kind primarily encapsulation dimensionality context information operators represented types gipsycontext dimension types mentioned please refer type system hoil support gipsy mokhov paquet common operators types include context switching querying operators well context calculus operators additional operators included depending intensional dialect used mentioned operators said baseline operators intensional language translated use icontextsetoperatorsprovider interface general implementation genericcontextsetoperatorsdelegate context calculus operators simple contexts include standard set operators union difference intersection issubcontext projection hiding override function kind function types represent either functional functions imperative procedures binary unary operators passed parameters returned results type system represented gipsyfunction gipsyoperator gipsyembed common operators include equality evaluation describing gipsy types properties show overview properties gipsy types get better complete understanding spectrum behavior cover light description comparison existential union intersection linear types existential types presented gipsy types existential types represent encapsulate modules abstract data types separate implementation public interface specified abstract gipsytype defined follows object object getenclosedtypeobject object getvalue implemented following manner boolt boolean object getenclosedtypeobject boolean getvalue intt long object getenclosedtypeobject long getvalue doublet double object getenclosedtypeobject double getvalue correspond subtypes abstract general existential type assuming value well typed regardless gipsytype may union types union two types produces another type valid range values valid either two however operators defined union types must valid types remain classical example range signed char range unsigned char thus signed char unsigned char union type roughly corresponds notion enforce operations possible union type must possible left right types type system hoil support gipsy mokhov paquet uniting operator class hierarchy gipsy union type among type parent parent class thus specific type system following holds gipsytype gipsytype gipsyarray gipsyobject gipsyobject gipsyvoid gipsyboolean gipsyboolean gipsyoperator gipsyfunction gipsyfunction concrete gipsy type collection types gipsy type system describing equivalently union two sibling types common parent class inheritance hierarchy interestingly enough explicitly expose kinds types still able following union type relationships defined based kind operators provide siblings common interface iarithmeticoperatorprovider iarithmeticoperatorprovider ilogicoperatorprovider ilogicoperatorprovider ibitwiseoperatorprovider ibitwiseoperatorprovider icontextoperatorprovider icontextoperatorprovider icompositeoperatorprovider icompositeoperatorprovider ifunctionoperatorprovider ifunctionoperatorprovider concrete gipsy type collection types provide arithmetic operators logic operators providers bitwise operators providers context operators providers composite operator providers function operator providers thus gipsyinteger gipsyfloat gipsydouble gipsyboolean gipsyinteger gipsyfloat gipsydouble gipsyboolean gipsycontext dimension gipsyobject gipsyarray gipsyembed gipsystring gipsyfunction gipsyoperator gipsyembed another particularity gipsy type system union string integer types dimension gipsyinteger gipsystring dimension dimension tag values allow either integers strings common union majority type system share common set tag set operators defined ordered finite tag sets next intersection types intersection type given two types range sets valid values overlap types safe pass methods functions expect either types intersection types restrictive compatible original types classical example intersection type implemented would signed char unsigned char intersection types also useful describing overloaded functions sometimes called refinement types class hierarchy intersection parent child classes derived type intersection sibling classes empty functionality offered type system hoil support gipsy mokhov paquet intersection types promising currently explicitly implicitly considered gipsy type system planned future work linear types linear uniqueness types based linear logic main idea types values assigned one one reference throughout types useful describe immutable values like strings hybrid objects see details useful operations object destroys creates similar object new values therefore optimized implementation mutation implicit examples types gipsy type system gipsystring internally relies java stringbuffer something similar well immutable gipsyobject jooip immutable gipsyfunction since either copy retain values warehouse therefore one violate referential transparency create side effects time efficient need worry synchronization overhead conclusion series discussions specification design implementation details presented type system used gipsy static dynamic type checking evaluation intensional hybrid languages highlighted particularities system attribute particular specific language traditionally type systems entire set languages hybrid paradigms linked type system pointed limitations type system projected work remedy limitations overall necessary contribution systems homogeneous environment statically dynamically intensional hybrid programs future work type system described paper implemented gipsy however due recent changes including introduction contexts values generic lucid well development hybrid language jooip implementation still needs thorough testing using complex program examples testing limits type system additional endeavours noted previous sections paper include exposing operators composite types lucid code segments allowing custom types extension existing operators operator overloading expose kinds first class entities allowing programs explicit manipulation types consider adding intersection types flexibility future development type system allowing type casting possibilities programming level acknowledgments work funded nserc faculty computer science engineering concordia university type system hoil support gipsy mokhov paquet references ashcroft faustini jagannathan wadge multidimensional declarative programming oxford university press london ashcroft wadge erratum lucid formal system writing proving programs siam grogono gipc increments technical report department computer science software engineering concordia university montreal canada apr grogono mokhov paquet towards jlucid lucid embedded java functions gipsy proceedings international conference programming languages compilers plc las vegas usa pages csrea press june ieee ieee standard arithmetic online http developing distributed component framework processing intensional programming languages phd thesis department computer science software engineering concordia university montreal canada mar mendelson introduction mathematical logic chapman hall edition mokhov paquet general imperative compiler framework within gipsy proceedings international conference programming languages compilers plc las vegas usa pages csrea press june mokhov paquet objective lucid first step intensional programming gipsy proceedings international conference programming languages compilers plc las vegas usa pages csrea press june mokhov towards hybrid intensional programming jlucid objective lucid general imperative compiler framework gipsy master thesis department computer science software engineering concordia university montreal canada isbn ostrum luthid manual department computer science university waterloo ontario canada paquet scientific intensional programming phd thesis department computer science laval university canada paquet grogono towards framework general intensional programming compiler gipsy proceedings annual acm conference programming systems languages applications oopsla vancouver canada acm paquet kropf gipsy architecture proceedings distributed computing web quebec city canada paquet mokhov tong design implementation context calculus gipsy environment proceedings annual ieee international computer software applications conference compsac pages turku finland july ieee computer society paquet gipsy platform investigation intensional programming languages proceedings international conference programming languages compilers plc pages las vegas usa june csrea press plaice mancilla ditu wadge sequential evaluation eager translucid proceedings annual ieee international computer software applications conference compsac pages turku finland july ieee computer society rahilly plaice multithreaded implementation translucid proceedings annual ieee international computer software applications conference compsac pages turku finland july ieee computer society tong design implementation context calculus gipsy master thesis department computer science software engineering concordia university montreal canada apr tong paquet mokhov context calculus gipsy unpublished vassev paquet generic framework migrating demands gipsy type system hoil support gipsy mokhov paquet tion engine proceedings international conference programming languages compilers plc pages las vegas usa june csrea press wadge ashcroft lucid dataflow programming language academic press london wan lucx lucid enriched context phd thesis department computer science software engineering concordia university montreal canada paquet mokhov intensional programming new concept objectoriented intensional programming domains unpublished paquet intensional programming gipsy preliminary investigations proceedings international conference programming languages compilers plc pages las vegas usa june csrea press
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dec note testing intersection convex sets sublinear time israela solomon february present simple sublinear time algorithm testing following geometric property let convex sets polytopes balls etc assume complexity set depends number sets test property exists common point sets intersection nonempty goal distinguish case intersection nonempty case even removing many sets intersection empty particular algorithm returns pass sets intersect returns sets fail probability least point belongs given call collection convex sets piq order elements tuple important motivation algorithm comes following well known theorem eduard helly helly theorem let finite collection convex sets intersection every sets nonempty whole collection nonempty intersection helly theorem inspires consider following algorithm algorithm tester nonempty intersection pick uniform distribution pij return fail end end return pass theorem algorithm returns pass intersection sets nonempty returns fail probability least sets empty running time algorithm intersection every depend order prove theorem show sets empty intersection probability picking whose intersection empty returning fail high purpose use following theorem katchalski liu fractional helly theorem let integers let family convex sets intersecting point least sets setting obtain following corollary corollary maximal number sets intersect less less intersect namely least empty intersection ready prove correctness algorithm proof theorem intersection sets nonempty intersection every sets nonempty algorithm returns pass assume intersection every sets empty lary less choices sets intersection nonempty means probability pick sets returning fail round smaller therefore probability algorithm returns fail least setting probability return fail least running time algorithm times time takes compute intersection convex sets note general complexity convex set bounded since assume complexity set depends running time depend acknowledgments would like thank ronitt rubinfeld interest suggesting write note references helly ber mengen konvexer krper mit gemeinschaftlichen punkte jahresbericht der deutschen url http katchalski liu problem geometry proceedings american mathematical society url http wikipedia helly theorem https url zhang notes helly theorem url http
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journal computer science engineering volume issue may metaheuristic approach projects portfolio optimization shashank pushkar abhijit mustafi akhileshwar mishra optimal selection interdependent projects implementation multi periods challenging framework real option valuation paper presents mathematical optimization model portfolio projects model optimizes value portfolio within given budgetary sequencing constraints period sequencing constraints due time wise interdependencies among projects metaheuristic approach well suited solving kind problem definition paper genetic algorithm model proposed solution optimization model solution approach help managers taking optimal funding decision projects prioritization multiple sequential periods model also gives flexibility managers generate alternative portfolio changing maximum minimum number projects implemented sequential period projects portfolio management optimization financial evaluation genetic algorithm real option introduction projects investment decisions crucial firm implement widely used discounted cash flow dcf method appropriate evaluate investments designed projects option features inability discounted cash flow analysis take care impact flexibility underlying investment decisions forced managers rely gut real option analysis alternative approach incorporates impact flexibility evaluating projects project evaluation method used based quantification projects benefits costs well risk volatility cash flows research scrutinized relevance real option analysis investments optimizing portfolio projects dickinson introduces optimization model interdependent technology projects framework real option valuation portfolio projects involve interdependencies create multiple options channes propose approach real option valuation portfolio real investment projects bardhan proposes integer programming model get optimal sequence implementation proposes approach evaluating software project chen propose portfolio selection model mixed give framework shashank pushkar birla institute technology mesra ranchi number mustafi birla institute technology mesra ranchi number mishra national institute technology department computer application jamshedpur jharkhand investment basis real option mean variance theory verhoef quantifies yields risk portfolio shashank proposes dynamic programming solution class problem works smaller portfolio increaded number projects variable becomes large curse dimensionality paper gives simplified mathematical model optimize projects portfolio projects sequentially interdependent important characteristic projects model dynamically calculates option values project due dependent projects implemented period subsequent genetic algorithms proven track record handling large search field problems simple act crossover mutation allows algorithm search large search space converge quickly optimal solution case exceptionally large search spaces algorithm provides option terminated fixed number iterations may provide best solution cases usually gives good approximation solution proposed solution yields optimum sequence implementation projects maximum overall portfolio value across multiple time periods help managers taking optimal funding decisions rest paper organized follows section specifies problem mathematical model defined section genetic algorithm solution given section illustrative numerical example given section finally conclude section jcse http problem specification problem undertaken comes category project valuation investment decisions project portfolio management problem applies firm decides implement wants invest multiple projects collection projects portfolio projects implemented sequential periods projects implemented separately two types dependency follows total dependency partial dependency dependency total dependency project project indicates capabilities developed project also required project creates option implement implemented either together implementation implementation implemented implementation call option value added dcf value project according real option analysis partial dependency project indicates capabilities developed project supports enhances capabilities required project strict requirement project implemented together implementation implemented without implementation benefit level would reduce fraction depending level dependency possible dependencies among projects fig investment projects solid arrow indicates total dependency totally dependent dashed arrow indicates partial dependency partially dependent project dependency projects independent without arrow projects portfolio could implemented simultaneously due constraints budget firm uncertainty regarding success projects customer response implementation success within firm etc projects generally implemented sequential periods period may ferent budget projects implemented initial stage period provides opportunities option remaining unimplemented projects dependent implemented ones thus initial implementation projects provide flexibility managers decide whether implement remaining projects portfolio seeing response customers success projects implementation within firm projects generally share technology firm share risk projects implemented early stages periods relax uncertainties remaining unimplemented projects according real option valuation methodologies projects create option dependent projects implemented subsequent periods hence option value dependent projects must added real option value total value project dcf value plus real option values due dependent projects implemented subsequent periods important note real option value dependent projects added option generating project either implemented together problem requires formulated mathematical model optimization projects portfolio across multiple sequential periods overall value portfolio maximized mathematical model notation cik set given projects number project total number periods completing implementation projects budget period rate interest present value cost project implemented period cost project funded period means project funded beginningof first period set projects directly dependent project dij level dependence project project defined follows dij directly dependent dij partially dependent dij totally dependent rik present value return project implemented period rit min return project end period rit xik xik expected number periods project going give return period implementation project binary value indicating implementation period project xik project implemented wik vij discounted cash flow dcf value project implemented period net present option value project attributed dependent projects implemented period option value project due project yijk vij implemented prior implementation vik otherwise min minimum number projects implemented period max number periods implemented period problem statement follows notation vik vij dij xik wik xik rik cik xik problem maximize number objective function indicates real option value project due project added implemented prior implementation observe additional sequencing constraint required maximizing objective function automatically take care sequencing genetic algorithm based solution implemented period otherwise max projects implemented period vik wik subject constraints budget constraint genetic algorithms optimization techniques based natural theory survival fittest operators involved tend heavily inspired natural selection consequently successive generations algorithm continue propagate best traits population leads rapid convergence search also introduction mutation operator ensures diversity neglected search trapped local maximum flow chart illustrating basics steps based optimization given fig chromosome structure chosen represent problem sequence bits projects periods representative bit sequence portfolio seven projects completed three periods would consequently bit sequence easy visualize case terms integer numbers number range would binary representation one possible chromosomes defined chromosome divided equilength sequences number periods consideration every set bit subsequences would represent project completed particular period introduced search search space optimize npv portfolio table option values due dependency among projects fig flowchart based optimization numerical example seven projects portfolio taken illustrative example interdependencies among projects shown figure section paper results simulated using genetic algorithm tollbox project planning horizon taken periods data portfolio listed tables table projects project present value cost period present value return period note present value costs returns projects implemented period taken shake simplicity table dependency level among projects note option values dependent projects taken simplicity calculated using nested option model option model benaroach project due dependent project budgets three periods qkmin qkmax table results selected projects funding costs projects budget portfolio value solution given section paper applied example portfolio results obtained tabulated table convergence algorithm shown fig results show projects provide infrastructure many projects thus high option values selected early periods funding option value deferred later periods also indicates maximization option component portfolio value selects projects maximum number dependent projects earlier projects less dependent projects dependent since projects types portfolio shares risk successful early implementation high tion projects would lower overall risk success portfolio model yields optimum value overall portfolio along periods funding projects portfolio method calculates option values project due dependent projects dynamically thus represents significant improvement existing models prioritization projects real option valuation framework approach help managers taking optimal funding decisions final demonstration best chromosome representative case used paper presented chromosome represents periods activity column set bits representing projects completed ideally particular period solution clearly demonstrates findings table fig best chromosome case study convergence best solution concluding remark developed multi period portfolio optimization model model uses calculate option values project due dependent projects implemented subsequently improvement existing works option values calculated statically projects due dependent projects dynamically time implementation decision making meta heuristic nature solution yields optimal sequence implementation projects multiple periods get maximum overall portfolio value proposed algorithm suitable solve problem modeled multistage optimization problem makes possible calculate option values project due subsequently implemented dependent projects maximize overall portfolio value model also gives flexibility managers generate alternative portfolio changing maximum minimum number projects funded period research work extended incorporating fuzziness model terms like dependency level among projects level benefits etc uncertain may change changing decision time references benaroach kauffman case using real options pricing analysis evaluate information technology project system research benaroach kauffman electronic network expansion using real option analysis mis quarterly fitchman real options platform adoption implications theory practice kambil henderson mohsenzadeh strategic management information technology investments banker kauffman mahmood strategic information technology management perspective organizational growth competitive advantage harrisburg idea group benaroach managing information technology investment risk real options perspective journal management information system schwartz uncertainty information technology acquisition development projects dickinson thomton graves technology portfolio management optimizing interdependent projects multiple time periods ieee transactions engineering management bardhan sougtad bagchi prioritization portfolio information technology projects journal management information systems bardhan sougtad bagchi real option approach prioritizing portfolio information technology projects case study utility sprague proc hawaii intl conf sys ieee los alamitos cobb charnes msimulations optimizations real option valuation proceedings winter simulation bardhan kauffman narapanawe optimizing project portfolio time wise interdependencies ieee computing society press los alamitos costa barros travassos evaluating software project portfolio risks journal systems software fang chen fukusima mixed projects scurities portfolio selection model european journal operation research ong management information technology investment framework based real options theory perspective technovation peters verhoef quantifying yield portfolios science computer programming shashank pushkar sharma mishra dynamic programming approach optimize portfolio interdependent international journal artificial intelligent systems machine learning trigeorrgis real options cambridge mit press goldberg genetic algorithms search optimization machine learning professional ist edition paulinas usinskas survey genetic algorithms applications image enhancement segmentation information technology control vol benaroch shah jeffery valuation multistage investments embeddingnested compound real options workingpaper syracuse university syracuse shashank pushkar lecturer department computer science engineering birla institute technology mesra ranchi research interest field information technology project management optimization technique abhijit mustafi mca university north bengal india currently senior lecturer department cse bit mesra india research interests include image processing meta heuristic algorithms web mining akhileshwar mishra phd iit kharagpur also professor computer applications national institute technology jamshedpur specializes computer applications optimizations field industrial management
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optimal design optimal control elastic structures undergoing finite rotations elastic deformations feb ecole normale cachan lmt gce avenue wilson cachan france email technologie compiegne lab roberval gsm compiegne france abstract work deal optimal design optimal control structures undergoing large rotations words show find corresponding initial configuration corresponding set multiple load parameters order recover desired deformed configuration desirable features deformed configuration specified precisely objective cost function model problem chosen illustrate proposed optimal design optimal control methodologies one geometrically exact beam first present formulation optimal design optimal control problems relying method lagrange multipliers order make mechanics state variables independent either design control variables thus provide general basis developing best possible solution procedure two different solution procedures explored one based diffuse approximation response function gradient method one based genetic algorithm number numerical examples given order illustrate advantages potential drawbacks presented procedures keywords structures finite rotation optimization control introduction modern structures often designed withstand large displacements rotations remain fully operational moreover construction phase also mastered placed control trying precisely guide large motion particular component total structural assembly ever increasing demands achieve economical design construction thus require problems kind placed sound theoretical computational basis one explored work namely optimization methods called upon guide design procedure achieve desired reduction mechanical geometric properties similarly control methods employed provide estimate loads minimal effort placing structure component directly optimal desired shape either tasks optimal design optimal control formally presented minimization chosen cost objective function specifying precisely desired goal main difference two procedures concerns choice variables defining cost function design variables typically related mechanical properties young modulus geometry structure particular coordinates initial configuration whereas control variables related actions forces applied structure order place desired position rather insisting upon difference treating optimal design optimal control quite different manners done number traditional expositions subject focus work common features allow unified presentation two problems development novel solution procedure applicable problems latter implies nonlinear mechanics model consideration geometrically exact beam placed central stage one show fully master variation chosen system properties loads order achieve optimal goal main contributions put forward presenting unconventional approach stated follows first present theoretical framework treating nonlinear structural mechanics optimization control coupled nonlinear problem problem optimization control nonlinear mechanics equilibrium equations reduced mere constraint respect admissibility given state structure displacements rotations using classical method lagrange multipliers see mechanics equilibrium equations promoted constraint one governing equations solved coupled problem kind intrinsic dependence state variables displacements rotations respect optimal design control variables eliminated turning variables independent variables clarity idea also developed within framework discrete approximation thus providing finite element model including degrees freedom pertinent displacements rotations also optimal design control variables detailed development presented chosen model problem geometrically exact beam see note passing proposed approach quite opposite traditional ones see recent review two fields directly concerned coupled problems kind nonlinear mechanics one side optimal design control studied developed separately resulting solution procedures one another applied sequential manner one typically employs two different computer programs one mechanics another optimization control toolbox matlab communication requirements reduced bear minimum sensitivity optimization code design control variables finite element code mechanics clear traditional approach analysis design control largely sacrifice computational efficiency cases practical interest cost function mechanics problem nonlinear require iterative procedures solved second aspects elaborated upon work pertains alternative method solving coupled problem analysis either design control two brought level lagrange multiplier method solved simultaneously words interdependence state variables displacements rotations one side design control variables another longer enforced one iterate simultaneously solve particular sensitivity analysis needs longer performed separately naturally integrated part simultaneous iterative procedure important note iterative intermediate values longer consistent equilibrium equations constraint except convergence basically solution obtained standard sequential solution procedure significantly reduced number iterations words simultaneous sequential solution procedures always yield result providing solution unique problems kind concern optimization control geometrically nonlinear response structures bifurcation phenomena occur solving nonlinear problems structural mechanics quite demanding task structures stiffness may differ significantly different deformation modes beam bending versus stretching adding control optimization problem top makes nonlinear problem much challenging different modifications basic solution procedure developed tested including genetic algorithms explore initial phase solution procedure gradient based acceleration near solution points response surface part solution constructed diffuse approximation see outline paper follows next section briefly present chosen model geometrically exact beam capable representing large displacements rotations theoretical formulation optimal control optimal design chosen mechanical model presented section along discrete approximations constructed finite element method given section proposed solution procedure described detail section several numerical examples presented section order illustrate performance various algorithms used computations closing remarks stated section model problem geometrically exact beam section briefly review governing equations chosen model problem structure undergoing large rotations curved beam thorough discussion chosen model refer among others assume initial configuration beam internal force free described position vector identifying position point neutral fiber inextensible fiber pure bending corresponding placement rod carried choosing local triad vectors vector triad kind obtained simply rotating global triad orthogonal usual choice normal coordinates first vector triad orthogonal remaining figure initial deformed configuration geometrically exact beam two placed plane cross section orthogonal tensor becomes known function initial configuration case curved beam studied chosen arc length parameter applying external loading parameterized pseudo implies inertia effects neglected obtain beam deformed configuration defined position vector orthogonal tensor latter accordance usual kinematic hypothesis beam would deform along hypothesis first vector triad remains orthogonal two within plane fully determines see figure initial deformed configuration beam presented short one state configuration space described model beam written spaces vectors special orthogonal tensors respectively main difficulty numerical solution structural mechanics problems featuring beams kind stems presence group configuration space see thorough discussion issues short performing standard task computing virtual work principle consistent linearization small rotation described tensor ought superposed large rotation described orthogonal tensor one must first make use exponential mapping exp exp cos sin cos axial vector tensor complexity last expression sharp contrast respect simple additive update virtual displacement field superposed deformed configuration results presented equivalent form stating tangent space chosen beam model defined strain measures employed beam theory written direct tensor notation form axial shear strains bending torsional strains subsequent equations denote superposed prime derivative respect coordinate initial configuration consider simplest case linear elastic material model beam allows express constitutive equations terms stress resultants diag diag illustration also consider simplest case circular cross section section diameter section area moment inertia polar moment order complete description chosen beam model state equilibrium equations weak form gext gext external virtual work virtual strains latter obtained derivative real strains taking results consideration particular leads also written equivalent form terms corresponding axial vectors external load derives given potential gext one also define governing equilibrium equation chosen model problem geometrically exact elastic beam principle minimum total potential energy defined according min implies denoted second variation total potential energy tangent operator obtained consistent linearization procedure see pertinent details consistent linearization refer quaternion parameterization finite rotations rotation parameters finite rotations alternative case external loading potential may defined weak form rather starting point solution procedure also encountered applications example case follower force follower moment follow motion particular beam node express contribution virtual work according gext follower force moment also contribute tangent operator according gext contribution also taken account computing solution trying ensure quadratic convergence rate coupled optimality problem presented beam model provides excellent basis master optimization problem well control problem geometrically nonlinear elastic structures although former deals geometric characteristics beam latter external loading sequence two problems formulated solved quite equivalent manner shown next optimal design optimal design problem addressed herein pertains selecting desired features mechanical model leaving free choice geometric properties beam model thickness variation chosen initial shape task often referred shape optimization mathematical standpoint shape optimization formulated problem minimization objective cost function specifying desired features latter considered functional depends mechanics state variables also design variables beam thickness shape shape optimization procedure interpreted herein minimization written cost objective functional min contrary minimization total potential energy functional mechanical design variables admissible candidates weak form equilibrium equations satisfied words need deal constrained minimization problem classical shape optimization procedure solving constrained minimization problem carried sequential manner iterative value design variables new iterative procedure must completed leading verifying equilibrium equations considerable computational cost procedure waisted iterating convergence equilibrium equations even values design variables drastically reduced formulating minimization problem using method lagrange multipliers max min lagrange multipliers inserted weak form equilibrium equations instead virtual displacements rotations accordance results presented previous section write explicitly gext main difference respect constrained minimization problem pertains fact state variables design variables considered independent iterated upon solved simultaneously optimality condition associated minimization problem written explicit form last term written denoted well moreover finally also illustration consider diameter circular chosen design variable allows express explicitly result diag diag order provide similar explicit result directional derivative cost function consider simple choice given beam mass equivalently volume constant density case contribution cost function optimality conditions written also consider complex case practical interest shape optimization carried respect beam axis form initial configuration reference configuration selected case see figure design variable given terms position vector describing beam initial configuration respect reference configuration cost function described case integrals must recomputed change variables like one presented typical isoparametric parent element mapping see zienkiewicz taylor also note passing derivatives respect coordinate ought computed making use chain rule example contribution cost function optimality conditions choice design variables written optimal control optimal control problem studied herein concerns external loading sequence chosen bring structure directly towards optimal desired final state may involve large displacements rotations precisely study mechanics problems introducing parameter describe particular loading program enough one also needs employ control variables latter contributes towards work external forces written ext contains fixed distribution external loading scaled chosen control optimal control presented following form find value control variables final value state variables close possible desired optimal fixed values formulated terms constrained minimization chosen cost function written min role constraint indicated last expression fix values state variables respect chosen value control given set equilibrium equations words given control finally obtain configuration desired state latter verifies equilibrium equations opposite case simply solution closest also verifies equilibrium equations order remove constraint able consider state variables independently control variables resort classical method lagrange multipliers namely introducing lagrange multipliers rewrite optimal control problem making use lagrangian functional allows obtain corresponding form unconstrained optimization problem max min one readily obtain optimality conditions problem according first equations precisely weak form equilibrium equation whereas second two provide basis computing control lagrange multipliers explicit form equilibrium equation similar one external load term variation lagrange multiplier replacing virtual displacement writing explicit form two equations choose particular form objective function desired beam shape scalar parameter specifying weighted contribution chosen control choice seek minimize distance desired shape computed admissible shape one satisfies equilibrium well force control needed achieve state explicit form first term thus written explicit result second term equations identical one potentially modified according case follower load finally second term equation written concludes description problem ingredients finite element discrete approximations section discuss several important aspects numerical implementation presented theory analysis design related issues arise numerical simulations analysis part problem state variables represented using standard isoparametric finite element approximations see particular implies element initial configuration represented respect parent element placed natural coordinate space corresponding fixed interval using nen position vector field respect reference configuration nodal values element nen nodes corresponding shape functions latter easily constructed beams using lagrange polynomials element nen nodes written using product monomial expressions nen nen nodal values natural coordinates isoparametric interpolations one chooses shape functions order approximate element displacement field allows construct finite element representation element deformed configuration nen nodal values position vector deformed configuration virtual incremental displacement field also represented isoparametric finite element interpolations nen nen latter enables new iterative guess deformed configuration easily obtained corresponding additive updates nodal values finite element approximation incremental displacement field point corresponding value linear combination nodal values referred continuum consistent see since allow commute finite element interpolation consistent linearization nonlinear problem therefore also choose isoparametric interpolations virtual incremental rotation field nen nen commutativity finite element discretization consistent linearization would also apply rotational state variables difference displacement field concerns multiplicative updates rotation parameters written nodal point exp combined analysis design procedure proposed herein one must also interpolate lagrange multipliers also done using isoparametric interpolations according nen corresponding integrals appearing governing lagrangian functional optimality conditions computed numerical integration gauss quadrature see illustrate ideas state single element contribution analysis part governing lagrangian functional given discrete approximation setting nip dna dna gext abscissas weights chosen numerical integration rule see nip total number integration points single element order complete discretization procedure one must also specify interpolations design variables latter thickness diameter chosen case circular element nodal coordinates possible use isoparametric finite element approximations however best results obtained reducing number design variables opposed chosen element level using concept design element implies increasing degree polynomial employing example curves representation beam shape reducing significantly number design parameters see detailed discussion ideas important note standpoint simultaneous solution procedure presented design variable point given linear combination design element interpolation parameters ndn consequently finite rather design element discretization consistent linearization commute observation already made earlier linear analysis problem chenais results hand write discrete approximation equilibrium equations nel nint dna dna faext denotes finite element assembly operator token discrete approximation optimality conditions presented nel nint dna dnds nen dnb dnds similarly simple choice objective function discrete proximation optimality condition written nel nint dna dnds ndn ndn summary discrete approximation optimal design problem reduces following set nonlinear algebraic equations unknowns nodal values displacements rotations corresponding values lagrange multipliers chosen values thickness parameters support curve thickness interpolations int ext int similar procedure leads discrete approximation optimal control problem fact first nonlinear algebraic equations slightly modified include control variables accordance expression int chosen control parameters particular choice objective function control problem given obtain discrete approximation optimality condition according discrete approximation last optimality condition also written explicitly note passing last two optimality conditions combined eliminate lagrange multipliers leading last result combined equilibrium equations providing reduced set equations nodal displacements rotations control variables unknowns kind form fully equivalent solution procedure used solving nonlinear mechanics problems presence critical points see among others one recognize particular choice supplementary condition used stabilize system solution procedure two novel solution procedures developed solving class problems optimal design optimal control described next diffuse approximation based gradient methods first solution procedure sequential one one first computes grid values cost function carry optimization procedure employing approximate values interpolated grid important note grid values provide design control variables along corresponding mechanical state variables displacements rotations must satisfy weak form equilibrium equation ensure requirement grid value design control variables one also solve associated nonlinear problem structural mechanics main goal subsequent procedure avoid solving nonlinear mechanics problems grid values simply assume interpolated values cost function sufficiently admissible respect satisfying equilibrium equations relaxed equilibrium admissibility requirements pick convenient approximation cost function simplify subsequent computation optimal value thus make much efficient interpolated values cost function visualized surface yet referred response surface trying approximate sufficiently well true cost function particular method used construct response surface kind method diffuse approximations see employing diffuse approximations approximate value cost function constructed following quadratic form constant reference value gradient hessian approximate cost function hij hij variables replaced either design variables case optimal design problem control variables case deal optimal control problem elaborate idea simple case design control variables used computational proposes case one uses polynomial approximation typical diffuse approximation see employing chosen quadratic polynomial basis particular point dependent coefficient values jappr comparing last two expressions one easily conclude approximation kind fitted known grid values cost function trying achieve coefficients remain smooth passing one another stated following minimization problem arg min weighting functions associated particular data point constructed using window function based cubic splines according max dist present case closest grid nodes given point see weighting functions take unit value closest grid nodes vanish outside given domain influence former assures continuity coefficients latter ensures approximation remains local character similar construction carried higher order problems requires increased number closest neighbors list keeping chosen point fixed considering coefficients diffuse approximation constants minimization amounts using pseudoderivative diffuse approximation see order compute yielding minimum japp according allows write set linear equations pwpt pwj diag note passing computed minimum value necessarily provide minimum true cost function also satisfy equilibrium equations however number applications solution quite acceptable latter sufficient ought explore alternative solution procedure capable providing rigorously admissible value computed minima cost function carrying simultaneous solution cost function minimization equilibrium equations proposed procedure based genetic algorithm described next genetic algorithm based method genetic algorithms belong popular optimization methods nowadays follow analogy processes run nature within evolution living organisms period many millions years unlike classic gradient optimization methods genetic algorithms operate called population set possible solutions applying genetic operators mutation selection principles genetic algorithm first proposed holland ever since genetic algorithms reached wide application domain see books goldberg michalewicz extensive review genetic algorithms original form operate population chromosomes binary strings represent possible solutions certain way engineering problems usually working real variables kind applications described optimized values load control design variables adaptation genetic algorithm idea problem made possible storn considering chromosomes vectors instead binary strings using differential operators affect distance chromosomes work employ improved version kind algorithm referred simplified atavistic differential evolution sade shown algorithm well suitable dealing fairly large number variables particular interest problem hands sade algorithm designed explore possible minima thus find global minimum even case cost function may steep gradients isolated peak values short description corresponding procedure given subsequently see elaborate presentation tradition evolutionary methods first step generate starting generation chromosomes choosing random values state variables along control design subsequently repeat convergence cycles containing creation new generation chromosomes mutation local mutation evaluation selection reduces actual number chromosomes initial number computations follow work population chromosomes total number unknowns problem population evolve following operations mutation let chromosome generation xin number variables cost function certain chromosome chosen mutated random chromosome generated function domain new one computed using following relation parameter constant algorithm equal number new chromosomes created operator mutation defined radioactivity another parameter algorithm constant value set calculations certain chromosome chosen locally mutated coordinates altered random value given usually small range aim utilize near neighborhood existing chromosomes make search improved solutions faster useful cost functions steep gradients case near optimal function value small change value variable may introduce large change function value aim cross operator create many new chromosomes last generation operator creates new chromosome according following sequential scheme choose randomly two chromosomes compute difference vector multiply coefficient add third also randomly selected chromosome every component exceeding defined interval changed appropriate boundary value domain parameter probably important effect algorithm behavior seems speed convergence needed parameter set lower value opposite case higher values parameter could improve ability solve local minimum computations value set selection represents kernel genetics algorithm goal provide progressive improvement whole population achieved reducing number living chromosomes together conservation better ones modified tournament strategy used purpose two chromosomes chosen randomly population compared worse cast conserves population diversity thanks good chance survival even badly performing chromosomes observed sade algorithm quite inefficient later stage analysis solution large number components almost converged reason tried improve performance forcing algorithm stick better converged values particular experimented modified form cross operator contrary one produce new chromosome building top best possible previous value according max sign change parameter supposed get correct orientation increase respect gradient moreover parameter constant values chosen randomly interval new parameter algorithm smaller influence behavior operator local mutation switched parameter operator mutation also constant chosen randomly interval algorithm modification subsequently referred grade numerical examples section present several illustrative examples dealing coupled problems mechanics either optimal control optimal design computations carried using mechanics model geometrically exact beam see developed either within matlab environment diffuse approximation based solution procedure within sade computer code genetic algorithms optimal control cantilever structure form letter final shape cga initial shape figure letter cantilever initial final intermediate configurations example study optimal control problem deploying initially curved cantilever beam final configuration takes form letter see figure initial final configurations indicated thick lines number intermediate deformed configurations indicated thin lines chosen geometric material properties follows diameter circular curved part length flat part cantilever equal beam unit square chosen values young shear moduli respectively deployment carried applying vertical force moment end curved part cantilever words chosen control represented vector desired shape cantilever takes form letter corresponds values force moment optimal control problem defined follows objective cost function defined imposing desired shape displacement degrees freedom recast discrete approximation setting uea uda uda uea computed uda desired nodal displacements note passing condition imposed cost function either rotational degrees freedom control vector nevertheless introduces difficulties solving problem first solution obtained diffuse approximation based gradient method calculation cost function first carried nodes following grids gradient type procedure started grid thanks smoothness diffuse approximation base representation approximate value cost function converged difficulty roughly iterations iterative values obtained gradient method computations different grids shown figure grid constructed following interval values force moment note different choices grid result different solutions since none solutions kind satisfy equilibrium equations quite large difference known optimal solution solutions response surface could explained different influence control variable value cost function see figure used grids able describe small influence force second solution method used problem employs genetic algorithm based computation computation used admissible intervals like previous case control variables force moment computations carried starting random values chosen interval stopped first value cost function found order able look statistics one hundred runs performed one converging exact solution three types procedures tried either tuning parameters controlling local mutation acceleration sign change results presented table computation sade letter problem number fitness calls minimum maximum mean value table cantilever sade algorithm performance see best performance achieved simple modification sade genetic algorithm accelerate convergence latest stage results obtained simple modification sade algorithm improved using described grade algorithm special role radioactivity parameter see table second solution procedure applied example simultaneous solution mechanics equilibrium optimal control equations written explicitly value total numb unknowns case grid solution evaluations grid solution evaluations grid solution evaluations grid solution evaluations figure letter cantilever gradient method iterative computation grid equal control variables force moment components nodal displacements rotations lagrange multiplier computational convenience problem solving set nonlinear algebraic equations recast minimization statement allows direct application genetic algorithm min solution efficiency proposed simultaneous procedure depends chosen upper lower bounds admissible interval initial guess negative control function netative control function moment moment force whole scale values force detailed value optimum figure letter cantilever contour cost function computation grade radioactivity radioactivity radioactivity radioactivity letter problem number fitness calls minimum maximum mean value table letter cantilever grade algorithm performance solution example mechanics state variables chosen featuring desired beam shape bounds controlled chosen parameter according equivalent bounds control variables obtained precious result obtained solving grid minimization problem results minimum response surface finally lagrange multipliers solved adopted values chosen one hundred computations performed indicated bounds unknowns value parameter set choice parameters grade algorithm radioactivity equal table summarizes statistics computation minimum maximum mean value number evaluations standard deviation table letter cantilever solution statistics last part analysis carried examples concerns attempt increase efficiency simultaneous solution procedure sense employ grade version genetic algorithm choice parameters radioactivity equal small value bounds chosen well computations performed hundred runs genetic algorithm summarized statistics given table minimal maximal mean value number evaluations standard deviation table letter cantilever solution statistics see table proper choice bounds force algorithm always converge solution latter consequence using simultaneous solution procedure assures computed solution also satisfies equilibrium equations moreover total cost simultaneous solution procedure reduced beyond one needed approximate solution computations either reducing interval done herein making modification algorithm order accelerate convergence rate optimal control cantilever structure form letter second example deal problem multiple solution regularized form restore solution uniqueness end cantilever beam used much similar one studied previous example except shorter straight bar length equal cantilever controlled moment couple follower forces follow rotation attached initial final configuration obtained zero couple moment shown figure along number intermediate configurations first computation performed cost function identical one imposing minimum difference desired computed deformed shape restriction control variables computation carried using grade genetic algorithm starting random values within selected admissible intervals force couple moment according algorithm performance illustrated table number different solutions obtained different computer runs performed see figure however solutions remain fact clearly related considering applied moment force couple play equivalent role controlling final deformed shape shown particular problem values force moment satisfy slightly perturbed version straight bar flexibility force equilibrium final shape shape loading cga initial shape figure letter cantilever initial final intermediate configurations letter problem number fitness calls minimum maximum mean value table letter cantilever grade algorithm performance admissible solution thus infinitely many solutions case final shape controlled choice cost function order eliminate kind problem perform regularization cost function requiring difference computed final shape minimized also control variables small possible namely modified form cost function uda uda chosen weighting parameter specifying contribution control set small value choose convergence tolerance carry computation yet hundred times whereas stringent value tolerance requires somewhat larger number cost function evaluation result runs remains always given found optimal value cost function results grade algorithm linear regression results values force moment figure letter cantilever different solutions letter problem extended number fitness calls minimum maximum mean value table letter cantilever grade algorithm performance optimal control deployment multibody system optimal control procedure deployment problem multibody system studied example initial configuration multibody system consists two flexible component units long connected revolute joints see single stiff component length equal placed parallel horizontal axis final deployed configuration multibody system take form letter stiff component completely vertical two flexible component considerably bent deployment system controlled five control variables three moments vertical horizontal force see figure cost function problem chosen one controls system would find configuration close possible desired configuration desired configuration system corresponds values forces moments solution computed using sade grade genetics algorithms starting random choice interval interest defined final shape cga shape loading initial shape figure multibody system deployment initial final intermediate configurations solution problem typically difficult obtain increase number control variables one reasons irregular form cost function sense refer illustrative representation cost function contours different subspaces control variables shown figures convergence tolerance cost function chosen equal sade algorithm performance simplest choice algorithm parameters presented table grade algorithm performance modified value radioactivity parameter presented table computation sade letter problem number fitness calls minimum maximum mean value table results sade algorithm control problem one notice order magnitude increase cost function evaluation brought elaborate form cost function see figures however latter reason particular problem role moments list control variables much important role horizontal vertical forces bringing system desired shape affects conditioning equations solved slow convergence rate complete system reality slow convergence single couple control components latter illustrated figure provide graphic control function control function force force force subspace subspace control function control function force moment moment force subspace force subspace control function control function moment moment force force subspace subspace figure multibody system deployment contours cost function different subspaces representation iterative values computed chromosomes every chromosome represented continuous line note population optimal values moments converges much quickly force values seek large number iteration order stabilize another point worthy negative control function control function moment moment moment moment subspace subspace negative control function control function moment moment moment subspace moment subspace control function control function moment moment moment moment subspace subspace figure multibody system deployment contours cost function different subspaces ploration best way accelerate convergence rate final computational phase computation grade radioactivity radioactivity radioactivity radioactivity letter problem number fitness calls minimum maximum mean value table results grade algorithm task figure multibody system deployment convergence iterative chromosome populations optimal design shear deformable cantilever last example study optimal design problem considers thickness optimization shear deformable cantilever beam shown figure beam axis initial configuration cantilever thickness considered variable chosen order assure optimal design specified cost function setting discrete approximation choose beam elements constant thickness results design variables imposed mass figure shear deformable cantilever beam optimal design initial deformed shapes beam mechanical geometric properties young modulus shear modulus rectangular cross section width mass density latter needed computr ing total mass beam used corresponding limitation computed solution assuring reasonable values optimal thickness vertical force order assure meaningful result computations performed chosen value mass limitation limitations also placed admissible values thickness element first computation performed using diffuse approximation based response function sequential solution procedure cost function selected shear energy beam problem cast maximization shear energy max shear strain component bounds thickness values chosen shown table diffuse approximation computations grid started thickness min max table shear deformable cantilever optimal design thickness admissible values initial guesses thickness took iterations converge solution given corresponding value shear energy solution jappr recall approximate solution since computed value correspond grid nodes solution next sought using grade algorithm chosen values grade parameters radioactivity genetic algorithm executed times leading computational statistics reported table comput minimum maximum mean value table shear deformable cantilever optimal design computation statistics algorithm yielded two different solutions essentially imposed chosen bounds namely runs converged corresponding value whereas settled close value hence two solutions leads improved value cost function second part example slightly modified version first one sense mechanics part problem kept new cost function defined seeking minimize euclidean norm computed displacement vector max choice cost function made well known result system expressing optimality conditions indeed type sequential solution procedure using diffuse approximation cost function needs iterations find converged solution starting number initial guesses final solution value given final stage computation recompute solution problem using genetic algorithm admissible value last element thickness also slightly modified reducing lower bound instead higher bound instead order avoid optimal value restricted bound first solution problem obtained using sequential procedure grade genetic algorithm employed last stage computed value displacement vector norm found solution mass computations carried hundred times starting random initial values statistics computations given table kind problem repeated using simultaneous solution procedure optimality condition treated equal solved simultaneously resulting thickness variables displacement rotation components many lagrange multipliers unknowns latter fact eliminated prior comput minimum maximum mean value standard deviation table shear deformable cantilever optimal design computation statistics solution making use optimality condition chosen upper lower bounds admissible interval chosen guess displacement obtained solving mechanics problem values thickness parameters given table limitation total mass added cost function choice grade algorithm parameters given radioactivity equal computation stopped fairly loose tolerance allows accelerate algorithm convergence always lead unique solution yet results table show standard deviation indeed remains small solution practically unique minimum maximum mean value standard deviation comput table shear deformable cantilever optimal design simultaneous computation statistics acknowledgements work supported french ministry research european student exchange program erasmus support gratefully acknowledged conclusions approach advocated herein dealing coupled problem nonlinear structural mechanics optimal design optimal control implies bringing optimality conditions level treating variables independent rather considering equilibrium equations mere constraint state variables dependent design control variables fairly unorthodox rather unexplored number applications proposed approach great potential particular problems interest work concern large displacements rotations structural systems key ingredient approach pertains geometrically exact formulation nonlinear structural mechanics problem makes dealing nonlinearity description devising solution schemes much easier model kind model problem geometrically exact beam explored detail herein one available classl refer work shells solids models sharing configuration space mechanics variables beam latter also allows directly exploit presented formulation solution procedures coupled problem nonlinear mechanics either shells solids optimal control optimal design two different solution procedures presented herein first one exploits response surface representation true cost function followed gradient type solution step leads approximate solution although quality solution always improved refining grid serves construct response surface exact solution never computed unless minimum corresponds one grid points second solution procedure solves simultaneously optimality conditions nonlinear mechanics equilibrium equations deliver exact solution although often appropriate care taken choose sufficiently close initial guess select admissible intervals variables accordingly probably best method sense combination sequential simultaneous procedure first serves reduce much possible admissible interval provide best initial guess whereas second furnishes exact solution number improvements made proposed methods kind help increase convergence rates accuracy computed solution one remember even mechanics component equilibrium equations problem kind difficult solve since large difference stiffness different modes bending versus stretching result poorly conditioned set equations adding optimality conditions top increases difficulty finding best possible way deal problem certainly worthy explorations references argyris excursion large rotations comput methods appl mech breitkopf rasineux villon efficient optimization strategy using hermite diffuse approximation mang editor proceedings fifth wccm vienna chenais design sensitivity code structure arch optimization opti southampton crisfield fast solution procedure handles snap computers structures goldberg genetic algorithms search optimization machine learning holland adaptation natural artificial systems university michigan ann arbor internal report hrstka improvements different types binary real coded genetic algorithms preventing premature convergence advances engineering software press hrstka zeman competitive comparison different types evolutionary algorithms computers structures stress resultant geometrically nonlinear shell theory drilling rotations part consistent formulation comput methods appl finite element implementation reissner geometrically nonlinear beam theory three dimensional curved beam finite elements comput methods appl choice finite rotation parameters comput methods appl mech frey finite element analysis linear planar deformations elastic initially curved beams international journal numerical methods engineering frey kozar computational aspects parameterization finite rotations international journal numerical methods engineering mamouri nonlinear dynamics flexible beams planar motions formulation scheme stiff problems comp mamouri rigid components joint constraints nonlinear dynamics flexible multibody systems imployin geometrically exact beam model comp methods appl mech kegl shape optimal design structures efficient shape representation concept int numer meth kleiber antunez hein kowalczyk parameter sensitivity nonlinear mechanics theory finite element computations john wiley sons hrstka homepage sade http luenberger linear nonlinear programming marsden hughes mathematical foundations elasticity michalewicz genetic programs nayroles touzot villon generalizing finite element method diffuse approximation diffuse elements computational mechanics ramm strategies tracing response near limit points wunderlich editor nonlinear finite element analysis structural mechanics pages berlin riks application newtons method problem elastic stability journal applied mechanics rousselet finite strain rod model design sensitivity mechanics structures machines simo symmetric hessian geometrically nonlinear models solid mechanics intrinsic definition geometric interpretation comput methods appl mech simo finite strain model part computational aspects comput methods appl mech storn usage differential evolution function optimization biennial conference north american fuzzy information processing society pages strang introduction applied mathematics press tortorelli michaleris design sensitivity analysis overview review inverse problems engineering zienkiewicz taylor finite element method volume vols iii butterworth london
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sep fast vehicle detection aerial imagery jennifer carlet keyw bernard abayowa sensors directorate air force research lab beavercreek wpafb abstract modified described section section gives results paper concluded section recent years several near object detectors developed however object detectors typically designed view images subject large image directly apply well detecting vehicles aerial imagery though detectors developed aerial imagery either slow handle imagery well popular detector modified vastly improve performance aerial data modified detector compared faster rcnn several aerial imagery datasets proposed detector gives near state art performance speed deep learning object detection algorithms since deep cnn easily imagenet competition cnns become state art object detection images whereas hand crafted features gradients color used detect objects cnns automatically learn features relevant detection work focused version look yolo detector however compared faster rcnn considered state art testing tensorflow versions detectors used faster rcnn introduction faster rcnn follow fast rcnn rcnn faster rcnn starts cnn adds region proposal network rpn create proposals bounding boxes features given cnn roi pooling classifier used classify score bounding box diagram net original paper given figure due speed accuracy faster rcnn heavily used since inception object detection ground view popular problem lot interest academic computer vision community detection aerial views interest significantly less studied consequently recent advancements primarily large image object detection classification mostly using deep convolutional neural networks cnns often neural networks work well directly applied small image objects however networks often modified improve performance type data particular interest detecting vehicles aerial platforms near speeds faster rcnn proven effective detecting vehicles aerial imagery unable reach anywhere near real time speeds desired many applications methods use sliding window techniques also slow newer detectors run whole images much faster yet proven aerial imagery paper fast deep cnn modified near detector aerial imagery rest paper organized follows section introduces deep cnns used paper section covers aerial imagery datasets used net figure faster rcnn data german aerospace center dlr images taken camera airplane feet munich germany surprisingly improved version yolo net reached near faster rcnn accuracy much greater speeds also starts cnn followed two cnns simultaneously produce bounding boxes object confidence scores class scores additionally includes route reorg layers allow net use features earlier net similar ssd helicopter set video clips taken camera helicopter flying los angeles california area afvid video clips taken small uav flying feet avon park florida flying feet camp atterbury indiana datasets train deep neural network object detector requires vast amount data hence coupling deep learning big data neural nets previous section provide pretrained network weights usually started imagenet pascal voc coco detailed table datasets provide thousands images detectors trained good general purpose object detectors however datasets largely contain imagery taken personal cameras ground level contain little aerial data additionally images relatively low resolution compared aerial imagery building camera handful images taken cameras tower wright patterson air force base dataset segmentation much aerial imagery given video clips meaning considered independent identically distributed iid unlike large detection image datasets means even though may thousands images may close appearance effectively less data none datasets anywhere near size even pascal voc ideally dataset separated train validation test sets typically done datasets independent images datasets consist short video sequences least two videos kept one validation one testing table public large scale detection datasets dataset imagenet coco pascal voc number training images modifying several publicly available imagery detection datasets additionally afrl house aerial imagery referred air force aerial vehicle imagery dataset afvid truthed yolo provides models pretrained weights coco pascal voc datasets capable detecting vehicles relatively large image detect objects small relative size image detect different numbers classes yolo net needs modified using appropriate data dataset descriptions aerial data surprisingly diverse different view points different ground sampling distances gsd different image sizes aspect ratios color etc vehicles different data may significantly different size appearance figure shows different images four different datasets obvious different voc data elevated data vedai afvid data somewhat similarities building camera data similar aerial satellite data detailed look aerial datasets available table number classes configuration file defines yolo net model classes definition used define number classes change number classes used classes setting need changed number filters last convolutional layer must altered reflect changed number classes number filters set num classes coords num number anchor boxes coords four corresponding four coordinates used define bounding box vedai vehicle detection aerial imagery vedai dataset consists satellite imagery taken utah orthonormal images rgb rgb used net resolution depth standard net input resolution pascal voc vedai afvid building camera figure sample imagery table aerial imagery datasets dataset vedai afvid afvid afvid afvid building camera image width height mean vehicle width height output feature resolution image would create feature resolutions corresponding pixels approximately size twice size vehicles vedai imagery meaning vehicle may correspond single point feature map ideally multiple points per vehicle increase net feature resolution consequently number feature points per vehicle two methods increase input resolution decrease net depth ratio target area percent targets overlapping ing width height first layer net previous example doubling width height leads output sizes pixels per feature four feature points average vehicle however significantly increases gpu memory usage decreases speed net decrease net depth convolutional max pooling layers removed net downsampled less making net shallower goes conventional wisdom deeper nets typically better shown work well aerial data increasing net input resolution simply means ing one max pooling layer associated cnn layers gives output resolution doubling input resolution without great effect memory usage speed net sample shallower yolo net given appendix table alongside typical net large somewhat diverse make good general purpose object detectors results detectors tested aerial imagery given table faster rcnn gives best precision recall slowest tested detectors detectors performed quite poorly without aerial data net shape yolo modifications yolo provide pretrained square shaped nets however data square shape net changed closer match aspect ratio input data faster rcnn automatically biggest increases performance come altering net size depth table compares performance results several modified yolo nets shallow faster rcnn net based nets using afvid vedai data yolo nets based net faster rcnn coco first yolo listed standard yolo modified one class vehicles five times faster faster rcnn performance across board much weaker doubling size described gives performance boost particularly afvid dataset halves speed removing several convolutional layers max pooling layer greatly improves precision recall speed standard input resolution shallower yolo approximately seven times faster shallower faster rcnn still lower recall lower precision afvid vedai datasets increasing size shallower net decreases speed net still boosts precision recall changing input shape shallow yolo net better match aspect ratio afvid data causes large increase precision data causes small decrease square vedai data building camera data none nets trained shallow yolo nets better faster rcnn one application use cnn network live data building cameras larger variance vehicle sizes aerial datasets nets trained afvid data images building cameras results table show rectangular shallow yolo outperforms faster rcnn data shallow faster rcnn still better afvid vedai data anchors faster rcnn anchors fixed bounding boxes refined fixed shape size detection better multiple anchor box sizes five anchor sizes determined approach using bounding boxes voc dataset due difference orientation scale vehicles anchors kept largest removed since expected large objects image experiments results calculate detectors performing two metrics used precision recall defense applications false alarm rate far often used instead precision precision recall sometimes referred detection rate another metric used frames per second fps match detections ground truth intersection union iou used iou ratio area two boxes overlap total area box including overlap detector desired aerial imagery small vehicles single class vehicle used training testing therefore average precision average recall refer average test images iou threshold deeper look results cases input net entire image object size relative size image important size object example mscoco dataset images typically benchmark define small object less medium object mean object sizes aerial imagery fall medium category definition image sizes larger resized input resolution nets equivalently fall small category hence pretrained object detectors open source detectors provide best trained pascal voc coco since datasets table pretrained object detectors tested aerial data net trained yolo yolo voc faster rcnn coco afvid vedai building camera fps table results afvid vedai data net shallow faster rcnn yolo yolo shallow yolo shallow yolo shallow yolo input size afvid vedai building camera fps irving isard jia jozefowicz kaiser kudlur levenberg monga moore murray olah schuster shlens steiner sutskever talwar tucker vanhoucke vasudevan vinyals warden wattenberg wicke zheng tensorflow machine learning heterogeneous systems software available egories redefined terms image size small objects less medium objects large objects greater original image looking results terms size building camera data table shows surprising result faster rcnn medium size actually worst performance shallow yolo performs expected irregularity medium size objects faster rcnn may due quirk small building camera dataset shown figure set images taken camera different times last two sets images taken minutes apart detectors well cars sparse visible separation however parking lot full cars densely packed faster rcnn sometimes fails detect vehicles yolo much better job irregardless illumination therefore poor performance medium sized objects faster rcnn net may due inability handle heavily overlapping objects images chen gupta implementation faster rcnn study region sampling corr everingham van gool williams winn zisserman pascal visual object classes voc challenge international journal computer vision june girshick fast international conference computer vision iccv girshick donahue darrell malik rich feature hierarchies accurate object detection semantic segmentation computer vision pattern recognition conclusions box yolo performs poorly aerial imagery modifications greatly improve performance first making net shallower increase output resolution second changing net shape closer match aspect ratio data modified yolo precision recall still typically bit worse faster rcnn increased speed makes good option near vehicle detectors aerial imagery required huster gale deep learning pedestrian detection aerial imagery mss passive sensors lin maire belongie bourdev girshick hays perona ramanan zitnick microsoft coco common objects context corr liu anguelov erhan szegedy reed berg ssd single shot multibox detector corr references redmon divvala girshick farhadi look unified object detection corr abadi agarwal barham brevdo chen citro corrado davis dean devin ghemawat goodfellow harp table results afvid building camera data net shallow faster rcnn shallow yolo shallow yolo input size afvid vedai building camera table building camera object size detector shallow faster rcnn shallow yolo map map large redmon farhadi better faster stronger arxiv preprint ren girshick sun faster towards object detection region proposal networks advances neural information processing systems nips trieu darkflow https sakla mundhenk deep multimodal vehicle detection aerial isr imagery ieee winter conference applications computer vision appendix map medium map small fps faster rcnn low density yolo low density faster rcnn high density high illumination yolo high density high illumination faster rcnn high density low illumination yolo high density low illumination figure object density illumination table nets layer input convolutional max pooling convolutional max pooling convolutional convolutional convolutional max pooling convolutional convolutional convolutional max pooling convolutional convolutional convolutional convolutional convolutional max pooling convolutional convolutional convolutional convolutional convolutional convolutional convolutional route convolutional reorg route convolutional convolutional filters kernal size net size layer input convolutional max pooling convolutional max pooling convolutional convolutional convolutional max pooling convolutional convolutional convolutional max pooling convolutional convolutional convolutional convolutional convolutional route convolutional reorg route convolutional convolutional filters kernal size shallow net standard net net size
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sparse coding spiking neural networks convergence theory computational results may ping tak peter tang lin mike davies intel corporation abstract spiking neural network snn individual neurons operate autonomously communicate neurons sparingly asynchronously via spike signals characteristics render massively parallel hardware implementation snn potentially powerful computer albeit non von neumann one one guarantee snn computer solves important problems reliably paper formulate mathematical model one snn configured sparse coding problem feature extraction moderate assumption prove snn indeed solves sparse coding best knowledge first rigorous result kind introduction central question computational neuroscience understand complex computations emerge networks neurons neuroscientists key pursuit formulate neural network models resemble researchers understanding physical neural activities functionalities precise mathematical definitions analysis models less important comparison computer scientists hand key pursuit often devise new solvers specific computational problems understanding neural activities serves mainly inspiration formulating neural network models actual model adopted needs much faithfully reflecting actual neural activities mathematically well defined possesses provable properties stability convergence solution computational problem hand paper goal computer scientist formulate two neural network models provably solve mixed optimization problem often called lasso problem lasso workhorse sparse coding method applicable across machine learning signal processing statistics work provide framework rigorously establish convergence firing rates spiking neural network solutions corresponding lasso problem network model namely spiking lca first proposed implement lca model using analog neuron circuit call lca model analog lca clarity next section introduce model configurations lasso constrained variant classo form hopfield network specific lasso configurations render convergence difficult establish outline recent results use suitable generalization lasalle principle show converges lasso solutions neurons communicate among real numbers analog values certain time intervals spiking lca neurons communicate among via spike digital signals encoded single bit moreover communication occurs specific time instances consequently much communication efficient section formulates auxiliary variables average soma currents instantaneous spike rates section subsequently provides proof instantaneous rates converge classo solutions proof built upon results obtained assumption neuron duration arbitrarily long unless stops spiking altogether finite time finally devise numerical implementation empirically demonstrate convergence classo solutions implementation also showcases potential power problem solving spiking neurons practice approximate implementation ran conventional cpu able converge solution modest accuracy short amount time convergence even faster fista one fastest lasso solvers result suggests specialized spiking neuron hardware promising parallelism sparse communications neurons fully leveraged architecture sparse coding analog lca neural network formulate sparse coding problem follows given vectors usually called vector consider input signal try code approximate well contains many zero entries possible solving sparse coding problem attracted tremendous amount research effort one effective way arrive solving lasso problem one minimizes distance regularization parameters reasons clear later consider problem additional requirement call classo constrained problem rozell presented first neural network model aims solving lasso neurons used represent dictionary atoms neuron receives input signal serves increase potential value neuron keeps time potential certain threshold send inhibitory signals aim reduce potential values list receiving neurons authors called kind algorithms expressed neural network mechanism locally competitive algorithms lcas paper call analog lca mathematically described set ordinary differential equations dynamical system form wij function thresholding also known activation function decides inhibition signal sent coefficients wij weigh severity inhibition signal general form instantiation hopfield network proposed given lasso classo problem configured wij lasso thresholding function set classo set defined def note normalized dynamical system vector notation vector function simply applies scalar function input vector component say solves lasso particular solution dynamical system converges vector optimal solution lasso convergence phenomenon demonstrated lca needs realized traditional computer via classical numerical differential equation solver one realize using example analog circuit may fact able solve lasso faster less energy point view establishing robust way solve lasso rigorous mathematical results convergence invaluable furthermore convergence theory bound bearings neural network architectures see section thresholding function strictly increasing unbounded standard lyapunov theory applied establish convergence dynamical system already pointed hopfield early work graded neuron model spiking neuron model nevertheless correspond lasso thresholding functions strictly increasing furthermore classo thresholding function bounded well rozell demonstrated convergence phenomenon two later works rozell colleagues attempted complement original work convergence analysis proofs among results works stated particular solution converges lasso optimal solution unfortunately detailed major gaps related proofs thus convergence claims doubt moreover case classo problem addressed one present authors established several convergence results summarize support development section interested reader refer complete details dynamical system form case function defined given starting point standard theory ordinary differential equations shows unique solution solutions also commonly called flows two key questions given starting point whether sense flow converges relationships exist limiting process lasso solutions lasalle invariance principle powerful tool help answer first question gist principle one construct function along flow one conclude flows must converge special largest positive invariant inside set points lie derivative zero crucial technical requirements possesses continuous partial derivatives radially unfortunately natural choice continuous first partial derivatives everywhere radially unbounded case classo failures due special form based generalized version lasalle principle proved establish flow lasso classo converges largest positive invariant set inside stationary set whenever established prove fact inverse image set optimal lasso solutions proof based kkt condition characterizes properties particular theorem convergence results given based one wants solve lasso classo let arbitrary starting point corresponding flow following hold let set lasso optimal solutions inverse image corresponding thresholding function arbitrary flow always converges set moreover optimal objective function value lasso finally lasso optimal solution unique unique furthermore sparse coding spiking lca neural network inherently communication efficient needs communicate others internal state exceeds threshold namely sparse coding problem expected internal state eventually stay perpetually threshold many neurons nevertheless entire duration neuron internal state threshold constant communication required furthermore value sent neurons real valued analog nature perspective spiking neural network snn model holds promise even greater communication efficiency typical snn various internal states neuron also continually evolving contrast however communication form one sent neurons certain internal state reaches level firing threshold internal state reset right spiking event thus cutting communication dist dist inf set positive invariant flow originated set stays set forever function radially unbounded whenever kuk immediately time internal state charged enough thus communication necessary certain time span single bit information carrier suffices snn admits mathematical descriptions hitherto rigorous results network convergence behavior particular unclear snn configured solve specific problems guarantees present mathematical formulation snn natural definition instantaneous spiking rate main result moderate assumption spiking rate converges classo solution snn suitably configured best knowledge first time rigorous result kind established snn neurons maintains time internal soma current configured receive constant input internal potential potential charged according configured bias current reaches firing threshold time neuroni resets potential simultaneously fires inhibitory signal preconfigured set receptive neurons whose soma current diminished according weighted exponential decay function wji zero otherwise let ordered time sequence spikes define soma current satisfies algebraic differential equations operator denotes convolution wij wij equation together definition spike trains describe spiking lca intuitive definition spike rate neuron clearly number spikes per unit time hence define instantaneous spiking rate average soma current def def parameter apply operator differential equation portion using also relationship obtain wij consider classo problem dictionary atoms normalized unit euclidean norm configure wij equation set configured shown soma currents magnitudes thus average currents well bounded consequently lim lim following relationship crucial equation moderate assumption duration arbitrarily long unless stops spiking altogether one prove complete proof result left appendix derive convergence follows since average soma currents bounded bolzanodef weierstrass theorem shows least one limit point point time sequence equation def equations must therefore since configured classo problem limit fact fixed point unique whenever classo solution case limit point average currents unique thus indeed must classo solution potential neuron neuron neuron neuron spike count current snn time snn time figure detail dynamics simple spiking network beginning neuron fires membrane potentials see neurons grow linearly rate determined initial soma currents see continues neuron becomes first reach firing threshold inhibitory spike sent neurons causing immediate drops soma currents consequently growths neurons membrane potentials slow neurons instantaneous spike rates decrease pattern membrane integration spike mutual inhibition repeats network rapidly moves steady state stable firing rates observed convergent firing rates yield classo optimal solution solution also verified running lars algorithm four subfigures evolution membrane potential evolution soma current spike raster plot solid lines cumulative spike count neuron dashed line depicts value corresponding close approximation indicates strong tie two formulations numerical simulations simulate dynamics conventional cpu one precisely solve spiking network formulation tracking order firing neurons consecutive spiking events internal variables neuron follow simple differential equations permit solutions method however likely slow requires global coordinator looks ahead future determine next firing neuron efficiency instead take approximate approach evolves network state discrete time steps every step internal variables neuron updated firing event triggered potential exceeds firing threshold simplicity approach admits parallel implementations suitable specialized hardware designs nevertheless approach introduces errors spike timings time neuron sends spike may delayed time step see section timing error major factor limits accuracy solutions spiking networks however may fact desirable certain applications machine learning illustration snn dynamics solve simple classo problem mina subject use network configured wij bias current firing threshold set figure details dynamics simple spiking network seen simple example network needs spike exchanges converge particular weak neuron neuron quickly rendered inactive inhibitory spike signals competing neurons raises important question many spikes network find question easy answer theoretically however empirically see number spikes approximated average average exponential kernel thresholded average current step step step snn time snn time figure convergence spiking network classo solution comparing convergence different formulations read solutions spiking neural network using positive equation gives fastest initial convergence using thresholded average current reaches highest accuracy quickest despite lack theoretical guarantee exponential kernel method yields acceptable though less accurate solution kernel easy implement hardware thus attractive snn computer built state variable equation solutions equation figure shows close approximation spike counts using example observe approximation consistently holds problems suggesting strong tie since configured sparse coding problem expect converge zero suggests total spike count small convergence spiking neural networks use larger spiking network empirically examine convergence spike rates classo solution neural network configured perform feature extraction image patch using dictionary learned image chosen optimal solution entries figure shows convergence objective function value spiking network solution comparing true optimal objective value obtained conventional classo solver indeed small step size spiking network converges solution close true optimum relationships among step size solution accuracy total computation cost noteworthy figure shows increasing step size sacrifices two digits accuracy computed total computation cost reduced factor takes times fewer time units converge time unit requires times fewer iterations multiplication effect cost savings highly desirable applications machine learning accuracy paramount note configuration also suitable problems whose solutions sparse total number spikes fewer thus total timing errors correspondingly fewer several ways read snn solution rigidly adhere equation practice picking better expect sparse solution resulting identically zero neurons spike time equation another alternative use solution likely deliver truly sparse solution finally one change definition impact spikes past decays quickly figure illustrates different read methods shows exponential kernel effective empirically although must point previous mathematical convergence analysis longer applicable case spikes avg execution time second image fista iteration sparsity error error fista error sparsity fista sparsity fista iteration spikes avg breakdown execution time second execution time second image figure cpu execution time spiking neural networks step size unknowns image case shown unknowns image case shown shows breakdown objective function image experiment error defined sparsity percentage entries values greater note spiking network finds optimal solution gradually increasing sparsity rather decreasing fista results spare spiking activities neurons cpu benchmark spiking network implementation earlier discussions suggest spiking network solve classo using spikes property important implications snn computational efficiency computation cost neuron spiking network two components neuron states update spiking events update neuron states update includes updating internal potential current values every neuron thus incurs cost every time step cost spiking events update proportional times average number connections spiking neuron updates soma currents neurons connects thus cost high networks connectivity two previous examples low networks local connectivity example nevertheless cost incurred spike may happen far fewer per time step practice observe computation time usually dominated update corroborating general belief spiking events relatively rare making spiking networks communication efficient report execution time simulating spiking neural network conventional cpu compare convergence time fista one fastest lasso solvers solve convolutional sparse coding problem image experiments ran intel xeon cpu using single core simd enabled exploit intrinsic parallelism neural network matrix operations shown figure spiking network delivers much faster early convergence fista despite solution accuracy plateauing due spike timing errors convergence trends figures similar demonstrating spiking networks solve problems various sizes fast convergence spiking networks attributed ability fully exploit sparsity solutions reduce spike counts asynchronous communication quickly suppress neurons firing fista conventional solvers communications variables similarly needed realized multiplications performed basis way exploit sparsity avoid computations involving variables gone zero one iteration comparison sparsity solutions evolves fista found figure discussion work closely related recent progress balanced network snn model differs slightly one internal state used former using input dimensions splitting image positive negative channels use patches stride dictionary language spike generated reaching threshold whose role eliminated altogether despite differences details neuron models spikes networks occur competitive process neurons serve minimize energy function work furthers understanding convergence property spiking networks additionally argued tightly balanced network spike codes highly efficient spike precisely timed keep network optimality work provides evidence high coding efficiency even network settles utilizing spikes neurons able collectively solve optimization problems minimum communications demonstrate insight translated practical value approximate implementation conventional cpu observe mathematical rigor focus statement tightly balanced network potential converges zero problematic taken literally spiking events eventually cease case stationary points loss function equation longer necessarily stationary points firing rates constrained general kkt condition used situation condition spike spike affect behavior loss function spikes essence guarantee trajectory variable generated snn descending loss function snn formulation established convergence properties easily extended incorporate additional term problem formulation handled modifying slope activation function follows def def corresponds setting bias current modifying firing thresholds neurons several works studying computation sparse representations using spiking neurons zylberberg show emergence sparse representations local rules provide energy function derive spiking network formulation minimizes modified lasso objective shapero first propose formulation yet provide analysis believe formulation powerful primitive future spiking network research computational power spikes enables new opportunities future computer architecture designs computational paradigm motivates architecture composed massively parallel computation units unlike von neumann architecture infrequent dispersed communication pattern units suggests decentralized design memory placed close compute communication realized dedicated routing fabrics designs potential accelerate computations without breaking limit appendices governing algebraic differential equations consider neural networking consisting neurons independent variables soma currents another variables potentials depedent currents described momentarily consider following configurations neron receives positive constant input current nonnegative current bias positive potential threshold set priori given time potential evolves according time time spike signal sent nerons connected weighted set weights potential reset zero immediately afterwards next spike generated moreover consecutive spike times finally receives spike time weight soma current changed additive signal heaviside function otherwise sign convention used means positive means spike always tries inhibit suppose initial potentials set spiking threshold dynamics system succintly described set algebraic equations convolution operator sequence spikes dirac delta function spike times determined turn evolution soma currents govern evolutions potentials one also express algebraicr equations set differential equations note heaviside function expressed hence thus differentiating equation yields note equations given terms spike trains governed turn soma currents well configuartions initial potentials spiking threshold bias current defining spike rates average currents suppose system spiking neurons initialized potentials thus least finite time soma currents remain constant neurons generate spikes furthermore consider inhibitory signals present let spike times neuron sequence could empty finite infinite empty potential never reaches threshold finite neuron stop spiking certain time onwards define spike rate average current neuron follows def def definitions section presents following results inhibition assumption leads fact soma currents bounded turns shows none neurons spike arbitrarily rapidly fact neurons spike arbitrarily rapidly implies soma currents bounded well main assumption needed something proved point neron spikes infinitely often duration consecutive spikes arbitrarily long using assumption previous established properties one prove important relationship spike rate average current terms familiar thresholding function proposition exists bounds def convention positive value whenever values exist proof spike signals inhibitory clearly equation def thus defining maxi leads given two consecutive exist note special case value hence min min thus whenever two spike times exist finally duration spikes arbitrarily small easy see def therefore def min indeed proposition shows among things lower bound duration consecutive spikes following assumption assumption assume positive number whenever numbers exist simple words assumption says unless neuron stop spiking althogether certain time duration consecutive spike become arbitrarily long assumption results proposition following important relationship established theorem let thresholding function neuron function proof let spikes infinitely often stands active stop spiking finite time stands inactive first consider let time final spike note always since obviously thus consider case let largest spike time bigger furthermore note assumption always lim inf otherwords time large enough moreover thus term eventually smaller magnitude spiking neural nets lca section shows spiking neural net snn corresponds lca limit points snn necessarily fixed points lca particular lca corresponds constrained lasso lasso parameters constrained nonnegative whose solution unique snn necessarily converges solution proof surprisingly straightforward following differential equation connecting spiking rates crucial derivation relationship straightforward first apply operation equation find expression left hand side note therefore consequently equation established observe bounded proposition average current means shown previously since bounded vectors must limit point theorem correpsonding moreover must hence matrix entries indeed correspond fixed point lca case lca corresponds lasso unique solution one fixed point implies also one possible limit point snn snn must converge lasso solution references balavoine romberg rozell convergence rate analysis neural networks sparse approximation ieee trans neural september balavoine rozell romberg convergence neural network sparse approximation using nonsmooth lojasiewicz inequality proceedings international joint conference neural networks dalla august barrett machens firing rate predictions optimal balanced networks nips beck teboulle fast iterative algorithm linear inverse problems siam journal imaging sciences boerlin machens deneve predictive coding dynamical variables balanced spiking networks plos comput biol 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optimal sparse approximation integrate fire neurons international journal neural systems tang convergence lca flows lasso solutions arxiv mar tibshirani regression shrinkage selection via lasso royal statist soc zeiler krishnan taylor fergus deconvolutional networks ieee cvpr zou hastie regularization variable selection via elastic net royal statist soc zylberberg murphy deweese sparse coding model synaptically local plasticity spiking neurons account diverse shapes simple cell receptive fields plos comput biol
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consideration publication theory practice logic programming arxiv may improving parma trailing tom bart demoen dept computer science belgium toms bmd maria garcia banda school computer science monash university australia mbanda peter stuckey nicta victoria laboratory department computer science university melbourne australia pjs submitted november revised october accepted may abstract taylor introduced variable binding scheme logic variables parma system uses cycles bindings rather linear chains bindings used standard wam representation hal dprolog languages make use parma representation herbrand constraint solvers unfortunately parma trailing scheme considerably expensive time space consumption aim paper present several techniques lower cost first introduce trailing analysis hal using classic parma trailing scheme detects eliminates unnecessary trailings analysis whose accuracy comes hal determinism mode declarations integrated hal compiler shown produce space improvements well speed improvements second explain modify classic parma trailing scheme halve trailing cost technique illustrated evaluated context dprolog hal finally explain modifications needed trailing analysis order combined modified parma trailing scheme empirical evidence shows combination effective techniques used isolation appear theory practice logic programming keywords constraint logic programming program analysis trailing introduction logic programming language mercury somogyi considerably faster traditional implementations prolog due two main reasons first research assistant fund scientific research flanders belgium vlaanderen mercury requires programmer provide type mode determinism declarations whose information used generate efficient target code second variables ground bound ground term new first time seen compiler hence unconstrained since neither aliased variables partially instantiated structures allowed mercury need support full unification assignment construction deconstruction equality testing ground terms required furthermore need perform trailing technique allows execution resume computation previous program state information old state logged forward computation used restore backtracking usually means recording state unbound variables right become aliased bound since mercury new variables representation need trailed hal demoen banda constraint logic language designed support construction extension use constraint solvers hal also requires type mode determinism declarations compiles mercury leverage sophisticated compilation techniques however unlike mercury hal includes herbrand constraint solver provides full unification solver uses taylor parma scheme taylor taylor rather standard wam representation unlike wam parma representation ground terms contain reference chains hence equivalent mercury thus calls herbrand constraint solver replaced calls mercury efficient routines whenever ground terms manipulated unfortunately increased expressive power full unification comes cost includes need perform trailing furthermore trailing parma scheme expensive wam terms time space present two techniques counter trailing penalty parma scheme first trailing analysis detects eliminates unnecessary trailings suitable system based classic parma trailing scheme without supporting information analysis rather inaccurate since little known way predicates used however mode determinism information available hal significant accuracy improvements obtained second technique modified parma trailing scheme considerably reduces required trail stack size technique applied system implemented dprolog demoen nguyen mercury hal system finally detail modifications required trailing analysis order combined modified trailing scheme empirical evaluation technique indicates combination modified trailing scheme trailing analysis results significant reduction trail size negligible time cost rest paper proceeds follows next section provides quick background trailing classic parma scheme trailing avoided section summarizes information used analyzer improve accuracy section presents notrail analysis domain section shows analyze hal body constructs section shows use analysis information avoid trailing section presents modified trailing scheme section shows changes required analysis deal modified scheme section presents results experimental evaluation technique finally future work discussed section background begin setting terminology bound variable variable bound nonvariable term aliased variable unbound equated variable free variable unbound unaliased also refer new variable variable hal mercury representation since yet constrained wam unbound variable represented linear chain variable free chain length one cell containing two free variables unified younger cell made point older cell see section discussion relative cell age two variables aliased series unifications free variables thus results linear chain references last one case variable becomes instantiated bound term representation implies testing whether source level variable bound unbound requires dereferencing dereferencing necessary unification thus performed quite often example consider execution goal variable initially represented using wam representation first unification points second unification points third unification must first dereference get dereference give point last unification dereference set changes heap states shown figure initially fig example binding chains using wam representation taylor taylor introduced different variable representation scheme suffer dereferencing need scheme unbound variable represented circular chain variable free chain length one wam unifying two variables scheme consists cutting circular chains combining one big circular chain variable bound cell circular chain replaced value bound dereferencing required verify whether cell bound tag cell immediately identifies cell bound however see later costs incurred scheme example consider execution goal variable initially represented using parma representation first unification points second unification points third unification must point final unification variable chain set changes heap states shown figure notice references remain final state opposed figure initially fig example binding chains using parma representation another difference wam parma binding schemes becomes apparent constructing new term containing unbound variable effectively aliasing new variable hence new variable must added variable chain example consider execution goal variable initially represented using wam representation first unification points second unification constructs heap term content namely points using parma representation first unification chains together second unification add copy chain resulting heap states shown figure mentioned trailing technique stores enough information regarding representation state variable able reconstruct state upon backtracking wam parma chains fig example constructing term containing unbound variable using wam parma representations change representation state occurs cell level variable represented cell associated variable unbound unaliased pointing another cell chain associated variable gets aliased pointing final bound structure variable bound directly indirectly thus needs trailed cells rest section discuss parma trailing scheme greater detail orthogonal issue trailing possible improvement based detection unnecessary trailings classic parma scheme value trailing classic parma trailing scheme uses value trailing described following valuetrail store contents cell store address cell takes address cell parma chain stores trail stack first old contents cell address global pointer top trail stack untrail operation value trailing straightforwardly defined address retrieve cell address address recover cell contents first pops address cell contents contrast trailing wam stores address cell reasons twofold first cell updated pointer code paper code implementation details obfuscate rather clarify concepts hand omitted either another cell linear chain structure second cell address content cell therefore cell updated old content cell one stored trailing always address allows wam value trailing optimized storing address cell reducing half space cost single cell trailing let discuss cells need trailed classic parma scheme seen trailing needed representation state variable changes happen variable unbound due unification becomes either aliased bound therefore need trail cells associated variables involved unification creating new term contains unbound variable following discussion distinguishes three cases cells associated variables involved unification unification new term construction trailing unification result aliasing two unbound variables belonging separate chains merging two chains single one done changing state two cells associated variables since associated cell appears different chain final chain formed simply interchanging respective successors one reconstruct previous situation remembering two cells changed initial value achieved unification following simplified code valuetrail valuetrail tmp tmp notice trailed independently associated cells need trailed refer kind trailing shallow trailing contrast kind unification wam update trail last cell one two linear chains hence space cost four times lower one value opposed four example consider parma trailing occurs first three unifications goal example variable initially represented first unification trail together initial value since also together initial value similarly second unification trail together value together value third unification trail together value together value resulting trail wam trail goal illustrated figure trails first finally resulting trail trailing unification unbound variable becomes bound every single cell chain set point nonvariable term thus reconstruct chain cells chain trailed combined simplified code unification term follows start next valuetrail term next start since cells chain unbound variable trailed refer kind trailing deep trailing contrast kind unification wam trail one cell linear chain hence space complexity wam compared parma number cells chain however time complexity due dereferencing wam example consider parma trailing happens last unification goal example binding variables chain adds trail elements contrast wam trailing adds single trail element trailing new term construction mentioned new term constructed heap copy unbound variable cell containing copy must added chain means must trail since value successor chain going change need trail new cell since clearly previous value need recover combined simplified code constructing unbound variable current top heap pointer valuetrail contrast construction wam need trail since simply points new cell old unbound variable either bound new variable complexity arise placed new structure pointing either nonvariable term trailing required case summary major advantage parma binding scheme requires dereferencing major disadvantages detailed account see lindgren parma trails cells per unification two unifications versus one trailing individual cell expensive two slots used versus one unlike wam cells trailed every time cell updated happen copying unbound variable structure involves trailing cell result trail stack usage expected much higher parma scheme wam demoen nguyen demoen nguyen indeed observed dprolog system maximal trail sizes parma scheme average twice large wam scheme techniques present paper attempt counter disadvantages trailing analysis reduces number trailings thereby counters disadvantages modified trailing scheme counters disadvantage conditional versus unconditional trailing cell changed requires trailing cell exist recent choice point since otherwise previous state restored backtracking property applies equally wam parma schemes systems simple test used verify whether cell older recent choice point younger cells require trailing cells heap kept order allocation test simply checks whether address cell smaller address top heap beginning recent choice point systems dprolog take advantage property use known conditional trailing let assume existence function older succeeds conditional trailing described following code valuetrail thus avoiding trailing cells newer recent choice point code unification described previous sections using unconditional valuetrail operation rewritten use conditional trailing simply substituting call valuetrail call cond valuetrail untrail operation remains unchanged systems order cells heap guaranteed unconditional trailing required mercury hal system example system since mercury uses boehm garbage collector preserve order cells heap garbage collections systems use unconditional trailing least unifications see instance van roy despain demoen nguyen shown global performance hardly affected choice conditional unconditional trailing since savings made avoided trailings balanced overhead tests differences conditional unconditional trailing affect proposed analysis thus analysis still used point conditional trailing becomes available mercury unnecessary trailing classic parma scheme considering trailing unbound variable appearing unification least two cases trailing avoided variable new previous value remember therefore trailing required fact subset cases exploited conditional trailing cells need trailed associated cell case variable cells case already trailed since recent upon backtracking earliest trailing important since one enables reconstruction state variable following sections see analysis information obtained detect two cases therefore used eliminate unnecessary trailing classical parma trailing scheme eliminate runtime tests performed conditional trailing variables known representation thus younger recent choice point language requirements analysis presented paper designed hal language however useful language uses parma representation provides accurate information regarding following properties instantiation state trailing analysis gain accuracy taking account instantiation state program variable whether variable new ground old state new corresponds program variables internal representation equivalent mercury free instantiation state ground corresponds program variables known bound ground terms case state old corresponding program variables might unbound representation chain length one bound term known ground program variables instantiation state new ground old called new ground old variables respectively note new variable becomes old ground never become new known ground remains ground thus three states considered mutually exclusive information available program point table associating variable scope instantiation state represent instantiation table information program point follows let arp denote set program variables scope program point function instp arp new ground old defines instantiation state program variable point function allows partition arp three disjoint sets ewp groundp oldp containing set new ground old variables respectively determinism trailing analysis also gain accuracy knowledge particular predicates one solution information available table associating predicate procedure precise determinism herein refer six main kinds determinism semidet set solutions det multi nondet erroneous failure purposes interested whether predicate return one answer represent determinism table function det red maps predicate maximum number solutions sharing trailing analysis exploit sharing information increase accuracy information available program point table associating variable scope set variables possibly share clearly variables may aliased together must possibly share represent sharing table program point function sharep oldp oldp assigns program variable oldp set program variables oldp share note program variables ewp groundp share definition notrail analysis domain aim notrail domain keep enough information able decide whether variables unification need trailed possible optimized versions perform trailing used instead order must remember variables unbound representation new need trailed suggests making use instantiation information mentioned previous section note since analysis works level program variables indirection required already established program variables ewp groundp represent variables need trailed thus variables oldp need represented notrail domain set new ground old program variables respectively assuming arp contains variables tree used implement underlying table sufficiently balanced size oldp complexity instp log recall oldp contains program variables representing variables unbound representation also variables bound terms analysis ensure ground necessary ensure correctness even though variables bound need trailed nonvariable terms bound might contain one unbound variables trailing state unbound runtime variables represented domain representation bound program variable decided program variables need represented domain decide represent saw unnecessary trail variable unification associated cell already trailed variable already shallow trailed since recent case nonvariable unification enough need ensure cells chain already trailed variable already deep trailed suggests domain distinguishes shallow deep trailed runtime variables easily done partitioning oldp three disjoint sets program variables different trailing state representing runtime variables might trailed yet representing variables least shallow trailed representing variables deep trailed sufficient keep track two sets able reconstruct third hence type elements notrail domain lnotrail oldp oldp first component contains set program variables representing variables already shallow trailed second component contains set program variables representing variables already deep trailed following use denote elements lnotrail program points already shallow deep trailed components corresponding elements also elements domain referred descriptions descriptions goal referred respectively note definition state variable already deep trailed also shallow trailed cells chain already trailed cell associated variable also trailed partial ordering relation lnotrail thus defined follows lnotrail implies deep trailing stronger information shallow trailing shallow trailing stronger trailing also note descriptions compared program point instantiation sharing information identical example trailing lattice shown fig clearly ssss kkkk ssss rrrr kkkk llll rrrr ssss kkkk llll ssss kkkk fig notrail lattice hasse diagram variables diagram lnotrail complete lattice top description bottom description oldp two important points need taken account considering domain first point component description used represent already deep trailed variables variable oldp whatever reason initialized since last choicepoint need part trailed second point soon deeply trailed program variable made share shallow trailed program variable also must become shallow trailed since cell newly merged chain might come thus might trailed sharing information program point used define following function makes trailing information consistent associated sharing information consistp sharep intuitively function eliminates every program variable shares variables adds assume lnotrail consistp use consist function preserve given hal implementation sharing analysis domain asub time complexity determining sharep variable furthermore since asub explicitly carries set ground variables program point use set rather computing new one groundp instantiation information thus increasing efficiency major cost consistp computation variables sharep set computed set operations negligible comparison hence overall time complexity see complexity function determines note notrail domain seen product domain also includes mode sharing information however simplicity consider different elements separately relating via associated program point complexity operations use thus use strictly necessary summary element domain interpreted follows consider program variable means cells chains represented already trailed needed therefore need trailed unification note could bound variable includes many different variable chains two possibilities known unbound associated cell shallow trailed therefore need trailed unification although practice consider optimizing unifications might bound cell one chains might trailed result optimization performed case could course represent bound variables accurately requiring domain keep track different chains contained structures program variables bound individual trailing state affected different program constructs known techniques see instance janssens bruynooghe van hentenryck mulkers lagoon stuckey based type information could used keep track constructor variable bound trailing state different arguments thereby making approach possible analyzing hal body constructs lnotrail section defines notrail operations required hal analysis framework bueno nethercote analyze different body constructs framework quite similar well known framework bruynooghe analyzing single module analysis framework handles analysis multiple module programs makes extra demands analysis domain thus paper simply treat program analyzed single module body construct hal show obtain postdescription information contained variable initialization init hal variable transits initial instantiation new instantiation old initialized since new variable need trailed simply add component recall represents already deep trailed variables also old variable need trailed formally let obtained unification several cases consider one variables say new simply assigned copy pointer unification performed trailing state becomes thus trailing state note never require call consist since new variable introduce sharing one variables ground one ground unification hence neither appear variables deep trailed cells associated chains trailed remain trailed unification obtained simply merging chains hence variables retain current trailing state remain unchanged variables already aliased belong chain nothing done unification hence retain current trailing state hence variables retain current trailing state remain unchanged otherwise least one variables deep trailed two unaliased variables considered variables unbound unification merge chains time performing shallow trailing necessary thus unification variables shallow trailed least one variable bound one become bound unification stated earlier bound variables treated way note either variable deep trailed unification shared variables must become shallow trailed well unification requires applying consist function formally let set ground variables program point unification obtained new remove ground ground unify min old unify otherwise otherwise remove ground min otherwise gives trailing state remove ground removes variables min distinguished three cases deep trailed nothing changed definitely aliased share ensures move shallow fig term construction example dashed line represents choicepoint trailed state otherwise description must remain unchanged since unification might done nothing thus might still untrailed adding would mistake note need apply consist since already share although sharing information might changed create sharing among variables already connected closure union performed consist worst case time complexity due consist unification two cases consider new unification simply constructs term otherwise treat purpose analysis two unifications new variable since unifications form discussed focus construction new variable following assume unification new term constructed represented parma chain argument cell structure representation inserted chain see requires shallow trailing cell term requires trailing newly created generalization variable term unification follows arguments deep trailed becomes deep trailed arguments remain deep trailed otherwise arguments become shallow trailed since argument least shallow trailed operation note least one argument deep trailed since argument shares unification must apply consist maintain information consistent formally let unification set variables set ground variables unification obtained remove ground otherwise worst case time complexity definition combined previous one overall definition unification implementation efficient complexity still predicate call let predicate call set variables first step project onto resulting description lproj note trivially defined onto proj second step consists extending lproj onto set variables local predicate call since variables known new thus appear extension operation domain trivially defined identity thus simply disregard extension steps required hal framework let lanswer answer description resulting analyzing predicate definition calling description lproj assume set variables local already projected lanswer identical remove ground time complexity order obtain make use determinism information thus derived combining lanswer using determinism predicate call follows predicate determinism multi nondet one answer variables become trailed possible introduction new choice point hence equal lanswer except fact apply consist function order take account changes sharing involving variables otherwise know trailing state variables unchanged except possibly new introduced sharing thus result combining lanswer follows trailing state variables taken lanswer variables taken deep trailed variables share trailed variables must course become shallow trailed formalized function defined comb lanswer sanswer danswer lanswer det otherwise obviously complexity consist example assume call predicate answer description call depends determinism predicate predicate one solution postdescription otherwise equal answer description note combination meet two descriptions specialized combination introduced banda assumes lanswer contains accurate information variables role combination propagate information rest variables clause disjunction disjunction reason trailing becomes necessary mentioned trailing might needed variables already old disjunction thus let entire disjunction except whose simply since disjunction implies backtracking last branch let goal assume set variables local already projected end result disjunction least upper bound lub defined remove ground intuitively variables deep trailed descriptions ensured remain deep trailed variables trailed descriptions always deep trailed ensured already least shallow trailed note variables known ground descriptions eliminated consistent view old variables represented descriptions avoids adding overhead abstract operations hal also includes switches disjunctions compiler detected one branch needs executed switches treated identically disjunctions except fact rather example let code fragment let assume sharing program point assuming old simply disjunction first branch element domain second branch since last branch disjunction identical entire disjunction respectively finally lub two postdescriptions results assume ground code fragment switch predescription first branch becomes post description note lub notrail domain alone product domain includes sharing groundness information finally lub two two branches time complexity joining branches simply lub operator fixed maximum number branches completely dominated function although could treated rather inaccurate since case switches one branch ever executed thus backtracking two branches hence better old variable exists bound aliased possibly requiring trailing backtracking condition fails harsh restriction since ensured whenever used logical way simply inspects existing variables change variable however general possible statically determine property instead safe approximation used treated condition contains old variables otherwise following stronger treatment used let also predescription let obtained also finally let obtained respectively obtained lub time complexity joining branches like operation disjunction example let following known ground assume variables share equal predescription thenand finally obtained lub example postdescription would since additional trailing required term construction involves creation partially evaluated predicate assuming predicate name arity equal higher construct created hal required new also often difficult even impossible know whether actually called thus hal follows conservative approach requires instantiation captured arguments remain unchanged calling also guarantees type mode checking terms ever unified requirements allow follow simple although conservative approach call trailing captured variables affected call might one solution thus may involve backtracking involved variables treated safely analysis call location still statically live call involve backtracking involve unifications trailing information might inferred correctly call location captured variables generally known call location keep trailing information safe potential unifications accounted unification since construction term involves backtracking unifications leave variables involve least shallow trailed sufficient demote captured deep trailed variables shallow trailed status together sharing deep trailed variables formally let term construction set variables obtained time complexity otherwise call call exact impact call difficult determine general fortunately even exact predicate associated variable unknown hal compiler still knows determinism help improve accuracy predicate might one solution variables must become trailed since called predicate typically unknown answer description available improve accuracy otherwise worst happen deep trailed arguments call become shallow trailed move deep trailed arguments set shallow trailed variables together variables share recall case captured variables already taken care constructing term sequence steps much predicate call first project onto set variables resulting lproj next answer description lanswer call computed indicated det lanswer otherwise combination lanswer computed obtain trailing optimization optimization phase consists deciding unification body clause variables need trailed decision based description unification inferred trailing analysis variables need trailed general unification predicate replaced variant trail particular variables thus need different variant possible combination variables need trailed unification two unbound variables trailing omitted either variable shallow trailed deep trailed binding unbound variable trailing omitted deep trailed construction term containing old unbound variable trailing omitted either shallow deep trailed unification two bound variables trailing chains structure either omitted deep trailed often known compile time whether variable bound general unification predicate required performs boundness tests selecting appropriate kind unification various optimized variants general predicate needed well experimental results analysis presented section improved trailing scheme let present trailing scheme sophisticated classic parma value trailing discussed section start considering improvements apply kind unification nonvariable finish showing combine modified scheme must able apply different untrail operations depending kinds trailing performed simple tagging scheme explained detail section used indicate kind untrailing required case unification swap trailing classic scheme value trailing cells takes four trail stack slots two addresses variable plus another two contents trailing unconditional undoing unification consists simply restoring old values cells separately however economic inverse operation undoes swapping happened unification simply swapping back swapping requires addresses involved cells respective old contents introduce new kind trailing named swap trailing exploits also corresponding untrailing operation swap trailing defined following code swaptrail addresses two cells pointer top trailing stack swap trail tag function set tag tags cell tag note swap trailing consumes two slots trail stack opposed four used unconditional value trailing classical scheme untrail operation swap trailing untag recover address recover address tmp swap contents tmp improvement assumes cells unconditionally trailed conditional value trailing available classic scheme would either consume zero two four slots respectively none one variables older recent choice point swap trailing used conjunction conditional trailing replace four slot case value trailing still needed two slot case result code conditional trailing looks like swaptrail trail using swaptrail else valuetrail trail else valuetrail trail important note potential gain space trail obtained operations comes cost execution time operations needed gain space guaranteed unification chain trailing seen unification pulls entire chain variable apart setting every cell chain nonvariable case classic value trailing every address cell stored twice address cell contents predecessor cell means quite redundancy obvious improvement store address name chain trailing length chain known marker needed indicate untrailing operation chain trailing ends last entry chain encountered untrailing first one actually trailed use chain end tag mark entry last address put trail tagged chain begin indicate kind trailing chains length one last first cell coincide chain end tag used mark single address chain trailing defined code chaintrail start true start trail cell address false one cell last tag last one last last untrail operation reconstructing chain straightforward dispatches appropriate untrailing action depending tag first cell encountered untrailing chain begin meaning corresponding code head untag previous head current current current previous previous current current current untag current current previous head current first tag chain end code untrailing cell untag cell cell example consider trailing occurs using improved scheme goal example first unification swaptrail trailing similarly second unification swaptrails third unification swap trails finally last unification chain trails resulting trail looks like use superscripts represent swap trail chain begin chain end tags respectively uses trail entries compared entries examples improvement assumes cells unconditionally trailed let assume chain consists cells older recent choice point conditional trailing available unconditional chain trailing consume space classic conditional value trailing fortunately conditional variant chain trailing also possible start first true true trail older cell chain first tag first first false else false start one older cell last tag last one last last conditional variant uses slots stack trail clearly improvement conditional value trailing whenever note untrail operation used unconditional chain trailing might look wrong first since cond chaintrail might trail cells chain however simply exploiting fact objective trailing able reconstruct bindings existed creation time choice point thus final state younger cells state cell intermediate steps untrailing irrelevant fact general better respect stack trail consumption principle behind old cells older recent choice point chain pointing old cells trailed old cell must made point new cell last kind trailing suitable insight special kind value trailing successive equal slots trail stack overlapped cond chaintrail operation approximates since implementation would incur undue time overhead extra runtime tests needed test age successors thus store addresses old cells even neither point pointed old cells example figure illustrates small example specified conditional chain trailing together previous trailings safely restores state variables older recent choice point consider following goal fail let assume older recent choice point newer three chains length depicted figure successive forward steps shown figures value trailed addresses stored trail stack conditional chain trailing side stack trail entries represent chain begin chain end tags respectively execution fails immediately backtracks initial state three steps first figure conditional chain trailing untrailed creating chain next figure value trailing undone finally figure value trailing reversed final state corresponds initial state except still bound however exist recent choice point content irrelevant point inaccessible reclaimed heap anyway forward execution resumes note illegal intermediate state illustrated figure important since occurs middle untrailing never execution combining improvements let first consider combination context modified unconditional trailing scheme mercury hal context addition swap unconditional chain trailing function trailing used allow custom trailings constraint solvers function trailing stores pointer untrailing function untrailing data thus need four different tags distinguish different trailing information appear trail fortunately two tag bits available aligned addressing bit machines one constraint allocation four different tags kinds could avoided known trailed already initially untrail untrail untrail fig conditional chain trailing example trailing chain end tag look tag intermediate addresses chain trail general untrail operation simply looks like untrail switch case break case break case break case note since assuming modified unconditional trailing scheme value trailing never used value trailing needed modified scheme whenever one two variables involved variablevariable unification newer recent choice point thus one trailed otherwise swap trailing used since conditional trailing allowed swap trailing always used unifications let consider combination context modified conditional trailing scheme dprolog context value swap conditional chain trailing used remarks application allocation tags unconditional case general conditional untrail operation looks identical except fact function trail case substituted value trail case call untrail functiontrail substituted call untrail valuetrail looking value trailings chains length one example previous section see figure obvious trailing alternative conditional system stores redundant information chain trailing indeed variable would chain trailed instead value trailed one instead two slots would used stack however would require tests implemented self pointer else experimental results conditional unconditional trailing scheme presented section analysis improved trailing scheme trailing analyses heavily depend details trailing scheme analysis presented section defined classic parma trailing scheme section present modifications needed analysis order applied improved trailing scheme see improved scheme gives rise fewer opportunities trail savings unnecessary trailing improved trailing scheme main difference two schemes terms unnecessary trailing appears considering cells trailed since recent choicepoint case cells need trailed since information stored first time allows reconstruct state right see later experimental evaluation allows previous analysis detect many spurious trailings assuming semantics function trailing rely intermediate state herbrand variable untrailing case swap trailing however cells need trailed even already trailed since recent swap trailing incremental kind trailing content cells stored trailing incremental change thus relies future trailings proper untrailing cells result untrailing process improved scheme later chain swap trailings undone swap trailing untrailed correctly thus opportunity avoid future trailings two choice points first trailing performed let illustrate counterexample counterexample let trail variables second time two choice points consider following code fail variables older recent choice point initially represented chains length one depicted figure first two steps four variables aliased swap trailed pairwise creating two chains length two see figure represent swap trail tags initially untrail fig counterexample incremental behavior swap trailing eliminate need trailing cells next aliased creating one large chain see figure step swap trailed since already swap trailed recent choice point assuming means trailing needed finally execution fails untrailing tries restore situation recent choice point however figure shows omission last swap trailing invalid untrailing fails restore correct situation thus cell involved swap trailing still needs trailing later segment execution ltrail analysis domain implications ltrail analysis domain simple needs distinguish variables trailed deep trailed rest words variables one two possible states particular program point deep trailed trailed hence type elements ltrail domain oldp ordering simply operations defined lnotrail domain adapted simplification adaptation rather straightforward every description ltrail treated description lnotrail new descriptions ltrail obtained first calculating descriptions using lnotrail operations setting optimization based analysis every unification used improve unification possible optimizations based ltrail domain limited lnotrail domain deep trailed variables represented descriptions unification two variables variant without swap trailing used variables binding unbound variable term variant unification without chain trailing used predescription addition swap trailing required appear unification two bound variables variables predescription one known ground trailing needed runtime means recursive unification process bound variables unbound variables unified bound nothing need trailed unbound variables experimental results first examine effect trailing analysis lnotrail associated optimizations classic parma trailing scheme hal look effect improved parma trailing scheme effect use trailing analysis ltrail improved parma trailing scheme finally examine improved parma trailing scheme context dprolog timing results obtained intel pentium ghz effect trailing analysis using lnotrail hal lnotrail analyzer implemented analysis framework hal applied six hal benchmarks use herbrand solver icomp hanoi qsort serialize warplan zebra table gives summary benchmarks benchmarks make use herbrand solver executed mercury programs without significantly modifying algorithm representation inferred unifications benchmarks table hal benchmark descriptions lines code benchmark icomp hanoi qsort serialize warplan zebra description cut version interactive bim compiler hanoi puzzle using difference lists quick sort algorithm using difference lists classic prolog palindrome benchmark war planner robot control classic five houses puzzle lines table compilation statistics notrail analysis benchmark icomp hanoi qsort serialize warplan zebra compilation time analysis total relative old unifications improved total relative size relative used optimize generated mercury code avoiding unnecessary trailing explained section table shows benchmark analysis time seconds compared total compilation time number improved unifications compared total number unifications involving old variables size generated binary executable binary size optimized program expressed number bytes relative unoptimized program high compilation times obtained benchmarks due existence predicates many different something analysis optimized yet deterministic nature hanoi qsort benchmarks allows analysis infer unifications replaced alternative benchmarks much smaller fraction unifications improved due heavy use predicates last table shows due specialization may considerable size particular icomp warplan size substantially increased various approaches limit number generated variants explored work apply work well example one approach use profiling information retain variants see ferreira damas another approach taken mazur generate least optimized variants latter would reproduce optimal result hanoi qsort table presents execution times seconds obtained executing benchmark number times loop iteration number table gives loop table benchmark timings classic parma unoptimized cparma optimized trailing analysis caparma benchmark icomp hanoi qsort serialize warplan zebra iterations cparma time caparma relative table benchmark trail sizes classic parma unoptimized cparma optimized trailing analysis caparma benchmark icomp hanoi qsort serialize warplan zebra maximum trail cparma caparma relative trailing operations cparma caparma relative count execution process iteration number also used obtain results shown hal benchmarks significant obtained hanoi qsort benchmarks explained effects replacing unifications version maximum size trail stack kilobytes total number trailing operations shown table benchmarks much smaller fraction trailing operations removed results smaller even slight shows optimization come without cost larger active code size due specialization impact instruction cache behavior table shows impact instruction references instruction cache misses obtained cachegrind skin valgrind memory debugger see nethercote seward number instruction references number times instruction retrieved memory instruction cache miss rate percentage instruction references main memory instead cache table clearly shows elimination trailing operations results considerable reduction executed instructions side spectrum specialization negative effect instruction cache miss rate explains warplan benchmark table benchmark instruction cache misses classic parma unoptimized cparma optimized trailing analysis caparma benchmark icomp hanoi qsort serialize warplan zebra instruction cache miss rate cparma caparma relative instruction references cparma caparma relative effect improved trailing scheme mercury hal improved unconditional parma trailing scheme also implemented mercury hal since mercury already tagged trail difficult aside discussed trailings unification system also requires trailing term constructed old variable argument term construction argument cell term structure inserted variable chain modifies one cell old variable chain classic scheme cell trailed value trailing avoid value trailing altogether replaced swap trailing improved trailing scheme table presents timing maximal trail classic improved trailing scheme six hal benchmarks used table timing maximal trail classic cparma improved iparma unconditional parma trailing scheme mercury hal benchmark icomp hanoi qsort serialize warplan zebra cparma time iparma relative maximal trail cparma iparma relative benchmarks improved trailing scheme faster classic scheme differences percentages though maximum difference slightly serialize benchmark much important effects improved trailing scheme maximal trail size maximal trail least smaller improved scheme classic scheme effect improved trailing scheme combined trailing analysis ltrail mercury hal trailing analysis presented section implemented hal modified proposed section deal improved trailing scheme table presents timing maximal trail hal benchmarks obtained improved scheme information inferred modified analysis compares results obtained scheme without analysis information table timing maximal trail improved unconditional parma scheme without iparma iaparma ltrail trailing analysis relative classic scheme without trailing benchmark icomp hanoi qsort serialize warplan zebra iparma time iaparma relative iparma maximal trail iaparma relative serialize warplan benchmarks analysis able reduce number actual trailing operations four benchmarks combination improved scheme analysis yields better results time maximal trail hanoi qsort benchmarks drastic improvement trailings avoided distinctive time improvement respectively two benchmarks icomp zebra maximal trail improvement together slightly reduced time better respectively overall combination improved scheme trailing analysis never makes results worse since drastically improves benchmarks shows modest improvement others fair conclude combination superior improved system without analysis effect improved trailing scheme dprolog let present experimental results improved conditional parma trailing scheme dprolog several small benchmarks one bigger program comp table shows timing maximal trail use benchmark time given seconds applies number runs iterations given maximal trail size given kilobytes applies single run time difference classic improved scheme negligible improved scheme slower zebra benchmark table parma dprolog classic cparma improved trailing iparma benchmark iterations boyer browse cal chat crypt ham meta qsort nrev poly queens queens reducer sdda send tak zebra relative average comp comp relative time cparma iparma maximal trail cparma iparma average equally fast price lower trail usage increase instructions executed net speedup differences maximal trail use however substantial swap trail chain trail halve trail stack consumption value trailing still used cases trailing yet experimental results show kind trailing occur often benchmarks maximal trail stack effectively halved eleven benchmarks average maximal trail use classical scheme results smaller benchmarks confirmed larger comp program execution time nearly classic improved trailing scheme maximal trail shows similar improvement almost related future work far know modifications suggested classic parma trailing scheme new somewhat similar analysis detecting variables trailed presented debray debray together corresponding optimizations debray analysis however wam variable representation traditional prolog setting without type mode determinism declarations also van roy despain trailing avoided variables new terminology setting basically wam representation taylor keeps track trailing state variables global analysis parma system classic parma trailing scheme see taylor taylor opposed lnotrail analysis presented taylor analysis less precise closer ltrail analysis presented trailing state variable trailed intermediary shallow trailing state exist also two technique preventing multiple value trailing two choice points first described works wam scheme introduces linear reference chains parma allow second described aggoun beldiceanu maintains timestamp every cell corresponds choicepoint last update however timestamp requires additional space even case cell never updated context parma timestamps would likely consume space actually saved avoiding trailing finally approaches reconstruction state backtracking trailing using either copying schulte recomputation van hentenryck ramachandran parma matter wam bindings keep enough information allow recomputation backtracking copying approach backtracking parma quite feasible remains interesting question future work little room left optimization trailing analysis improved unconditional trailing scheme course analysis improved adopting refined representation bound variables currently parma chains structure bound variable represented trailing state bound variables could represented accurately requiring domain keep track different chains contained structures program variables bound individual trailing state affected different program constructs known techniques see instance janssens bruynooghe van hentenryck mulkers lagoon stuckey lagoon based type information could used keep track constructor variable bound trailing state different arguments thereby making approach possible applies equally analysis classical scheme additionally would interesting see much extra gain analysis add improved conditional trailing scheme implemented dprolog mercury hal supports conditional trailing analysis would certainly improve maximal trail would remove overhead test likely also result small though experimental results show improved scheme analysis better classic scheme analysis need true programs recall two choice points value trailings cell first eliminated classic scheme swap trailings could eliminated improved scheme hybrid scheme would possible using analysis decide single unification basis either swap trailing value trailing better minimizing amount trailing cost untrailing analysis would require global view trailings two choice points moreover trailings could common different pairs choice points optimality would depend execution spends time also untrailing operation improved analysis able determine instance trailing happened swap trailing tags need set tested acknowledgements would like thank referees detailed insightful reports significantly improved paper references aggoun beldiceanu time stamps techniques trailed data constraint logic programming systems splt programmation logique bourgault dincbas eds cnet france warren abstract machine tutorial reconstruction mit press bruynooghe practical framework abstract interpretation logic programs journal logic programming bueno banda hermenegildo marriott puebla stuckey model analysis optimizing compilation lopstr selected papers form international workshop logic based program synthesis transformation lau lecture notes computer science vol springer verlag london banda demoen marriott stuckey gates hal hal tutorial flops proceedings international symposium functional logic programming eds lecture notes computer science vol springer verlag aizu japan banda marriott stuckey differential methods logic program analysis journal logic programming debray simple code improvement scheme prolog journal logic programming may demoen banda harvey marriott stuckey overview hal proceedings international conference principles practice constraint programming jaffar lecture notes computer science vol springer verlag alexandria virginia usa demoen nguyen many wam variations little time proceedings international conference computational logic lloyd dahl furbach kerber lau palamidessi moniz pereira sagiv stuckey eds lecture notes artificial intelligence vol alp springer verlag london ferreira damas controlling code expansion multiple specialization prolog compiler proceedings ciclops colloquium implementation constraint logic programming systems lopes ferreira eds university porto mumbai india technical report dcc liacc univeristy porto december janssens bruynooghe deriving descriptions possible value program variables means abstract interpretation journal logic programming lagoon mesnard stuckey termination analysis types accurate iclp proceedings international conference logic programming palamidessi lecture notes computer science vol springer verlag mumbai india lagoon stuckey framework analysis typed logic programs flops proceedings international symposium functional logic programming kuchen ueda eds lecture notes computer science vol springer verlag tokyo japan lindgren mildner bevemyr taylor scheme unbound variables tech computer science department uppsala university mazur garbage collection declarative language mercury thesis department computer science leuven belgium mulkers winsborough bruynooghe dataflow analysis prolog acm transactions programming languages systems nethercote analysis framework hal thesis university melbourne nethercote seward valgrind program supervision framework electronic notes theoretical computer science elagage contexte retour superficiel modifications autres une approfondie wam thesis rennes schulte comparing trailing copying constraint programming proceedings sixteenth international conference logic programming schreye mit press las cruces usa somogyi henderson conway execution algorithm mercury efficient purely declarative logic programming language journal logic programming application abstract interpretation logic programs occur check reduction esop proceedings european symposium programming robinet wilhelm eds lecture notes computer science vol springer verlag germany taylor removal dereferencing trailing prolog compilation proceedings internation conference logic programming levi martelli eds mit press lisbon portugal taylor high performace prolog implementation thesis basser department computer science taylor parma bridging performance gap imperative logic programming journal logic programming van hentenryck cortesi charlier type analysis prolog using type graphs journal logic programming van hentenryck ramachandran backtracking without trailing clp rlin acm transactions programming languages systems july van roy despain logic programming aquarius prolog compiler ieee computer
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optimal algorithm hitters insertion streams related problems mar arnab palash david indian institute science bangalore ibm research almaden arnabb palash dpwoodru abstract give first optimal bounds returning hitters data stream insertions together approximate frequencies closing long line work problem stream items parameters let denote frequency item number times item occurs stream arbitrarily large constant probability algorithm returns items returns items returns approximations item returns algorithm uses log log log log bits space processes stream update time report output time linear output size also prove lower bound implies algorithm optimal constant factor space complexity modification algorithm used estimate maximum frequency additive error amount space resolving question iitk workshop algorithms data streams case hitters also introduce several variants heavy hitters maximum frequency problems inspired rank aggregation voting schemes show techniques applied settings unlike traditional heavy hitters problem variants look comparisons items rather numerical values determine frequency item introduction data stream model emerged standard model processing massive data sets sheer size data traditional algorithms longer feasible may hard impossible store entire input algorithms need run linear even sublinear time algorithms typically need randomized approximate moreover data may physically reside device internet traffic data stored algorithm may impossible recover hence many algorithms must work given single pass data applications data streams include data warehousing network measurements sensor networks compressed sensing refer reader recent surveys data stream model one oldest fundamental problems area data streams problem finding hitters simply heavy hitters also known popular items frequent items elephants iceberg queries algorithms used subroutines network flow identification routers association rules frequent itemsets iceberg queries iceberg datacubes survey presents overview problem theoretical practical standpoints formally define heavy hitters problem focus paper definition hitters problem hitters problem given parameters stream items let denote number occurrences item frequency algorithm make one pass stream end stream output set item algorithm output estimate frequency satisfies note natural definitions heavy hitters possible sometimes used example heavy hitters items generally hitters items fip fjp sense definition corresponds hitters hitters relax hitters algorithms many interesting applications focus direct common formulation heavy hitters notion interested algorithms use little space bits possible solve hitters problem also interested minimizing update time reporting time algorithms update time defined time algorithm needs update data structure processing stream insertion reporting time time algorithm needs report answer processed stream allow algorithm randomized succeed probability least make assumption ordering stream desirable often applications one assume even random order also interested case length stream known advance give algorithms general setting first algorithm hitters problem given misra gries achieved log log bits space algorithm rediscovered demaine karp algorithms deterministic also number randomized algorithms countsketch sketch sticky sampling lossy counting sample hold bloom filters sampling berinde show using log bits space one achieve stronger guarantee reporting item res res denotes sum frequencies items excluding frequencies frequent items emphasize prior work best known algorithms hitters problem used log log bits space two previous lower bounds known first lower bound log log bits comes fact output set contain items takes many bits encode second lower bound follows folklore reduction randomized communication complexity index problem problem two players alice bob alice bit string length bob index alice creates stream length consisting one copy copies dummy item fill rest stream runs heavy hitters streaming algorithm stream sends state algorithm bob bob appends copies item stream continues execution algorithm holds moreover differs additive factor depending whether therefore randomized communication complexity index problem hitters problem requires bits space although proof better lower bound known thus upper bound hitters problem log log bits best known lower bound log bits constant log represents nearly quadratic gap upper lower bounds given limited resources devices typically run heavy hitters algorithms internet routers quadratic gap critical applications problem related hitters problem estimating maximum frequency data stream also known iitk workshop algorithms data streams open question asks algorithm estimate maximum frequency item additive error using little space possible best known space bound still log bits stated original formulation question note question corresponds space complexity problem hitters upper bound log log log log theorem log log log log theorem log log log log theorem log log log log theorem log log theorem lower bound log log log log theorem log log log log theorem log log theorem log log log log theorem log log log theorem table bounds hold constant success probability algorithms sufficiently large terms hitters problem problem also achieve update time reporting time linear size output upper bound resp returning every item borda score resp maximin score additive resp additive lower bound resp returning approximate borda score resp maximin score approximate maximum note one find item whose frequency largest additive error one solve problem latter problem independently interesting corresponds finding approximate plurality election winners voting streams refer problem problem finally note many variants hitters problem one consider one simple variant output item frequency within minimum frequency item universe refer problem makes sense small universes otherwise outputting random item typically works useful one wants count number dislikes anomaly detection see motivation settings one may numerical scores associated items rather stream update consists ranking total ordering stream items may case ranking aggregation web see voting streams see one may consider variety aggregation measures borda score item asks sum rankings number items ranked ahead ranking alternatively one may consider maximin score item asks minimum items number rankings ranked ahead aggregation measures one may interested finding item whose score approximate maximum analogue problem one may interested listing items whose score threshold analogue hitters problem give motivation variants heavy hitters section precise definitions section contributions results summarized table note independently work nearly parallelly improvements space complexity hitters problem insertion streams time complexity hitters problem streams works use different techniques turnstile stream updates modify underlying vector initialized zero vector update form standard unit vector insertion stream updates form allowed first contribution optimal algorithm lower bound hitters problem namely show randomized algorithm constant probability success solves problem using log log log log bits space prove lower bound matching constant factors ram model log bit words algorithm update time reporting time linear output size standard assumptions length stream universe size least poly log furthermore achieve nearly optimal space complexity even length stream known advance although results achieve stronger error bounds terms tail useful skewed streams focus original formulation problem next turn problem estimating maximum frequency data stream additive give algorithm using log log log log bits space improving previous best algorithms required space least log bits show bound tight example setting parameters log log log log space complexity log log log bits improving previous bits space algorithm also prove lower bound showing algorithm optimal constant factors resolves open question iitk workshop algorithms data streams case insertion streams case hitters algorithm also returns identity item approximate maximum frequency solving problem focus number variants problems first give nearly tight bounds finding item whose frequency within minimum possible frequency solved using new algorithm hitters problem would incur log log log bits space whereas give algorithm using log log log log bits space also show nearly matching log log bits space lower bound note problem dependence necessary since number possible items sufficiently large outputting identity random item among first say items correct solution large constant probability finally study variants heavy hitter problems setting stream update consists total ordering universe items problem give algorithm using log log log log log bits space report borda score every item additive also show nearly optimal proving log log log bit lower bound problem even case one interested outputting item maximum borda score additive problem give algorithm using log log bits space report maximin score every item additive prove log log bits space lower bound even case one interested outputting maximum maximin score additive shows finding heavy hitters respect maximin score significantly expensive respect borda score motivations variants heavy hitters hitters problem data stream literature variants introduced provide additional motivation problem formulation item frequency zero one occur stream valid solution problem certain scenarios might make sense stream containing small fraction addresses however scenarios argue natural problem instance consider online portal users register complaints products minimum frequency items correspond best items frequencies arise context voting generally making choice cases one strong preference item definitely like certain items problem applies since frequencies correspond number dislikes problem may also useful anomaly detection suppose one known set sensors broadcasting information one observes field broadcasted packets sensors send small number packets may defective algorithm problem could find sensors finding items maximum minimum frequencies stream correspond finding winners plurality veto voting rules respectively context streaming aspect voting could crucial applications like online polling recommender systems voters providing votes streaming fashion every point time would like know popular items elections political positions scale election may large enough require streaming algorithm one key aspect latter votingbased problems useful numerical scores available orderings naturally arise several applications instance website multiple parts order user visits parts given clickstream defines voting data mining recommendation purposes website owner may interested aggregating orderings across users motivated connection define similar problems two important voting rules namely borda maximin borda scoring method finds applications wide range areas artificial intelligence example machine learning image processing information retrieval etc maximin score often used spread best worst outcome large see maximin scoring method also used frequently machine learning human computation etc preliminaries denote disjoint union sets denote set permutations set positive integer denote set places ignore floors ceilings sake notational simplicity model input data input data stream elements universe context voting input data stream universe possible rankings permutations communication complexity use lower bounds communication complexity certain functions prove space complexity lower bounds problems communication complexity function measures number bits need exchanged two players compute function whose input split among two players restrictive communication model alice first player sends one message bob second player bob outputs result protocol method players follow compute certain functions input also protocols randomized case protocol needs output correctly probability least probability taken random coin tosses protocol randomized communication complexity function error probability denoted standard reference communication complexity model computation model computation ram model words size log capable generating uniformly random words performing arithmetic operations one unit time note model computation used store integer using variable length array allows read update time log bits space fact first work formally pose heavy hitters problem couched context voting universal family hash functions definition universal family hash functions family functions called universal family hash functions picked uniformly random know exists universal family hash functions every positive integer every prime moreover size problem definitions formally define problems study suppose definition heavy hitters given stream length universe size find items frequency along frequencies additive error report items frequency less definition given stream length universe size find maximum frequency additive error next define minimum problem definition given stream length universe size find minimum frequency additive error next define related heavy hitters problems context rank aggregation input stream rankings permutations item set problems borda score item sum rankings number items ranked ahead ranking definition borda given stream universe find items borda score along borda score additive error report items borda score less definition given stream universe find maximum borda score additive error maximin score item minimum items number rankings ranked ahead definition maximin given stream universe find items maximin score along maximin score additive error report items maximin score less definition given stream universe find maximum maximin score additive error notice maximum possible borda score item maximum possible maximin score item justifies approximation factors definition note finding item maximum borda score within additive maximum maximin score within additive corresponds finding approximate winner election precisely known algorithms section present upper bound results omitted proofs appendix describing specific algorithms record claims later use lemma follows checking whether get heads log tosses fair coin lemma suppose power algorithm choosing item probability space complexity log log bits time complexity ram model proof generate integer uniformly random record sum digits choose item sum digits remark algorithm lemma optimal space complexity shown proposition appendix second claim standard result universal families hash functions lemma universal family hash functions proof every since universal family hash functions apply union bound get third claim folklore also follows celebrated dkw inequality provide simple proof works constant lemma let frequencies item stream random sample size respectively log probability every universe item simultaneously proof constant follows chebyshev inequality union bound indeed consider given frequency suppose sample occurrences probability parameter expected number sampled occurrences variance var applying chebyshev inequality var union bound sample setting constant makes probability element stream independently probability probability exists desired assume length stream known advance show section remove assumption algorithms whenever pick item probability assume without loss generality power two replace largest power two less affect correctness performance algorithms list heavy hitters heavy hitters problem present two algorithms first slightly suboptimal simple conceptually already constitutes large improvement space complexity known algorithms expect algorithm could useful practice well second algorithm complicated building ideas first algorithm achieves optimal space complexity upto constant factors note algorithms proceed sampling stream items updating data structure stream progresses cases time update data structure bounded standard assumption length stream least poly time perform update spread across next stream updates since large probability items sampled among next stream updates therefore achieve update time simpler algorithm theorem assume stream length known beforehand randomized algorithm heavy hitters problem succeeds probability least using log log log log log log bits space moreover update time reporting time linear output size overview overall idea follows sample many items stream uniformly random well hash word short identifier sampled elements space size stream length well universe size poly lemma suffices solve heavy hitters problem sampled stream lemma hash function chosen universal family sampled elements distinct hashed feed elements standard data structure counters incurring space log want return unhashed element heavy hitters also use log space recording top items according data structure output asked report proof theorem pseudocode heavy hitters algorithm algorithm lemma select subset size least log uniformly random stream frequencies item input stream respectively first show choice line algorithm number items sampled least probability least let indicator random pmvariable event item sampled total number items sampled since following inequality follows chernoff bound value onwards assume number items sampled use modified version algorithm estimate frequencies items length table algorithm pick hash function uniformly random universal family hash functions size note picking hash function uniformly random done using log bits space lemma shows collisions hash function probability least onwards assume collision among ids sampled items hash function modify algorithm follows instead storing item table table line algorithm store hash also store ids emphasize stronger amortized guarantee every insertion cost algorithm heavy hitters input stream length let frequency output set function every initialize log hash function uniformly random universal family empty table key value pairs length key entry store integer value entry store integer table sorted order value throughout empty table length entry store integer entries correspond ids keys highest values procedure insert probability continue otherwise return perform update using maintaining sorted values value among highest valued items currently contains many items among highest valued items replace else put ensure elements ordered according corresponding values procedure report return items along corresponding values hash items highest values another table moreover always maintain table consistent table sense ith highest valued key hash ith upon picking item probability create entry corresponding make value one space available decrement value every item one table already full increment entry table corresponding already present table decrement value every item table remains consistent need anything else otherwise three cases consider case among highest valued items case need anything else case among highest valued items among highest valued items case last item longer among highest valued items replace case among highest valued items stream finishes output ids items table along values corresponding table correctness follows correctness algorithm fact collision among ids sampled items optimal algorithm theorem assume stream length known beforehand randomized algorithm heavy hitters problem succeeds constant probability using log log log log bits space moreover update time reporting time linear output size note section sake simplicity ignore floors ceilings state results constant error probability omitting explicit dependence algorithm heavy hitters input stream length universe let frequency output set function every initialize hash functions log uniformly random universal family empty table key value pairs length key entry store element value entry store integer empty table rows log columns entry store integer empty table size log log entry store integer upper bounds allowed cells actually used procedure insert probability increment continue else return perform update log probability increment min probability increment procedure report key nonzero value log log min median log return overview simpler algorithm sample many stream elements solve list heavy hitters problem sampled stream also algorithm heavy hitters returns candidate set items containing items frequency least remains count frequencies items upto additive error remove whose frequency less fix item let count sampled stream natural approach count approximately increment counter probabilistically instead deterministically every occurrence suppose increment counter probability whenever item arrives stream let value counter let see var follows var hence unbiased estimator additive error constant probability call counter accelerated counter probability incrementing accelerates increasing counts maintain log accelerated counters independently take median drive probability deviating constant probability frequency items data structure estimated within error desired however two immediate issues approach first problem may need keep counts distinct items costly purposes get around use hash function universal family hash universe space size work throughout hashed show space complexity iteration also accelerated counters estimate frequencies hashed instead actual items universality expected frequency hashed desired error bound second issue need constant factor approximation set algorithm needs first compute one pass run accelerated counter another divide stream epochs stays within factor use different epoch particular set pti log want keep running estimate count within factor know current epoch incremented subsample element stream probability independently maintain exact counts observed hashed easy see requires bits expectation consider prefix stream upto let frequency prefix let frequency among samples prefix see moreover show approximation constant probability repeating log times independently taking median error probability driven every hashed need one accelerated counter log many one corresponding epoch element hash arrives position decide based epoch belongs increment accelerated counter probability pti storage cost still also iterate whole set accelerated counters log times making total storage cost log let bei count accelerated counter hash epoch let clearly variance epoch var log wanted issue fixed change sampling probabilities defined formal proof proof theorem pseudocode appears algorithm note numerical constants chosen convenience analysis optimized also sake simplicity pseudocode optimal reporting time modified achieve see end proof details standard chernoff bounds probability least length sampled stream let fsamp frequency sampled stream lemma probability least fsamp fix log let fsamp random expected value fsi samp since universal mapping space size hence using markov inequality fsamp fsamp lemma show log error probability line estimates additive error hence estimating fsi additive error taking median log repetitions line makes error probability using standard chernoff bounds hence union bound probability least keys nonzero values estimate within additive error thus showing correctness lemma fix log let quantity computed line proof index sampled stream elements let frequency items hash restricted first elements sampled stream let denote value procedure insert called first items sampled stream claim probability least within factor proof fix note incremented rate var chebyshev inequality break stream chunks apply inequality chunk take union bound conclude namely integer define first exists probability least every within factor since every within factor claim follows assume event claim henceforth ready analyze particular first observe line position stream time must standard markov chernoff bounds probability least assume event claim var proof stream element position causes increment probability line must case highest count increments slot number occurrences claim applied since equation var elements inserted probability obviously contribute variance conditioning events mentioned probability deviates removing conditioning yields wanted next bound space complexity claim probability least algorithm uses log log log log bits storage proof expected length sampled stream number bits stored log note lines given storing total elements expectation hashed counts summing accounting empty cells gives bits storage total space requirement log probability hashed gets counted table line definition moreover claim therefore expected value cell first coordinate taking account many number epochs associated log log line get total space required log log first term inside summation since expected space bound obtain space bound error probability markov bound space required sampling additional log log using lemma note space bound made worst case aborting algorithm tries use space remaining aspect theorem time complexity observed section update time made per insertion standard assumption stream sufficiently long reporting time also made linear output changing bookkeeping bit instead computing reporting time maintain every insertion although apparently makes insert costlier true fact spread cost future stream insertions space complexity grows constant factor maximum tweaking algorithm slightly get following result problem theorem assume length stream known beforehand randomized onepass algorithm problem succeeds probability least using min log log log log log log bits space moreover algorithm update time proof instead maintaining table algorithm store actual item maximum frequency sampled items minimum theorem assume length stream known beforehand randomized algorithm problem succeeds probability least using log log log log bits space moreover algorithm update time overview pseudocode provided algorithm idea behind problem follows easily explained looking report procedure starting line lines ask universe size significantly larger note outputting random item likely solution otherwise next point number distinct elements stream smaller log could store items together frequencies bits space indeed first sample stream elements relative frequencies preserved additive thereby ensuring frequency stored log bits also since universe size item identifiers also stored log bits part algorithm starts taking much space stop know number distinct elements least log means minimum frequency log implemented steps algorithm also ensure minimum frequency least log indeed randomly sampling log stream elements maintaining bit vector whether item universe occurs bits space since item frequency least log sampled entry bit vector empty output solution implemented steps algorithm finally know minimum frequency least log log point randomly sample stream elements chernoff bounds item frequencies preserved relative error factor particular relative minimum frequency guaranteed preserved additive point maintain exact counts sampled stream truncate exceed poly log bits since know counts correspond minimum thus need log log bits represent counts implemented step step algorithm proof theorem pseudocode algorithm algorithm size universe least return item chosen uniformly random note many items frequency least hence every item among remaining many items frequency less thus correct output instance thus probability answer correctly least let assume value follows proof theorem assume happens probability least first show every item frequency least sampled probability least let xij indicator random variable event sample item item frequency least let set items frequencies least following xij xij exp applying union bound get following xij hence probability least output line correct show frequency item probability least algorithm input stream length let frequency output item every initialize log log log bit vector procedure insert put probability updating bit vector number distinct items stream far log pick probability put initialize corresponding counter increment counter corresponding pick probability put initialize corresponding counter increment counter corresponding truncate counters procedure report return item first items ordered arbitrarily uniformly random return item number distinct items stream log return item minimum counter value return item minimum frequency hence onwards assume frequency every item least number distinct elements line outputs minimum frequency item additive factor due chernoff bound note need bits space storing ids hence stored space assume number distinct elements least hence frequency item minimum frequency let frequency item counter value applying chernoff bound following fixed exp exp applying union bound get following using fact applying chernoff bound union bound get following hence items frequency approximated multiplicative factor counters items may truncated items frequency approximated relative error thus additive error counters items would get truncated hence item minimum counter value item minimum frequency additive need bits space bit vector set need bits space set bits space set choice truncation threshold need additional bits space sampling using lemma moreover using data structure section algorithm performed time alternatively may also use strategy described section spreading update operations several insertions make cost per insertion borda maximin theorem assume length stream known beforehand randomized onepass algorithm borda problem succeeds probability least using log log log log log log bits space proof let log insertion vote select probability store every number candidates candidate beats vote keep exact counts counter length follows proof theorem probability least moreover straightforward application chernoff bound see follows denotes borda score candidate restricted sampled votes space complexity exactly storing counts log log log log log space sampling votes log log lemma theorem assume length stream known beforehand randomized onepass algorithm maximin problem succeeds probability least using log log log log bits space proof let put current vote set probability follows proof theorem probability least suppose let set votes sampled let total number votes beats number votes choice chernoff bound see follows every pair candidates note vote stored log bits space hence simply finding every storing returning items maximin score least requires log log log log log bits memory additive log log due lemma unknown stream length consider case length stream known beforehand present algorithm heavy hitters problems setting length stream known beforehand theorem randomized algorithm heavy hitters problems space complexity log log log log bits update time even length stream known beforehand proof describe randomized algorithm heavy hitters problem may assume length stream least otherwise use algorithm theorem get result guess length stream run instance algorithm log line choice size sample log outputs correctly probability least length stream length stream exceeds run another instance algorithm log line choice size sample outputs correctly probability least length stream stream length exceeds discard free space uses run instance algorithm log line point time two instances algorithm running stream ends return output older instances currently running use approximate counting method morris approximately count length stream know morris counter outputs correctly probability using log log bits space point time also since morris counter increases item read outputs correctly factor four every position outputs correctly positions call event choosing applying union bound positions correctness algorithm follows correctness algorithm fact discarding many items stream discarding run instance algorithm space complexity update time algorithm follow theorem choice fact two instances algorithm currently running point time algorithm problem algorithm except use algorithm theorem instead algorithm note proof technique seem apply optimal algorithm similarly theorem get following result problems theorem randomized algorithms problems space complexity log log log log log log log log log log log log log log bits respectively even length stream known beforehand moreover update time hardness section prove space complexity lower bounds hitters problems present reductions certain communication problems proving space complexity lower bounds let first introduce communication problems necessary results communication complexity definition indexingm let positive integers alice given string bob given index bob output following well known result lemma indexingm log constant defines communication problem called perm generalize follows definition alice given permutation partitioned many contiguous blocks bob given index output block belongs lower bound matches lower bound perm lemma proof reader may find useful information theory facts described appendix lemma perm log constant proof let assume permutation alice uniformly distributed set permutations let denotes block item let alice message bob random variable depending randomness private coin tosses alice perm hence enough lower bound following chain rule number ways partition items blocks hence log consider correctness algorithm fano inequality log hence following log finally consider problem definition alice given integer bob given integer bob output otherwise following result due provide simple proof seems literature lemma log every proof reduce problem problem thereby proving result alice runs protocol input number whose representation binary bob participates protocol input number whose representation binary reductions observe trivial log bits lower bound heavy hitters borda maximin follows fact algorithm may need output many items universe also trivial log lower bound borda maximin stream item permutation hence requiring log bits read show space complexity lower bound log bits hitters problem similar proof appears gives weaker lower bound theorem suppose size universe least randomized one pass hitters algorithm success probability least must use log bits space constant proof consider problem alice given string bob given index assume stream generate possible since let large positive integer alice generates stream length inserting copies alice sends memory content algorithm bob bob resumes run algorithm generating another stream length inserting copies length stream frequency item frequency every item hence output hitters algorithm bob knows probability least result follows lemma since log log use idea proof theorem prove log space complexity lower bound problem theorem suppose size universe least randomized one pass algorithm success probability least must use log bits space constant proof consider problem alice given string bob given index stream generate possible since let large positive integer alice generates stream length way frequency every item least frequency item alice sends memory content algorithm bob bob resumes run algorithm generating another stream length way frequency every item least frequency item frequency item least frequency every item hence algorithm must output probability least result follows lemma prove space complexity lower bound bits theorem suppose universe size least randomized one pass algorithm must use bits space proof reduce thereby proving result let inputs alice bob index respectively alice bob generate stream universe alice puts two copies item every runs algorithm alice sends memory content algorithm bob bob resumes run algorithm putting two copies every item stream bob also puts one copy suppose size support since following algorithm must output probability least algorithm must output probability least result follows lemma show next log bits space complexity lower bound theorem one pass algorithm must use log bits space proof reduce suppose alice permutation bob index item set reduced election alice generates vote item set follows vote defined follows alice runs algorithm vote sends memory content bob let arbitrary fixed ordering items reverse ordering bob resumes algorithm generating two votes form let call resulting election number votes borda score item least borda score every item hence algorithm must output item moreover follows construction additive approximation borda score item reveals block belongs instance next give lower bound problem theorem algorithm requires memory bits storage proof reduce indexing let suppose alice string length partitioned blocks length bob index indexing problem return lower bound lemma initial part reduction follows construction proof theorem encapsulate following lemma lemma theorem given alice construct matrix using public randomness rows respectively probability least let alice construct according lemma adjoin bitwise complement matrix form matrix note column exactly interpret row candidate column vote following way vote candidates ascending order top positions rest candidates ascending order bottom positions alice inserts votes stream sends state algorithm bob well hamming weight row bob inserts votes candidate comes first candidate comes second rest candidates arbitrary order note bob votes maximin score number votes among ones casted alice defeats since columns candidate beats candidate thus votes defeats size set therefore bob estimate maximin score upto additive error find upto additive error bob knows enough lemma solve indexing problem probability least finally show space complexity lower bound depends length stream theorem one pass algorithm hitters must use log log memory bits even stream universe size every proof enough prove result hitters since three problems reduce hitters universe size suppose randomized one pass hitters algorithm uses bits space using algorithm show communication protocol problem whose communication complexity thereby proving statement universal set alice generates stream many copies item alice sends memory content algorithm bob resumes run algorithm generating stream many copies item item whereas item acknowledgments would like thank jelani nelson helpful conversation led discover error previous version paper references arvind arasu shivnath babu jennifer widom cql language continuous queries streams relations database programming languages international workshop dbpl potsdam germany september revised papers pages javed aslam mark montague models metasearch proceedings annual international acm sigir conference research development information retrieval pages acm rakesh agrawal ramakrishnan srikant fast algorithms mining association rules large databases vldb proceedings international conference large data bases september santiago chile chile pages gediminas adomavicius alexander tuzhilin toward next generation recommender systems survey possible extensions knowledge data engineering ieee transactions daniel blandford guy blelloch compact dictionaries keys data applications acm transactions 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monwar marina gavrilova multimodal biometric system using fusion approach systems man cybernetics part cybernetics ieee transactions gurmeet singh manku rajeev motwani approximate frequency counts data streams proceedings international conference large data bases pages vldb endowment peter bro miltersen noam nisan shmuel safra avi wigderson data structures asymmetric communication complexity comput syst strother moore journal algorithms june robert morris counting large numbers events small registers communications acm andrew mao ariel procaccia yiling chen social choice human computation proceedings fourth workshop human computation andrew mao ariel procaccia yiling chen better human computation principled voting proceedings conference artificial intelligence aaai mullen roth decision making logic practice savage rowman littlefield publishers shanmugavelayutham muthukrishnan data streams algorithms applications publishers inc nikos mamoulis man lung yiu kit hung cheng david cheung efficient aggregation ranked inputs acm trans database rabia nuray fazli automatic ranking information retrieval systems using data fusion information processing management jelani nelson sketching streaming algorithms processing massive data xrds crossroads acm magazine students adarsh prasad harsh pareek pradeep ravikumar distributional rank aggregation axiomatic analysis proceedings international conference machine learning pages paul resnick hal varian recommender systems communications acm nisheeth shrivastava chiranjeeb buragohain divyakant agrawal subhash suri medians beyond new aggregation techniques sensor networks proceedings international conference embedded networked sensor systems sensys baltimore usa november pages smirnov shannon information methods lower bounds probabilistic communication complexity master thesis moscow university ashok savasere edward omiecinski shamkant navathe efficient algorithm mining association rules large databases vldb proceedings international conference large data bases september zurich pages xiaoming sun david woodruff tight bounds graph problems insertion streams approximation randomization combinatorial optimization algorithms techniques august princeton usa pages hannu toivonen sampling large databases association rules vldb proceedings international conference large data bases september mumbai bombay india pages dirk van gucht ryan williams david woodruff qin zhang communication complexity distributed applications matrix multiplication proceedings acm symposium principles database systems pages acm maksims volkovs richard zemel new learning methods supervised unsupervised preference aggregation journal machine learning research gang wang frederick lochovsky feature selection conditional mutual information maximin text categorization proceedings thirteenth acm international conference information knowledge management pages acm lirong xia computing margin victory various voting rules proceedings acm conference electronic commerce pages acm andrew yao complexity questions related distributive computing preliminary report proceedings eleventh annual acm symposium theory computing pages acm appendix information theory facts discrete random variable possible values shannon entropy defined let denote binary entropy function two random variables possible values respectively conditional entropy given defined let denote mutual information two random variables let denote mutual information two random variables conditioned following summarizes several basic properties entropy mutual information proposition let random variables takes value log independent similarly independent given chain rule mutual information pni generally random variables thus fano inequality let random variable chosen domain according distribution random variable chosen domain according distribution reconstruction function error log refer readers nice introduction information theory appendix remark algorithm lemma optimal space complexity shown proposition may independent interest proposition algorithm chooses item set size probability unit cost ram model must use log log bits memory proof algorithm generates bits uniformly random number bits generates uniformly random may also depend outcome previous random bits finally picks item say consider run algorithm chooses item smallest number random bits getting generated say generates random bits run means run algorithm item chosen algorithm must generate least many random bits let random bits generated let memory content algorithm immediately generates ith random bit run first notice probability item chosen would contradiction hence claim must different indeed otherwise let assume algorithm chooses item generating many random bits strictly less random bits generated contradicts assumption run started chooses item smallest number random bits generated
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beyond provable guarantees sampling distributions using simulated tempering langevin monte carlo nov rong holden andrej november abstract last several years provable guarantees iterative optimization algorithms like gradient descent settings become topic intense research machine learning community works shed light practical success algorithms many unsupervised learning settings matrix completion sparse coding learning latent variable bayesian models another elementary task bayesian settings besides model learning sampling distributions specified partition function constant proportionality concrete example models sampling posterior latent variables model used learned similar theoretical issues plague task learning one without assumptions sampling even approximately however works provided beyond guarantees settings analogue convexity inference distributions classical results going back show natural markov chains called langevin diffusions mix polynomial time salient feature violated practice commonly distributions wish sample multimodal presence multiple deep modes langevin diffusion suffers torpid mixing address problem combining langevin diffusion simulated tempering result markov chain mixes rapidly transitioning different temperatures distribution analyze markov chain canonical distribution mixture gaussians equal variance algorithm based markov chain provably samples distributions close mixtures gaussians given access gradient analysis use spectral decomposition theorem graphs markov chain decomposition technique duke university computer science department rongge princeton university mathematics department holdenl massachusetts institute technology applied mathematics idss risteski contents introduction results prior work preliminaries markov chains overview langevin dynamics overview simulated tempering algorithm overview proof decomposing simulated tempering chain existence partitions mixing highest temperature discretizing langevin diffusion estimating partition functions spectral gap simulated tempering defining partitions proving eigenvalue gap scaled temperature approximates mixture gaussians inequalities small subsets existence partition making arbitrarily fine partitions restriction large compact set mixing highest temperature discretizing continuous chains proof main theorem acknowledgements background markov chains discrete time markov chains restricted projected markov chains conductance clustering continuous time markov processes examples pertubation tolerance mixing time tempering chain mixing time highest temperature discretization putting things together another lower bound simulated tempering introduction recent years one fruitful directions research providing theoretical guarantees optimization settings particular routine task unsupervised supervised learning use training data fit optimal parameters model parametric family theoretical successes context range analyzing approaches using iterative techniques like gradient descent variational inference variety models models include topic models dictionary learning gaussian mixture models bayesian networks finding maximum likelihood values unobserved quantities via optimization reasonable many learning settings number samples large maximum likelihood converge true values quantities however bayesian inference problems given document topics number samples limited maximum likelihood may cases would prefer sample posterior distribution generality typical scenario sampling posterior distribution latent variables latent variable bayesian model whose parameters known models observable variables follow distribution simple succinct form given values latent variables joint factorizes factors explicit hence posterior distribution form even though numerator easy evaluate without structural assumptions distributions often hard sample exactly approximately difficulty evaluating denominator even approximately even simple models like topic models sampling analogues convex functions arguably widest class functions optimization easy distributions distributions form convex function recently renewed interest analyzing popular markov chain sampling distributions given gradient access natural setup posterior sampling task described particular markov chain called langevin monte carlo see section popular bayesian practitioners proven work various rates depending properties distributions necessarily density functions one local maximum must global maximum fails capture many interesting scenarios many simple posterior distributions neither instance posterior distribution means mixture gaussians practical direction complicated posterior distributions associated deep generative models variational believed multimodal well figure mixture two gaussians left two gaussians local sampling algorithm move modes right distribution high temperature possible move modes goal work initiate exploration provable methods sampling beyond parallel optimization beyond convexity results prohibited hardness results must make assumptions distributions interested first step paper consider prototypical multimodal distribution mixture gaussians results formalize problem interest follows wish sample distribution allowed query point start focus problem density function mixture gausp sians given centers weights variance gaussians spherical covariance matrix function defined exp log furthermore suppose show efficient algorithm sample distribution given access theorem main defined equation algorithm informal given running time poly outputs sample distribution within note algorithm direct access even sampling mixture gaussians distribution sampling algorithms based making local steps langevin monte carlo move different components gaussian mixture gaussians see figure left algorithm use simulated tempering see section technique considers distribution different temperatures see figure right order move different components appendix give examples show simple heuristics work assumption gaussians covariance removed particular show note expression inside log essentially probability density mixture gaussians except normalization factor missing however normalization factor introduce constant shift really change random initialization enough find modes also give example mixture two gaussians even covariance differs constant multiplicative factor simulated tempering known take exponential time course requiring distribution exactly mixture gaussians strong assumption results generalized functions close mixture gaussians precisely function satisfies following properties log exp intuitively conditions show density distribution within multiplicative factor unknown mixture gaussians theorem generalized case theorem general case informal function equations algorithm runs time poly outputs sample distribution prior work algorithm use two classical techniques theory markov chains langevin diffusion chain sampling distributions form given gradient access simulated tempering heuristic technique used tackling multimodal distributions recall briefly known techniques langevin dynamics convergence stationary distribution classic result understanding mixing time continuous dynamics distributions also classic result show distributions satisfy logsobolev inequality characterize rate convergence course algorithmically one run discretized version langevin dynamics results approaches much recent obtained algorithm sampling distribution gave algorithm sample distribution restricted convex set incorporating projection step give nonasymptotic analysis langevin dynamics arbitrary distributions certain regularity decay properties course mixing time exponential general spectral gap chain small furthermore long known transitioning different modes take exponentially long time phenomenon known folklore result guarantees mixing extend distributions nice function close convex function distance however address global deviations convexity clear distributions far many deep modes additional techniques necessary among many proposed heuristics situations simulated tempering effectively runs multiple markov chains corresponding different temperature original chain mixes different markov chains intuition markov chains higher temperature move modes easily one mix points lower temperature chains mixing time ought improve well provable results heuristic however far spectral gap generic simulated tempering chains crucial technique paper shares markov chain decomposition technique due however scenario section interested spectral gap bound exponentially small function number modes result remedy preliminaries section first introduce notations markov chains details deferred appendix briefly discuss langevin monte carlo simulated tempering markov chains paper use discrete time continuous time markov chains section briefly give definitions notations discrete time markov chains continuous time markov chains follow intuition defer formal definitions appendix definition discrete time markov chain measure space probability measure defines random process follows stationary distribution equivalently chain reversible markov chain finite number states represented weighted graph transition probabilities proportional weights edges reversible markov chain represented undirected graph variance dirichlet form spectral gap important quantity markov chain spectral gap definition markov chain let operate functions suppose unique stationary distribution let define dirichlet form variance write define eigenvalues eigenvalues respect norm define spectral gap gap inf case finite undirected graph function corresponds vector dirichlet form corresponds normalized laplacian matrix variance squared norm spectral gap controls mixing markov chain define distance let initial distribution distribution running markov chain steps min restrictions projections later also work continuous time markov chains langevin dynamics see section proof also need consider restrictions projections markov chains intuitively restricting markov chain subset states denote removes states replaces transitions merges projecting markov chain partition denote parts partition individual states formal definitions see appendix conductance clustering finally define conductance clusters markov chains familiar concepts undirected graphs definition let markov chain unique stationary distribution let drawn stationary distribution next state markov chain define external conductance cheeger constant min clustering markov chain analogous partition vertices undirected graphs good clustering require large small definition let markov chain finite state space say disjoint subsets overview langevin dynamics langevin diffusion stochastic process described stochastic differential equation henceforth sde wiener process crucial folklore fact langevin dynamics langevin dynamics converges stationary distribution given substituting gives langevin diffusion process inverse temperature stationary distribution equivalently also possible consider temperature changing magnitude noise course algorithmically run process run discretized version process namely run markov chain random variable time described step size reason scaling running brownian motion time scales variance works analyzed convergence properties bias stationary distribution convergence rate distributions give convergence rates distributions course latter case rates depend spectral gap often exponential dimension overview simulated tempering simulated tempering technique converts markov chain new markov chain whose state space product original state space temperature new markov chain allows original chain change temperature maintaining correct marginal distributions given discrete time markov chain consider temperatures let denote set define simulated tempering chain follows definition let sequence markov chains state space unique stationary distributions let define simulated tempering markov chain relative probabilities follows states suppose current state probability transition keep fixed update according call type probability following step draw randomly transition probability min stay otherwise call type transition remark type two transitions instead pick slightly improve bounds mixing time ratio bounded exponential otherwise simplicity stick traditional definition simulated tempering markov chain typical setting follows markov chains come smooth family markov chains parameter markov chain parameter using terminology statistical physics inverse temperature interested sampling distribution large small however chain suffers torpid mixing case distribution peaked simulated tempering chain uses smaller larger help mixing stationary distribution inverse temperature course langevin dynamics introduced previous section continuous time markov chain algorithm change discrete time markov chain fixing step size another difficulty running simulated tempering chain directly access know partition function make use flexibility fix issue details see section crucial fact note stationary distribution mixture distributions corresponding different temperatures namely proposition folklore reversible markov chains stationary distributions simulated tempering chain reversible markov chain stationary distribution algorithm algorithm run simulated tempering chain polynomial number temperatures running discretized langevin dynamics various temperatures full algorithm specified algorithm mentioned obstacle running simulated tempering chain access partition function solve problem estimating partition function high temperature low temperature adding one temperature time see algorithm note simulated tempering chain mixes produce good samples standard reductions easy estimate ratios partition functions main theorem following algorithm simulated tempering langevin monte carlo step size time interinput temperatures partition function estimates val number steps output random sample approximately distribution let probability keep fixed update according repeat times probability make type transition acceptance ratio min end final state return otherwise chain algorithm main algorithm input function satisfying assumption gradient access output random sample let sequence inverse temperatures satisfying let run simulated tempering chain algorithm temperatures estimates step size time interval number steps given lemma return sample repeat get samples let end theorem main theorem suppose exp algorithm parameters given lemma produces sample distribution time poly simplicity stated theorem distributions exactly mixtures gaussians theorem robust perturbations give general theorem appendix overview proof first briefly sketch entire proof subsequent sections expand individual parts key part proof new technique bounding spectral gap simulated tempering chain using decompositions section temperature make partition large pieces inside partition done every temperature difference temperature small enough chain mixes highest temperature show simulated tempering chain also mixes quickly general theorem mixing simulated tempering chains may useful settings figure mixture gaussians partition space regions langevin monte carlo mixes well show mixture gaussians indeed partition exists section use spectral clustering techniques developed finite graphs main technical difficulty transferring discrete continuous cases finally complete proof showing markov chain mixes highest temperature section discretized markov chain approximates continuous time markov chain section partition functions estimated correctly allows run simulated tempering chain section last appendix prove arguments tolerant perturbations algorithm works distributions exactly mixtures gaussians decomposing simulated tempering chain first show exists partition temperature markov chain mixes rapidly within set partition gap large sets partition small chain mixes highest temperature simulated tempering chain mixes rapidly theorem theorem let sequence markov chains state space stationary distributions consider simulated tempering chain min probabilities let max let partition ground set define overlap parameter min min define min spectral gap tempering chain satisfies gap min gap prove use techniques similar existing work simulated tempering precisely similar proof apply decomposition theorem theorem analyzing mixing time simulated tempering chain note using theorem analysis need existential algorithmic result order apply theorem show exist good partitions spectral gap gap within set large set partition size poly remark appendix theorem also give different incomparable criterion spectral gap improves bound cases exponential factor theorem requires partitions successive refinements advantage depending parameter larger unlike even polynomial exponentially many pieces theorem necessary proof main theorem existence partitions show existence good partitions using theorem theorem shows singular value large possible clustering parts high inside conductance within clusters low outside conductance clusters definition theorem spectrally partitioning graphs theorem let reversible markov chain states let eigenvalues markov chain exists sets mixture gaussians using inequality gaussians show continuous langevin chain bounded away therefore one would hope use theorem obtain clustering however difficulties especially theorem holds discrete time discrete space markov chain solve discrete time problem fix time consider discrete time chain step running langevin time solve discrete space problem note apply theorem markov chain projected partition see definition definition projected markov chain series technical lemmas show eigenvalues conductances change much pass discrete time space chain another issue although theorem guarantees good immediately give lowerbound size clusters use inequality show small set must large therefore clustering guaranteed theorem small clusters thus assumptions theorem satisfied get partition large internal conductance small external conductance projected chain lemma letting size cells partition show gap projected chain approaches gap continuous chain lemma lemma works compact sets also need show restricting large ball change eigenvalues much lemma mixing highest temperature next need show mixing highest temperature bounded domains could set highest temperature infinite would correspond uniform sampling domain since working unbounded domain instead compare langevin dynamics strictly convex function close distance lemma use fact langevin dynamics mixes rapidly strictly convex functions perturbation affects spectral gap factor discretizing langevin diffusion point though subdivided time discrete intervals size time interval ran continuous langevin chain time however algorithmically run discretization langevin diffusion need bound drift discretization continuous chain follow usual pattern discretization arguments run continuous chain time step size drift discretized chain continuous bound provides bound drift precisely show lemma lemma let distributions running simulated tempering chain steps temperature type transitions taken according discrete time markov kernel running langevin diffusion time type transitions taken according running steps discretized langevin diffusion using discretization granularity max prove consider two types steps separately type steps tempering chains increase divergence continuous discretized version chains type steps increase divergence basis bounded using existing machinery discretizing langevin diffusion see along decomposition theorem divergence mixture distributions lemma make brief remark since means satisfy easy characterize location conclude bounding quantity essentially requires mass initial distributions concentrated ball size namely following holds lemma let estimating partition functions finally filter type step simulated tempering chain requires estimate partition functions temperatures sufficient estimate partition functions within constant factor gap tempering chain depends ratio probability given temperature run chain temperatures obtain good samples use estimate use lemma show high probability good estimate spectral gap simulated tempering section prove lower bound spectral gap simulated tempering given partition assumptions let sequence markov chains state space stationary distributions consider simulated tempering chain min probabilities let max let ground set define overlap parameter min min allow overlaps sets measure theorem suppose assumptions hold define min spectral gap tempering chain satisfies gap min gap proof let stationary distribution first note easily switch using note define partition theorem gap gap min gap second term gap related gap gap abuse notation considering sets states identify union sets corresponding set states bound gap bounding conductance using cheeger inequality theorem suppose first intuitively means highest temperature set top bottom find partition set interaction part temperature already provide large enough cut let minimal probability proposing switch level min min min min max defining definition consider case case highest temperature set try find part highest temperature set note define thus cheeger inequality gap therefore proved projected markov chain partitions good spectral gap left prove inside partition markov chain good spectral gap note gap gap chain state transitions according probability plugging gives bound remark suppose type transition instead pick follows probability let probability let let instead becomes proof get improved gap gap min gap defining partitions section assemble ingredients show exists partition langevin chain gap large part partition also significant probability hence partition sufficient partitioning technique discussed previous section plan use theorem find partitions temperature indeed mixture gaussians hard show eigenvalue large lemma eigenvalue gap mixtures let probability distributions let suppose inequality holds constant eigenvalue satisfies defer proof section however still many technical hurdles need deal apply theorem spectral partitioning temperature different distribution proportional longer mixture gaussians show still close mixture gaussians sense density function within fixed multiplicative factor density mixture gaussians section kind multiplicative guarantee allows relate constants two distributions show section inequality small sets serves two purposes proof shows large shows set small conductance small also deal problem continuous time taking fixed time running markov chain multiples prove lemma shows discretize markov chain exists good partitioning resulting markov chain section show restrict langevin chain large ball discretize space large ball finely enough limit spectral gap discretized chain spectral gap markov chain section finally section show restrict langevin chain restricted large ball proving eigenvalue gap prove lemma main idea use variational characterization eigenvalues show bad directions proof use variational characterization eigenvalues min max subspace dim suffices produce subspace choose span take since inequality holds thus needed scaled temperature approximates mixture gaussians following lemma shows changing temperature approximately changing variance gaussian state generally arbitrary mixtures distributions form lemma approximately scaling temperature let probability distribur tions let let define distribution inverse temperature define distribution proof inequality hand given setting gives implies gives inequalities small subsets section prove inequalities small sets fact need prove property true robustly order transform continuous time markov chain discrete time markov chain definition given measure say inequality constant holds sets measure whenever supp robust following sense condition satisfied still satisfies inequality even completely supported small set lot mass small set main reason need robust version transform continuous time markov chain discrete time markov chain even initialized small set time probability mass going spill slightly larger set lemma robustly small sets let subset suppose inequality constant holds sets measure showing implies much mass comes small much mass comes intersection set large set means use inequality sliced version proof scaling may assume let suffices show slice portion large translate note take discontinuous compare proposition noting supp lower bound follows let used fact putting everything together var var lemma conductance let set suppose inequality constant holds sets measure let conductance set running langevin time min inequality thought giving lower bound instantaneous conductance show implies good conductance finite time could wrong rate mass leaving set large time quickly goes show using lemma happen significant mass escaped proof want bound since semigroup respect definition let minimal let never happens however show lemma differential inequality implies exponential decay fact known gronwall inequality max last inequality follows decreasing lemma easy direction relating laplacian projected chain let reversible markov chain projection respect partition let proof action functions action subspace functions constant set partition denoted means action embeds action let projection note variational characterization eigenvalues min min max dim lemma inequality mixtures keep setup lemma suppose holds gaussians equal variance suppose inequality holds constant inequality holds constant sets size proof let supp lemma using using lemma lemma satisfies inequality constant existence partition ready prove main lemma section gives partitioning discrete space discrete time markov chain later subsections connect back continuous space markov chain lemma let probability density function let langevin chain satisfies inequality holds constant sets size let partition let markov chain step running continuous langevin time markov chain restricted projected chain respect suppose satisfies following exists partition sets following hold every set partition measure least proof first show eigenvalue large note eigenvalues exponentials eigenvalues min lemma assumption let small constant chosen let largest integer parameters theorem exists clustering consider set partition let suppose way contradiction lemma noting conductance conductance satisfies assumption chosen small enough together give contradiction making arbitrarily fine partitions section show discretization fine enough spectral gap discrete markov chain approaches spectral gap markov chain need following fact topology lemma continuity implies uniform continuity compact set compact metric space continuous every know gap projected chain least gap original chain gap gap lemma show size cells goes reverse inequality also holds moreover convergence uniform size cells lemma let reversible markov chain kernel continuous everywhere stationary distribution continuous function fix compact set lim inf gap gap infimum compact sets partitions composed sets diam proof lemma exists also choose small enough sets diameter let denote kernel stationary distribution let notation remind transition succeeds let consider partition whose components diameter let projected chain stationary distribution let capital letters denote elements corresponding subsets denote function let probability distribution given similarly define also write rej rej acc inf gap inf inf inf gap inf relate two quantities consider denominator pythagorean theorem equals variation sets partition given denominator plus variation within sets partition also decompose numerator first show approximate distribution independent given set containing using let distribution defined follows rej use pythagorean theorem letting rej thus using min decompositions min first ratio minimum least gap bound second ratio numerator min acc min acc claim acc gap consider gap acc acc putting everything together gap min gap gap combined lemma gap gap letting finishes proof restriction large compact set finally show restrict large compact set intuitively clear gaussian density functions highly concentrated around means lemma let pdf gaussian mean let exists supp note improve arbitrary careful analysis using martingale concentration bounds weaker version suffices proof let let lemma let consider change variable since linear lemma reduces usual change variables suppose point norm martingale property integral markov inequality large enough shows first part note restricted operates functions supp without loss generality may assume unchanged whether take large enough taking large enough made arbitrarily close inequality eigenvalues follows variational characterization proof lemma mixing highest temperature definition function define convex envelope sup abbreviation strongly convex proposition convex note use following definition strongly convex valid necessarily differentiable function proof let check sup sup sup sup lemma let suppose exists convex function proof show sce works let convex proposition show max min min rhs convex function equals everywhere therefore sce lemma keep setup lemma langevin diffusion satisfies inequality constant proof let lemma since convex theorem satisfies inequality constant lemma satisfies inequality constant discretizing continuous chains notational convenience section follow denote lemma let distributions running simulated tempering chain steps temperature type transitions taken according discrete time markov kernel running langevin diffusion time type transitions taken according running steps discretized langevin diffusion using discretization granularity max proving statement make note location make sense namely show lemma location minimum let proof since follows however holds min hence implies statement lemma prove technical lemmas first prove continuous chain essentially contained ball radius precisely show lemma reach continuous chain let markov kernel corresponding evolving langevin diffusion defined time proof let lemma claim indeed last inequality follows lemma together get integrating get taking expectations using martingale property integral get claim lemma next prove technicall bound drift discretized chain discrete steps proofs follow similar calculations first need bound hessian lemma hessian bound proof notational convenience let note log hessian satisfies max log need lemma bounding interval drift setting section let let proof let random variable distributed lemma lemma similarly proof corollary implies finally prove convenient decomposition theorem divergence two mixtures distributions terms divergence weights components mixture concretely lemma let distributions domain full support let distributions arbitrary domain proof overloading notation use two measures even necessarily probability distributions obvious definition log first inequality holds due convexity divergence mind prove main claim proof let denote distribution corresponding running langevin diffusion chain time steps coordinate starting keeping remaining coordinates fixed let define analogous distribution except running discretized langevin diffusion chain time steps coordinate let denote distribution running markov transition matrix corresponding type transition simulated tempering chain starting proceed induction towards obviously write similarly note transition matrix change discretized continuous version convexity divergence lemma similarly lemma together lemma max lemmas hence inductively putting together max induction hence max need proof main theorem putting everything together show estimate partition functions apply following lemma estimating partition function within constant factor suppose probability distributions suppose distribution variable given samples define random let suppose probability proof chernoff bound gives combining using triangle inequality dividing using gives result lemma suppose let suppose min min choosing quantity min proof let let lemma using tail bound thus using min min min max min lemma probability distributions sum proof calculate lemma suppose algorithm run temperatures satisfying partition function estimates parameters satisfying max let distribution distribution running steps satisfies setting taking samples probability estimate also satisfies proof first consider case consider simulated tempering type transitions running continuous langevin time type transition would leave instead stay location let distribution steps starting triangle inequality note approach concentrate two terms let chain inverse temperature consider first lemma choose thus ensure theorem inequality holds constant generator langevin letting lemma lemma lemma lemma inequality holds constant sets size min lemma choose partition every compact consisting union sets gap gap conditions lemma satisfied least obtain partition projected chain set partition measure chose partition fine enough cheeger inequality set gap gap highest temperature lemma gap lemma since condition satisfied assumption defined assumption theorem spectral gap simulated tempering chain gap min min min choosing get note chain somewhat lazy stays probability calculate first note distance lemma lemma together gives term use pinsker inequality lemma get min gives second part setting gives lemma noting lemma gives collecting samples probability set consider general transform problem problem change variables changes note running discretized chain transformed problem step size corresponds running discretized chain problem step size step corresponds step prove main theorem theorem proof theorem choose number temperatures use lemma ductively probability estimate satisfies estimating final distribution within accuracy gives desired sample remark one reason large powers appear lemma going conductance spectral gap multiple times time lose square cheeger inequality care spectral gap within sets partition theorem controls conductance rather spectral gap may possible tighten bound proving variant theorem controls spectral gap directly acknowledgements work done part authors visiting simons institute theory computing thank matus telgarsky maxim raginsky illuminating conversations early stages work references agarwal learning sparsely used overcomplete conference learning theory colt arora moitra learning topic models going beyond svd proceedings annual ieee symposium foundations computer science focs anandkumar spectral algorithm latent dirichlet allocation advances neural information processing systems awasthi risteski provably correct cases variational inference topic models advances neural information processing systems arora practical algorithm topic modeling provable guarantees international conference machine learning arora simple efficient neural algorithms sparse coding arora provable learning networks symposium theory computing stoc bakry simple proof inequality large class probability measures including case electron commun probab bakry diffusions hypercontractives xix springer belloni escaping local minima via simulated annealing optimization approximately convex functions conference learning theory bubeck eldan lehec sampling distribution projected langevin monte carlo arxiv preprint bovier gayrard klein metastability reversible diffusion processes precise asymptotics small eigenvalues journal european mathematical society bakry gentil ledoux analysis geometry markov diffusion operators vol springer science business media bhattacharya criteria recurrence existence invariant measures multidimensional diffusions annals probability bovier metastability low lying spectra reversible markov chains communications mathematical physics bovier metastability reversible diffusion processes sharp asymptotics capacities exit times journal european mathematical society dalalyan theoretical guarantees approximate sampling smooth densities journal royal statistical society series statistical methodology dalalyan stronger analogy sampling optimization langevin monte carlo gradient descent arxiv preprint durmus moulines bayesian inference via unadjusted langevin algorithm diaconis comparison theorems reversible markov chains annals applied probability gharan trevisan partitioning expanders proceedings annual symposium discrete algorithms society industrial applied mathematics hsu kakade learning mixtures spherical gaussians moment methods spectral decompositions proceedings conference innovations theoretical computer science acm kingma welling variational bayes arxiv preprint laurent massart adaptive estimation quadratic functional model selection annals statistics risteski algorithms matching lower bounds approximatelyconvex optimization advances neural information processing systems simonovits random walks convex body improved volume algorithm random structures algorithms madras randall markov chain decomposition convergence rate analysis annals applied probability rezende mohamed wierstra stochastic backpropagation approximate inference deep generative models international conference machine learning raginsky rakhlin telgarsky learning via stochastic gradient langevin dynamics nonasymptotic analysis arxiv preprint sontag roy complexity inference latent dirichlet allocation advances neural information processing systems vempala geometric random walks survey combinatorial computational geometry woodard schmidler huber sufficient conditions torpid mixing parallel simulated tempering electronic journal probability woodard schmidler huber conditions rapid mixing parallel simulated tempering multimodal distributions annals applied probability zheng swapping simulated tempering algorithms stochastic processes applications background markov chains discrete time markov chains definition discrete time markov chain measure space probability measure defines random process follows stationary distribution equivalently simplicity notation appendix consider chains absolutely continuous respect use notation rather rather results definitions apply modified notation case chain reversible definition markov chain let operate functions suppose unique stationary distribution let define dirichlet form variance write define eigenvalues eigenvalues respect norm define spectral gap gap inf note gap inf remark normalized laplacian graph defined adjacency matrix diagonal matrix degrees change scale turns transition matrix random walk graph eigenvalues equal eigenvalues markov chain defined spectral gap controls mixing markov chain define distance let initial distribution distribution running markov chain steps min restricted projected markov chains given markov chain define two markov chains associated partition definition markov chain set define restriction markov chain words proposes transition transition rejected would leave suppose unique stationary distribution given partition define projected markov chain respect words total probability flow omit superscript clear following theorem gap original chain terms gap projected chain minimum gap restrictioned chains theorem theorem let markov chain stationary distribution let partition gap min gap gap gap conductance clustering definition let markov chain unique stationary distribution let drawn stationary distribution next state markov chain define external conductance cheeger constant min definition let markov chain finite state space say disjoint subsets theorem spectrally partitioning graphs let reversible markov chain states let eigenvalues markov chain exists sets proof theorem except use different notion restriction markov chain reconcile consider markov chain associated graph consider cheeger constant induced graph definition clustering recast definition language follows construct weighted graph weight edge given restricted chain corresponds graph induced graph except take edges leaving vertex redraw surface seems cause problem define volume set sum weights vertices volume larger yet amount weight leaving increase however examining proof see every lower bound form obtained first vol exactly thus theorem works equally well instead cheeger inequality relates conductance spectral gap theorem cheeger inequality let reversible markov chain finite state space conductance gap continuous time markov processes continuous time markov process instead defined natural object consider generator definition continuous time markov process given define random proces define stationary distributions reversibility variance discrete case define generator lim unique stationary distribution define spectral gap defined discrete case definition eigenvalues defined eigenvalues note discrete case corresponds continuous case note order valid markov process must case forms markov semigroup definition continuous markov process satisfies inequality constant another way saying gap langevin diffusion stationary distribution since depends natural way also write inequality langevin diffusion thus takes form following classical result theorem let convex differentiable satisfies inequality particular holds giving inequality gaussian distribution spectral gap equivalently inequality implies rapid mixing gap examples might surprising sampling mixture gaussians require complicated markov chain simulated tempering however many simple strategies seem fail langevin restarts one natural strategy try simply run langevin polynomial number times randomly chosen locations time escape mode enter different one could exponential may hope different runs explores individual modes somehow stitch runs together difficulty means gaussians difficult quantify far individual runs reach thus combine various runs recovering means gaussians another natural strategy would try recover means gaussians mixture performing gradient descent polynomial number random restarts hope would maybe local minima correspond means gaussians enough restarts able find unfortunately strategy without substantial modifications also seems work instance dimension consider mixture gaussians means corners simplex substantially smaller diameter considering one center simplex order discover mean gaussian center would starting point extremely close center simplex high dimensions seems difficult additionally address issue robustness perturbations though algorithms optimize approximately convex functions typically handle small perturbations gaussians different covariance result requires gaussians variance necessary even variance gaussians differ factor examples simulated tempering chain takes exponential time converge intuitively illustrated figure figure left shows distribution low temperature case two modes separate significant mass figure right shows distribution high temperature note although case two modes connected volume mode smaller variance much smaller exponentially small therefore high dimensions even though modes connected high temperature probability mass associated small variance mode small allow fast mixing figure mixture two gaussians different covariance different temperature pertubation tolerance previous sections argued sample distributions form exp section goal argue sample distributions form exp main theorem following theorem main theorem perturbations suppose satisfy algorithm parameters given lemma produces sample distribution time poly theorem follow immediately lemma straightforward analogue precisely lemma suppose algorithm run temperatures satisfying partition function estimates parameters satisfying min min max min max let distribution distribution running steps satisfies setting estimate taking samples probability also satisfies way prove theorem prove tolerance proof ingredients perturbations mixing time tempering chain first show mixing time tempering chain uses continous langevin transition exp comparable exp keeping mind statement lemma following lemma suffices lemma suppose following hold probability distributions letting generators langevin diffusion inequality holds constant inequality holds constant note given probability distributions conditions lemma satisfied proof ratio use first part along variational characterization max closed subspace dim min use second part inequality lower bound mixing time highest temperature show use highest temperature corresponding well cost mixing time namely since lemma immediately lemma satisfy exists function consequence proof lemma implies lemma satisfy langevin diffusion satisfies inequality constant discretization proof lemma combined fact gives lemma perturbed reach continuous chain let markov kernel corresponding evolving langevin diffusion defined time proof proof proceeds exactly lemma noting implies furthermore since lemma get lemma perturbed hessian bound consequence analogue lemma gives lemma bounding interval drift setting lemma let let putting together get analogue lemma lemma let distributions running simulated tempering chain steps temperature type transitions taken according discrete time markov kernel running langevin diffusion time type transitions taken according running steps discretized langevin diffusion using discretization granularity max putting things together finally prove theorem proof proof analogous one lemma combination lemmas previous subsections analysis simulated tempering chain consider partition used lemma lemma gap highest temperature lemma gap min furthermore lemma since condition satisfied lemma spectral gap simulated tempering chain gap lemma since min min max triangle inequality proof lemma bounds get term use pinsker inequality lemma get min gives proof second part lemma proceeds exactly another lower bound simulated tempering theorem comparison theorem using canonical paths let finite markov chain stationary distribution suppose pair associated path define congestion max gap definition say partition refines written every exists define chain partitions refinement theorem suppose assumptions hold furthermore suppose chain partitions define chain partitions min min gap min gap proof let stationary distribution first note easily switch using define partition theorem gap gap min gap gap abuse notation considering sets states identify union sets corresponding set states consider tree nodes edges connecting designate root define canonical path unique path tree note given edge consider union children including follows paths exactly one subset upper bound definition max min next lower bound probability level proposing switch min min min min max putting together using theorem gap min gap min gap gap min gap taking partitions except first see theorem improvement bound simulated tempering theorem gives bound gap min min gap gap number sets partition notably bound exponential bound dependence
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generative models spoken dialog systems chatting capability tiancheng zhao allen kyusong lee maxine eskenazi language technologies institute carnegie mellon university pittsburgh pennsylvania usa tianchez arlu kyusongl jun abstract generative models offer great promise developing domaingeneral dialog systems however mainly applied conversations paper presents practical novel framework building dialog systems based models framework enables models accomplish independent interact external databases moreover paper shows flexibility proposed method interleaving chatting capability slotfilling system better recovery models trained data bus information system chat data results show proposed framework achieves good performance offline evaluation metrics task success rate human users introduction spoken dialog systems transformed interaction enabling people interact computers via spoken language raux young bohus rudnicky sds usually system creators first map user utterances semantic frames contain slots intents using spoken language understanding slu mori set dialog state variables tracked retain context information turns williams lastly dialog policy decides next move list dialog acts covers expected communicative functions system although approach successfully applied many practical systems limited ability generalize ood requests scale new domains example even within simple domain real users often make requests included semantic specifications due proper error handling strategies guide users back conversation crucial dialog success bohus rudnicky past error handling strategies limited set predefined dialog acts request repeat clarification constrained system capability keeping users engaged moreover increased interest extending systems multiple topics lee multiple skills grouping heterogeneous types dialogs single system zhao cases require system flexible enough extend new slots actions goal move towards sds framework flexible enough expand new domains skills removing assumptions dialog state dialog acts bordes weston achieve goal neural encoderdecoder model cho sutskever suitable choice since achieved promising results modeling conversations vinyals sordoni encodes dialog history using deep neural networks generates next system utterance via recurrent neural networks rnns therefore unlike traditional sds pipeline model theoretically limited vocabulary ive implementation system would use rnns encode raw dialog history generate next system utterance using separate rnn decoder however implementation might achieve good performance offline evaluation closed dataset would certainly fail used humans several reasons real users mention new entities appear training data new restaurant name entities however essential delivering information matches users needs system sds obtains information knowledge base constantly updated today weather different every day simply memorizing results occurred training data would produce false information instead effective model learn query constantly get information users may give ood requests say day system must handled gracefully order keep conversation moving intended direction paper proposes effective encoderdecoder framework building sdss propose entity indexing tackle challenges oov entities query moreover show extensibility proposed model adding chatting capability sds better ood recovery approach assessed let bus information data dialog state tracking challenge williams report performance offline metrics real human users results show model attains good performance metrics matically learn distributed vector representation dialog state accumulating observations turn williams zweig zhao eskenazi dhingra williams learned dialog state used dialog policy select next action second branch research develops action space dialog policy prior work replaced dialog acts natural language semantic schema action space dialog managers eshghi lemon dynamic syntax kempson recently wen shown feasibility using rnn decoder generate system utterances word word dialog policy proposed model fine tuned using reinforcement learning furthermore deal challenge developing dialog models able interface external prior work unified special query actions via deep reinforcement learning zhao eskenazi soft attention database dhingra third branch strives solve problems time building model maps observable dialog history directly word sequences system response using model successfully applied conversational models serban zhao well task oriented systems bordes weston yang eric manning order better predict next correct system action branch focused investigating various neural network architectures improve machine ability reason user input model longterm dialog context related work past research developing dialog systems broadly divided three branches first one focuses learning dialog state representation still using dialog act system actions researchers proposed idea extracting independent statistics dialog state wang dialog state representation shared across systems serving different knowledge sources another approach uses rnns paper closely related third branch differs following ways models independent leveraging domaingeneral entity recognizer extensible oov entities models emphasize interactive nature dialog address handling interleaving chatting taskoriented conversations instead testing synthetic dataset approach focuses real world use testing system human users via spoken interface proposed method proposed framework consists three steps shown figure entity indexing independent sied system utterance lexicalization intuition leverage named entity recognition ner tjong kim sang meulder techniques extract salient entities raw dialog history convert lexical values entities entity indexes model trained focus solely reasoning entity indexes dialog history make decisions next utterance produce including query way model unaffected inclusion new entities new maintaining interface easy extension new types conversation skills lastly output decoder networks lexicalized replacing entity indexes special tokens natural language following sections explain step detail entity indexing utterance lexicalization entity indexing two parts first utilizes existing ner extract entities user system utterances note entity assumed slots domain example system system may contain two slots tolocation departure arrival city respectively however extracts every mention location utterances leaves task distinguishing departure arrival model furthermore step replaces search result search query weather cloudy kbsearch second step involves constructing indexed entity table entity indexed order occurrence conversation figure shows example two location mentions properties entity indexing section several properties assumptions addressed first entity indexed uniquely entity type index note index associated entity value rather solely order appearance dialog despite actual words hidden figure example entity indexing utterance lexicalization human still easily predict entity system confirm search based logical reasoning therefore alleviates oov problem deploying model real world also forces model focus learning reasoning process dialogs instead leveraging much information language modeling moreover sdss apart informing concepts kbs usually introduce novel entities users instead systems mostly corroborate entities introduced users assumption every entity mention system utterances always found users utterances dialog history therefore also found indexed entity table property reduces grounding behavior conventional dialog manager selecting entity indexed entity table confirming user utterance lexicalization reverse since deterministic process effect always reversed finding corresponding entity indexed entity table replacing index word search simple string matching algorithm search special kbsearch token take following generated entities argument actual results replace original query figure shows example utterance lexicalization models model read dialog history predict system next utterance format specifically dialog history turns represented respectively system user utterance asr confidence score turn utterance dialog history encoded fixedsize vectors using convolutional neural networks figure proposed pipeline dialog systems cnns proposed kim specifically word utterance mapped word embedding utterance represented matrix size word embedding filters size conduct convolutions obtain feature map features window size passed nonlinear relu glorot layer followed layer obtain compact summary salient features maxpool relu using cnns capture information crucial able distinguish differences entities example simple embedding approach fail distinguish two location entities leave cnn encoder capture context information two entities obtaining utterance embedding turnlevel dialog history encoder network similar one proposed zhao eskenazi used turn embedding simple concatenation system user utterance embedding confidence score long memory lstm hochreiter schmidhuber network reads sequence turn embeddings dialog history via recursive state update lstm output lstm hidden state decoding attention vanilla decoder takes last hidden state encoder initial state decodes next system utterance word word shown sutskever assumes hidden state expressive enough encode important information history dialog however assumption may often violated task complex reasoning entire source sequence attention mechanism proposed bahdanau machine translation community helped encoderdecoder models improve performance various tasks bahdanau attention allows decoder look every hidden state encoder dynamically decide importance hidden state decoding step significantly improves model ability handle dependency experiment decoders without attention attention computed similarly multiplicative attention described luong denote hidden state decoder time step hidden state outputs encoder turn predict next word aji softmax hti aji sej tanh softmax sej decoder next state updated lstm sej leveraging chat data improve ood recovery past work shown simple supervised learning usually inadequate learning robust sequential policy williams young ross model exposed expert demonstration examples recover mistakes users ood requests present simple yet effective technique leverages extensibility model order obtain robust policy setting supervised learning specifically artificially augment dialog dataset chat data conversation corpus shown effective improving performance systems let original dialog dataset dialogs dialog turns furthermore assume access chat dataset common adjacency pairs appear chats hello create new dataset repeating following process certain number times randomly sample dialog randomly sample turn randomly sample adjacency pair replace user utterance insert new turn figure illustration data augmentation turn dashed line inserted original dialog step error handling system utterance system answers user ood request study experimented simple case system repeat previous prompt responding via figure shows example augmented turn eventually train model union two datasets discussion several reasons data augmentation process appealing first model effectively learns ood recovery strategy first gives chatting answers users ood requests tries pull users back conversation second chat data usually larger vocabulary diverse natural language expressions reduce chance oovs enable model learn robust word embeddings language models experiment setup dataset domain cmu let bus information system raux spoken dialog system contains bus information combined datasets dstc williams contain total dialogs average dialog length turns dialogs randomly splitted proportions data data noisy since dialogs collected real users via telephone lines furthermore version let used inhouse database containing port authority bus schedule current version database replaced google directions api reduces human burden maintaining database opens possibility extending let cities pittsburgh connecting google directions api involves post call url given access key well parameters needed departure place arrival place departure time travel mode always set transit obtain relevant bus routes distinct dialog acts available system system utterance contains one dialog acts lastly system vocabulary size user vocabulary size process sizes become respectively chat data use publicly available chat corpus used total chatting adjacency pairs control number data injections number turns original dtsc dataset leads user vocabulary size system vocabulary size training details experiments word embedding size sizes lstm hidden states encoder decoder layer attention context size also tied cnn weights encoding system user utterances cnn filter windows feature maps trained model using adam kingma learning rate batch size combat overfitting apply dropout zaremba lstm layer outputs cnn outputs maxpooling layer dropout rate responses computed precision recall slots metric measures model performance generating correct mostly occur grounding utterances confirm queries compute precision recall queries although slots metric already covers queries system utterances contain queries also explicitly measured due importance specifically action measures whether system able generate special kbquery symbol initiate query well accurate corresponding query arguments bleu papineni compares ngram precision length penalty popular score used evaluate performance natural language generation wen dialog models reported experiments results approach assessed offline online evaluations offline evaluation contains standard metrics test encoderdecoder dialog models serban system performance assessed three perspectives essential taskoriented systems dialog acts query online evaluation composed objective task success rate number turns subjective satisfaction human users offline evaluation dialog acts system utterance made one dialog acts leaving want request arrival place evaluate whether generated utterance dialog acts ground truth trained multilabel dialog tagger using support vector machines svm tsoumakas katakis features dialog act label since natural language generation module let handcrafted dialog act tagger achieved average label accuracy dataset used dialog act tagger tag ground truth generated text api metrics vanilla slot bleu table performance model automatic measures four systems compared basic models without vanilla basic model model attentional decoder model trained dataset augmented chatting data comparison carried exactly test dataset contains dialogs table shows results seen four models achieve similar performance dialog act metrics even vanilla model confirms capacity models learn shape conversation since achieved impressive results challenging settings modeling conversations furthermore since data collected several months minor updates made dialog manager therefore inherent ambiguities data dialog manager may take different actions situation conjecture near upper limit data modeling system next dialog act given dialog history hand proposed methods significantly improved metrics related slots queries inclusion alone able improve slots relative confirms crucial developing independent models modeling dialogs likewise inclusion attention improved prediction slots system utterances adding attention also improved performance predicting queries overall slot accuracy expected since queries usually issued near end conversation requires global reasoning entire dialog history use attention allows decoder look history make better decisions rather simply depending context summary last hidden layer encoder good performance achieved models attentional decoder attention weights equation every step decoding process two example dialogs test data visualized figures vertical axes show dialog history flowing top bottom row turn format system utterance user utterance top horizontal axis shows predicted next system utterance darkness bar indicates value attention calculated equation first example shows attention grounding new entity previous turn attention weights become focus previous turn predicting implicit confirm action second dialog example shows challenging situation model predicting query see attention weights generating input argument query clearly focus specific mention dialog history visualization confirms effectiveness attention mechanism dealing pendency discourse level figure visualization attention weights generating implicit confirm top query bottom surprisingly model trained data augmented chat achieved slightly better slot accuracy performance even though augmented data directly related dialogs furthermore model trained chataugmented data achieved better scores query metrics several reasons may explain improvement since chat data exposes model significantly larger vocabulary resulting model robust words seen original training data augmented dialog turn seen noise dialog history adds extra regularization model enables model learn robust reasoning mechanisms human evaluation although model achieves good performance offline evaluation may carray real user dialogs system must simultaneously deal several challenges automatic speech recognition asr errors ood requests etc therefore real user study conducted evaluate performance proposed systems real world due limited number real users two best performing system compared users able talk web interface dialog systems via speech google chrome speech api served asr tts modules done via chrome voice activity detection vad plus finite state detector zhao lastly hybrid named entity recognizer ner trained using conditional random field crf mccallum rules extract types entities location hour minute process experiment setup follows user logs website system prompts user goal randomly chosen combination departure place arrival place time leave cmu airport system also instructs user say goodbye thinks goal achieved wants give user begins conversation one two evaluated systems chance choosing either system visible user user session finished system asks give two scores correctness naturalness system respectively subjects study consist undergraduate graduate students however many subjects follow prompted goal rather asked bus routes therefore dialog manually labeled dialog success dialog successful systems give least one bus schedule matches three slots expressed users table shows metrics dialog slot precision precision success rate avg turns avg correctness avg naturalness table performance model automatic measures standard deviations subjective scores parentheses results overall systems achieved reasonable performance terms dialog success rate model achieves slightly higher success subjective naturalness metrics although difference statistically significant due limited number subjects precision grounding correct slots predicting correct query also manually labelled model performs slightly better model slot precision latter model performs significantly better query precision addition leads slightly longer dialogs sometimes generates chatting utterances users understand users utterances last investigated log files identified following major types sources dialog failure rnn decoder invalid output occasionally rnn decoder outputs system utterances okay going get right found indexed entity table invalid output confuses users occurred dialogs system utterances contain invalid symbols imitation suboptimal dialog policy since models trained imitate suboptimal dialog policy limitations show original dialog manager handle situation failing understand slots appeared compound utterances future plans involves improving models perform better suboptimal teacher policy conclusions conclusion paper discusses constructing dialog systems using generative encoder decoder models effective solving oov entity query challenges sdss additionally novel data augmentation technique interleaving dialog corpus chat data led better model performance online offline evaluation future work includes developing advanced models better deal dialog history complex reasoning challenges current models furthermore inspired success mixing chatting dialogs take full advantage extensibility models investigating make systems able interleave various conversational tasks different domains chatting turn create versatile conversational agent references dzmitry bahdanau kyunghyun cho yoshua bengio neural machine translation jointly learning align translate arxiv preprint dan bohus alexander rudnicky ravenclaw dialog management using hierarchical task decomposition expectation agenda dan bohus alexander rudnicky error handling ravenclaw dialog management framework proceedings conference human language technology empirical methods natural language processing association computational linguistics pages antoine bordes jason weston learning dialog arxiv preprint 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alternating generator network tian han yang zhu ying nian dec department statistics university california los angeles usa abstract convolutional neural network convnet cnn lecun krizhevsky sutskever hinton dosovitskiy springenberg brox shown recently generator network capable generating realistic images denton radford metz chintala generator network fundamental representation knowledge following properties analysis model disentangles variations observed signals independent variations latent factors synthesis model synthesize new signals sampling factors known prior distribution transforming factors signal embedding model embeds manifold formed observed signals euclidean space latent factors linear interpolation lowdimensional factor space results interpolation data space paper proposes alternating algorithm learning generator network model model nonlinear generalization factor analysis model mapping continuous latent factors observed signal parametrized convolutional neural network alternating algorithm iterates following two steps inferential infers latent factors langevin dynamics gradient descent learning updates parameters given inferred latent factors gradient descent gradient computations steps powered share code common show alternating algorithm learn realistic generator models natural images video sequences sounds moreover also used learn incomplete indirect training data introduction paper studies fundamental problem learning inference generator network goodfellow generative model become popular recently specifically propose alternating algorithm learning inference model factor analysis model learned algorithm rubin thayer dempster laird rubin based multivariate linear regression inspired algorithm propose alternating algorithm learning generator network iterates following inferential training example infer continuous latent factors langevin dynamics gradient descent learning update parameters given inferred latent factors gradient descent langevin dynamics neal stochastic sampling counterpart gradient descent gradient computations steps powered convnet structure gradient computation step actually gradient computation step terms coding given factors learning convnet supervised learning problem dosovitskiy springenberg brox accomplished learning backpropagation factors unknown learning becomes unsupervised problem solved adding inferential inner loop learning factor analysis generator network generalization factor analysis factor analysis prototype model unsupervised learning distributed representations two directions one pursue order generalize factor analysis model one direction generalize prior model prior assumption latent factors led methods independent component analysis karhunen oja sparse coding olshausen field matrix factorization lee seung matrix factorization completion recommender systems koren bell volinsky etc direction generalize factor analysis model generalize mapping continuous latent factors observed signal generator network example direction generalizes linear mapping factor analysis mapping defined alternating equal contributions process shall show alternating algorithm learn realistic generator models natural images video sequences sounds alternating algorithm follows tradition alternating operations unsupervised learning alternating linear regression algorithm factor analysis alternating least squares algorithm matrix factorization koren bell volinsky kim park alternating gradient descent algorithm sparse coding olshausen field unsupervised learning algorithms alternate inference step learning step case alternating generative adversarial network gan goodfellow denton radford metz chintala assisting network discriminator network plays adversarial role generator network unlike alternating vae perform explicit inference gan avoids inferring latent factors altogether comparison alternating algorithm simpler basic without resorting extra network difficult compare methods directly illustrate strength alternating learning incomplete indirect data need explain whatever data given may prove difficult less convenient vae gan meanwhile alternating complementary vae gan training may use vae initialize inferential result may improve inference vae inferential may help infer latent factors observed examples gan thus providing method test gan explain entire training set generator network based convnet one also obtain probabilistic model based bottomup convnet defines descriptive features xie zhu inference inferential solves inverse problem process latent factors compete explain training example following advantages inference latent factors latent factors may follow sophisticated prior models instance textured motions wang zhu dynamic textures doretto latent factors may follow dynamic model vector inferring latent factors explain observed examples learn prior model observed data may incomplete indirect instance training images may contain occluded objects case latent factors still obtained explaining incomplete indirect observations model still learned factor analysis beyond let observed data vector image let vector continuous latent factors traditional factor analysis model matrix error vector observational noise assume stands identity matrix also assume observational errors gaussian white noises three perspectives view basis vectors write column vector basis vectors coefficients loading matrix write row hwj components respectively loading factors vector loading weights indicating factors important determining called loading matrix matrix factorization suppose observe whose factors factor analysis model learned rubinthayer algorithm involves alternating regressions steps powered sweep operator rubin thayer liu rubin factor analysis model prototype many subsequent models generalize prior model independent component analysis karhunen oja assumed follow independent heavy tailed distributions sparse coding learning incomplete indirect data venture propose main advantage generative model learn incomplete indirect data uncommon practice generative model evaluated based well recovers unobserved original data still learning model generate new data learning generator network incomplete data considered generalization matrix completion also propose evaluate learned generator network reconstruction error testing data factor analysis convnet contribution related work main contribution paper propose alternating algorithm training generator network another contribution evaluate generative models learning incomplete indirect training data existing training methods generator network avoid inference latent factors two methods recently devised accomplish methods involve assisting network separate set parameters addition original network generates signals one method variational vae kingma welling rezende mohamed wierstra mnih gregor assisting network inferential recognition network seeks approximate posterior distribution latent factors method olshausen field assumed redundant sparse vector small number significantly different zero matrix factorization lee seung assumed recommender system koren bell volinsky vector customer desires different aspects vector product desirabilities aspects algorithm dempster laird rubin completedata model given log log const model obtained integrating posterior distribution given function training data logpi likelihood log assume given learning inference accomplished maximizing obtained alternating gradient descent algorithm iterates following two steps inference step update running steps gradient descent learning step update one step gradient descent rigorous method maximize log log dzi likelihood takes account uncertainties inferring see appendix understanding gradient calculated according following fact underlies algorithm log log convnet mapping addition generalizing prior model latent factors also generalize mapping paper consider generator network model goodfellow retains assumptions traditional factor analysis generalizes linear mapping mapping convnet collects connection weights bias terms convnet model becomes reconstruction error may assume sophisticated models colored noise texture binary emit probability map exp sigmoid transformation bernoulli sampling carried may assume multinomial logistic emission model ordinal emission model although mapping convnet parameterization makes particularly close original factor analysis specifically write convnet follows expectation respect approximated drawing samples computing monte carlo average langevin dynamics sampling iterates layer matrix connection weights vector bias terms layer convnet considered recursion original factor analysis model factors layer obtained linear superposition basis vectors basis functions column vectors factors layer serving coefficients linear superposition case convnet basis functions versions one another like wavelets see appendix understanding model denotes time step langevin sampling step size denotes random vector follows langevin dynamics process latent factors compete explain away current residual explain langevin dynamics continuous time version sampling exp dynamics stationary distribution shown wellbehaved testing function alternatively given suppose stochastic gradient algorithm younes used learning iteration single copy sampled running finite number steps langevin dynamics starting current value warm start sampled alternating observe training set data vectors corresponding share convnet intuitively infer learn minimize reconstruction error plus regularization term corresponds prior formally model written adopting language close prior small posterior may evolving energy landscape may help alleviate trapping local modes practice tune value instead estimating langevin dynamics extended hamiltonian monte carlo neal sophisticated versions girolami calderhead manner update parameter based gradient whose monte carlo approximation log kyi experiments code experiments based matconvnet package vedaldi lenc training images sounds scaled intensities within range adopt structure generator network radford metz chintala dosovitskiy springenberg brox topdown network consists multiple layers deconvolution linear superposition relu tanh radford metz chintala make signals fall within also adopt batch normalization ioffe szegedy fix standard deviation noise vector use steps langevin dynamics within learning iteration langevin step size set run learning iterations learning rate momentum learning algorithm produces learned network parameters inferred latent factors signal end synthesized signals obtained sampled prior distribution algorithm describes details learning sampling algorithm algorithm alternating require training examples number langevin steps number learning iterations ensure learned parameters inferred latent factors let initialize initialize repeat inferential run steps langevin dynamics sample warm start starting current step follows equation learning update computed according equation learning rate let gaussian noise langevin dynamics removed algorithm becomes alternating gradient descent algorithm possible update simultaneously joint gradient descent inferential learning guided residual inferential based whereas learning based gradients efficiently computed computations two gradients share steps specifically convnet defined share code chain rule computation thus code part code algorithm langevin dynamics samples gradually changing posterior distribution keeps changing updating collaborate reduce reconstruction error kyi parameter plays role annealing tempering langevin sampling large posterior qualitative experiments figure modeling texture patterns example left observed image right generated image experiment modeling texture patterns learn separate model texture image images collected internet resized synthesized images figures shows four examples factors top layer form image pixel following independently image transformed convnet use learning stage texture experiments order obtain synthesized image randomly sample expand learned network generate synthesized image training network follows starting image network layers deconvolution kernels linear superposition basis functions factor layer basis functions pixels apart number channels first layer translation invariant basis functions decreased factor layer langevin steps step size figure modeling object patterns left synthesized images generated method generated learned discretized values right synthesized images generated using deep convolutional generative adversarial net dcgan discretized values within figure modeling sound patterns row waveform training sound range seconds row waveform synthesized sound range seconds figure modeling object patterns left image generated method obtained first sampling generating image learned middle interpolation images four corners reconstructed inferred vectors four images randomly selected training set image middle obtained first interpolating vectors four corner images generating image right synthesized images generated dcgan dimension vector sampled uniform distribution experiment modeling sound patterns sound signal treated texture image mcdermott simoncelli sound data collected internet training signal second clip sampling rate hertz represented vector learn separate model sound signal latent factors form sequence follows network consists layers deconvolution kernels size factor number channels first layer decreases factor layer synthesis start longer gaussian white noise sequence generate synthesized sound expanding learned network figure shows waveforms observed sound signal first row synthesized sound signal second row experiment modeling object patterns model object patterns using network structure essentially network texture model except include fully connected layer latent factors vector images use relu leaking factor maas hannun langevin steps step size first experiment learn model two components training data images tigers lions training model generate images using learned convnet discretize equally spaced values left panel figure displays synthesized images panel second experiment learn model face images randomly selected celeba dataset liu left panel figure displays images generated learned model middle panel displays interpolation results images four corners generated vectors four images randomly selected training set images middle obtained first interpolating four corner images using sphere interpolation dinh bengio generating images learned convnet also provide qualitative comparison deep convolutional generative adversarial net dcgan goodfellow radford metz chintala right panel figure shows generated results dataset using right panel figure displays generated results trained aligned faces celeba dataset use code https tuning parameters radford metz chintala run iterations method experiment modeling dynamic patterns model textured motion wang zhu dynamic texture doretto dynamic system azt assume latent factors follow vector model matrix innovation model direct generalization linear dynamic system doretto reduced principal component analysis pca via singular value decomposition svd learn model two steps treat independent examples learn infer treat training data learn doretto synthesize new dynamic texture start generate sequence according learned model discard period frames figure shows experiments set first row segment sequence generated model second row generated method doretto dimensionality possible generalize model recurrent network may also treat video sequences images learn generator networks filters basis functions puted summing pixels partially observed image compute summing observed pixels continue use alternating backpropagation algorithm infer learn inferred learned image automatically recovered end able accomplish following tasks recover occluded pixels training images synthesize new images learned model recover occluded pixels testing images using learned model experiment error table recovery errors experiments learning occluded images figure learning incomplete data columns belong experiments respectively row original images observed learning row training images row recovered images learning want emphasize experiments training images partially occluded experiments different vincent training images fully observed noises added matter regularization prior model regularization already learned given task mentioned tasks learning incomplete data difficult gan vae occluded pixels different different training images evaluate method images randomly selected celeba dataset design experiments two types occlusions experiments salt pepper occlusion randomly place masks image domain cover roughly pixels respectively experiments denoted respectively pepper experiments single region mask occlusion randomly place mask image domain experiments denoted respectively mask set table displays recovery errors experiments error defined per pixel difference relative range pixel values original image recovered image occluded pixels emphasize recovery errors training errors intensities occluded figure modeling dynamic textures row segment synthesized sequence method row sequence method doretto rows two sequences method quantitative experiments experiment learning incomplete data method learn images occluded pixels task inspired fact images contain occluded objects considered generalization matrix completion recommender system method adapted task minimal modification modification involves computation fully observed image experiment abp pca pixels observed training figure displays recovery results experiment pixels occluded still learn model recover original images experiment error table reconstruction errors testing images learning training images using method abp pca table recovery errors experiments learning compressively sensed images figure comparison method pca row original testing images row reconstructions pca eigenvectors learned training images row reconstructions generator learned training images methods figure learning indirect data row original images projected onto white noise images row recovered images learning experiment learning indirect data learn model compressively sensed data romberg tao generate set white noise images random projections project training images white noise images learn model random projections instead original images need replace ksy given white noise sensing matrix observation treat fully connected layer known filters continue use alternating infer learn thus recovering image end able recover original images projections learning synthesize new images learned model recover testing images projections based learned model experiments different traditional compressed sensing task tasks moreover image recovery work based dimension reduction instead linear sparsity evaluate method face images randomly selected celeba dataset images projected onto white noise images pixel randomly sampled random projection image size becomes vector show recovery errors different latent dimensions table recovery error defined per pixel difference relative range pixel values original image recovered image figure shows recovery results experiment model evaluation reconstruction error testing data learning model training images assumed fully observed evaluate model reconstruction error testing images randomly select face images training images testing celeba dataset learning infer latent factors testing image using inferential reconstruct testing image using inferred learned inferential inferring initialize run langevin steps step size table shows reconstruction errors alternating backpropagation learning abp compared pca learning different latent dimensions figure shows reconstructed testing images pca learn eigenvectors training images project testing images learned eigenvectors reconstruction experiments may used evaluate generative models general experiments appear new found comparable methods accomplish three tasks simultaneously conclusion paper proposes alternating algorithm training generator network recognize generator network generalization factor analysis model develop alternating backpropagation algorithm generalization alternating regression scheme algorithm fitting factor analysis model alternating algorithm iterates inferential backpropagation inferring latent factors learning updating parameters backpropagation steps share computing steps chain rule calculations learning algorithm perhaps canonical algorithm training generator network based maximum likelihood theoretically accurate estimator maximum likelihood learning seeks explain charge whole dataset uniformly little concern biased fitting unsupervised learning algorithm alternating algorithm natural generalization original algorithm supervised learning adds inferential step overall bias term depends distribution essentially piecewise gaussian generator model considered explicit implementation local linear embedding roweis saul embedding local linear embedding mapping implicit generator model mapping explicit relu convnet mapping piecewise linear consistent local linear embedding except partition linear pieces generator model learned automatically inferential langevin dynamics energy function belongs piece defined inferential backpropagation seeks approximate basis via ridge regression keeps changing langevin dynamics may also changing algorithm searches optimal reconfigurable basis approximate may solve methods iterated ridge regression computationally expensive simple gradient descent ing step minimal overhead coding affordable overhead computing inferential seeks perform accurate inference latent factors worthwhile tasks learning incomplete indirect data learning models latent factors follow sophisticated prior models unknown parameters inferential may also used evaluate generators learned methods tasks reconstructing completing testing data method variants applied matrix factorization completion also applied problems components aspects factors supervised code images sounds videos http acknowledgement thank yifei jerry help experiments summer visit thank jianwen xie helpful discussions work supported nsf dms darpa simplex onr muri darpa aro density mapping density shifting suppose training data come data distribution pdata understand alternating algorithm idealization maps prior distribution latent factors data distribution pdata learned define appendix relu piecewise factor analysis generator network modern convnet usually linearity rectified linear unit relu krizhevsky sutskever hinton leaky relu maas hannun relu unit corresponds binary switch case relu following analysis pascanu montufar bengio write diag diagonal matrix indicator function case leaky relu values diagonal replaced leaking factor forms classification according network specifically factor space divided large number pieces hyperplanes piece indexed instantiation write make explicit dependence piece indexed assuming simplicity thus piece defined corresponds linear factor analysis whose basis multiplicative recomposition basis functions multiple layers recomposition controlled binary switches multiple layers hence convnet amounts reconfigurable basis representing model piecewise linear factor analysis retain bias term pdata pdata pdata pdata pdata pdata obtained averaging posteriors observed data pdata pdata considered data prior data prior pdata close true prior sense pdata pdata pdata right hand side minimized maximum likelihood estimate hence data prior pdata especially close true prior words posteriors data points pdata tend pave true prior rubin multiple imputation point view rubin algorithm infers number multiple imputations multiple guesses multiple guesses account uncertainty inferring maximizes obtain log data point seeks reconstruct inferred latent factors words seeks map pooling pdata hence unknown obtained averaging diagonal elements computation done sweep operator szz pivotal matrix based multivariate linear regression given updates multivariate linear regression steps accomplished sweep operator use notation gram matrices highlight analogy two steps algorithm considered alternating linear regression alternating sweep operation serves prototype alternating seeks map pdata data distribution pdata course mapping exact fact maps patch around local patches patch manifold form observed examples interpolations algorithm process density shifting pdata shifts towards thus maps pdata factor analysis alternating regression alternating algorithm inspired algorithm factor analysis observed data model posterior distribution available closed form algorithm factor analysis interpreted alternating linear regression rubin thayer liu rubin factor analysis model joint distribution references romberg tao robust uncertainty principles exact signal reconstruction highly incomplete frequency information ieee transactions information theory dempster laird rubin maximum likelihood incomplete data via algorithm journal royal statistical society denton chintala fergus deep generative image models using laplacian pyramid adversarial networks nips denote szy dinh bengio density estimation using real nvp corr doretto chiuso soatto dynamic textures ijcv posterior distribution obtained linear regression dosovitskiy springenberg brox learning generate chairs convolutional neural networks cvpr szy szz szy girolami calderhead riemann manifold langevin hamiltonian monte carlo methods journal royal statistical society goodfellow mirza wardefarley ozair courville bengio generative adversarial nets nips computation carried sweep operator pivotal matrix suppose observations compute compute szz szy karhunen oja independent component analysis john wiley sons ioffe szegedy batch normalization accelerating deep network training reducing internal covariate shift icml kim park nonnegative matrix factorization based alternating nonnegativity constrained least squares active set method siam journal matrix analysis applications kingma welling variational bayes iclr use denote conditional expectations regress obtain coefficient vector residual matrix szz szy koren bell volinsky matrix factorization techniques recommender systems computer krizhevsky sutskever hinton imagenet classification deep convolutional neural networks nips lecun bottou bengio haffner learning applied document recognition proceedings ieee lee seung algorithms nonnegative matrix factorization nips liu luo wang tang deep learning face attributes wild iccv liu rubin parameter expansion accelerate algorithm biometrika zhu learning frame models using cnn filters aaai maas hannun rectifier nonlinearities improve neural network acoustic models icml mcdermott simoncelli sound texture perception via statistics auditory periphery evidence sound synthesis neuron mnih gregor neural variational inference learning belief networks icml neal mcmc using hamiltonian dynamics handbook markov chain monte carlo olshausen field sparse coding overcomplete basis set strategy employed vision research pascanu montufar bengio number response regions deep feed forward networks linear activations radford metz chintala unsupervised representation learning deep convolutional generative adversarial networks iclr rezende mohamed wierstra stochastic backpropagation approximate inference deep generative models nips roweis saul nonlinear dimensionality reduction locally linear embedding science rubin thayer algorithms factor analysis psychometrika rubin multiple imputation nonresponse surveys volume john wiley sons vedaldi lenc matconvnet convolutional neural networks matlab int conf multimedia vincent larochelle bengio manzagol extracting composing robust features denoising autoencoders icml wang zhu modeling textured motion particle wave sketch iccv xie zhu theory generative convnet icml wang chen empirical evaluation rectified activations convolutional network corr younes convergence markovian stochastic algorithms rapidly decreasing ergodicity rates stochastics international journal probability stochastic processes
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delimited continuations natural language quantification polarity sensitivity harvard university oxford street cambridge usa arxiv apr ccshan abstract making linguistic theory like making programming language one typically devises type system delineate acceptable utterances denotational semantics explain observations behavior via connection programming language concept delimited continuations help analyze natural language phenomena quantification polarity sensitivity using logical metalanguage whose syntax includes control operators whose semantics involves evaluation order analyses expressed direct style rather style phenomena thought computational side effects sentence ambiguous least two readings one reading speaker must decline run spot fails substantiate claims whatsoever another reading exist certain claims ones say speaker must decline run spot fails substantiate categories subject descriptors programming languages language constructs structures linguistics languages theory keywords delimited continuations control effects natural language semantics quantification polarity sensitivity introduction paper computational linguistics sense applying insights computer science linguistics linguistics strives scientifically explain empirical observations natural language semantics particular concerned phenomena following sentences left entail counterparts right others 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 copy otherwise republish post servers redistribute lists requires prior specific permission fee continuation workshop venice italy copyright acm must decline run spot fails substantiate certain finally among four sentences acceptable used idealized conversation unacceptability rest notated asterisks general terms every student passed every diligent student passed student passed diligent student passed student passed diligent student passed students passed diligent students passed student liked course every student liked course student liked course students liked course linguistic entailments facts english speaker english make judgments nevertheless presumably corresponding logical entailments english speaker judges every student passed entails every diligent student passed mandarin speaker judges meige xuesheng dou entails meige xuesheng dou rely knowing every student passed every diligent student passed thus typical linguistic theory specifies semantics natural language translating declarative sentences logical statements truth conditions linguistic entailments hold goes theory two sentences model verifies former also verifies latter much work natural language semantics aims way depicted figure explain horizontal positing vertical approach reminiscent programming language research language perhaps one complicating feature like sentence part statement made cable television company comcast cnn channel rejected commercial hours scheduled air january every student passed student passed every diligent student passed student diligent passed hsome truth condition modelsi hsome truth condition modelsi figure approach natural language semantics ceptions studied translation simpler language without exceptions better understood translation target posited natural language semantics often combination predicate logic example verb passed might translated passed paper argues example translation target logical metalanguage delimited continuations examples two natural language phenomena quantification words like every polarity sensitivity part words like quantification first analyzed explicitly using continuations barker building insight paper makes following two contributions first analyze natural language direct style rather style words logical metalanguage used one includes control operators delimited continuations rather pure denotations need handle continuations explicitly natural language thus endowed operational semantics computer science richer second propose new analysis polarity sensitivity improves upon prior theories explaining student liked course acceptable student liked course analysis crucially relies notion evaluation order programming languages thus elucidating role control effects natural language supporting broader claim linguistic phenomena fruitfully thought computational side effects rest paper organized follows introduce simple grammatical formalism describe linguistic phenomenon quantification show straw man analysis deals cases others introduce programming language delimited continuations use improve straw man analysis quantification position treated inverse scope covered turn linguistic phenomenon polarity sensitivity show computationally motivated notion evaluation order improves upon previous analyses place examples broader context conclude grammatical formalism section introduce simple grammatical formalism use rest paper notational variant categorial grammar introduced carpenter chapter instance verb like usually requires object right subject left alice liked alice liked alice liked bob intuitively like function takes two arguments sentences unacceptable due type mismatch model formally assigning types denotations alice liked take atomic expressions jalicek alice thing thing jlikedk liked thing thing bool thing type individual objects bool type truth values propositions following justifiable standard practice linguistics let liked take object first argument subject second argument example first argument liked second argument alice shows two ways combine expressions function take argument either right combining liked left combining alice liked denote two cases two infix operators forward combination backward combination tick marks depict direction function leans argument derive sentence prove type bool derivation written tree term alice liked jalicek jlikedk liked alice bool convention infix operators associate right parentheses optional unfortunately system set far derives acceptable sentence also unacceptable sentence meaning alice liked reason system derives direction function application unconstrained derivation liked takes first object argument left usually disallowed english alice liked jalicek jlikedk liked alice bool rule derivation type system split function type constructor two type structors one direction application using new type constructors change denotation liked specify first argument right second argument left jlikedk liked thing thing bool also revise combination rules require different function type constructors system rejects continuing accept desired grammar derive sentence every student liked jeveryk jstudentk jlikedk existential determiner analyzed similarly let denote jsomek thing bool thing bool bool derive sentence student liked quantification student liked linguistic phenomenon quantification illustrated following sentences every student liked student liked every course alice consulted bob meetings summarize treat determiners like every functions two arguments restrictor scope quantifier functions thing bool functions popular analysis natural language determiners known semanticists since montague generalized quantifiers however simplistic account presented handles quantificational noun phrases subject position example neither forward backward combination apply join verb liked type thing thing bool object every course type thing bool bool yet empirically speaking sentence acceptable fact ambiguous two available readings problem prompted great variety supplementary proposals linguistics literature barwise cooper hendriks may inter alia next section presents solution using delimited continuations previously encountered sentences natural language semanticist wants translate english logical formulas account entailment properties precisely problem posit translation rules map sentences thus instance would like map formula like student liked bool constants thing bool bool bool bool bool drawn abstract syntax predicate logic end subject noun phrase every student denote unlike alice nothing type thing quantificational noun phrase every student denote still allow desired translation generated time would like retain denotation previously computed verb phrase liked namely liked taking considerations account one way translate determiner every denote jeveryk thing bool thing bool bool restrictor scope variables intended receive respectively denotations noun student type thing bool verb phrase liked type thing bool precisely variables tick mark signifies direction function application sentence like verb phrase takes subject argument contrast quantificational sentence subject takes verb phrase argument extended lexical entry every assuming student denotes jstudentk student thing bool jsomek jstudentk jlikedk student liked bool delimited continuations continuations represent entire default future computation kelsey clinger rees refining concept felleisen introduced delimited continuations encapsulate prefix future paper uses shift reset danvy filinski popular choice control operators delimited continuations review briefly shift operator notated captures current context computation removing making available program function example evaluating term variable bound function multiplies every number thus expression evaluates via following sequence reductions reduced subexpression step underlined term reductions performed deterministically applicative order reset operator notated square brackets delineates far shift reach shift captures current context computation closest dynamically enclosing reset hence reach shift reset come operational semantics illustrated reductions well denotational semantics via cps transform kinds semantics available important linguistics meanings natural language expressions studied semantics need related humans process studied psycholinguistics quantificational expressions natural language thought phrases manipulate context sentence like alice liked context function mapping thing proposition alice liked compared proper noun special quantificational expression like every course captures surrounding context used alice liked every course thus loosely speaking meaning sentence longer overall shape alice liked occurrence every course considered much meaning program longer overall shape shift expression evaluated let add shift reset target language translation english translate every course jevery coursek course thingbool bool type notation indicates control effect cps transform maps denotation every course behaves locally thing requires current context answer type bool maintains answer type see new denotation action let derive sentence type every course thingbool bool similar type thing derivation analogous alice liked every course jalicek jlikedk jevery coursek alice liked course course alice liked course alice liked bool like straw man analysis denotation generalizes determiners every abstract noun course every course deal student similarly jeveryk jsomek thing bool thingbool bool require restrictor type thing bool type form thing incur control effects applied control effect restrictor induced quantificational noun phrase company sentence every representative company left must contained within reset importantly unlike straw man analysis new analysis works uniformly quantificational expressions subject object positions intuitively shift captures context expression matter deeply embedded adding control operators delimited continuations logical metalanguage arrive analysis quantification greater empirical coverage figure shows logical metalanguage formalizes basic ideas presented language denotations page written reduced refining danvy filinski original language distinguish pure impure expressions impure expression may incur control effects evaluated whereas pure expression incurs control effects contained within reset danvy hatcliff nielsen thielecke distinction reflected typing judgments impure judgment gives type also specifies two answer types contrast pure judgment gives type seen lift rule pure expressions polymorphic answer type mentioned use directionality function types control word order new linguistics use delimited control operators analyze quantification turns tie potential presence control effects function bodies directionality directional whose types decorated tick potentially impure functions need deal including contexts captured shift pure another link directionality control effects rules directional function application merely mirror images answer types chained differently premises due evaluation made distinction pure impure expressions require shift rule body shift expression pure change danvy filinski original system simplifies type system cps transform shift expression language may need rewritten cps transform metalanguage follows typing rules standard supplies denotational semantics operational semantics metalanguage specifies computation relation complete terms also standard shown figure present analysis almost quite directstyle analogue barker cps analysis put terms barker function bodies always pure whereas function bodies harbor control effects matter deep closest dynamically enclosing reset control delimiters correspond islands natural language barker directions thing bool types antecedents terms constants thing bool bool thing bool bool bool bool bool bool bool bool liked thing thing student thing bool pure expressions const var reset impure expressions lift shift figure logical metalanguage directionality delimited control operators values unknowns contexts metacontexts computations dhch dhch dhchv dhch eii dhche dhche dhche dhchv dhe chxi figure reductions logical metalanguage words function bodies allowed shift determiner denotations contrast barker uses choice functions assign meanings determiners quantifier scope ambiguity course natural language phenomena never simple couple programming language control operators quantification exception speak example sentence student liked every course ambiguous following two readings student course liked course student liked surface scope reading takes scope every inverse scope reading every takes scope given evaluation takes place left right shift student evaluated shift every course grammar thus predicts surface scope reading inverse scope reading prediction seen first reductions unique derivation jsomek jstudentk jlikedk jeveryk jcoursek student liked course student liked course student liked course regardless evaluation order specify long rules semantic translation remain deterministic generate one reading sentence hence theory fails predict ambiguity sentence better account data need introduce sort nondeterminism theory two natural ways proceed first allow arbitrary evaluation order change would render term calculus nonconfluent result unwelcome programming language researchers welcome light ambiguous natural language sentence route pursued success barker groote however empirical reasons maintain evaluation one appears second way introduce nondeterminism maintain evaluation generalize shift reset hierarchy control operators barker danvy filinski barker leaving unspecified level hierarchy quantificational phrase shifts following danvy filinski extend logical metalanguage superscripting every shift expression pair reset brackets nonnegative integer indicate level control hierarchy level highest level lowest shift expression level evaluated captures current context computation closest dynamically enclosing reset level higher smaller example whereas expression evaluates expression evaluates superscripts thought strength levels shifts resets danvy filinski give denotational semantics multiple levels delimited control using continuations type take advantage work quantificational denotations letting shift level ambiguity predicted follows suppose student shifts level every course shifts level somem student liked everyn course surface scope reading results inverse scope reading results general quantifier shifts higher level always scopes another shifts lower level regardless one evaluated first way evaluation order determine scoping possibilities among quantifiers sentence unless two quantifiers happen shift level summarize discussion far whether introduce nondeterministic evaluation order hierarchy delimited control operators account ambiguity sentence well complicated cases quantification english mandarin example nondeterministic evaluation order approach control hierarchy approach predict correctly sentence three quantifiers ambiguous every representative company saw samples despite fact three quantifiers sentence sentence readings company occurs within restrictor every representative company incoherent every scope reason neither approach generates reading seen denotation every located immediately abstract syntax intervening control delimiter control operator insert material samples exist computational linguistics literature algorithms computing possible quantifier scopings sentence like hobbs shieber followed lewin moran related quantifier scoping control operators gain denotational understanding algorithms accords theoretical intuitions empirical observations extended logical metalanguage infinite hierarchy control operators shown figure system complex one figure two ways first instead making binary distinction pure impure expressions use number measure pure expression expression pure level incurs control effects levels evaluated pure expressions special case purity level expression reflected typing judgment judgment states expression pure level computation type levels defined figure consists value types together specify computation pure level affects answer types levels special case computation type familiar form previous system figure directional functions always impure pure level nondirectional functions always pure pure level current system kinds functions declare types level bodies pure example determiners every allowed shift level type bool thing bool thingbool also type thing thing see second technical complication argument type thing shows first argument determiners restrictor yet impure pure level traverse control hierarchy add new reset rule makes expression pure new lift rule makes less pure consecutive nested expression resets like abbreviated without loss coherence second complication system contrast figure longer encode logical quantification using constant like thing bool bool constant requires logical formula pure function requirement problematic exactly impurity quantified logical formulas underlies account quantifier scope ambiguity one hand want quantify logical formulas impure hand want rule expressions like logical variable leaks illicitly surrounding context issue precisely problem classifying open closed terms staged programming see taha nielsen references therein types thing bool really represent individuals truth values staged programs compute individuals truth values paper adopt simplistic solution adjoining types set free logical variables directions value types computation types antecedents terms thing bool constants thing bool bool bool bool bool bool bool bool bool bool liked thing thing bool student thing bool expressions const var lift reset shift reset lift figure extending logical metalanguage hierarchy control operators tracking purposes denoted unfortunately also need stipulate logical variables freshly created gensym occurrence quantifier sentence cps transform reductions metalanguage standard latter appears figure present analysis almost quite directstyle analogue barker cps analysis even though use control hierarchy level barker hierarchy intuitively staged computation produced one level higher concretely computation type levels system shape rather shape issue encode logical quantification impure formulas receives satisfactory treatment barker system stipulation necessary analogue prohibit relation system staged programming effects yet explored polarity sensitivity analysis far focuses meaning quantifiers equates determiners existential quantifiers type denotation furthermore sentences like anyone arrived suggest determiner also means thing contrary though determiners always interchangeable existential usage sentences readings show take scope differently relative negation cases quantifier student liked course unambiguous student liked course ambiguous student liked course unambiguous student liked course unambiguous student liked course ambiguous student liked course unacceptable determiner negative polarity item first approximation occur contexts scope monotonically decreasing quantifier ladusaw quantifier type thing bool bool monotonically decreasing case quantificational noun phrases student course monotonically decreasing since instance student liked course general student liked computer science course particular whereas negative polarity item positive polarity item roughly speaking allergic values unknowns contexts hch metacontexts level computations hch hche hch hche hchv hche hch hchv hdn eii iii hdn hchxii figure reductions extended logical metalanguage contexts especially overtly negative word like generalizations regarding polarity items cover data principle goes theory sentences ambiguous two scopings polarity sensitivity rule one scoping four sentences thus predicted unambiguous remains unclear downright unacceptable tradition linguistics polarity sensitivity typically implemented splitting answer type bool several types different functor applied bool related subtyping bernardi bernardi moot fry instance differentiate determiners formalism add types boolpos boolneg alongside bool supertypes bool sub reset boolpos reset boolpos finally refine types determiners bool jnok thing bool thingboolneg janyk thing bool chain transitions one quantificational expression next acts automaton shown figure states automaton three supertypes bool two subtyping relations transitions determiners boolpos bool boolneg figure automaton transitions machine enforces polarity constraints follows valid derivation sentence assigns quantifiers shift certain level control hierarchy level quantifiers order form add side condition reset requiring produced answer type bool boolpos boolneg boolneg jak thing bool bool boolneg bool boolpos also extend subtyping relation value computation types usual closure rules allow implicit coercion subtype supertype boolpos jsomek thing bool either path boolpos bool state machine words string determiners matching regular expression path bool bool state machine words string determiners matching regular expression furthermore levels hierarchy must higher levels every assignment quantifiers levels satisfies conditions gives reading sentence quantifiers higher levels scope wider among quantifiers level ones evaluated earlier scope wider consider two alternative ways characterize scope ambiguity suggested first approach allow arbitrary evaluation order use degenerate control hierarchy one level take route account acceptability ambiguity judgments distinguish acceptable sentence unacceptable words would mystery acceptability sentence hinges linear order quantifiers appear mystery noted ladusaw fry defect current accounts polarity sensitivity second approach using control hierarchy multiple levels fares better stick evaluation must preceded scopes intervening indeed variations ambiguity acceptability among sentences completely captured intuition imagine hearer sentence must first process trigger context like makes sense process negative polarity item like intuition aside notion evaluation order provides syntactic hacker formal types new tool capture observed regularities natural language linguistic side effects paper outlines quantification polarity sensitivity natural language modeled using delimited continuations two examples support claim formal theory computational intuition continuations help construct understand maintain linguistic theories sure work far first time insights programming languages applied natural language long noted intensional logic montague grammar couched understood computationally hobbs rosenschein hung zucker dynamic semantics groenendijk stokhof relates anaphora discourse natural languages nondeterminism mutable state programming languages van eijck applied variety natural language phenomena ellipsis van eijck francez gardent hardt kind jumps labels one second kind mutable state analogy use term linguistic side effects refer aspects natural language either unclear denotational semantics look like obvious denotational semantics making clause denote whether true turns break referential transparency besides quantification polarity sensitivity examples bob thinks alice likes man walks whistles star alice see alice saw venus king france whistles intensionality variable binding interrogatives focus presuppositions study linguistic side effects propose draw analogy computational side effects computer scientists want express computational side effects uniform modular framework study control interacts mutable state felleisen hieb linguists want investigate properties common linguistic side effects study quantification interacts variable binding furthermore computer scientists want relate operational notions like evaluation order parameter passing denotational models like continuations monads linguists want relate dynamics information language processing static definition language generative device whether analogy yields linguistic theory empirically adequate open scientific question find attractive pursue acknowledgments thanks stuart shieber chris barker raffaella bernardi barbara grosz pauline jacobson aravind joshi william ladusaw fernando pereira avi pfeffer chris potts norman ramsey dylan thurston yoad winter anonymous referees work supported united states national science foundation grants references however link natural language continuations recently made explicit paper use control operators analysis novel analyses presented part larger project relating computational side effects linguistic side effects term computational side effect covers programming language features either unclear denotational semantics look like obvious denotational semantics making arithmetic expression denote number turns break referential transparency computational side effect first barker chris notes continuations manuscript university california san diego although paper uses danvy filinski control hierarchy polarity sensitivity expressed equally well barker system syntactic distinction among types bool boolpos boolneg may even semantically interpretable via correspondence potential connection briefly explored bernardi nilsen examine connection krifka others proposed pragmatic grounds determiners like negative polarity items indicate extreme points scale bernardi raffaella richard moot generalized quantifiers declarative interrogative sentences journal language computation continuations nature quantification natural language semantics barwise jon robin cooper generalized quantifiers natural language linguistics philosophy bernardi raffaella reasoning polarity categorial type logic thesis utrecht institute linguistics ots utrecht university bernardi raffaella nilsen polarity items type logical grammar connection dmg slides talk learning logic grammar workshop amsterdam carpenter bob semantics cambridge mit press danvy olivier andrzej filinski functional abstraction typed contexts tech diku university copenhagen denmark http abstracting control proceedings acm conference lisp functional programming new york acm press danvy olivier john hatcliff strictness analysis acm letters programming languages systems transformation direct continuation semantics mathematical foundations programming semantics international conference stephen brookes michael main austin melton michael mislove david schmidt lecture notes computer science berlin van eijck programming dynamic predicate logic report centrum voor wiskunde informatica amsterdam also research report institute logic language computation universiteit van amsterdam van eijck jan nissim francez ellipsis dynamic semantics applied logic logical approaches natural language michael masuch dordrecht kluwer felleisen matthias theory practice firstclass prompts popl conference record annual acm symposium principles programming languages new york acm press felleisen matthias robert hieb revised report syntactic theories sequential control state theoretical computer science fry john proof nets negative polarity licensing semantics syntax lexical functional grammar resource logic approach mary dalrymple chap cambridge mit press gardent claire dynamic semantics logics european workshop jelia jan van eijck lecture notes artificial intelligence berlin groenendijk jeroen martin stokhof dynamic predicate logic linguistics philosophy groote philippe type raising continuations classical logic proceedings amsterdam colloquium robert van rooy martin stokhof institute logic language computation universiteit van amsterdam hardt daniel dynamic interpretation verb phrase ellipsis linguistics philosophy hendriks herman studied flexibility categories types syntax semantics thesis institute logic language computation universiteit van amsterdam hobbs jerry stanley rosenschein making computational sense montague intensional logic artificial intelligence hobbs jerry stuart shieber algorithm generating quantifier scopings computational linguistics hung jeffery zucker semantics pointers referencing dereferencing intensional logic lics proceedings symposium logic computer science washington ieee computer society press kelsey richard william clinger jonathan rees report algorithmic language scheme symbolic computation also acm sigplan notices krifka manfred semantics pragmatics polarity items linguistic analysis ladusaw william polarity sensitivity inherent scope relations thesis department linguistics university massachusetts reprinted new york garland lewin ian quantifier scoping algorithm without free variable constraint coling proceedings international conference computational linguistics vol may robert logical form structure derivation cambridge mit press montague richard proper treatment quantification ordinary english formal philosophy selected papers richard montague richmond thomason new yale university press moran douglas quantifier scoping sri core language engine proceedings annual meeting association computational linguistics somerset association computational linguistics nielsen lasse selective cps transformation proceedings mfps conference mathematical foundations programming semantics stephen brooks michael mislove electronic notes theoretical computer science amsterdam elsevier science quantifier strengths predict scopal possibilities mandarin chinese draft manuscript harvard university http chris barker explaining crossover superiority evaluation draft manuscript harvard university university california san diego http taha walid michael florentin nielsen environment classifiers popl conference record annual acm symposium principles programming languages new york acm press thielecke hayo control effects typed continuation passing popl conference record annual acm symposium principles programming languages new york acm press
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psyphy psychophysics driven evaluation framework visual recognition traditional natural evaluation brandon richardwebster student member ieee samuel anthony student member ieee walter scheirer senior member ieee visual psychophysics sep natural providing substantial amounts data standardized evaluation protocols datasets computer vision helped fuel advances across areas visual recognition even light breakthrough results recent benchmarks still fair ask recognition algorithms well think vision sciences large make use different evaluation regime known visual psychophysics study visual perception psychophysics quantitative examination relationships controlled stimuli behavioral responses elicit experimental test subjects instead using summary statistics gauge performance psychophysics directs construct curves made individual stimulus responses find perceptual thresholds thus allowing one identify exact point subject longer reliably recognize stimulus class article introduce comprehensive evaluation framework visual recognition models underpinned methodology millions procedurally rendered scenes images compare performance convolutional neural networks results bring question recent claims humanlike performance provide path forward correcting newly surfaced algorithmic deficiencies index recognition visual psychophysics neuroscience psychology evaluation deep learning ntroduction often attribute understanding cognitive predicates metaphor analogy cars adding machines artifacts nothing proved attributions john searle imagine following scenario marvelous black box algorithm appeared purportedly solves visual object recognition ability good scientist might falsifying claim accounts algorithm achieves superior performance established benchmarks computer vision internal workings opaque external observer situation far fetched familiar studying machine learning visual recognition many computer vision might realize setup happens classic chinese room problem proposed philosopher john searle searle thought experiment person speak chinese alone locked room following instructions computer program generate chinese characters respond chinese messages slipped door message passer outside room person inside richardwebster scheirer department computer science engineering university notre dame notre dame corresponding author anthony department psychology harvard university perceptive automata rendered gaussian blur linear occlusion rotation fig article concept applying psychophysics recognition model introduced figure two models compared top traditional evaluation summary statistics generated large sets data little consideration given specific conditions lead incorrect recognition instances bottom psychophysics set experimental concepts procedures psychology neuroscience helps plot exact relationships perturbed test images resulting model behavior determine precise conditions models fail instead comparing summary statistics compare itemresponse curves representing performance versus dimension image manipulated understands chinese however case person inside room simply following instructions complete task real replication competency knowing chinese language linking back computer vision summary statistics performance algorithms look good benchmark tests enough believe close human performance cases algorithms really solving general problem visual object recognition simply leveraging instructions provided form labeled training data solve dataset datasets computer vision intended controlled testbeds algorithms task difficulty modulated facilitate measurable progress research dataset could made images specifically acquired experimentation publicly available images crawled web regime strong advancements demonstrated number problems notably object recognition deep learning mainstay computer vision thanks part imagenet challenge alexnet reduced object classification error previously best reported result algorithms evaluated common footing possible track meaningful improvements artificial intelligence like one however increases error different datasets used training testing make wonder way comes natural intelligence neuroscientists psychologists evaluate animals people way computer vision scientists evaluate algorithms good reason collection images crawled web straightforward way determine exact condition caused subject fail recognizing stimulus presented experiment natural image product physics instant sensor acquired scene latent parameters largely unknown instead behavioral experiments meant discover perceptual thresholds average point subjects start fail vision sciences outside computer vision use concepts procedures discipline visual psychophysics psychophysics quantitative study relationships controlled stimuli behavioral responses elicit subject way probe perceptual processes presentation incremental many cases extremely perturbations visual stimuli properties stimulus varied along one physical dimensions thus controlling difficulty task result fig curve performance accuracy plotted dimension manipulated gaussian blur point curve reflects individual stimulus letting map performance back causal conditions precise manner psychophysics indispensable tool vision science deployed discover minimum threshold stimulation retinal photoreceptor single photon confirm helmholtz assertions color absorption retina establish criteria diagnose prosopagnosia inability recognize face discoveries biological vision submit psychophysics holds much promise discovering new aspects inner workings machine learning models article introduce comprehensive evaluation framework visual recognition underpinned principles psychophysics regime stimulus object drawn purely rendered data natural scene data varying physical parameter control amount transformation subsequent set manipulated images derived original stimulus key difference traditional benchmarks computer vision instead looking summary statistics average accuracy auc precision recall compare algorithm performance compare resulting curves complete control underlying parameter space find procedural graphics useful way generate stimuli manipulated way desire procedure rendered scene find model failing parametric level see using framework explore artificial vision systems like psychologists many interesting new findings surfaced strengths limitations computer vision models summarize main contributions follows general evaluation framework developed performing visual psychophysics computer vision models framework strong grounding work psychology neuroscience behavioral experimentation investigation procedural graphics psychophysics experiments applied models parallelized implementation psychophysics framework deployable python package case study consisting battery experiments incorporating millions procedurally rendered images images perturbed performed set convolutional neural network cnn models elated ork methods evaluation vision sciences respect work computer vision directly using psychophysics related establishing human baselines comparison algorithmic approaches riesenhuber poggio described series psychophysical comparisons humans hmax model visual cortex using limited set stimuli rendered computer graphics similarly eberhardt designed experiment measure human accuracy reaction time visual categorization tasks natural images compared different layers cnn models geirhos undertook similar study image degradations respect features gerhard introduced new psychophysical paradigm comparing human model sensitivity local image regularities psychophysics also used performance evaluation scheirer introduced notion perceptual annotation machine learning whereby psychophysical measurements used weights loss function give training regime priori notion sample difficulty using accuracy reaction time measured via online psychophysics testing platform perceptual annotation shown enhance face detection performance along lines vondrick devised method inspired psychophysics estimate useful biases recognition computer vision feature spaces outside work specifically invoking psychophysics one find related methods psychology neuroscience model testing natural images common type data computer vision form good basis algorithmic evaluation mode toole philips designed controlled datasets natural images compare human face recognition performance algorithms focus algorithmic consistency human behavior explicit model model comparison methods control experimentation achieved use rendered scenes cadiue yamins hong make use rendered images parametrized variation compare representations models found primate brain pramod arun describe set perceived dissimilarity measurements humans used study systematic differences human perception large number handcrafted learned feature representations need control object parts latent parameters scenes procedural graphics introduced tenenbaum study oneshot learning using probabilistic generative models use procedural graphics generative models developed yildirim kulkarni studies vary conditions stimuli using procedures psychophysics use renderings order millions scenes manipulations stimuli visual recognition evaluations work coming directly computer vision also addresses stimulus generation purpose isolating model weaknesses hoiem suggest systematically varying occlusion size aspect ratio visibility parts viewpoint localization error background identify errors object detectors wilber systematically apply noise blur occlusion compression textures warping effects scenes assess face detection performance finally whole host approaches found manipulate inputs cnns order highlight unexpected classification errors include noise patterns introduced szegedy imperceptible humans fooling images produced via evolutionary algorithms explored nguyen bendale boult level control evaluation procedures varies approaches common starting point based model preference class missing object configuration produces highest score suggest article use computer graphics helps address iii ethod ramework procedure performing psychophysics model largely follows established procedures found psychology key adaptations accommodate artificial perception purposes article focus two tasks yield interpretable curve descriptions procedures psychophysics see first let consider forced choice task common psychological testing procedure observer shown sample stimulus followed two alternate stimuli one positive matching stimulus negative stimulus observer asked choose alternate stimuli stimulus best matches sample match criterion may may provided observer observer repeats task different perturbed stimulus levels either adaptive pattern like gradient descent humans via method constants predetermined set perturbed stimulus levels regardless method task two presented alternate stimuli thus analysis experiment would utilize mean median accuracy humans achieved stimulus level mean median human response time recorded models tested precisely way algorithm dfm binary decision softmax layer cnn used preferred view calculation mafc input single network model input input image input expected class softmax vector find class label incorrect class negate response end return decision score input images arranged described accuracy used performance measure second consider mapping difficult classification procedure machine learning general version procedure call mapped classification mafc mafc probe image classification equivalent sample stimulus classification rarely two classes model choose thus value becomes number labeled training classes imagenet learned classes making likewise number presented alternate stimuli changes number images used training set images model implicitly matching imagenet training images testing model psychophysics procedure need special process selection stimuli default state perturbation blanz show humans natural inclination towards recognizing certain view object called canonical view assume human trials object configuration close canonical view chosen maximizing probability observers problems performing least part task however simple task involving model necessarily know follows similar canonical view say model preferred view view produces strongest positive response determined decision score note one preferred view hence use term preferred ties often observed strongest response especially quantized decision score spaces choosing preferred view crucial guaranteeing stimulus perturbed model response already maximum class perturbation cause decline possibly change strength model response increase psyphy framework mafc inspired software frameworks subject testing like psychopy implemented mafc procedures described using python framework model testing called psyphy describe details component framework basic steps stimuli selection preferred view selection perturbation algorithm point generation function supporting tasks image transformation function input decision function input vector preferred views set classes input stimulus level pick negative max return coordinate pair stimulus level accuracy trials one point curve generation apply psychophysics procedure specific mafc procedures may viewed pluggable modules within framework psyphy flexible respect tasks support first step select initial set stimuli class natural images set chosen images class rendered scene set image specifications provided rendering function implemented work using mitsuba render single object centered image view parameter set coordinates real numbers range representing scale real number range second step find individual model preferred view class natural images preferred view function used second preferred view function uses create rendered images classification search space almost infinite thus find absolute global maximum rather approximation argmax dfm argmax dfm decision function classification dfm alg normalizes score output model value range gives decision confidence associated decision value range incorrect decision correct decision parameter input stimulus expected class natural preferred view single selected image dfm strongest positive response preferred view single selected set dfm strongest positive response major difference use render image prior measuring response dfm invoking class create vector preferred views preferred views selected classes whether natural rendered next step apply perturbations procedure set preferred views perturbed specific stimulus level amount perturbation using function could image transformation function gaussian blur rotation parameter one preferred view either algorithm point generation function supporting mafc tasks image transformation function input dfm decision function mafc input vector preferred views set classes input pthe stimulus level max ddf return coordinate pair stimulus level accuracy trials one point image format rendered stimuli stimulus level function perturbs set preferred views given makes decision image using decision function specific implementation described alg mafc described alg procedure specific decision functions required alg used dfm alg used mafc individual image evaluation trial value returned represents one point curve computed accuracy trials one trial per class curve set coordinates represent model behavior set stimuli value represents perturbation level accuracy model performance note traditional psychophysics experiments live test subjects often apply psychometric function interpolate points generate curve approximate psychometric function better interpretability applied rectangular smoothing unweighted smoothing window size padding curve repeated edge values final step generates curves using function procedure simple requires repeated execution stimulus level steps shown alg procedure create set stimulus levels starting lower bound ending upper bound closest stimulus level preferred view stimulus level farthest away parameter number stimulus levels use typically visual pyschophysics stepping used evaluation near canonical view strategy used preferred view xperiments first goal experiments demonstrate psyphy psychophysics evaluation framework processed millions procedurally rendered scenes images perturbed second goal demonstrate utility procedural graphics largescale psychophysics experiments thus broke data two sets natural scenes rendered scenes final goal evaluate strengths weaknesses cnn models looked model behavior mafc tasks behavior dropout test time perturbations comparisons algorithm best match decision final feature layer cnn used input single network model input input image input expected positive image input expected negative image gather activations final feature layer pearson correlation incorrect selection negate response else end return decision score algorithm curve generation function type decision function input point generator input input model input vector preferred views input number stimulus levels input lower upper bound values stimulus levels let stimulus levels return curve human behavior experiments chose use five convolutional neural network models pretrained imagenet alexnet caffenet googlenet complete set plots details methods found supplemental material data generation natural scene experiments perturbed images imagenet training dataset consists million images classes using training set instead testing set gives model intentional bias towards expert performance following transformations applied gaussian blur linear occlusion salt pepper noise brightness contrast sharpness condition created perturbed images starting preferred view stepped towards increasing difficulty result images per class per network images per network images per condition total images evaluated experiments rendered images selected objects blend swap library corresponded classes imagenet see supp material list classes objects randomly rendered uniformly distributed rotations scales supplemental material accessible http ing images preferred view selected set following transformations applied graphics engine rotations dimensions scale applied positive negative direction addition transformations natural image experiment repeated using rendered preferred views transformations rendered images starting preferred view stepped towards increasing difficulty result images per class per network images per network images per transformation additional transformations resulted total images brought rendered image total evaluated images experiments motivation experiments twofold test decision making fundamental level via activation matching look class labels test precise implementation wellknown task given setting two instances initially expected models perform relatively well variety perturbations experiments included use natural scenes rendered scenes stimuli model behavior stable rotation fig center supp fig contrast sharpness supp figs transformations rest transformations induced erratic behavior accuracy declining fig left right gaussian blur detrimental model accuracy even limited matching setting tells something receptive fields convolutional layers networks large enough tolerate even modest levels blur also interesting results scale perturbation let isolate specific failure modes example scale decreases original size object structure still clearly visible accuracy drops lower networks supp fig observation could made looking summary statistics differences behavior across networks significant examining curves confidence interval plotted supp figs network behavior consistently demonstrates trends transformation type across perturbations commonly believed architectural innovation advancing deep learning finding indicates model behavior influenced training data models experiments vgg networks suggests additional layers beyond certain point imply better performance degrading conditions likewise switching order pooling normalization layers caffenet alexnet imply better performance degrading conditions mafc experiments motivation mafc experiments evaluate task closely aligned classification task models originally trained given choices setting instead two expected models perform much worse transformations exactly observed results fig supp figs fig selection curves task natural scene experiment left reflects classification accuracy across imagenet classes rendered scene experiments center right reflect accuracy across classes experiment used five cnns perfect curve would flat line top plot images bottom curve show perturbations increase right left starting perturbation original image conditions experiments indicate models tolerate different perturbations settings cases gaussian blur red dot indicates mean human performance selected stimulus level plots well next sets figs best viewed color instance compare plots increasing brightness fig positive rotation fig corresponding plots fig transformation types networks experienced moderate dips performance extreme perturbations case fall accuracy points caveat mafc decision function patterned classification task computer vision uses class labels make decisions thus leaves activation information used case highlights important occur designing decision functions psychophysics evaluations task fidelity versus task difficulty curiously large asymmetries transformations increasing decreasing perturbation levels see plots fig supp figs brightness contrast sharpness contrast particularly intriguing case transformation contrast single operation applied globally image positive direction contrast increased performance network degrades rapidly negative direction contrast decreased performance network remains relatively stable objects low contrast suggests contrast sensitivity problem mafc decision function opposite human patients vision deficits struggle positive aspect finding diminished contrast sensitivity may induce human driver autonomous driving systems expected operate effectively dark dropout experiments experiments looked thus far assume deterministic outputs settings stochastic outputs support uncertainty decision making gal ghahramani introduced dropout testing time implement bayesian inference neural networks sort variability lead various transformations perturbation levels tell certainty experiments setup experiments identical setup mafc experiments including preferred views except evaluation applied dropout test time caffe version alexnet also trained dropout deploying model test dropped model large neurons layers uniformly randomly setting activations zero repeated different random seeds transformation except salt pepper noise linear occlusion performed due randomness underlying perturbation functions anticipated variability base model performance introduced fig supp figs importantly runs still demonstrated large measure consistency fig across ranges perturbations indicating higher degrees model certainty good stability property model fails expected way across different dropout configurations lending credibility initial characterizations behavior earlier experiments variability observed rendered objects versus natural scenes attributed use objects outside training set models maximum difference observed points two runs transformation sharpness applied objects supp fig half cases maximum difference human comparisons using psychophysics model performance directly compared human performance obtain human data points red dots figs conducted study participants participant performed task described mitigate fatigue trials fixed psychophysical parameter setting given transformations fig participants also performed mafc task limited choices instead full classes make task tractable experiments participants performed trials fixed psychophysical parameter setting transformation fig original images trial chosen randomly preferred views class used one time participant order fig selection curves mafc task top natural scenes bottom rendered scenes plots directly comparable corresponding plots fig perturbations except linear occlusion human model behavior consistent prevent participants learning performing task trials tasks sample images presented subjects unlimited time answer even without generating full psychometric curve human subjects apparent two nine experiments showed consistency human model performance fig center fig cases human model behavior wildly divergent salt pepper noise fig human performance superior model performance cases one case humans slightly worse models increasing brightness fig left brightness adjustment image processing constant multiplicative change pixel values preserves edges allows networks recognize geometry object almost saturation reached humans also good task still worse perturbation level analyzed iscussion visual psychophysics convenient practical alternative traditional dataset evaluation however use psychophysics testing datasets mutually exclusive one needs datasets form training basis datadriven model moreover major utility large amount data essential making machine learning capture enough intraclass variance generalize well unseen class instances data augmentation obvious strategy leveraging rendered images problematic model psychophysics testing expand scope training set however diminishing returns datasets grow sizes exceed available memory even disk space training using limited training data reinforcement learning optimizes curves correct recognition errors likely better path forward recent research shown cnns able predict neural responses visual cortex primates coupled excellent benchmark dataset results across multiple recognition domains suggests good progress made towards reaching performance strong counterpoint psychophysics experiments show current popular cnn models sometimes fail correctly classify images humans make mistakes missing models causing behavioral discrepancy certainly capturing operation ventral stream particularly filtering stages early vision today cnn architectures lack intricate structure function biological neural networks advances connectomics subfield neuroscience attempting map circuits brain functional imaging allows vivo study neuronal responses likely inform new architectures near future psychophysics guide performance candidate models easily accepted dismissed representative form making harder fooled person inside chinese room acknowledgment authors thank lucas parzianello helping import blendswap models psyphy brian turnquist providing feedback early draft work funding fig curves five different runs alexnet model dropout applied test time rotation transformation black line indicates mean five alexnet curves maximum difference points two curves plot provided iarpa contract nsf dge nsf sbir award hardware support generously provided nvidia corporation eferences dosher visual psychophysics laboratory theory mit press kingdom prins psychophysics practical introduction academic press searle minds brains programs behavioral brain sciences vol russakovsky deng krause satheesh huang karpathy khosla imagenet large scale visual recognition challenge ijcv vol imagetnet large scale visual recognition challenge http index accessed krizhevsky sutskever hinton imagenet classification deep convolutional neural networks nips torralba efros unbiased look dataset bias ieee cvpr tommasi patricia caputo tuytelaars deeper look dataset bias ser lncs gall gehler leibe eds springer vol embretson reise item response theory psychologists lawrence erlbaum associates hecht shlaer pirenne energy quanta vision journal general physiology vol bowmaker dartnall visual pigments rods cones human retina journal physiology vol duchaine nakayama cambridge face memory test results neurologically intact individuals investigation validity using inverted face stimuli prosopagnosic participants neuropsychologia vol tenenbaum kemp griffiths goodman grow mind statistics structure abstraction science vol yildirim kulkarni freiwald tenenbaum efficient robust vision computational framework behavioral tests modeling neuronal representations annual conference cognitive science society cogsci kulkarni kohli tenenbaum mansinghka picture probabilistic programming language scene perception ieee cvpr zhang xue freeman tenenbaum learning probabilistic latent space object shapes via generativeadversarial modeling nips jia shelhamer donahue karayev long girshick guadarrama darrell caffe convolutional architecture fast feature embedding arxiv preprint szegedy liu jia sermanet reed anguelov erhan vanhoucke rabinovich going deeper convolutions ieee cvpr simonyan zisserman deep convolutional networks image recognition corr vol riesenhuber poggio individual nothing class everything psychophysics modeling recognition object classes mit tech october hierarchical models object recognition cortex nature neuroscience vol eberhardt cader serre deep feature analysis underlying rapid visual categorization nips geirhos janssen rauber bethge wichmann comparing deep neural networks humans object recognition signal gets weaker arxiv preprint gerhard wichmann bethge sensitive human visual system local statistics natural images plos computational biology vol scheirer anthony nakayama cox perceptual annotation measuring human vision improve computer vision ieee vol august germine nakayama duchaine chabris chatterjee wilmer web good lab comparable performance web lab experiments psychonomic bulletin review vol vondrick pirsiavash oliva torralba learning visual biases human imagination nips toole phillips jiang ayyad penard abdi face recognition algorithms surpass humans matching faces changes illumination ieee vol toole dunlop natu phillips comparing face recognition algorithms humans challenging tasks acm transactions applied perception tap vol phillips toole comparison human computer performance across face recognition experiments image vision computing vol cadieu hong yamins pinto majaj dicarlo neural 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reconstruction codes dna sequences uniform errors jan yonatan yehezkeally student member ieee moshe schwartz senior member ieee data storage medium several advantages including far greater data density compared electronic media propose schemes data storage dna living organisms may benefit studying reconstruction problem applicable whenever multiple reads noisy data available strategy uniquely suited medium inherently replicates stored data multiple distinct ways caused mutations consider noise introduced solely uniform utilize relation constantweight integer codes manhattan metric bounding intersection hyperplanes prove existence reconstruction codes greater capacity known codes determine analytically set parameters index storage reconstruction systems errors ntroduction dna attracting considerable attention recent years medium data storage due high density longevity data storage dna may provide integral memory methods required offer protected medium data storage particular storage dna living organisms becoming feasible varied usages including watermarking generically modified organisms research material even afford concealment sensitive information naturally therefore data integrity media great interest several recent works studied inherent constraints storing retrieving data dna desired sequences quaternary alphabet may synthesized albeit suffering substitution noise generally data read observation subsequences quite possibly incomplete observation moreover nature dna current technology results asymmetric errors depend upon dataset medium also introduces types errors atypical electronic storage adjacent transpositions possibly inverted finally purely combinatorial problem recovering sequence multiset subsequences including numbers incidence also studied well coding schemes involving multisets profile vectors describing incidence frequency subsequence version paper submitted isit work supported part isf grant authors department electrical computer engineering university negev beer sheva israel yonatany schwartz works concerned data storage dna living organism affords level protection data even propagation dna replication also exposed specific noise mechanisms due mutations examples noise include symbol insertions deletion substitutions duplication including therefore schemes data storage live dna must address data integrity effort better understand typical noise mechanism potential generate diversity observed nature studied classified capacity expressiveness systems sequences finite alphabet generated four distinct substring duplication rules endduplication duplication fully characterized expressiveness bounded systems proved bounds capacity cases even exact values later showed act together process may actually increase capacity generated system looked typical duplication distance binary sequences number generating binary sequence root proven exponentially small number sequences number proportional sequence length combined within duplicated string shown frequency substitutions governs whether distance becomes logarithmic generative properties also studied probabilistic point view showed assumption uniformity frequencies incidence subsequence converge limit achieved source thus reinforcing notion interspersedduplication generating diversity specifically looked found exact capacities case duplication length generalization urn model applies strings also tightly bounded capacity complement process duplicated symbol complemented using binary alphabet finally codes data affected tandemduplication studied presented construction codes correcting number errors uniform fixed duplication length computing thus capacity also presented framework construction optimal codes correction fixed number errors next studies bounded characterization capacity codes made small constants general characterized cases process traced back uniquely however classical coding ignores properties dna storage channel namely stored information expected replicated even mutated lends quite naturally reconstruction problem assumes data simultaneously transmitted several noisy channels decoder must therefore estimate data based several distinct noisy versions solutions problem studied several contexts solved sequence reconstruction finite alphabets several error models considered substitutions transpositions deletions moreover framework presented solving reconstruction problem general cases interest coding theory utilizing graph representation error model developed problem also studied context permutation codes transposition reversal errors partially solved therein later applications found storage technologies since modern application might preclude retrieval single data point favor requests however problem addressed yet data storage dna living organisms may applicable paper study reconstruction problem dna sequences uniform error paper organized follows section present notations definitions section iii demonstrate reconstruction codes codes find requisite function reconstruction parameters section study bounds sizes codes isometric embedding codes manhattan metric finally section show considering reconstruction codes improve capacity known codes conclude closing remarks section reliminaries throughout paper though human dna composed four nucleotide bases observe general case sequences finite alphabet since alphabet elements immaterial discussion denote throughout observe set finite sequences also words two words denote concatenation word denote length also take special note set words length higher equal denote ease notation let stand set integers throughout paper integer assumed strictly positive make special note fact define mappings uvvw uvw otherwise occurs whenever say descendant denote given sequence say denote completeness also denote finally exists also denote denote set dkt also denote ease notation define set given dkt similarly say irreducible descendant word exclude definition shorter words condition vacuously holds denote irrk set irreducible words irrk irrk shown word unique irreducible word exists descendant call root denote induces equivalence relation also follow defining pref prefix suff suffix pref suff using notation define embedding zkq pref suff pref seen mapping indeed injective shown one also defines zkq zkq otherwise simplicity comparison motivates analysis using images sequence composed subsequences define mod mod mod lemma holds may note wth wth wth hamming weight nwth nwth also observe recoverable proven cor unit vector thus moreover irrk finally znq define min dkt dkt dkt dkt shown lem hence finite dkt particular furthermore lem shows holds proof let denote note exists occurs exists turn equivalently stated unit vector corollary holds proof using lemma observe case thus defines metric znq iii econstruction odes reconstruction problem context uniform errors stated follows suppose data encoded znq suppose later able read distinct dkt specific uniquely identify seen successful reconstruction ensured code satisfies following requirement znq definition take say uniform reconstruction code abbreviate code max dkt unit vectors ejt claim follows theorem poset isomorphism definition let defined noted previous section isometry definition define partial order every coordinate holds product order proof using corollary partially ordered set observe also poset poset homomorphism already noted injective isometry finally note every element decomposed eju eju throughout section fix irrk denote noted hence particular wth wth therefore denote wth make following definition eju section find size intersection descendant cones thereby evaluate size codes structure descendant cones unit vectors ejt lemma get sequence eju thus also surjective thus endows lattice structure denote join meet following immediate results corollary lemma proof lemma know dkt since bijective equals number distinct integer solutions equivalently number distinct ways distribute identical balls bins corollary znq utr code reconstruction uniform errors proof claim follows lemma lemma words intersection reconstruction code descendant cone irreducible word simply code suitable minimal distance note however hold general minimal distance dependent also znq depends start rephrasing properties descendant cones seen previous section corollary suppose irrk following conditions equivalent holds holds exist proof follows theorem implications follow trivially show note previous parts particular lemma implies observations allow classify intersection size lemma znq dkt dkt dkt dkt otherwise proof already noted dkt dkt otherwise corollary note dkt dkt therefore dkt dkt corollary allows using following definition characterize codes definition take denote min min size reconstruction codes section aim estimate maximal size codes definition denote call theorem take positive integers znq irrk denote define moreover code irrk image satisfies min proof follows definitions since necessarily furthermore dkr hence seen proof theorem dkr eju conclude proof recall irrk bijective isometry therefore see finding codes uniform essentially amounts finding codes manhattan metric start notating maximal size codes definition define max min note znq irrk therefore practical assume results following corollary concludes section corollary znq code irrk holds since size ball invariant also denote ease notation irrk rll wth proof first trivially observe irrk satisfies wth rest follows theorem odes simplex anhattan metric corollary motivates estimate optimal size codes manhattan metric simplex section dedicated question key component evaluation requisite minimal distance sphere size section evaluate size spheres establish bound asymptotic regime lemma positive integers holds binary entropy function defined proof hence thus claim trivially follows lemma definition denote hyperplane omit lemma denote min proof may count following manner partition distance origin partition number strictly positive coordinates min choose positive coordinates may done distinct ways find number solutions xij xij remaining coordinates sum distinct distributions thus min suffices note strictly increasing function thus claim satisfied following lemma make asymptotic evaluation lemma take fix suppose given sequence dimensions mnn lemma integers holds also denote ball radius proof sufficiently large find thus using lemma straightforward show lim recalling lemma proof completed quickly state bounds lemma next order bound first establish upper bound proof proof application lemma typical argument maximal code thus following theorem take integer sequences mnn rnn also take fixed lim proof use lemma find lim proof using lemma mayqbound get reordering inequality hand since lemma implies reverse inequality yielding claim reconstruction codes determine capacity asymptotic regimes define rate family codes znq lim sup logq minimal distance reconstruction codes next given establish bounds min seen definition ease notation follows make notation lemma proof may verify substitution satisfies using strict monotonicity done lemma holds min proof assumption satisfies inequality therefore may restrict minimum giving lemma implies completes proof theorem take max apacity using lemma know proof claim restated substitution known inequality proof follows elementary calculus omitted finally lim sup lim inf lemma holds capacity supremum rates families codes since showed irrk correct number errors trivially section prove reconstruction codes strictly higher capacity first denote rll wth recall arg maxr corollary znq hand every holds therefore focus maximizing lim sup logq choice follows take set shall assume exist hence infinitely many exist refer indinces irrk irrk recall denoted rll shall build reconstruction code descendant cones denote lemma exists system rll cap lim logq zlq cap rll holds wth proof let strongly connected deterministic digraph generating system rll adjacency matrix well known case see characteristic polynomial qxk hence perron eigenvalue unique qxk greater positive root fact readily confirmed either using elementary calculus methods since rll positive associated denote respectively specifically terminates first node conclude proof note may verify incidentally follows particular next recall given path path denoted edges appears path times satisfying let system induced paths well known cap cap rll note length generated path wth precisely number edges terminate first node since wth note hence every entry indeed positive denoting follows see sec stochastic represents transition matrix stationary markov chain probability measure edges set satisfying logq cap rll stationary distribution ofp markov chain positive given defined holds sum probabilities edges terminating jth node hence take observe satisfy proposition lemma implies exists subset irrk cap cap irrk every length rest section build codes descendant cones roots note denote resulting codes rate lim sup logq cap irrk lim sup logq theorem denote lim logq dnn following two regimes fixed psfrag replacements constants satisfying hence infinitely many indices proof note sufficiently large resulting lemma dnn note rnn mnn hence theorem claim proven fixed theorem dnn max max sufficiently large hence dnn since fixed may apply argument used previous part moving show generally may made exceed cap irrk careful choice look following example example perron eigenvalue cap addition less satisfies lemma alternatively may set special case human dna perron eigenvalue given hence cap may choose less shown function cases figure assumptions asymptotic regime made theorem figure demonstrates capacity reconstruction codes bounded maximum curve greater cap irrk attempt maximize proper choice theorem motivates following definition figure rate cases value equals cap irrk previously best known construction define definition take analysis simpler using following change variable definition define observe decreasing diffeomorphism lemma one logq logq proof observe log log log log particular logq logq logq hence logq logq logq logq logq logq show always exists choice get cap irrk theorem proof observe continuously differentiable satisfies find logq logq logq logq logq logq logq logq thus equation unique solution since rhs monotonic increasing function vanishing unbounded grows moreover since hence rhs greater thus unique local extremum suffices show concave indeed logq logq proof fixed define satisfies therefore egx case satisfies equation hence simplify thus first upper bound proven second require holds readily shown differentiation hand equation implies therefore proves lower bound next show may tighten bounds derived previous lemma lemma let unique solution equation denote follows thus main result paper established remains section show analytically find maximizes begin establishing bounds following lemma proof assumption hence implying similarly proposition trivially follows lemma let unique maximum denote finally show may found following limiting process theorem unique solution equation given proof denote unique solution take note lemma implies prove contraction indeed recalling find next last inequality may directly verified small done utilize banach theorem deduce unique fixed point necessarily defining get onclusion proposed reconstruction codes applied dna living organisms due channel inherent property data replication showed assumption uniform noise reconstruction codes errorcorrecting codes minimal distance dependent reconstruction parameters proved existence codes rates surpassing know codes particular theorem allows find parameters required real application resulting code optimal rate believe research focus explicit code constructions also desirable examine problem broader noise models bounded tandemduplication perhaps inversed well combinations multiple error models eferences acharya das milenkovic orlitsky pan string reconstruction substring compositions siam discrete vol alon bruck hassanazadeh jain duplication distance root binary sequences ieee transactions information theory vol arita ohashi secret signatures inside genomic dna biotechnology progress vol balado capacity dna data embedding substitution mutations ieee trans inform theory vol cassuto blaum codes read channels ieee transactions information theory vol dec church gao kosuri digital information storage dna science vol clelland risca bancroft hiding messages dna microdots nature vol elishco farnoud schwartz bruck capacity string models proceedings ieee international symposium information theory barcelona spain july farnoud schwartz bruck stochastic model genomic interspersed duplication proceedings ieee international symposium information theory hong kong china jun gabrys kiah milenkovic asymmetric lee distance codes storage ieee trans information theory vol aug gabrys yaakobi milenkovic codes damerau distance dna storage proceedings ieee international symposium information theory barcelona spain july hassanzadeh schwartz bruck capacity stringduplication systems ieee trans inform theory vol feb heider barnekow watermarks using dnacrypt algorithm bmc bioinformatics vol howell statistical properties selected recording codes ibm journal research development vol jan jain hassanzadeh bruck capacity expressiveness genomic tandem duplication ieee transactions information theory vol oct jain hassanzadeh schwartz bruck duplicationcorrecting codes data storage dna living organisms ieee trans information theory vol noise uncertainty systems proceedings ieee international symposium information theory aachen germany june jupiter ficht samuel qin figueiredo dna watermarking infectious agents progress prospects plos pathog vol kiah puleo milenkovic codes dna sequence profiles ieee trans inform theory vol june konstantinova reconstruction permutations distorted single reversal errors discrete appl vol reconstruction signed permutations distorted reversal errors discrete vol konstantinova levenshtein siemons reconstruction permutations distorted single transposition errors arxiv preprint arxiv leupold mitrana uniformly bounded duplication languages discrete appl vol levenshtein konstantinova konstantinov molodtsov reconstruction graph vertices discrete appl vol levenshtein siemons error graphs reconstruction elements graphs combin theory ser vol levenshtein efficient reconstruction sequences ieee trans inform theory vol jan liss daubert brunner kliche hammes leiherer wagner embedding permanent watermarks synthetic genes plos one vol macwilliams sloane theory codes marcus roth siegel introduction coding constrained systems oct unpublished lecture notes online available raviv schwartz yaakobi rank modulation codes dna storage proceedings ieee international symposium information theory aachen germany june shipman nivala macklis church crisprcas encoding digital movie genomes population living bacteria nature vol jul shomorony courtade tse fundamental limits genome assembly adversarial erasure model ieee transactions molecular biological communications vol wong wong foote organic data memory using dna approach commun acm vol yaakobi bruck siegel decoding cyclic codes read channels proceedings ieee international symposium information theory cambridge usa july yaakobi bruck uncertainty information retrieval associative memories proceedings ieee international symposium information theory cambridge usa jul zehavi wolf runlength codes ieee trans inform theory vol jan
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locally compact groups intermediate subfactors dec boutonnet arnaud brothier abstract study actions locally compact groups von neumann factors associated von neumann algebras setting totally disconnected groups provide sufficient conditions action ensuring inclusion irreducible every intermediate subfactor form closed subgroup partially generalizes result choda moreover show one hope use strategy groups introduction theory von neumann algebras construction associates action locally compact group von neumann algebra new von neumann algebra denoted encodes action extent construction goes back murray von neumann case state preserving actions countable groups abelian von neumann algebras called group construction thus algebras appear one basic examples von neumann algebras case actions discrete groups elementary properties algebras quite well understood instance abelian corresponding essentially free factor action ergodic factor action properly outer meaning corresponding automorphism inner moreover settings one completely describe intermediate subalgebras hybrid cases combining aspects cases considered recently case groups picture nice main difference fourier decomposition elements namely every element represented nevertheless setting actions abelian algebras satisfying results sauvageot showed section equivalence action essentially free still holds moreover case state preserving actions unimodular groups powerful tool available crossedsection equivalence relation relies observation appropriate corners described explicit equivalence relation refer details references article interested setting actions factors action locally compact group factor called strictly outer known case properly outer actions need strictly outer see assuming first author partially supported peps grant insmi second author partially supported nsf grant boutonnet arnaud brothier action strictly outer allows deduce conclusions instance see implies normalizer inside product see corollary particular derive discrete case strictly outer actions pairs completely characterize actions cocycle conjugacy refer theorem examples strictly outer actions also give new criterion providing examples proposition main goal prove intermediate subfactor theorem namely provide examples strictly outer actions groups satisfy following property definition say strictly outer action locally compact group factor satisfies intermediate subfactor property subfactor containing form closed subgroup case outer actions discrete groups choda proved intermediate subfactor property extra assumption intermediate subfactor range normal conditional expectation able show existence conditional expectation automatic discrete groups result known discrete groups unfortunately discuss hope adapt strategy general locally compact groups fact general situation analytically much harder handle precisely conditional expectations need exist general main result relies different approach inspired techniques restrict attention actions totally disconnected groups allow use fourier decomposition arguments statement theorem consider action totally disconnected locally compact group factor assume properly outer relative compact open subgroup whose action minimal satisfies intermediate subfactor property refer section definition minimal action definition notion relatively properly outer action theorem applies bernoulli shifts arbitrary compact open subgroup see section instance closed subgroup automorphisms locally finite tree acts transitively bernoulli shift action satisfies intermediate subfactor property refer section examples mentioned result simple generalization discrete case even tracial general intermediate subalgebra behave well hilbert theory perspective nevertheless still manage use hilbert techniques perform proof approach relies two ingredients first one averaging argument assumption used see remark technique also allows deal actions type iii factors need make assumption state large centralizer second ingredient extension notion support defined eymard language quantum groups support element spectrum dual action setting totally disconnected groups notion particularly well suited see instance proof proposition key fact use proof theorem element whose support contained closed subgroup actually belongs subalgebra result certainly known experts locally compact groups quantum groups able find explicit reference provide self contained proof section view theorem remark make following general conjecture conjecture strictly outer action factor type satisfies intermediate subfactor property remark conjecture holds full generality compact groups indeed compact group strictly outer action arbitrary von neumann algebra pair identified basic construction inclusion result showed generally integrable strictly outer actions locally compact groups vaes theorem moreover theorem every intermediate subfactor inclusion form form closed subgroup combining two facts indeed yells satisfies intermediate subfactor property going back general necessarily compact groups one prove conjecture even tracial arbitrary locally compact group mentioned remark allows produce actions type iii factors condition large centralizers need never fulfilled case actions factors type raises following question question one provide explicit example strictly outer action group type satisfies intermediate subfactor property discuss section possibility solve conjecture generalizing work namely completely characterize case arbitrary strictly outer actions existence normal conditional expectation operator valued weight inclusion closed subgroup theorem consider strictly outer action arbitrary von neumann algebra take closed subgroup following characterizations exists normal faithful operator valued weight modular functions coincide inclusion expectation open inside parts theorem parts follow easily modular theory namely conditions easily seen imply existence operator valued weight conditional expectation respectively main contribution show actually necessary note outer actions exist conditional expectations although open inside instance consider product trivial action assume second countable discrete open inside faithful normal state gives rise conditional expectation onto hand know whether characterization regarding existence normal faithful operator valued weights holds arbitrary actions necessarily strictly outer next show nevertheless strategy choda applied intermediate subfactors yielding following result also mention applications hecke pairs groups see corollary theorem consider arbitrary strictly outer action compact open subgroup subfactor containing form intermediate subgroup boutonnet arnaud brothier going back conjecture let finally mention another partial result intermediate subfactors globally invariant dual action namely factor intermediate subfactor form closed subgroup see instance chapter question consider strictly outer action locally compact quantum group factor one show von neumann subalgebra containing globally invariant dual action refer definitions mention remark also applies quantum setting namely vaes result theorem valid general quantum setting correspondence result theorem generalized tomatsu quantum setting acknowledgement work initiated trimester von neumann algebras held hausdorff institute spring parts work also due discussions conference memory uffe haagerup held copenhagen june oberwolfach meeting august warmly thank organizers events second author invited mathematical institute bordeaux september gratefully acknowledges kind hospitality received mathematically grateful sven raum interesting discussions pointing regarding notion support also indebted amaury freslon explaining part history quantum groups related work providing references theorem motivated discussions first author hiroshi ando cyril houdayer many thanks thank narutaka ozawa yoshimichi ueda helpful comments previous version preprint contents introduction preliminaries support applications actions totally disconnected groups existence conditional expectations operator valued weights references preliminaries general notations article letter refers locally group denotes haar measure modular function consider always meant respect measure integrating functions sometimes use notation instead dmg left regular representation denoted follow french convention according locally compact assumption also contains hausdorff axiom locally compact groups letter refers arbitrary von neumann algebra acts action denoted called generically action mean ultraweakly continuous homomorphism automorphism group given von neumann algebra represented hilbert space denotes commutant unit ball operator norm unitary group aut automorphism group denote set normal faithful states set normal faithful weights nfs weights short respectively weight consider left ideal defines norm denote hilbert space completion write inclusion map group actions von neumann algebras let give precise definition main object study refer chapter details facts definition fix action represent hilbert space crossedproduct von neumann algebra denoted von neumann algebra generated operators defined formulae notational simplicity often omit identify definition throughout article always assume standardly represented conjugation operator positive cone case abuse notations denote canonical implementation action see operator defined follows lies commutant denote compactly supported continuous functions endowed product involution given formulae algebra also two sided actions map defines embedding way viewed ultraweakly dense subgroups given subgroup one restrict action action case closed inside von neumann subalgebra generated unitaries isomorphic seen using induced representations see chapter tools one also compute commutant inside theorem theorem commutant inside von neumann algebra generated commutant subalgebra consisting functions equivalently boutonnet arnaud brothier remark using theorem also allows compute basic construction inclusion fact description gives isomorphism acts diagonally modular theory operator valued weights given von neumann algebra modular flow weight denoted aut centralizer subalgebra elements fixed flow denoted another weight denotes connes derivate defined section need following simple lemma lemma consider von neumann algebra two weights automorphism aut proof denote canonical basis put define formula note weight associated similar manner definition connes derivative lem kms condition see chapter implies lemma easily follows notions modular group connes derivative defined normal faithful conditional expectations need extended definition operator valued weights defined haagerup let fix notations recall known facts operator valued weights refer precise definitions proofs inclusion von neumann algebras denote set nfs operator valued weights given set one define composition weight nfs weight resulting weight normal faithful generally makes sense compose operator valued weights von neumann algebras one naturally define way modular automorphism associated weight leaves von neumann subalgebra globally invariant restriction depend choice restriction called modular flow denoted moreover another operator valued weight connes derivative element depend denoted called connes derivative time particular case well denotes jones basic construction inclusion see better expectation exists locally compact groups called dual jones projection associated see lemma operator valued weight finally consider action theorem exists unique operator valued weight call plancherel operator valued weight weight one defines dual weight see different construction outer actions von neumann algebras definition say group action von neumann algebra properly outer element acts inner automorphism strictly outer relative commutant trivial minimal faithful fixed point subalgebra irreducible subfactor observed strictly outer action properly outer converse true general lemma proposition action compact group strictly outer minimal let record basic example later use example consider diffuse factor faithful action group finite set corresponding bernoulli action strictly outer hence minimal setting totally disconnected groups following lemma useful proof much inspired proposition input galois correspondence theorem theorem lemma consider minimal action compact group open subgroup proof recall represented hilbert space subalgebra generated operators defined picture take since commutes acts minimally deduce note also write linear functional note functions left let prove functions generate von neumann algebra see sufficient check separate points take theorem exists particular distinct elements find linear functional separates arrived conclusion commutes particular orthogonal projection onto means leaves invariant boutonnet arnaud brothier fourier algebra dual action multipliers locally compact group denote fourier algebra introduced eymard definition set functions form denotes convolution product function note equality set algebra pointwise multiplication norm defined minimal value functions satisfy norm banach algebra set compactly supported continuous functions contained densely inside banach space isometric predual duality pairing given well defined formula abuse notations write mean notation somewhat consistent fact function namely using product define multipliers precisely gives rise normal completely bounded map defined formula generally action element one construct multiplier following way consider unitary operator denote associated automorphism identify way restriction definition notations fourier multiplier associated element normal completely bounded map defined formula practice multiplier characterized formula aug aug way one easily checks case two constructions multipliers coincide support applications section give generalities spectrum dual action defined instance chapter adopt point view eymard rather talk support believe transparent reader familiar actions discrete groups much quantum group language goal prove theorem regarding elements whose support contained subgroup certainly known experts able find explicit reference although used theorem convenience tried keep section definition first properties let fix arbitrary action von neumann algebra definition support element denoted supp set elements satisfying also describe support explicitly terms interactions copy inside measurable subset write orthogonal projection onto proposition take following equivalent locally compact groups supp open set proof consider open set take open subset open neighborhood fix function supported indeed formula clear form auh follows arbitrary linearity density note spans dense subset indeed see defined follows extend arbitrarily linear functional notations section since commutes deduce particular fix pick open neighborhood vanish take put particular suffices show since pick open set claim linearity density suffices check formula form auh auh auh formula obvious auh sides formula equal indeed since multiplier two terms scalar multiple suffices check vanishing side since deduce leads auh auh wanted proves claimed equality hence sequel sometimes appear one two descriptions better suited work freely switch two points view reach simplest arguments let record properties support lemma take following assertions true support closed subset belongs supp coincides support function supp proof consider net elements supp converges take since continuous large enough also hence desired continuous function observe function defined statement easily follows result obvious boutonnet arnaud brothier lemma consider open subsets satisfying relation supp particular open set supp proof first treat special case supp compact exists open set conjugating equation assume open neighborhood recall right action defined satisfies compactness fixed exists finite open cover define open neighborhood get gwg since smaller supremum projections obtain gwg therefore since gwg suppose open relatively compact exists compact neighborhood supp proof proves desired equality second part statement follows taking supp mentioning interesting consequences lemma let give essentially equivalent form involving multipliers lemma consider compact support take function equal neighborhood supp proof take open set equal supp proceeding proof proposition one checks supp supp since supp supp lemma equality imply moreover set satisfies condition equality holds place since open inside get equality follows corollary consider following assertions true adjoint supp supp sum supp supp supp product supp supp supp vanishing criterion supp locally compact groups proof results fact defined consider complementary supp supp exists particular product satisfies summary hence supp consider element supp supp take open neighborhoods identity supp supp particular supp supp closures supp supp intersect besides lemma implies altogether following equality shows supp xyp supp lemma implies theorem consider action locally compact group arbitrary von neumann algebra take closed subgroup element belongs subalgebra support contained proof first assume find open set lemma deduce supp hence supp conversely assume element support contained order show use theorem reduces task check commutes inside subalgebra consisting left functions given left open set whose boundary measure equality projections lemma get since also support moreover shows commutes claim set functions generates denote canonical projection measure coset space map normal isomorphism identification indicator function set identified indicator function moreover open subset set open boundary haar measure left check span functions ultraweakly dense classical fact borel measures locally compact spaces see proof references therein deduce claim commutes wanted use theorem deduce boutonnet arnaud brothier special case trivial subgroup theorem yields beurling theorem particular stress following corollary use several times corollary given element crossed product von neumann algebra support singleton exists aug proof supp supp particular theorem applications moving proof main theorems let mention classical results follow easily properties support first application concerns generalization theorem theorem start analyzing support behaves map given action put consider notations introduced section view algebra von neumann algebra associated action way makes sense talk support element inside lemma notations following fact hold support equal supp supp supp supp proof function belongs fourier algebra one easily checks formula particular find take functions disjoint supports get hence belong support thus deduce support contained diagonal easily implies belongs support supp conversely take supp show since continuous vanish open neighborhood multiplying local inverse necessary may assume actually equal pick function supported deduce since supp term hence take supp get deduce supp supp converse inclusion treated using local inverse alternatively use description support follows take supp supp take open set find open sets assumption know hence locally compact groups clearly follows proves supp prove following well known generalization theorem theorem initially deals case group algebras case trivial actions proof given rather involved elementary proof relying implicitly support already appears case general actions predual identified algebra general hence notion character apply anymore nevertheless still provide easy proof relying notion support corollary given action put denote map defined assume element form elements exists yug proof may assume lemma equality implies supp supp contained diagonal way happen supp singleton case supp result follows corollary immediate corollary deduce following result corollary given strictly outer action normalizer inside equal aug particular two strictly outer actions locally compact groups cocycle conjugate pairs isomorphic proof take use notation identity map hence one easily checks commutes words exists conclude corollary aug second part statement routine part always true even actions strictly outer see corollary part follows adapting proposition case actions general locally compact groups actions totally disconnected groups notations tools several advantages working totally disconnected groups fix locally compact group action von neumann algebra put firstly given compact open subgroup one define projection note net projections increases compact open subgroups decrease let mention two elementary properties projections lemma fix weight denote corresponding dual compact open subgroup kapk proof immediate consequence boutonnet arnaud brothier lemma given compact open subgroup map apk onto isomorphism von neumann algebras proof particular case proposition give complete proof simpler case convenience reader since commutes clear map normal computations given lemma see moreover injective check onto need prove dense image aug apk conditional expectation hence aug belongs range map proving lemma secondly open subgroup action always exists faithful normal conditional expectation case compact open one sees multiplier associated function gives desired expectation general case open subgroups necessarily belong still positive definite one use theorem construct associated multiplier alternative way construct expectation considering modular flows point view becomes obvious preserves plancherel operator valued weight totally disconnected one checks support element described follows supp compact open subgroups notation given compact open subgroup group set denote lift set representatives inside means lift exists unique element lift lemma consider compact open subgroup given element compact support map right compactly supported moreover proof fix finite set support contained lift function equal neighborhood support lemma moreover decomposed one easily checks corresponding multiplier satisfies leave rest proof reader although use fact let mention general element kdecomposition still makes sense case sum appears infinite converges bures topology associated inclusion expectation refer section original book definition bures topology locally compact groups strictly outer actions following proposition combines fourier coefficient approach used setting discrete groups lemma actions compact groups proposition consider properly outer action totally disconnected locally compact group action strictly outer admits compact open subgroup acts minimally case unitary normalizing form aug proof assume strictly outer one easily checks definitions properly outer moreover restriction compact subgroup strictly outer thus minimal proposition conversely assume properly outer admits compact open subgroup acts minimally note subgroup acts minimally well hence strictly outer way put take element proposition need show support singleton support corollary take supp compact open subgroups put since supp elements minimality relation tells multiple unitary element implies unitaries proportional particular proportional conclude contained hence support equal implying moreover satisfies since properly outer gives desired statement normalizer follows corollary although could checked directly similar computations support intermediate subfactors turn question determining intermediate subfactors order establish main result theorem need able compute relative commutants form small compact open subgroups forces strengthen assumptions action definition given subgroup say action properly outer relative following holds elements exists elements note action properly outer properly outer relative trivial subgroup provide examples relatively properly outer actions next section lemma consider action totally disconnected group properly outer relative compact open subgroup put proof take show supp fix supp since net converges strongly find compact open subgroup boutonnet arnaud brothier define since open finite index inside hence intersection defining fact finite conclude open subgroup moreover contained particular since normal inside projection commutes lemma sum implies exists since qgkg two elements commute follows hence lemma exists unique apl uniqueness see satisfies conclude hence supp theorem proposition consider action totally disconnected group properly outer relative compact open subgroup whose action minimal open subgroup proof fix lemma applied see since acts minimally result follows lemma order prove theorem use convex combination argument following lemma needed lemma lemma consider von neumann algebra weight closed convex subset bounded operator norm proof theorem fix intermediate subfactor set take show support contained conclude theorem denote normal faithful trace denote corresponding dual weight step compact open subgroups finite trace projection exists qpk qpk qpk fix qpk qpk put qpk qpk lemma exists unique apk note qqq element satisfies apk lemma implies ypk denote ultraweak closure convex hull conv qqk locally compact groups proceed proof theorem show cpk bounded except slightly weaker assumptions since contained centralizer clear triangle inequality kxpk kypk arbitrary take net converges ultrastrongly converges ultraweakly since lower ultraweakly see theorem iii get kxpk xpk lim inf kypk note moreover kypk since qpk thus ultraweakly closed convex set cpk bounded operator norm apply lemma particular find zpk unique element cpk minimal let check element satisfies zpk first since qpk qpk bounded normal linear functional constant hence constant follows qpk zpk qpk ypk ypk indeed find zpk note also cpk globally invariant affine action qqk given since centralizes action fixes zpk equivalently zpk qqk moreover zpk belongs proposition qqk qlk zpk qlkpk since projection central minimal inside exists scalar zpk obtain qpk unfortunately priori control small could large operator norm get around issue would like identify polar parts need extra commutations properties apply convex combination argument second time arguing one find element ultraweak closure conv qqk zpk unique element zpk minimal enjoys following properties zpk qqk zpk zpk particular zpk qqk uniqueness minimizer inside zpk hence proposition gives zpk qpk scalar particular facts give zpk qpk follows positive hence equality qpk qpk shows commutes write polar decomposition partial isometry note uqpk partial isometry since qpk commutes recall uqpk combining get qpk zpk uqpk hence qpk uqpk proportional partial isometries coincide proves step qpk however needs unique nothing ultraweakly closed convex hull qqk boutonnet arnaud brothier step compact open subgroups exists fix subgroup since acts minimally expectation inside lemma implies trace still denote increasing net projections finite trace converges exists step deduce exists denote ultraweak limit net taking corresponding limit gives ypk desired step support contained take supp fix compact open subgroup net projections converges strongly identity small enough compact open subgroups take normal open subgroup commutes lemma write small enough compact open subgroups normal inside since sum find applying step ghl find element ughl note exists net compact open subgroups normal form neighborhood basis comes fact open subgroup finite index inside open normal subgroup contained compactness weak operator topology exist subnets hli zli converge elements respectively taking ultraweak limits get ugh lim ughli pli lim zli pli net projections converges ultrastrongly identity hence conclude arbitrarily small closed conclude finishes proof step theorem follows theorem examples actions proposition let totally disconnected group compact open subgroup let call subgroup eventually malnormal take trace following satisfy assumptions theorem strictly outer satisfy intermediate subfactor property bernoulli action obtained shifting indices free bernoulli action proof let first check separately minimality condition case since eventually malnormal inside acts faithfully put hence since compact open commensurated hence acts locally compact groups finite orbits let denote orbits example shows fixed point algebra irreducible subfactor since contains irreducible inside acts minimally put since normal inside acts faithfully free product situation clear copy located label irreducible inside moreover algebra contained acts faithfully check relatively properly outer condition simultaneously situations take decompose product copies position tensor product free product remaining copies tensor situation free situation one easily checks nets converge weakly one lim avn trace assuming exists take net unitaries converges weakly set get lim aun lim hence desired generally one easily check relative outerness condition theorem action large commutant thanks following fact lemma consider action closed subgroup whose action minimal assume centralizer aut satisfies exists action properly outer relative elements acting trivially elements proof part trivial conversely assume precisely set elements act trivially take exists satisfying since acts minimally assume unitary automorphism moreover using commutes see previous equation reads particular find hence assumption leads hence fixes pointwise thus assumption note condition centralizer lemma fulfilled soon admits subgroup preserves state moreover trace invariant automorphism hence second condition needs verified deduce following result spirit vaes examples theorem boutonnet arnaud brothier corollary fix totally disconnected group compact open subgroup consider faithful action diagonal action satisfies assumptions theorem proof faithful theorem implies strictly outer hence minimal moreover centralizer diagonal action contains shift automorphisms obtained permuting indices hence condition appearing lemma satisfied see comment lemma thus result follows lemma remark note bernoulli shift action lemma sometimes special case diagonal action corollary instance happens hyperfinite factor however clear case prime factor hence even strict outerness actions follow theorem moving next section let briefly explain adapt argument cover actions type iii factors remark let totally disconnected group let eventually malnormal compact open subgroup see lemma let arbitrary diffuse factor admitting faithful normal state large centralizer meaning bernoulli shift satisfies intermediate subfactor property let briefly explain denote centralizer irreducible inside invariant one show compact open subgroups finite intersection conjugates fixed point algebra satisfies done following proof lemma noting action properly outer relative therefore one use averaging argument proof theorem instead average elements small groups note moreover case since state weight one need bother projection appearing step proof theorem fact remark also applies free bernoulli actions weaker assumptions elaborate existence conditional expectations operator valued weights section discuss various results existence conditional expectations operator valued weights connection izumi longo popa paper let start discussion investigating existence conditional valued weights pairs form associated closed subgroups locally compact groups proof theorem let start two lemmas rely notion support lemma consider two actions diagonal action action strictly outer proof embed diagonally identified subalgebra note particular element viewed element support contained diagonal subgroup element support contained trivial group corollary get wanted lemma consider action arbitrary von neumann algebra put take weight denote corresponding dual weight support supp positive haar measure inside proof use description dual weight relying hilbert algebra approach according approach given weight exists left hilbert algebra satisfying following properties hilbert completion isomorphic left von neumann algebra identified dual weight corresponds canonical weight associated hilbert algebra bounded vector function defined right bounded vector corresponding operator given denotes operator associated right bounded vector take facts chapter exists left bounded vector operator extending left multiplication claim support function contained support fact equality holds clearly need inclusion deduce lemma take function support take open neighborhood identity element show bounded vector check suitable choice quantity since function support exists positive measure since set bounded dense inside may find two bounded vectors sufficiently close set positive haar measure follow use uniqueness standard form identify canonically boutonnet arnaud brothier particular function defined satisfies put denote function function representation defined since continuous representation continuous function get supported small enough neighborhood definition get indeed exists bounded vector function supported proof theorem prove two facts separately first assume modular functions coincide theorem weight dual weight satisfy therefore exists nfs operator valued weight theorem conversely assume theorem remark deduce exists operator valued weight isomorphic basic construction intermediate goal deduce exists nfs weight unfortunately know priori get around issue exploit fact action strictly outer use modular theory let consider following operator valued weights plancherel operator valued weight plancherel operator valued weight tensor product operator valued weight associated weight identity map see theorem fix lemma hence connes cocycle sense definition takes values flow construction weight weight locally compact groups simply dual weight associated hence since abelian conclude one parameter subgroup unitaries exists nfs weight claim weight denote generically letter actions fix show connes derivative equal lemma hence becomes show right hand side equal computing terms equality denoted resp operator valued weight resp take weight definition connes derivative operator valued weights third equality follows lemma hence becomes theorem easily seen imply altogether rewritten see right hand side equal proving claim recall subalgebra right functions inside denote quotient map formula defines measure borel measure faithful claim measure finite every compact set since faithful exists borel set take compactly supported function function defined follows continuous dmg boutonnet arnaud brothier theorem key equation dmg dmg dmg key equation tells first continuous function exists open set key equation also tells particular since compact set covered finitely many translates claim follows corollary deduce existence modular functions must coincide proves mentioned earlier paper open inside indicator function continuous positive definite associated multiplier see theorem gives desired conditional expectation onto conversely assume open inside modular functions coincide part ensures nfs operator valued weight onto particular conditional expectation assume modular functions coincide fix nfs weight denote associated dual weights saw proof theorem implies exists operator valued weight inclusion irreducible suffices show unbounded theorem fix element note since open inside measure inside particular since supp lemma implies contrast hence expression defines normal positive linear functional hence bounded proof complete conditional expectations exist applications hecke pairs proving theorem let mention argument choda applies beyond setting discrete groups theorem theorem consider strictly outer action locally compact group take von neumann subalgebra contains range faithful normal conditional expectation form open subgroup proof usual consider closed subgroup defined let show converse inclusion also holds since contains action strictly outer deduce scalar multiple scalar multiple positive measure borel set inside product contains neighborhood identity locally compact groups means hence scalar question must obtain following computation aug aug since normal deduce linearity density thus equality fact open follows theorem mention lemma provides existence conditional expectations follows main technical result lemma consider compact open subgroup locally compact group let strictly outer action put intermediate von neumann algebra range normal faithful conditional expectation proof show satisfy assumptions corollary observe inclusion irreducible since action strictly outer mentioned earlier paper since open inside exists conditional expectation onto remark basic construction isomorphic acts diagonally embedding given aug note discrete since open lemma hence relative commutant isomorphic von neumann algebra maps represent faithfully obvious way picture jones projection orthogonal projection onto dirac mass coset consider dual operator valued weight lemma implies characteristic function kgk since compact open subgroup index finite thus operator valued given weight theorem weight states restriction modular flow associated equal modular flow restriction therefore since commutative hence pair indeed satisfies assumptions corollary implies lemma proof theorem follows immediately combining theorem lemma let derive applications hecke pairs groups definition hecke pair pair groups subgroup commensurated almost normal sense finite index refer details facts typical example hecke pair arises subgroup automorphism group locally finite connected graph subgroup elements stabilize given vertex fact example somewhat generic see theorem new hecke pair associated schlichting completion hecke pair totally disconnected group compact open subgroup precise construction goes follows view subgroup permutation group endow topology pointwise converge viewed resp closure resp inside discrete space define boutonnet arnaud brothier idea using schlichting completion study operator algebras hecke pairs goes back tzanev key observation proposition say action action hecke pair extends continuously let define von neumann action schlichting completion algebra associated action construction generalizes associated hecke pairs correspond trivial action originate also refer general treatment actions hecke pairs defined von neumann algebraic version section let space continuous functions hgk induced functions finitely supported note fixed define multiplication involution follows lift system representatives space endowed operations unital contains copy fixed point von neumann algebra via map characteristic function assume standardly represented hilbert space denote canonical implementation action see let hilbert space viewed discrete space endowed counting measure consider subspace hgh similar proof proposition gives following lemma map defined bounded representation call standard representation hecke algebra denote bicommutant call von neumann algebra next proposition relates hecke pairs group schlichting completion generalizes lemma mentioned idea goes back tzanev proposition consider action hecke pair von neumann algebra schlichting completion pairs denote phe isomorphic phe averaging projection associated compact open subgroup defined locally compact groups naturally isomorphic since action proof note coset spaces pairs extends continuously action equal replacing isomorphic moreover schlichting completion necessary may assume compact open inside consider isometry given formula put observe map representation equivalent acting jph conjugation operator standard representation particular faithful representation consider map defined formula observe range space functions kgl range precisely corner therefore isomorphism von neumann algebras onto lemma one easily checks observe intermediate closed group natural identification subalgebra functions supported identification extends respective von neumann algebras see little effort use proposition follows definition schlichting completion isomorphic closure inside schlichting completions respect subgroup mentioned section isomorphic weak closure algebraic crossedproduct inside proposition later fact imply exists injective morphism von neumann algebras sends weak closure inside theorem together proposition implies following result hecke pairs corollary consider action hecke pair von neumann algebra intermediate von neumann algebra exists intermediate group isomorphic identified subalgebra recently shown hecke pairs appear subfactor theory consider finite index subfactor symmetric enveloping inclusion see type iii finite depth setting cases exists hecke pair actions isomorphic see theorem hence last corollary gives information lattice intermediate subfactor symmetric enveloping inclusion boutonnet arnaud brothier references invariant proper metrics coset spaces topology appl approximation properties coset spaces operator algebras proceedings operator theory timisoara bekka harpe valette kazhdan property new mathematical monographs cambridge university press cambridge bisch nicoara popa continuous families hyperfinite subfactors standard invariant internat math bost connes hecke algebras type iii factors phase transitions spontaneous symmetry breaking number theory selecta math boutonnet houdayer vaes strong solidity free factors preprint brothier fixed point planar algebras math phys bures abelian subalgebras von neumann algebras memoirs american mathematical society american mathematical society providence choda galois correspondence von neumann algebra tohoku math journ cameron smith intermediate subalgebras bimodules crossed products general von neumann algebras appear internat combes delaroche groupe modulaire une conditionnelle dans une von neumann bull connes une classification des facteurs type iii ann sci norm sup eymard fourier groupe localement compact bull haagerup standard form von neumann algebra math scand haagerup dual weight von neumann algebras math scand haagerup dual weight von neumann algebras math scand haagerup operator valued weights von neumann algebras funct anal haagerup operator valued weights von neumann algebras funct anal houdayer isono unique prime factorization bicentralizer problem class type iii factors adv math houdayer raum locally compact groups acting trees type conjecture nonamenable von neumann algebras preprint izumi longo popa galois correspondence compact groups automorphisms von neumann algebras generalization kac algebras funct anal jones index subfactors invent math kosaki extension jones theory index arbitrary factors funct anal kyed vaes numbers locally compact groups cross section equivalence relations trans amer math soc longo rehren nets subfactors rev math phys murray von neumann rings operators ann math nakagami takesaki duality crossed products von neumann algebras lecture notes mathematics springer berlin palma crossed products hecke pairs mem math soc published popa symmetric enveloping algebras amenability afd properties subfactors math res lett sauvageot sur type produit dune von neumann par groupe localement compact bull soc math france takesaki duality crossed products structure von neumann algebras type iii acta math takesaki theory operator algebras encyclopaedia mathematical sciences operator algebras geometry berlin tomatsu galois correspondence compact quantum group actions reine angew math tzanev hecke amenability operator theory locally compact groups vaes unitary implementation locally compact quantum group action funct anal vaes strictly outer actions groups quantum groups reine angew math boutonnet institut bordeaux bordeaux cours talence cedex france address arnaud brothier department mathematics university rome tor vergata via della ricerca scientifica roma italy address https
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mar dimension quotients fox subgroups limits functors roman mikhailov inder bir passi abstract paper presents description fourth dimension quotient using theory limits functors category free presentations given group category abelian groups functorial description quotient third fox subgroup given consequence identification involving isolator third fox subgroup obtained shown limit category free representations third fox quotient represents composite two derived quadratic functors introduction given group let integral group ring augmentation ideal dimension quotients defined subquotients nth dimension subgroup nth term lower central series evaluation dimension quotients challenging problem theory group rings subject investigation since quotients trivial free groups case groups odd groups first shown rips general subsequently structure fourth dimension quotients described tahara narain gupta instances dimension quotients dimensions known however precise structure still remains open problem another challenging problem concerning normal subgroups determined twosided ideals group rings fox subgroup problem page problem asks identification normal subgroup rfn free group normal subgroup solution problem given yunus narain gupta chapter iii turns identification given isolator subgroup instance identification essentially amounts one coefficients group ring field rational numbers rather ring integers thus raises question precise determination involved torsion roman mikhailov inder bir passi aim paper present entirely different approach problems via derived functors limits functors category free presentations approach motivated connections theory limits functors homology groups derived functors sense doldpuppe cyclic homology group rings instance even dimensional integral homology groups turn limits lim nth tensor power relation module via group action conjugation diagonal certain derived functors sense doldpuppe turn limits gab lim gab lim first derived functors symmetric square cube functor respectively gab description derived functors spn gab limits given application theory limits cyclic homology given shown cyclic homology algebras defined limits category free presentations certain simply defined functors work direction consider category groups approach brings fresh context study dimension subgroups fox subgroups describe main results present work let free group normal subgroup natural short exact sequence observe first two terms viewed functors category free presentations category abelian groups limit functor lim known left exact sends monomorphisms monomorphisms however right exact therefore short exact sequences presentations induce long exact sequences involving higher derived lim instance short sequence induces following long exact sequence lim lim lim main result dimension quotients describes cokernel left monomorphism exact sequence precise theorem natural short exact sequence lim lim dimension quotients fox subgroups limits functors thus present description fourth dimension quotient purely functorial terms involving group ring surprising result next give functorial description quotient together complete identification involving isolator third fox subgroup theorem let free group normal subgroup natural isomorphism subgroup generated elements finally give description lim may noted sition derived functors appears theorem natural isomorphism lim gab particular monomorphism gab paper organized follows section recall basic properties results concerning limits functors derived functors needed sequel section generalized dimension quotients devoted proving several lemmas subgroups determined ideals free group rings theorems proved sections respectively refer reader details limits identification normal subgroups determined idelas group rings preliminaries elementary properties limits begin recalling basic facts limits functors let arbitrary categories let category abelian groups recall chapter limit lim functor object together universal collection morphisms lim morphism roman mikhailov inder bir passi universality means object collection morphisms morphism exists unique morphism lim limit functor always exist however exists unique unique isomorphism commutes morphisms category said strongly connected objects homset homc standard argument implies strongly connected category functor object natural embedding lim precisely lim identified follows lim morphisms category pairwise coproducts category objects category exists coproduct pairwise coproducts objects functor natural map functor called monoadditive injective pair objects following lemma due ivanov gives way define limit functor equalizer lemma used proofs main statements paper lemma let strongly connected category pairwise coproducts functor exact sequences lim lim map lim given lim structure morphism lim proof since strongly connected map lim phism image equal subgroup elements lim identify lim subgroup dimension quotients fox subgroups limits functors claim ker lim consider two arbitrary arrows commutative diagram assume ker ker hence take idc get follows thus lim let lim take obtain ker since monomorphism image contains kernel kernel equal lim well corollary let strongly connected category pairwise coproducts let functor functor monoadditive lim main concern category free presentations given group objects surjective homomorphisms free group morphisms homomorphisms category coproducts given category strongly connected category pairwise coproducts therefore lemma applies functors thus particular limits lim identified corresponding equalizers representation functor called grepresentation natural inclusion lim isomorphism depends limit lim defined right adjoint diagonal functor left exact right exact short exact sequence representations induces long exact sequence derived functors lim lim lim lim lim lim lim roman mikhailov inder bir passi see section details paper use one property higher limits namely triviality higher limits lim defined ith cohomology category coefficients representation since category pairwise products contractible see example lemma hence lim lemma let representation lives exact sequence also proof let first consider case case natural surjection assertion case follows following diagram lim lim lim lim since quotient lim zero dually quotient since epimorphic image assertion case follows following commutative diagram lim lim lim let assume map monomorphism epimorphism since lim obtain following diagram lim result follows lim lim dimension quotients fox subgroups limits functors quadratic functors use following basic quadratic functors tensor square symmetric square exterior square antisymmetric recall abelian group square abelian group natural exact sequence left hand map given derived functors sense defined follows abelian group endofunctor category abelian groups derived functor given projective resolution transform inverse moore normalization functor simplicial abelian groups chain complexes first derived functor used many places paper natural quotient tor diagonal elements first derived functor subfunctor tor generated diagonal elements natural short exact sequence tor abstract value abelian group easily computed following data tor abelian groups derived functors naturally appear homology spaces example abelian natural short exact sequences split see tor exterior cube refer thesis jean structure derived functors higher symmetric powers also need natural exact sequences like following tor tor roman mikhailov inder bir passi general sequences note map obtained deriving sequences respectively following sequence used proofs main results several times free abelian group subgroup natural exact sequence image element proof see theorem section proof directly follows result saying complex represents element derived category abelian groups generalized dimension quotients analogy dimension subgroups normal subgroup free group normal subgroups ideal called generalized dimension subgroups set example description generalized dimension subgroup connection derived functor use later following shown natural isomorphism gab see examples type need identification certain generalized dimension subgroups recall normal subgroup free group basis ideal viewed left resp right free basis theorem lemma free group finite rank ordered basis normal subgroup generated xemm integers satisfying generalized dimension subgroup frf frf generated modulo commutators lcm dimension quotients fox subgroups limits functors proof easy see elements lcm lie frf let hxei frf observe frf fsf since basis modulo aijk fsf aijk modulo aijk left differentiating sense free differential calculus see respect aiji aijk right differentiating respect gives hence implies aiji aijk since second sum involve conclude aiji aijk implies aiji therefore similarly implies thus reduces aijk fsf roman mikhailov inder bir passi yields eqs imply lcm hence element lies subgroup generated commutators lcm claimed corollary free group normal subgroup frf proof easy see frf observe reverse inclusion suffices consider case finitely generated lemma applies free group subgroup lemma lemma proof let modulo aijk aijk modulo every aijk since follows aijk implies every aijk follows thus see dimension quotients fox subgroups limits functors hence hence reverse inclusion easily seen hold lemma free group normal subgroup modulo proof first equality immediate consequence canonical induced easy check conversely let analyze may clearly assume finitely generated lemma therefore lemma modulo bijk bijk collecting terms therefore differentiating respect yields fkek hence fkek consequently observe every free group normal subgroup roman mikhailov inder bir passi see equality consider map exterior squares fab induced inclusion fab fab denotes abelianization map monomorphism since induced monomorphism free abelian groups needed equality follows following identifications fab thus particular view known structure chapter section tor abelian groups preceding lemma immediately yields following result lemma natural epimorphism tor gab proof theorem first observe cokernel natural map naturally identified fourth dimension quotient see example obvious short exact sequence right hand map fact surjective let quotient relation module natural epimorphism find element thus follows every element dimension quotient preimage dimension quotients fox subgroups limits functors consider natural diagram exact rows columns observe lim follows monoadditivity easily checked top two horizontal exact sequences imply natural isomorphisms lim lim lim lim lim therefore left hand vertical exact sequence diagram implies following long exact sequence lim lim lim next consider following diagram exact rows columns observe right hand horizontal map general surjective generalized dimension quotient first derived functor symmetric square gab identified roman mikhailov inder bir passi see lemmas dimension quotient therefore lemma using upper horizontal exact sequence diagram conclude lim looking left hand vertical epimorphism diagram conclude natural map lim lim epimorphism map zero lim asserted short exact sequence follows proof complete proof theorem let set since subgroup observe view following natural diagram exact rows columns let set dimension quotients fox subgroups limits functors clearly observe invoking exact sequence following sequence natural isomorphism observe omit last term recall see hence monomorphisms since coker natural monomorphism free abelian see sequence implies following diagram thus obtain following identification simple way pick representatives also follows sequence subgroup generated elements one easily check pair assert horizontal arrow let call diagram epimorphism let roman mikhailov inder bir passi consider element clearly since working modulo hence map epimorphism next show left hand vertical map diagram namely monomorphism clearly next use following identification see denotes augmentation ideal group ring recall quote preceding lemma corollary implies right hand vertical arrow monomorphism hence proof part theorem complete part follows part together lifting elements described proof theorem theorem lim lim dimension quotients fox subgroups limits functors order study right hand limit consider following diagram exact rows columns gab gab gab gab middle vertical sequence sequence fab gab fab fab gab gab right hand vertical map monomorphism since let set following short exact sequence gab observe monomorphism gab implies induced map tor gab gab tor also monomorphism therefore induced maps gab gab also monomorphisms since subfunctors since functor tor sequence implies induced map gab roman mikhailov inder bir passi monomorphism sequence implies inclusion tor gab tor gab gab assert lim tor tor gab gab natural consequence vanishing lim phism gab lim theorem follow remains establish vanishing result observe exact sequence tor gab gab tor gab tor gab since gab fab natural inclusion tor gab tor gab gab fab short exact sequence fab fab fab gab fab tensoring gab gives inclusion tor gab gab fab tor gab gab fab fab since representation tor gab gab fab fab monoadditive one easily check representation representation fab follows lim tor gab consequently tor gab gab lim tor gab similar argument shows lim tor hence lim tor gab lim tor since lim tor gab gab conclude remark theorem implies natural exact sequence quotient dimension quotients fox subgroups limits functors representation hence lim gab remark middle vertical sequence implies gab group well since quotient subgroup conclude gab theorem acknowledgement research supported russian science foundation grant authors thankful research institute allahabad warm hospitality provided visit february references joan birman braids links mapping class groups annales mathematics studies vol princeton univ press breen functorial homology abelian groups pure appl alg cohn generalizaion theorem mangnus proc london math dold puppe homologie nicht additiver funktoren anwendugen annales institut fourier tome emmanouil mikhailov limit approach group homology algebra fox free differential calculus derivation free group ring ann math karl gruenberg cohomological topics group theory lnm vol narain gupta free group rings contemporary mathematics vol american mathematical society hartl mikhailov passi dimension quotients indian math new ser spec centenary jean foncteurs derives lalgebre symetrique application calcul certains groupes dhomologie fonctorielle des espaces doctoral thesis university paris available http karan kumar vermani intersection theorems subgroups determined certain ideals integral group rings algebra colloq computing homology koszul complexes trans amer math losey dimension subgroups trans amer math mac lane categories working mathematicians secon edition springer magnus beziehngen zwischen gruppen einem speziellen ring math magnus beziehungen zwischen kommutatoren reine angew roman mikhailov inder bir passi lower central dimension series groups lnm vol springer mikhailov passi generalized dimension subgroups derived functors pure appl algebra mikhailov passi free group rings derived functors roman mikhailov inder bir passi rips fourth integer dimension subgroup israel math ivanov mikhailov higher limit approach homology theories pure appl algebra passi dimension subgroups algebra passi group rings augmentation ideals lnm vol tahara fourth dimension subgroups polynomial maps algebra tahara structure fourth dimension subgroup japan math quillen cyclic cohomology algebra extensions yunus problem fox soviet math witt treue darstellung liescher ringe reine angew dimension quotients fox subgroups limits functors roman mikhailov petersburg department steklov mathematical institute chebyshev laboratory petersburg state university line saint petersburg russia school mathematics tata institute fundamental research mumbai india email romanvm inder bir passi centre advanced study mathematics panjab university sector chandigarh india indian institute science education research mohali punjab india email ibspassi
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lie bracket approximation approach distributed optimization extensions limitations feb simon bahman christian institute systems theory automatic control university stuttgart germany michalowsky department mathematics statistics queen university canada bahman relax assumptions present work want extend results show approach applied large class optimization problems mild assumptions communication network optimization problems linear constraints objective functions form sum separable functions considered present work enhance methodology general convex optimization problems main idea approach use lie bracket approximation techniques find distributed approximations dynamics previous works certain lie bracket approximations used show whole class applicable abstract consider problem solving smooth convex optimization problem equality inequality constraints distributed fashion assuming group agents available capable communicating communication network described directed graph derive distributed agent dynamics ensure convergence neighborhood optimal solution optimization problem following ideas introduced previous work combine dynamics lie bracket approximation techniques methodology previously limited linear constraints objective functions given sum strictly convex separable functions extend result show applies general class optimization problems mild assumptions communication topology introduction preliminaries last decades distributed optimization active area research high practical relevance see applications type problems goal cooperatively solve optimization problem using group agents communicating network algorithms distributed optimization constitute majority existing literature focus algorithms regained interest last decades algorithms require strong assumptions either structure optimization problem communication network recently novel approach distributed optimization based lie bracket approximations proposed potential notation denote set real vectors write set continuously differentiable functions gradient function respect argument denoted often omit subscript clear context denote entry matrix aij sometimes denote aij use denote vector ith entry equal entries equal specify dimension expect clear context vector let diag denote diagonal matrix whose diagonal entries entries given two continuously differentiable vector fields lie article extended version additionally including proofs results discussion choice vector fields remark overview table bracket evaluated defined set problem unique solution goal design optimization algorithms converge arbitrarily small neighborhood unique global optimizer implemented distributed fashion precisely assume group agents available capable interchanging information communication network described directed graph graph laplacian gij set nodes edge set nodes present setup node represents agent edges define existing communication links agents edge node node agent access information provided agent say algorithm distributed agent uses information well provided agents let rneq rnineq neq neq nineq nineq denote lagrangian associated slight abuse notation sometimes also write vector finite set denote ordered stacked vector example basics graph theory recall basic notions graph theory refer reader standard references information directed graph simply digraph ordered pair set nodes set edges edge node setup edges encode agents agent access means node receives information node say node outneighbor node edge node node adjacency matrix aij associated defined aij otherwise stacked vectors respectively rneq rnineq associated lagrange multipliers sequel assume state ith agent comprises well dual variables associated constraints given exist without loss generality put following assumption indexing constraints also define matrix dij associated aik dij otherwise assumption ieq iineq exists finally call gij laplacian directed path sequence nodes connected edges write path node node vir say path simple sequence contains node ieq iineq following dynamics adapted known converge saddle point lagrangian see proof thus providing solution consider following convex optimization problem nutshell assumption newguarantees ith constraints functions lagrangian newa saddle point optimizer say point saddle point nineq rneq problem setup min rneq rnineq neq nineq strictly convex functions ieq affine iineq convex assume feasible diag diag notation define index set however general distributed aforementioned sense since side composed admissible vector fields vector fields computed locally nodes example vector field admissible recently novel approach distributed optimization proposed employs lie bracket approximation techniques derive distributed approximations idea write side means lie brackets admissible vector fields achieved rewrite form neq neq neq nineq nineq associating components complete state ith agent meaning part state agent hence state vector ith agent given based define set indices associated agent jth index note set lie brackets admissible vector fields employ lie bracket approximation techniques derive distributed approximations precisely find family functions parametrized trajectories lie brackets admissible vector fields following first want discuss kind vector fields written terms lie brackets admissible vector fields define observe admissible exist next newresult consider lie brackets admissible vector fields form uniformly converge increases given initialized equally general algorithm compute suitable functions presented present modified version thereof tailored problem hand present paper discuss second step design functions focus rewriting side terms lie brackets admissible vector fields lemma consider graph nodes let denote simple path vir main results consider dynamics observe side sum vector fields form neq nineq neq nineq side lie bracket admissible vector fields proof given section lemma vector field takes form side written terms lie bracket admissible vector fields worth mentioning classify whole set vector fields written lie bracket admissible vector fields since limited single path next discuss special case particular importance application hand complete state ith unit vector vector fields might either admissible depending communication graph well problem structure following wish discuss write vector fields form means lie brackets admissible vector fields purpose proposition consider graph nodes let denote simple path vir let set indices suppose hence see proof bounded vector field leads another unbounded vector field hence half vector fields bounded particular choose functions follows guarantee holds cos even odd sin even cos odd example consider graph shown fig assume sake simplicity write vector fields form well sums thereof terms lie brackets admissible vector fields long admit analytic expression antiderivatives fact cos however choice lead functions globally continuous hence interval need choose appropriately sufficiently small choice functions even bounded functions odd unbounded next discuss make use previous results rewrite general vector fields lemma enables write products functions variables nodes lie path lie brackets admissible vector fields directly use result rewrite functions fulfill property make clearer consider following example also bounded since disjunct however proof suppose exists set bounded vector fields holds bounded function remark holds true assume structure fact structure required variables except cancel lemma suppose assumptions proposition fulfilled exists set bounded vector fields holds hence leading however view given admissible vector fields form often desired admissible vector fields certain properties boundedness order simplify calculation approximating inputs improve transient behavior however see next possible render vector fields bounded thus strictly monotone contradicts boundedness assumption thus concluding proof proof given section view constraint ensures terms depending cancel equation particular interest since vector fields often take form side according exists whole class vector fields equivalently functions holds particularly simple choice utilized previous works take side lie bracket admissible vector fields note bounded away zero exists constant observe following proposition choosing figure graph considered example ith agent associated state using finally managed rewrite terms admissible vector fields remark instead realizing multiplication means lie brackets another way augment agent state estimates respective state agent precisely example augment state agent estimates respectively let however directly use rewrite nonadmissible vector field form next result wish overcome limitation show write sums vector fields form side terms lie brackets admissible vector fields also products thereof sufficiently large hence resulting vector fields complete augmented system written terms lie brackets admissible vector fields using proposition however application hand alters dynamics necessitates stability analysis augmented system proposition let define dzi hence suitable assumptions communication graph allows write vector fields whose components sums products arbitrary functions terms lie brackets admissible vector fields observation gives rise next lemma lemma consider strongly connected graph nodes let analytic function vector field written possibly infinite sum lie brackets admissible vector fields proof given section vector fields defined general thus side lie bracket admissible vector fields however observing vector fields take form side make use proposition write side lie bracket admissible vector fields long exists path ith node node associated state illustrate means example proof since analytic series expansion written possibly infinite sum monomials components using proposition monomials written terms lie brackets vector fields form strong connectivity vector fields written terms lie brackets admissible vector fields thus concluding proof example continued reconsider example suppose want rewrite vector field following proposition let result might theoretical nature application hand nevertheless shows proposed approach principle applies large class problems vector field admissible rewritable gij pij ieq gik aki pik aki iineq nck gij jck pij jck ieq assumption holds pki aki iineq nck assumption holds pkj jck aki table overview vector fields appearing distributed optimization via lie brackets functions sequel apply results last section problem hand rewrite dynamics means lie brackets admissible vector fields sake simpler notation assume following neq ieq nineq iineq assume agent associated equality inequality constraint ieq iineq always achieved augmenting optimization problem constraints alter feasible set emphasize necessary methodology apply illustrate example section write dynamics equivalently strictly convex functions iineq jck nci nck convex observe nck infinite monomials includes analytic functions jck obtain particularly important special case objective function constraints sum separable functions hence also case linear constraints considered covered assumption vector fields appearing summed table depending communication graph well structure constraints objective function vector fields either admissible particular vector fields admissible constraints compatible communication topology defined graph following assumption holds nck assumption gij motivated previous discussions assume following objective function well inequality constraints sums products separable well point vector fields written terms lie brackets admissible vector figure communication graph example considered section vector field corresponding path lie bracket representation table overview vector fields lie bracket representations index sets fields appropriate assumptions communication graph see also last column table specifically graph strongly connected vector fields rewritten independent objective function well constraints given admit structure cases however much less restrictive requirements communication graph sufficient explicitly discuss rewrite vector fields using proposition proposition general illustrate means example section point dynamics given next rewrite vector fields using proposition thereby follow choice make sure holds discuss rewrite vector field detail limit vector field comparing example section illustrate previous results means example consider following optimization problem following proposition require path node node given thus obtain min side lie bracket admissible vector fields vector fields treated similarly sum resulting lie brackets table simulation let largest index respective index set choice inject less perturbation primal dual variables new also visible simulation results depicted fig seen distributed algorithm approximates trajectories dynamics converges neighborhood optimizer newwe also assume communication topology described graph fig observe constraints compatible graph topology hence assumption fulfilled corresponding bracket approximation approach distributed optimization proposed discussed kind vector fields principle written terms lie brackets admissible vector fields discuss construction approximating inputs postpone present modified version general algorithm exploits structure problem hand references michalowsky gharesifard ebenbauer lie bracket approximation approach distributed optimization extensions limitations european control conference ecc accepted boyd parikh chu peleato eckstein distributed optimization statistical learning via alternating direction method multipliers foundations trends machine learning vol bullo distributed control robotic networks ser applied mathematics series princeton university press zhao distributed control optimization microgrids automatica vol supplement feijer paganini stability gradient dynamics applications network optimization automatica vol wang elia control perspective centralized distributed convex optimization ieee conference decision control european control conference zeng ebenbauer saddle point seeking convex optimization problems ifac proceedings volumes vol gharesifard distributed convex optimization weightbalanced digraphs ieee transactions automatic control vol distributed coordination separable convex optimization coupling constraints ieee conference decision control cdc dec time figure simulation results example section upper plot shows primal variable lower left one dual variable lower right one dual variable newin plot dotted black lines marked squares depict trajectories saddlepoint dynamics thinner oscillating ones depict trajectories distributed approximation corresponding thick lines depict distributed approximation additional filters dashed black lines indicate desired equilbrium included simulation results additional filters distributed effect primal variables small since already reduced oscillations design choice dual variables show significantly less oscillations better approximate trajectories rigorous stability analysis augmented distributed dynamics design filters future work conclusions outlook considered convex optimization problem showed distributed optimization algorithms designed quite general class problems little structural requirements mild assumptions communication network therefore extended lie touri gharesifard dynamics distributed convex optimization general directed graphs ieee conference decision control cdc induction hypothesis ebenbauer michalowsky grushkovskaya gharesifard distributed optimization directed graphs help lie brackets proc ifac world congress michalowsky gharesifard ebenbauer distributed extremum seeking directed graphs ieee conference decision control cdc unit vector used simple path hence since disjunct also using definition obtain finally since gik admissible hence side lie bracket admissible vector fields lie bracket approximation approach distributed optimization directed graphs arxiv https biggs algebraic graph theory cambridge university press ebenbauer class smooth optimization algorithms applications control ifac proceedings volumes vol ifac conference nonlinear model predictive control proof proposition proof observe first liu approximation algorithm nonholonomic systems siam journal control optimization vol sussmann liu limits highly oscillatory controls approximation general paths admissible trajectories ieee conference decision control cdc using applying lemma result immediately follows appendix proof lemma proof proposition proof prove result induction first trivially true definition suppose claim holds consider define proof first show induction clear holds suppose holds slight abuse notation used last step proves finally using obtain dzi finishes proof
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published conference paper iclr dynamic eural rogram mbeddings gram epair feb university california davis usa kbwang rishabh singh microsoft research redmond usa risin zhendong university california davis usa bstract neural program embeddings shown much promise recently variety program analysis tasks including program synthesis program repair codecompletion fault localization however existing program embeddings based syntactic features programs token sequences abstract syntax trees unlike images text program semantics difficult capture considering syntax syntactically similar programs exhibit vastly different behavior makes syntaxbased program embeddings fundamentally limited propose novel semantic program embedding learned program execution traces key insight program states expressed sequential tuples live variable values capture program semantics precisely also offer natural fit recurrent neural networks model evaluate different syntactic semantic program embeddings task classifying types errors students make submissions introductory programming class codehunt education platform evaluation results show semantic program embeddings significantly outperform syntactic program embeddings based token sequences abstract syntax trees addition augment program repair system predictions made semantic embedding demonstrate significantly improved search efficiency ntroduction recent breakthroughs deep learning techniques computer vision natural language processing led growing interest applications programming languages software engineering several areas include program classification similarity detection program repair program synthesis one key steps using neural networks tasks design suitable program representations networks exploit existing approaches neural program analysis literature used program representations mou proposed convolutional neural network abstract syntax trees asts program representation classify programs based functionalities detecting different sorting routines deepfix gupta synfix bhatia singh recent neural program repair techniques correcting errors student programs mooc assignments represent programs sequences tokens even program synthesis techniques generate programs output robustfill devlin also adopt program representation output decoder exception piech introduces novel perspective representing programs using pairs however representations accurately capture program properties programs behavior may different syntactic characteristics consequently embeddings learned pairs precise enough many program analysis tasks although pioneering efforts made significant contributions bridge gap deep learning techniques program analysis tasks program representations fundamentally limited due enormous gap program syntax static expression work done internship microsoft research published conference paper iclr int insertionsort int int left int right int bubblesort int int left int right int right left int left int tmp int left right int left int tmp instrumentation line instrumentation line tmp instrumentation line tmp instrumentation line return return bubble insertion figure bubble sort insertion sort code highlighted shadow box syntactic differences two algorithms execution traces input vector displayed right brevity values variable shown static int max int arr variable trace state trace max val max val item item max val item max val max val item item max val item max val max val item item max val item int foreach int item arr item item return figure example illustrating program dependency table variable state traces obtained executing function max given arr semantics dynamic execution gap illustrated follows first program executed runtime statements almost never interpreted order corresponding token sequence presented deep learning models exception straightline programs ones without statements example conditional statement executes one branch time token sequence expressed sequentially multiple branches similarly iterating looping structure runtime unclear order two tokens executed considering different loop iterations second program dependency data control exploited token sequences asts despite essential role defining program semantics figure shows example using simple max function line assignment statement means variable max val item addition execution statement depends evaluation condition line max val also item well third pure program analysis standpoint gap program syntax semantics manifested similar program syntax may lead vastly different program semantics example consider two sorting functions shown figure functions sort array via two nested loops compare current element successor swap order incorrect however two functions implement different algorithms namely bubble sort insertion sort therefore minor syntactic discrepancies lead significant semantic differences intrinsic weakness inherited deep learning technique adopts program representation published conference paper iclr tackle aforementioned fundamental challenge paper proposes novel semantic program embedding learned program runtime behavior dynamic program execution traces execute program set test cases program states comprising variable valuations introduce three approaches embed dynamic executions variable trace embedding consider variable independently state trace embedding consider sequences program states comprises set variable values hybrid embedding incorporate dependencies individual variable sequences avoid redundant variable values program states novel program embeddings address aforementioned issues syntactic program representations dynamic program execution traces precisely illustrate program behaves runtime values variable program point precisely models program semantics regarding program dependencies dynamic execution traces expressed sequential list tuples represents value variable certain program point provides opportunity recurrent neural network rnn establish data dependency control dependency program monitoring particular value patterns interacting variables rnn able model relationship leading precise semantic representations reed freitas recently proposed using program traces sequence training neural network learn execute algorithm addition sorting notion program traces different dynamic execution traces consisting program states variable valuations notion offers following advantages sequence program states viewed sequence pairs executed statement words sequences program states provide robust information sequences executed statements although sequence executed statements follows dynamic execution still represented syntactically therefore may adequately capture program semantics example consider two sorting algorithms figure according reed freitas identical representation statements modify variable repetition tmp eight times representation hand capture semantic differences terms program states also considering valuation variable evaluated dynamic program embeddings context automated program repair particular use program embeddings classify type mistakes students made programming assignments based set common error patterns described appendix dataset experiments consists programming submissions made module assignment two additional problems microsoft codehunt platform results show dynamic embeddings significantly outperform program embeddings including trained token sequences abstract syntax trees addition show dynamic embeddings leveraged significantly improve efficiency searchbased program corrector arf wang algorithm presented appendix importantly believe dynamic program embeddings useful many program analysis tasks program synthesis fault localization similarity detection summarize main contributions paper show fundamental limitation representing programs using features propose dynamic program embeddings learned runtime execution traces overcome key issues syntactic program representations evaluate dynamic program embeddings predicting common mistake patterns students make program assignments results show dynamic program embeddings outperform syntactic program embeddings show dynamic program embeddings utilized improve existing production program repair system background dynamic rogram nalysis section briefly reviews dynamic program analysis ball influential program analysis technique lays foundation constructing new program embeddings unlike static analysis nielson analysis program source code dynamic analysis focuses program executions execution modeled set atomic actions events currently integrated feedback generator production use published conference paper iclr organized trace event history simplicity paper considers sequential executions opposed parallel executions lead single sequence events specifically executions statements program detailed information executions often readily available separate mechanisms needed capture tracing information often adopted approach instrument program source code adding additional monitoring code record execution statements interest particular inserted instrumentation statements act monitoring window values variables inspected instrumentation process occur fully automated manner common approach traverse program abstract syntax tree insert write statements right program statement causes changing values variables consider two sorting algorithms depicted figure assume variable interest subject monitoring instrument two algorithms program location code whenever lines marked comments given input vector execution traces two sorting routines shown right figure one key benefits dynamic analysis ability easily precisely identify relevant parts program affect execution behavior shown example despite similar program syntax bubble sort insertion sort dynamic analysis able discover distinct program semantics exposing execution traces since understanding program semantics central issue program analysis dynamic analysis seen remarkable success past several decades resulted many successful program analysis tools debuggers profilers monitors explanation generators overview pproach present overview approach given program execution traces extracted variables introduce three neural network models learn dynamic program embeddings demonstrate utility embeddings apply predict common error patterns detailed section students make submissions online introductory programming course variable trace embedding shown table row denotes new program point variable gets entire variable trace consists variable values program points subsequent step split complete trace list one variable use one single rnn encode independently perform max pooling final states rnn obtain program embedding finally add one layer softmax regression make predictions entire workflow show figure state trace embedding variable trace handled individually previous approach variable precisely captured address issue propose state trace embedding depicted table program point introduces new program state expressed latest variable valuations entire state trace sequence program states learn state trace embedding first use one rnn encode program state tuple values feed resulting rnn states sequence another rnn note assume order variables values encoded rnn program state rather maintain consistent order throughout program states given trace finally feed softmax regression layer final state second rnn shown figure benefit state trace embedding ability capture dependencies among variables program state well relationship among program states dependency enforcement variable trace embedding although state trace embedding better capture program dependencies also comes challenges significant redundancy consider looping structure program iteration whenever abstract syntax trees enable complete automation regardless structure programs ignore input variable arr since similarly state trace later published conference paper iclr figure variable trace program embedding figure state trace program embedding figure dependency enforcement embedding dotted lines denoted dependencies one variable gets modified new program state created containing values variables even unmodified loop issue becomes severe loops larger numbers iterations tackle challenge propose third final approach dependency enforcement variable trace embedding hereinafter referred dependency enforcement embedding combines advantages variable trace embedding compact representation execution traces state trace embedding precise capturing program dependencies dependency enforcement embedding program represented separate variable traces variable handled different rnn order enforce program dependencies hidden states different rnns interleaved way simulates needed data control dependencies unlike variable trace embedding perform average pooling final states rnns obtain program embedding build final layer softmax regression figure describes workflow published conference paper iclr dynamic rogram mbeddings formally define three program embedding models variable race odel given program variable set variable trace sequence values variable assigned execution let denote value variable trace fed rnn encoder gated recurrent unit time input resulting rnn hidden state compute variable trace embedding equation follows denotes last hidden state encoder gru gru maxpooling evidence whp softmax evidence compute representation program trace performing max pooling last hidden state representation variable trace embedding hidden states denotes size hidden layers rnn encoder evidence denotes output linear model program embedding vector obtain predicted error pattern class using softmax operation tate race odel key idea state trace model embed program state numerical vector first feed program state embeddings sequence another rnn encoder obtain program embedding suppose value variable program state resulting hidden state program state encoder equation computes program state embedding equations encode sequence program state embeddings another rnn compute program embedding gru gru gru gru gru gru denote respectively sizes hidden layers first second rnn encoders ependency nforcement variable race mbedding motivation behind model combine advantages previous two approaches representing execution trace compactly enforcing dependency relationship among variables much possible model variable trace handled different rnn potential issue addressed variable words variables may named differently different programs processing variable single rnn among programs dataset cause memory issues importantly loss precision solution execute programs collect traces variables perform dynamic time wrapping vintsyuk variable traces across programs find used variables account vast majority variable usage rename used variables consistently across programs rename variables special variable presentation simplicity assume program take inputs published conference paper iclr given set variables among programs mechanism dependency enforcement top ones fuse hidden states multiple rnns based new value variable produced example figure line new value max val item item time step new value max val produced latest hidden states rnns encode variable item well together determine previous state rnn upon new value max val produced value produced without dependencies mechanism take effect words rnn act normally handle data sequences work enforce assignment statement declaration statement method calls control statements statements equations expose inner workflow hlt denotes latest hidden state rnn encoding variable trace point time input rnn encoding variable trace denotes matrix product hlt hlt hlt gru given depends averagepooling valuation train dynamic program embeddings programming submissions obtained assignment introduction offered edx two problems microsoft codehunt platform print chessboard print chessboard pattern using represent squares shown figure count parentheses count depth nesting parentheses given string generate binary digits generate string binary digits given integer xoxoxoxo oxoxoxox xoxoxoxo oxoxoxox xoxoxoxo oxoxoxox xoxoxoxo oxoxoxox figure desired output chessboard exercise regarding three programming problems errors students made submissions roughly classified technical issues list indexing branching conditions looping bounds conceptual issues mishandling corner case misunderstanding problem requirement misconceptions underlying data structure test inputs order sufficient data training models predict error patterns convert incorrect program multiple programs new program one error mutate correct programs generate synthetic incorrect programs exhibit similar errors students made real program submissions two steps allow set dataset depicted table based set training data evaluate dynamic embeddings trained three network models compare program embeddings error prediction task testing data models include one trained rnn encodes syntactic traces programs reed freitas another trained rnn encodes token sequences programs third trained rnn abstract syntax trees programs socher problem print chessboard count parentheses generate binary digits program submissions correct incorrect synthetic data training validation testing table dataset experimental evaluation please refer appendix detailed summary error patterns problem published conference paper iclr models implemented tensorflow encoders trace model two stacked gru layers hidden units layer except state encoder state trace model one single layer hidden units adopt random initialization weight initialization vocabulary unique tokens values variables time step embedded vector networks trained using adam optimizer kingma learning decay rates set default values learning rate size variable trace dependency enforcement models trace padded length across batch state trace model number variables program state well length entire state trace padded training dependency enforcement model observed dependencies become complex network suffers optimization issues diminishing exploding gradients likely due complex nature fusing hidden states among rnns echoing errors back forth network resolve issue truncating trace multiple last feedforwarding rest regarding baseline network trained syntactic sequences use encoder architecture two layer gru hidden units processing embedding vector ast model learn embedding type syntax node propagating leaf simple look root learned production rules finally use root embeddings represent programs programming problem print chessboard count parentheses generate binary digits variable trace state trace dependency enforcement syntactic trace token ast table comparing dynamic program embeddings program embedding predicting common error patterns made students shown table embeddings trained execution traces significantly outperform trained program syntax greater accuracy compared less embeddings conjecture fact minor syntactic discrepancies lead major semantic differences shown figure dataset large number programs distinct labels differ number tokens ast nodes causes difficulty syntax models generalize even simpler errors buried large number syntactic variations size training dataset relatively small models learn precise patterns contrast dynamic embeddings able canonicalize syntactical variations pinpoint underlying semantic differences results models learning correct error patterns effectively even relatively smaller size training data addition incorporated dynamic program embeddings arfgen wang program repair system demonstrate benefit producing fixes correct students errors programming assignments given set potential repair candidates arfgen uses enumerative technique find minimal changes incorrect program use dynamic embeddings learn distribution corrections prioritize search repair establish baseline obtain set corrections arfgen real incorrect program three problems enumerate subset find minimum fixes contrary also run another experiment prioritize correction according prediction errors dynamic embeddings worth mentioning one incorrect program may caused multiple errors therefore predict error time repair program corresponding corrections program still incorrect repeat procedure till program fixed comparison two approaches based long takes repair programs corrections merely syntactic discrepancies change program semantics modifying order provide precise fixes false positives would need eliminated published conference paper iclr number fixes enumerative search variable trace embeddings state trace embeddings dependency enforcement embeddings table comparing enumerative search guided dynamic program embeddings finding minimum fixes time measured seconds shown table fixes required speedups dynamic program embeddings yield order magnitude speedups number fixes four greater number fixes greater seven performance gain drops significantly due poor prediction accuracy programs many errors words dynamic embeddings viewed network capturing incorrect execution traces rather new execution traces therefore predictions become unreliable note ignored incorrect programs greater errors experiments run memory baseline approach elated ork significant recent interest learning neural program representations various applications program induction synthesis program repair program completion specifically neural program repair techniques none existing techniques deepfix gupta synfix bhatia singh considered dynamic embeddings proposed paper fact dynamic embeddings naturally extended new feature dimension existing neural program repair techniques piech notable recent effort targeting program representation piech explore possibility using pairs represent program despite new perspective direct mapping input output programs usually precise enough pair may correspond two completely different programs two sorting algorithms figure often observe dataset programs error patterns also result different pairs approach clearly ineffective scenarios reed freitas introduced novel approach using execution traces induce execute algorithms addition sorting examples differences work use sequence instructions represent dynamic execution trace opposed using dynamic program states goal synthesize neural controller execute program sequence actions rather learning semantic program representation deal programs language primitives function stack actions rather programming language learning representations several related efforts modeling semantics sentence symbolic expressions socher zaremba bowman approaches similar work spirit target different domains programs onclusion presented new program embedding learns program representations runtime execution traces used new embeddings predict error patterns students make online programming submissions evaluation shows dynamic program embeddings significantly outperform learned via program syntax also demonstrate via additional application dynamic program embeddings yield speedups compared enumerative baseline program repair beyond neural program repair believe dynamic program embeddings fruitfully utilized many neural program analysis tasks program induction synthesis published conference paper iclr eferences thoms ball concept dynamic analysis proceedings european software engineering conference held jointly acm sigsoft international symposium foundations software engineering sahil bhatia rishabh singh automated correction syntax errors programming assignments using recurrent neural networks corr samuel bowman recursive neural tensor networks learn logical reasoning arxiv preprint jacob devlin jonathan uesato surya bhupatiraju rishabh singh abdel rahman mohamed pushmeet kohli robustfill neural program learning noisy proceedings international conference machine learning rahul gupta soham pal aditya kanade shirish shevade deepfix fixing common language errors deep learning proceedings aaai conference artificial intelligence diederik kingma jimmy adam method stochastic optimization url http corr lili mou zhang tao wang zhi jin convolutional neural networks tree structures programming language processing proceedings thirtieth aaai conference artificial intelligence flemming nielson hanne nielson chris hankin principles program analysis chris piech jonathan huang andy nguyen mike phulsuksombati mehran sahami leonidas guibas learning program embeddings propagate feedback student code proceedings international conference machine learning yewen karthik narasimhan armando regina barzilay neural program corrector moocs companion proceedings acm sigplan international conference systems programming languages applications software humanity splash companion scott reed nando freitas neural arxiv preprint richard socher alex perelygin jean jason chuang christopher manning andrew christopher potts recursive deep models semantic compositionality sentiment treebank proceedings conference empirical methods natural language processing taras vintsyuk speech discrimination dynamic programming cybernetics wang rishabh singh zhendong feedback generation introductory programming exercises corr url http wojciech zaremba karol kurach rob fergus learning discover efficient mathematical identities advances neural information processing systems published conference paper iclr ppendix rror patterns print chessboard misprinting printing lower case instead upper case characters switching across rows supposed way around printing oxoxoxox odd number rows xoxoxoxo even number rows printing first row correctly failed make switch across rows printing entire chessboard printing chessboard correctly extra unnecessary characters printing incorrect number rows printing incorrect number columns printing characters correctly wrong format correctly seperated spaces form rows others count parentheses miss corner case empty strings mistakenly consider parenthesis symbols rather mishandling string unmatched parentheses counting number matching parentheses rather depth incorrectly assume nested parentheses always present miscounting characters ignored others generate binary digits miss corner case integer misunderstand binary digits underlying bytes string mistakes arithmetic calculation regrading shift operations adding binary digits rather concatenating string miss one significant bit others published conference paper iclr arf lgorithm algorithm arfgen feedback generation procedure incorrect program correct solutions function fixgeneration begin among identify pcs reference programs fix pcs candidatesidentification initialize minimum number fixes inifinity initialize minimum set fixes null pcs generates syntactic discrepencies discrepenciesgeneration selecting subsets size one itll csubs attemp subset csub csubs patchapplication csub update necessary iscorrect return algorithm incorporate model arfgen feedback generation procedure learned model function fixgeneration begin among identify pcs reference programs fix pcs candidatesidentification initialize minimum number fixes inifinity initialize minimum set fixes null pcs generates syntactic discrepencies discrepenciesgeneration executing extract dynamic execution trace dynamictraceextraction prioritizing subsets model csubs prioritization csub csubs patchapplication csub iscorrect return
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jul regularity edge ideals associated bipartite graphs oscar philippe gimenez abstract focus paper edge ideals associated bipartite graphs give combinatorial characterization regularity regularity strictly bigger determine first step minimal graded free resolution exists minimal generator degree show step highest degree minimal generator determine value corresponding graded betti number terms combinatorics associated bipartite graph results easily extended case polarization also study family ideals regularity play important role main result whose graded betti numbers completely described closed combinatorial formulas introduction studying homological invariants monomial ideals polynomial ring looking combinatorial properties discrete objects graphs hypergraphs simplicial complexes associated well known technique fruitfully exploited last decades see example surveys references fact provides quite complete dictionary two algebraic combinatorial classes classical objects used relate combinatorics monomial ideals ideals simplicial cellular resolutions facet ideals monomial ideal generated quadrics viewed facet ideal graph graph simple loops ideals called edge ideals first introduced homological invariants monomial ideal interested encoded minimal graded free resolution ideal namely graded betti numbers regularity considering standard polynomial ring graded betti number number minimal generators degree syzygy module ideal denote resp maximal resp minimal degree minimal generator syzygy module fact resolution minimal implies authors partially supported ministerio ciencia spain oscar philippe gimenez minimal degree generator regularity ideal defined reg max interesting situation minimal generators degree case reg say ideal resolution nice combinatorial characterization edge ideals resolution regularity terms complement associated graph due new proof result recently obtained topology edge ideals studied another proof characterization found moreover least characterized combinatorial way edge ideal linear resolution result obtained independently also shown value sygygies first appear moreover determined terms complement graph results recalled section theorem together required definitions notations also show section graded betti numbers arbitrary edge ideal satisfy following property theorem every implies particular corollary refinement theorem aim paper characterize edge ideals associated bipartite graphs regularity determine regularity first step minimal resolution done section also prove value one show number induced subgraphs bipartite complement isomorphic cycles minimal length fundamental role played subgraphs reason previously devote section study graded betti numbers edge ideal associated bipartite complement even cycle show edge ideal regularity give closed combinatorial formulas graded betti numbers finally prove last section similar results monomial ideals generated quadrics squarefree dependence betti numbers edge ideals characteristic field even case edge ideals associated bipartite graphs prevents possibility obtaining similar results higher regularity preliminaries graphs simplicial complexes start recalling elementary concepts graphs handle along paper see terminology included regarding graphs simplicial complexes respectively regularity edge ideals associated bipartite graphs consider finite simple graph denote edge vertex sets respectively say subgraph induced subset write induced subgraph unspecified subset vertices complement graph vertex set given vertex denote set vertices adjacent subset degree denoted deg number elements connected graph whose vertices degree two called denoted say length cycle induced subgraph also cycle called induced cycle graph said chordal induced cycle graph whose vertices degree one necessarily vertices consists disconnected edges denote definition induced matching number graph maximal denote number consider simplicial complex given subset vertex set induced subcomplex recall one two subcomplexes long exact sequence reduced homologies called sequence whenever definition given simplicial complex consider following two simplicial complexes cone base appex suspension vertices following results cones suspensions see theorems useful sequel proposition associated graph one independence complex defined simplicial complex vertex set face edge subset observe graph arbitrary subset one oscar philippe gimenez remark flag complex simplicial complex every pair elements face also face particular flag complex containing pairs vertices necessarily simplex moreover independence complex graph always definition flag complex definition independence complex graph consider three induced subcomplexes featured paper note independence complex induced subgraph vertex cone appex hence acyclic proposition since apply whenever get let focus bipartite graphs recall graph bipartite vertex set splitted two disjoint sets way edge one vertex one deals bipartite graphs usually convenient use different symbols variables variables denote variables variables biadjacency matrix bipartite graph defined otherwise bipartite complement bipartite graph bipartite graph gbc vertex set gbc gbc one gbc matrix whose entries note bipartition may unique graph connected notions biadjacency matrix bipartite complement depend bipartition reason section restrict connected bipartite graphs next lemma useful handle homology independence complex bipartite graph last three items rules one apply reduction simpler case removing vertices biadjacency matrix satisfies properties lemma let bipartite graph bipartition biadjacency matrix independence complex row column whose entries regularity edge ideals associated bipartite graphs exist rest entries row column zeros one row resp column entries row resp column resp row resp column unique zero entry say another zero entry column resp row resp two rows resp two columns resp resp proof vertex corresponding row column zero entries isolated hence cone appex acyclic proposition one since result follows sequence particular cases follows applying vietoris sequence observing acyclic example independence complex dimk otherwise shown induction follows since consists two disjoint vertices connected applying lemma one gets result follows properties graded betti numbers edge ideal given minimal graded free resolution homogeneous ideal regularity sense defined terms graded betti numbers reg max graded betti numbers usually arranged table called betti diagram placed column row oscar philippe gimenez table note index last nonzero row betti diagram regularity monomial ideal provide polynomial ring usual minimal multigraded free resolution define multigraded betti numbers number minimal generators degree syzygy module correspondence squarefree monomial ideals generated degree simple graphs associated simple graph one edge ideal generated monomials form edge ideal indeed ideal independence complex multigraded betti numbers edge ideal expressed terms reduced homology using hochster formula theorem recall one monomial squarefree dimk otherwise graded betti numbers given following formula dimk hochster formula powerful tool one wants get information betti numbers edge ideals used example prove following property graded betti numbers edge ideal theorem proof denote independence complex assume exists dimk flag complex exist remark consider following decomposition empty since invoking hochster formula one using sequence one gets contradiction note theorem easily extended monomial ideals generated degree two may squarefree polarization see section following direct consequence answers question aldo regularity edge ideals associated bipartite graphs conca asked theorem could improved direction avramov conca iyengar proved bounds syzygies koszul algebras question arose context corollary let monomial ideal generated degree two denote maximal degree minimal generator syzygy module edge ideal linear resolution nonzero entries betti diagram located first row proved edge ideal linear resolution graph chordal rephrase nice combinatorial characterization follows theorem edge ideal regularity induced cycles authors one step show reg syzygies appear first time step resolution minimal length induced cycle result contained following stronger statement theorem edge ideal reg let minimal length induced cycle induced one otherwise observe previous theorem induced cycles play important role previously focused particular family edge ideals associated complements cycles gave proposition closed combinatorial formulas graded betti numbers following philosophy focus graphs bipartite complement even cycle since induced even cycles bipartite complement arbitrary graph play fundamental role later main theorem family graphs describe graded betti numbers associated edge ideals theorem bipartite complement cycle even length following result direct consequence propositions prove section theorem edge ideal associated bipartite complement even cycle length regularity betti diagram oscar philippe gimenez nonzero entries located shadowed area moreover given respectively following closed combinatorial formulas bipartite complement even cycle let least vertices vertices edges even cycle respectively along section sometimes refer two subsets bipartition biadjacency matrix exactly two zero entries row column order use hochster formula determine graded betti numbers need compute reduced simplicial homologies subsets case solved proposition proof requires following lemma lemma every acyclic proof without loss generality let choose observed definition independence complex whose biadjacency matrix obtained removing first row observe first last columns satisfy condition lemma hence removed first last rows new matrix satisfy condition remove recursively odd respectively even reduce computation homology case independence complex graph whose biadjacency matrix respectively matrix whose central column respectively row two entries equal zero acyclic lemma regularity edge ideals associated bipartite graphs proposition dimk otherwise proof since connected dimk using sequence previous lemma implies every vertex biadjacency matrix transpose matrix exactly two zero entries one row located two different columns applying lemma many times necessary one gets result follows example let proper subset set denote number connected components graph isolated vertices note proposition acyclic dimk otherwise dimk otherwise remark observe denotes complete graph set vertices thus kwx kwy since kwx kwy connected one empty connected connected otherwise kwx kwy connected components thus condition proposition satisfied connected denote biadjacency matrix easy check zero entries proof proposition simplex follows assume zero entries remark lemma follows example assume least one zero entry first observe number zero entries row column two least two columns rows one lemma dimension reduced homologies change remove row column zero entry words since row column zero entry corresponds isolated vertex subset formed elements isolated vertices one oscar philippe gimenez moreover using lemma one remove row resp column exactly one zero entry column resp row zero entry located another zero entry row column corresponds vertex degree one whose unique neighbor degree two removing vertex change number connected components creates vertex kind reach vertex degree one whose neighbor also degree one thus result direct consequence following technical lemma lemma dimk otherwise proof biadjacency matrix graph matrix whose entries except ones principal diagonal zero denote independence complex since connected dimk order determine homology consider family subcomplexes whose elements index recall nerve simplicial complex vertex set whose faces subsets intersection corresponding elements non empty simplicial complex facets since applying nerve theorem see example theorem one gets hand hence proposition result follows straightforward consequence one gets last row betti nonzero entry one indexed regi diagram nonzero entry row otherwise proposition proof putting together propositions one every subset dimk moreover dimk dimk result follows hochster formula one order complete description betti diagram determine graded betti numbers first two rows regularity edge ideals associated bipartite graphs start first row using hochster formula proposition one needs determine proper subsets connected indeed number induced subgraphs vertices non connected let denote cycle vertex set whose edges note edges correspond pairs elements lemma assume connected exists comp set connected components proof ncx thus one implies connected remark follows prove induction statement holds consider assume statement holds subsets know exists one two possibilities connected case otherwise cases applying inductive hypothesis one gets nonzero entries first row betti diagram given following result proposition proof consider proper subset already obbc hochster served contribute formula connected remark connected thus implies lemma since hence oscar philippe gimenez thus connected follows count many subsets satisfy connected choice elements order must choose elements possible choices lemma fix number connected components denote possible subsets according lemma result follows corollary first last nonzero entries first row coincide betti diagram proof one case hence complete description betti diagram give graded betti numbers located second row next result proposition proof proposition contribute hochster formula proper subset least connected components isolated vertices precisely denoting number proper subsets connected components isolated vertices particular since subset elements one connected follows denote set proper subsets isolated vertices connected components isolated vertices reduced compute possible regularity edge ideals associated bipartite graphs proof lemma observe subset represented vector length whose entry element belongs otherwise using correspondence number nonzero entries vector number vertices number blocks nonzero entries related number connected components order avoid distinguishing cases vector allow modify starting vertex focus vectors whose first entry last entry denote set vectors length entries whose first entry last entry whose nonzero entries located different blocks let subset formed vectors blocks length strictly bigger blocks length whose first block nonzero entries length strictly bigger element corresponds elements one choice vertex vertex corresponding first entry element corresponds distinct elements one connected component choose one gives first block nonzero entries vector thus finally order determine note element comes vector adding block inserting times two zero entries already observed proof lemma moreover element block zero entries length give places one add two zero entries since element zero entries located different blocks element provide elements putting together one gets done corollary first last nonzero entries second row coincide betti diagram proof hence done hand remark recall induced matching number graph regularity edge ideal satisfy reg case bipartite complement even cycle one easily determine induced matching number since hence formed two hence matching edges oscar philippe gimenez regularity number related follows reg difference cases indeed even complete intersection would gorenstein impossible since betti diagram symmetric regularity bipartite edge ideals section focus edge ideals associated bipartite graphs call bipartite edge ideals consider connected graphs betti numbers edge ideal associated disconnected graph computed betti numbers edge ideals associated connected components see lemma bipartite edge ideals regularity characterized using theorem shown edge ideals associated ferrer graphs theorem aim prove main results theorems first one analogous classical theorem provides combinatorial characterization bipartite edge ideals regularity second one analogous theorem gives extra information bipartite edge ideal regularity determine first step minimal graded free resolution syzygies contributing graded betti number located outside first two rows betti diagram also show syzygies concentrated degree compute corresponding graded betti number theorem let connected bipartite graph edge ideal regularity least one induced cycle length gbc induced cycle length theorem let connected bipartite graph set assume reg let minimal length induced cycle gbc induced gbc one gbc otherwise prove results let recall construction results useful given simplicial complex vertex set whose facets denoted consider new regularity edge ideals associated bipartite graphs vertices define new simplicial complex vertex set denotes vertex set simplicial complex given theorem states let connected bipartite graph vertex set set denote subset one set ngbc simplicial complex isolated vertices gbc definition say subset relevant remark relevant exist lemma relevant lemma relevant proof denote simplicial complex associated graph let set facets set defined since relevant isolated vertex hence vertex set moreover ngbc implies ngbc consider also otherwise one ngbc ngbc thus lemma let ideal associated simplicial complex let monomial minimal generating set one oscar philippe gimenez relevant deg max deg proof relevant thus ngbc hence therefore must dimension strictly greater since minimal generators correspond minimal first claim follows xid minimal generator xid ngbc xgi hence every xik ngbc element denote xil ngbc equivalently xil xik xil set consisting disconnected edges implies proof theorems first prove equivalence theorem show extra information contained theorem follows quite easily first assume reg theorem contains induced cycle length moreover exists subset gbc even since one hand hochster formula clbc theorem one gets reg contradiction conversely assume reg reg induced cycle theorem result holds reg theorem exists denote smallest integer property lemma number induced subgraphs isomorphic notice obtain gbc contains induced cycle length hand items theorem follow case theorem theorem states monomial one collects step minimal multigraded free resolution minimal generators whose multidegree divides one gets minimal multigraded free resolution edge ideal whose minimal generators divide hochster formula tells exists dimk case done using theorem theorem show subsets satisfying ones satisfies satisfies proposition take satisfying consider simplicial complex note regularity edge ideals associated bipartite graphs using remark relevant subset vertices minimality applying lemma one dimk dimk moreover dimk since dimk dimk reach contradiction minimality size hence lemma thus lemma generated degree edge ideal hence write simple graph vertex set thus dimk dimk applying theorem necessarily therefore clc together fact ngbc ngbc deggbc ngbc deggbc implies moreover gbc connected hence gbc remark one find several examples edge ideals whose regularity depending characteristic field shows theorem bipartite hypothesis removed since information provided depends combinatorics graph restricted bipartite edge ideals work observe even bipartite edge ideals hopeless try extrapolation results higher values regularity example shows case let ideal generated monomials degree two squarefree assume without loss generality minimally generated squarefree define isqf ipol ideal ipol called polarization following useful property provide deg deg corollary ipol ideals isqf ipol edge ideals first one vertex set second call non simple graph associated denote squarefree case oscar philippe gimenez denote gsqf gpol simple graphs associated isqf ipol respectively observe gsqf gpol obtained removing loops substituting whiskers loops respectively definition say two edges totally disjoint provided assume simple graph gsqf connected bipartite case say non simple graph bipartite define bipartite complement bipartite complement simple graph gsqf gbc gsqf also define complement complement simple graph gsqf gsqf complete characterization ideals associated bipartite graphs regularity case follows proposition let monomial ideal generated degree two assume non simple graph associated bipartite regularity either two totally disjoint edges three edges pairwise totally disjoint gbc induced cycle length proof reg reg ipol using theorem occurs gpol induced cycle length gpol induced cycle length rewriting properties graph gpol terms graph result follows reg claims theorem remain valid contain three edges pairwise totally disjoint since gsqf gsqf coincide gpol gpol respectively provided however three edges pairwise totally disjoint number induced subgraphs isomorphic three pairwise totally disjoint edges considering one consists three totally disjoint edges otherwise example ideal satisfies bipartite graph gbc induced three pairwise disjoint edges references avramov conca iyengar free resolutions commutative koszul algebras math res lett regularity edge ideals associated bipartite graphs topological methods handbook combinatorics vol elsevier amsterdam corso nagel monomial toric ideals associated ferrers graphs trans amer math soc dalili kummini dependence betti numbers characteristic diestel graph theory graduate texts mathematics vol springer heidelberg fourth edition eisenbud green hulek popescu restricting linear syzygies algebra geometry compos math gimenez first nonlinear syzygies ideals associated graphs comm algebra rings topics algebra part warsaw banach center publ vol gasharov hibi peeva resolutions ideals algebra herzog hibi monomial ideals graduate texts mathematics vol london hochster rings combinatorics simplicial complexes ring theory proc second univ oklahoma norman lecture notes pure appl math vol dekker new york van tuyl resolutions monomial ideals via facet ideals survey algebra geometry interactions contemp math vol amer math providence jacques katzman betti numbers forests katzman betti numbers graph ideals combin theory ser morey villarreal edge ideals algebraic combinatorial properties munkres elements algebraic topology publishing company menlo park nevo regularity edge ideals graphs via topology lcmlattice combin theory ser villarreal graphs manuscripta math villarreal monomial algebras monographs textbooks pure applied mathematics vol marcel dekker new york university valladolid address oscarf university valladolid address pgimenez
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geometrically exact finite element formulations curved slender beams theory theory christoph meiera wolfgang wallb alexander poppb mechanosynthesis institute group massachussets institute technology massachusetts avenue cambridge usa computational mechanics technical university munich boltzmannstrasse garching germany sep abstract present work focuses geometrically exact finite elements highly slender beams aims proposal novel formulations type detailed review existing formulations simoreissner type well careful evaluation comparison proposed existing formulations two different rotation interpolation schemes strong weak kirchhoff constraint enforcement respectively well two different choices nodal triad parametrizations terms rotation tangent vectors proposed combination schemes leads four novel finite element variants based hermite interpolation beam centerline essential requirements representability general dynamic problems involving slender beams arbitrary initial curvatures anisotropic shapes preservation objectivity consistent convergence orders avoidance locking effects well conservation energy momentum employed spatial discretization schemes also range practically relevant secondary aspects investigated analytically verified numerically different formulations shown geometrically exact beam elements proposed work first ones type fulfill essential requirements contrary type formulations fulfilling requirements found literature well however argued formulations provide considerable numerical advantages lower spatial discretization error level improved performance time integration schemes well linear nonlinear solvers smooth geometry representation compared formulations applied highly slender beams concretely several representative numerical test cases confirm proposed formulations exhibit lower discretization error level well considerably improved nonlinear solver performance range high beam slenderness ratios compared two representative element formulations literature keywords geometrically exact beams finite elements objectivity introduction highly slender components represent essential constituents mechanical systems countless fields application scientific disciplines mechanical engineering biomedical engineering material science molecular physics examples industrial webbings composite materials fibrous materials tailored porosity synthetic polymer materials also cellulose fibers determining characteristics paper entirely different time length scales slender components relevant analyzing supercoiling process dna strands characteristics carbon nanotubes brownian dynamics within cytoskeleton biological cells biopolymer network highly slender filaments crucially influences biologically relevant processes cell division cell migration often slender components modeled cosserat continua based geometrically nonlinear beam theory corresponding author email address chrmeier christoph meier september mentioned cases mechanical contact interaction crucially influences overall system behavior authors recent works novel contact formulation proposed faces challenges arising high beam slenderness ratios complex contact configurations present contribution focuses development geometrically exact beam element formulations suitable accurate efficient robust modeling highly slender beams vast majority geometrically exact beam element formulations available literature based theory thick rods current work proposes geometrically exact finite elements based theory thin rods taylored high beam slenderness ratios basically two essential motivations applying beam theory instead continuum mechanics theory modeling slender bodies identified early days beam theories accessibility analytic solutions example euler elastica even large deformation problems motivated development application theories nowadays knowledge modeling highly slender bodies via beam theories yields considerably efficient also numerical formulations would case continuum theories induced beam theories regarded reduced continuum theories consistently derived theory continuum mechanics consideration basic kinematic constraint reflects deformation states expected slender bodies reasonable manner beam theories typically allow describe motion deformation slender bodies space basis proper kinematic kinetic constitutive resultant quantities case induced beam theories resultant quantities example derived via integration stress measures beam stress measures typically result constrained displacement field well standard strain measures constitutive relations context beam represents collection material points sharing beam length coordinate configuration contrary intrinsic beam theories directly postulate resultant quantities theories internally consistent sense resultant quantities well relations connecting quantities still fulfill essential mechanical principles equilibrium forces moments conservation energy rather existence work conjugated pairs observer invariance conservative problems nevertheless intrinsic beam theories decoupled continuum mechanics theory typically postulated constitutive constants relating stress strain measures determined experimentally constitutive constants induced beam theories follow directly corresponding constitutive laws postulated constitutive laws based experimentally determined constants favorable applications continuum foundation exists considering low number discrete molecules distributed thickness macromolecules occurring example biopolymer networks dna strands carbon nanotubes come back applications mentioned continuum theory applied reasonable manner nevertheless slender components described good approximation continuum theories associated experimentally determined constitutive constants compromise induced intrinsic theories beam theories constitutive law postulated remaining kinetic kinematic relations consistently derived theory based bernoulli hypothesis undeformed work euler kirchhoff beam proposed kirchhoff heidelberg first formulation allowing arbitrary initial curvatures large deformations including states bending torsion theory enhanced love also account small axial tension comprehensive historic overview early developments given work dill reissner case general case completed theory two additional deformation measures representing shear deformation beam problem statement reissner still based additional approximations simo extended work reissner yield formulation truly consistent sense beam theory thus starting basic kinematic assumption kinetic kinematic quantities relations consistently derived continuum theory constitutive law postulated originally theory denoted geometrically exact beam theory nowadays also referred beam theory according definition simo also work beam theory denoted geometrically exact relationships configuration strain measures consistent virtual work principle equilibrium equations deformed state regardless magnitude displacements rotations strains reason also notation beams applied original work however later argued several authors see accordance derivations literature see also section work consistency geometrically exact beam theory theory continuum mechanics sense fully induced beam theory assumed long small strains considered fulfillment basic kinematic assumption rigid underlying geometrically exact beam theory requires pointwise six translational rotational degrees freedom order uniquely describe centroid position orientation consequently beam theory identified cosserat continuum derived boltzmann continuum pointwise three translational degrees freedom exists variety beam theories also consider well distortion contribution focuses geometrically exact beam formulations basis rigid assumption applied simo reissner furthermore throughout work notion theory preferred since notion geometrically exact beam theory following definition presented also applies consistently derived formulations basis theory remainder work notion represents opposite thus equivalent vanishing shear strains course vanishing shear stresses unfortunately absolute consensus concerning naming different beam models found literature reason following nomenclature trying consistent important representatives literature applied name anisotropic straight isotropic nonlinear timoshenko inextensible theory nonlinear nonlinear nonlinear tension shear torsion bending nonlinear nonlinear nonlinear linear linear table nomenclature applied within current contribution different geometrically exact beam theories models geometrically nonlinear beam models capturing modes axial tension torsion bending appropriate initially straight beams isotropic shapes identical moments inertia area denoted nonlinear beams extension anisotropic shapes different moments inertia area arbitrary initial curvatures referred theory order simplify subsequent comparison beam formulations literature refined nomenclature isotropic straight iii anisotropic formulations introduced theories capturing neither anisotropic shapes initial curvatures anisotropic shapes initial curvatures iii anisotropic shapes initial curvatures supplementation shear deformation modes covered theory formulations neglect mode axial tension denoted inextensible variants recently sheardeformable beam formulations proposed also neglect mode torsion formulations capable accurately modeling ropes cables providing mechanically consistent bending stabilization chains also general continua full bending torsional stiffness restrictions concerning external loads torsional moment loads initial geometry fulfilled eventually restriction theories geometrically linear regime yields linear timoshenko beam models overview beam models given table identifying configuration space underlying geometrically exact beam theory nonlinear differentiable manifold lie group structure pointing important algorithmic consequences resulting nonadditivity associated group elements original work simo quent work simo laid foundation abundant research work topic following years static beam theory extended dynamics cardona geradin simo contributions kondoh dvorkin well iura atluri regarded pioneering works field contributions mark starting point development large variety geometrically exact beam element formulations basically differ type rotation interpolation interpolation incremental iterative total rotational vectors choice nodal rotation parametrization via rotation vectors quaternions etc type iterative rotation updates multiplicative additive time integration scheme applied rotational degrees freedom based additive multiplicative rotation increments also extensions geometrically exact beam theory arbitrary shapes shear centers differing centroid found overview important developments time exemplarily given text books crisfield well geradin cardona break development given works crisfield shown none rotation field discretizations formulations existent time could preserve important properties objectivity see also discussion topic furthermore new objective orthogonal interpolation scheme proposed directly acts rotation manifold rotation vector parametrizations done works formulation starting point development many alternative rotation interpolation strategies geometrically exact beams also preserve properties among others orthogonal interpolations relative rotation vectors see quaternions see interpolation strategies combination modified beam models see interpolation strategies subsequent orthogonalization see identified reported original work rotation interpolation scheme proposed crisfield exactly represent state constant curvature thus interpreted geodesic shortest connection two configurations rotation manifold consequently geodesic rotation interpolation schemes represent counterpart linear interpolations translational quantities works well recent contributions identified geometrically exact beam element formulations based geodesic interpolations rotational translational primary variable fields extension helicoidal interpolations elements given formulation smooth centerline representation based isogeometric collocation scheme proposed besides purely elements also interpolation schemes directly acting strain level combined subsequent derivation position rotation field via integration see well mixed formulations proposed furthermore variety contributions considering time integration rotational variables found context formulations presented based finite element method fem also discrete representatives beam formulations based finite difference schemes found literature often denoted discrete elastic rods based concept discrete differential geometry ddg context finite element formulations geometrically nonlinear beam problems variety alternatives geometrically exact formulations considered last two paragraphs found maybe prominent representatives alternatives corotational method well absolute nodal coordinate anc solid beam element formulations corotational technique initially introduced wempner well belytschko shows strong similarities natural approach argyris basic idea split overall deformation contribution stemming large rotations part stemming local deformations expressed local corotated frame often local deformation modeled basis order theories entire degree nonlinearity represented rotation local frame basic idea anc beam element formulations employ standard shape functions known solid finite element formulations order interpolate displacement field within beam instead introducing kinematic constraint deriving resultant model different polynomial degrees typically applied interpolation beam length direction transverse directions numerical comparisons performed romero bauchau advocate geometrically exact beams general orthogonal triad interpolation schemes see particular regard computational accuracy efficiency especially arguments stated terms possible benefits compared beam element formulations range high beam slenderness ratios pronounced even stronger comparing geometrically exact beam element formulations anc beam element formulations latter additionally exhibiting highly stiff deformation modes dealing slender beams hand clear distinction corotational geometrically exact formulations always possible example interpolation scheme geometrically exact formulation proposed crisfield also based definition corotated reference triad consequently interpreted corotational formulation time stated geometrically exact finite element formulations become meantime arguably regarded methods computational treatment geometrically nonlinear beam problems overview different fem discretiation techniques given figure figure overview different discretization techniques nonlinear problems slender continua based finite element method context geometrically nonlinear beam theory several discrete realizations based finite difference schemes recently proposed contrast theory also several works based analytic treatment beam problems exist modern literature interestingly approaches two categories found field molecular physics although theoretical basis beam formulations much longer tradition theory beams geometrically nonlinear finite element representations reached excellent properties geometrically exact formulations far recent works armero valverde gave historic overview existing kirchhoff finite elements pointed drawbacks accordingly first kirchhoff type element formulations applied different interpolations lagrange polynomials axial displacements hermite polynomials transversal displacements different led loss objectivity later works objectivity could preserved employing trigonometric shape functions corresponding formulations limited investigation plane circular arches see different approach development framework corotational beams category formulations naturally preserves objectivity continuous problem pointed types kirchhoff type formulations often exhibit comparatively poor accuracy fact directly traced back lack exact representation kinematic quantities critical issue relevant context thin kirchhoff beams membrane locking locking phenomenon given distinction stolarski belytschko general membrane locking denotes inability curved structural elements beams shells represent inextensibility constraint vanishing membrane axial strains thin kirchhoff beams one first contributions effect investigated relating beam slenderness ratio condition number stiffness matrix without explicitly using term locking diverse methods proposed literature order avoid membrane locking kirchhoff rods amongst others approaches reduced selective integration see assumed strains based functional see assumed stresses based functional see penalty relaxation stabilization techniques combination membrane correction factors see works limited beam problems historic overview concerning development kirchhoff beam elements general key issues objectivity membrane locking particular given recent works armero valverde beam elements presented typically based additional kinematic assumptions thus consistent concept geometrically exact beams besides contributions considered far also finite element finite difference discretization approaches kirchhoff beams found literature fact geometrically exact rely global interpolation strategies typically based rotation curvature interpolation strategy subsequent integration rotation field along entire beam length order yield explicit beam centerline representation unfortunately global approaches typically rely serial finite element evaluation yield dense system matrices desirable sparse system matrices small bandwidths typical standard fem approaches consequently approaches suitable computing considered following number existing geometrically exact locally supported finite element formulations basis theory limited example recent contribution sansour proposes method initially straight geometrically exact elements based nonlinear theory hand armero valverde developed plane beam element formulations accounting arbitrarily curved initial geometries well anisotropic shapes guarantee fundamental properties objectivity geometrical exactness however beam elements cover geometrically linear case infinitesimal deformations general case geometrically exact beam elements found literature arguably first formulations kind proposed boyer weiss recent work boyer extended original formulation modeling undersea cables also essential requirement objectivity fulfilled approaches boyer weiss however geometrically nonlinear formulations treat special case beams circular straight initial configuration case beams rotationally symmetric reference geometry following denoted isotropic case see first column table limitation simplifies theory considerably already modeling simple piecewise straight frames difficult since variables available determine orientation required kinematic constraints joints recent contributions zhang zhao allow anisotropic still focus initially straight beams denoted straight case second column table however later become clear twist angle interpolation underlying formulations might general allow optimal spatial convergence rates considering employed centerline interpolation recent works greco first steps towards geometrically nonlinear isogeometric beam elements accounting initial curvature anisotropic made however formulation applied geometrically linear examples bauer adapted ideas greco facilitated application geometrically nonlinear examples formulations denoted anisotropic case table unfortunately important properties objectivity considered works greco bauer also question membrane locking yet consistently treated within geometrically nonlinear realm textitisotropic straight anisotropic formulations considered paragraph nevertheless considered elaborate geometrically exact beam element formulations literature since ones accounting numerical treatment general problems consequently formulations focus detailed evaluations comparisons performed throughout work concluded none existing geometrically exact beam elements type comparable existing formulations type terms generality fulfillment essential properties far also detailed comparisons geometrically exact beam elements range high slenderness ratios still missing backlog motivation development novel geometrically exact beam elements fulfilling following essential requirements geometrical exactness already mentioned proposed beam element formulations geometrically exact sense derived deformation measures consistent virtual work principle equilibrium equations deformed state independent magnitude displacements rotations strains representability general geometries external loads general case problems thin beams anisotropic shapes well arbitrary initial geometries curvatures represented proposed rotation interpolation schemes capable representing general scenarios without exhibiting singularities practically relevant configurations fulfillment essential mechanical principles essential mechanical principles objectivity observer frame invariance well preserved employed discretization schemes fulfillment general requirements spatial finite element discretization finite element formulations presented work accuracy verified terms convergence towards proper reference solutions furthermore following criteria fulfilled optimal spatial convergence rates measured appropriate error norms achieved furthermore effective methods avoidance membrane locking required remaining deterioration spatial convergence behavior observed resulting finite elements even range high slenderness ratios properties conservation linear momentum conservation angular momentum conservation energy arbitrarily rough spatial discretizations desirable proposed spatial interpolation schemes also beneficial beam element formulations fulfill simple patch tests exactly represent states constant axial tension torsion bending may preferable finite element solution independent chosen node numbering fulfillment general requirements temporal discretization main focus work lies development evaluation spatial finite element discretizations geometrically nonlinear beam problems nevertheless least required dynamics represented general time integration achieved often conserving time integration schemes may favorable simple realization essential boundary conditions joints choices nodal primary variables especially respect rotation parametrization demanded enable formulation practically relevant dirichlet boundary conditions also joints several beams without need additional constraint equations standard existing geometrically exact beam element formulations avoidance lagrange multipliers saddle point systems also existing geometrically exact beam elements type subjected kirchhoff constraint via additional lagrange multiplier fields procedure typically results saddle point systems need special class linear solvers requires global condensation strategies beam elements considered neither rely lagrange multipliers yield saddle point systems required calculations feasible manner suitability computing literature several finite difference finite element discretizations beams proposed rely global strategies rotation field construction even though schemes show otherwise desirable properties typically suffer two elementary drawbacks mostly schemes result dense discrete system matrices depend successive serial evaluation individual finite elements within discretization two properties make formulations virtually impossible computing finite elements considered required result sparse system matrices small bandwidths suitable parallel computing element evaluation routines authors recent contribution first geometrically exact beam element formulation fulfills essential properties objectivity capable representing arbitrary initial curvatures anisotropic shapes proposed subsequent work also important question membrane locking successfully addressed current type source formulation geometrically exact initial curvature anisotropic objective optimal convergence order locking avoided conservation properties patch test passed symmetric discretization suitable dynamics conservation properties realization dbcs joints lagrange multipliers avoided computing isotropic literature authors straight anisotropic current literature table fulfillment essential requirements geometrically exact beam elements comparison existing proposed formulations tribution extends methodologies providing considerable improvements terms accuracy practical applicability generalization dynamic problems well theoretical numerical comparisons existing formulations table fulfillment essential requirements stated verified existing categories geometrically exact beam elements identified far isotropic straight anisotropic formulations literature author recent contributions well formulations proposed current work course stated requirements explicitly investigated mentioned references however evaluation results presented table derived best authors knowledge explanations derivations required evaluation individual categories found sections form paragraphs beinning expression review general symbols table mean requirement fully partly fulfilled question mark indicates required information could definitely extracted literature according table formulations newly proposed current work indeed close gaps left existing geometrically exact formulations however several existing beam element formulations already fulfill stated requirements fact might indicate increased complexity consistently incorporating additional constraint vanishing shear strains formulations order underline statement also type formulation proposed crisfield first one type preserving objectivity represented one possible example table right column given excellent properties existing geometrically exact finite element formulations question arises benefits gained applying instead beam element formulations quite obvious range low beam slenderness ratios negligible requires application beam element formulations taking fact account also underlying continuum theory based unconstrained cosserat continuum pointwise six degrees freedom resulting discrete problem statement seem easier formulated case however range high beam slenderness ratios thus exactly scenarios prevalent many practically relevant applications mentioned beginning contribution beam element formulations type may exhibit considerable advantages following potential benefits shall outlined lower discretization error level essential difference proposed ing beam element formulations lies neglect shear deformation beam theory underlying former category property independent chosen discretization strategy consequently otherwise comparable interpolation strategies applied assumed kirchhoff type formulations require less degrees freedom order yield polynomial approximation quality eventually discretization error level since additional primary variables required order represent shear deformation numerical benefits range high slenderness ratios range high slenderness ratios influence shear modes overall beam deformation negligible also beneficial abstain high stiffness contributions numerical point view mechanical problems slender beams typically lead stiff differential equations numerical problems deteriorating performance time integration schemes nonlinear solvers linear solvers avoidance stiff shear mode contributions considerably improve situation concretely detailed numerical investigations several numerical test cases involving highly slender beams considered test cases reveal considerably improved performance nonlinear solution schemes kirchhoff type instead reissner type discretizations employed similar trends predicted least theoretically behavior linear solvers time integration schemes iii smooth geometry representation proposed beam elements based centerline interpolations interpolations eventually result smooth contact kinematics property highly desirable order yield efficient robust contact algorithms see derivation reduced models special reduced model denoted beam theory derived general theory reduced model shown valid special problem classes concerning beam geometry external loads however present many fields application finite elements resulting reduced model typically feature simplified numerical implementation increased computational efficiency although theory lie focus work proposed formulations lie foundation reduced models possible benefits detailed section verified numerically section motivated intention current work main scientific contributions shall briefly highlighted two novel rotation interpolation strategies geometrically exact beam element formulations proposed first represents orthonormal interpolation scheme fulfills kirchhoff constraint strong manner regarded generalization scheme proposed allows exact conservation energy momentum alternative sets nodal degrees freedom simplify prescription essential boundary conditions theory second variant based weak enforcement kirchhoff constraint discrete realization kirchhoff constraint relies properly chosen collocation strategy combination employed smooth centerline interpolation also strategy completely avoid additional lagrange multipliers two element formulations two different sets nodal rotation parametrizations proposed one based nodal rotation vectors one nodal tangent vectors different choices shown yield identical fem solutions differ resulting performance nonlinear solvers effort required prescribe essential boundary conditions joints four finite element formulations resulting combination two interpolation schemes two choices nodal primary variables subject detailed comparisons respect resulting discretization error levels performance nonlinear solution schemes resulting finite element formulations combined finite difference time integration scheme large rotations recently proposed literature cardona implicit scheme allows accurate time integration basis optimized numerical dissipation identified extension scheme best author knowledge current work represents first application lie group time integration scheme based optimized numerical dissipation geometrically exact beam elements one first applications scheme geometrically nonlinear beam element formulations furthermore current work intends review evaluate existing geometrically exact beam element formulations compare formulations newly proposed work concretely fulfillment essential requirements according table studied analytically three identified categories isotropic straight anisotropic formulations identified general elaborated geometrically exact formulations previously proposed literature one exemplary representative formulations literature well kirchhofflove formulations presented current former works authors original contribution work given detailed systematic numerical comparison performed proposed geometrically exact two representative beam element formulations literature specifically resulting spatial convergence rates discretization error levels well performance nonlinear solution schemes compared different beam slenderness ratios eventually organization remainder contribution shall briefly given one distinctive property geometrically exact beam formulations presence large rotations within associated configuration space order provide theoretical basis subsequent derivations following section group large rotations well possible parametrizations introduced section general type geometrically exact beam formulations considered work theory thick beams presented subsequently section general theory restricted theory thin shearfree beams different methodologies imposing kirchhoff constraint vanishing shear strains strong weak sense investigated afterwards beam problem discretized section recently proposed extension method vector spaces lie groups applied beam formulations proposed work afterwards section spatial discretization performed represents core topic development geometrically exact kirchhoff beam elements considered work sections specific finite element interpolations employed translational rotational primary variable fields considered work proposed section already stated essential requirements employed spatial discretizations resulting finite element formulations detailed also detailed explanations derivations evaluation results table presented sections subsequent sections different finite element realizations resulting proposed interpolation schemes rotation parametrizations presented fulfillment basic requirements stated confirmed analytically concretely section presents basics geometrically exact simoreissner type formulation proposed serve reference several numerical comparisions performed section subsequent section benefits applying instead simoreissner element formulations range high slenderness ratios quantified afterwards two major developments work element formulation based strong constraint enforcement well element formulation based weak constraint enforcement proposed sections formulation two different variants nodal rotation parametrization one based nodal rotation vectors one based nodal tangent vectors derived eventually section proposed concepts resulting finite element formulations verified means proper numerical test cases reader primarily interested practical implementation newly proposed geometrically exact beam element formulations referred sections well appendices work rotation group category beam theories considered contribution assumes beam rigid consequently kinematics uniquely defined six degrees freedom three translational ones representing position vector centroid three rotational ones describing crosssection orientation thereto orthonormal triad consisting base vectors attached beam furthermore inertial cartesian frame associated material configuration corresponding inertial cartesian frame spatial configuration introduced nevertheless simplicity assumed frames coincide thus rotation global frame onto local frame described via orthogonal transformation following summation convention repeated indices holds throughout work index near matrix example index equation denotes basis associated tensor represented context geometrically exact beam theories tensor acts operator see material spatial objects rather mathematical point view rotation tensor identified element special orthogonal group orthogonal transformations according det action multiplication inverse element identity element det determinant transpose inverse identity matrix nonlinear manifold classified lie group tangent space denotes set skew symmetric tensors isomorphism enables unique expression vector denoted axial vector inserting special choice easily verified represents tangent space identity lie group lie algebra related exponential map exp according exp rotation group introduced without stating specific parametrization rotation tensor following sections two possible parametrizations useful development beam element formulations according theory presented parametrization via rotation vectors exist various parametrizations rotation tensor rotation vectors euler angles rodrigues parameters based minimal set three parameters also representations rotation tensor quaternions proven useful practical purposes within work two different parameterizations employed one presented section based rotation vectors section alternative parametrization especially suited kirchhoff formulations presented rotation vector parametrization explicitly given via rodrigues formula exp sin cos represents scalar rotation angle axis rotation throughout work denotes euclidean norm indicated notation exp equation represents representation exponential map initially introduced last section rotation vector given rotation tensor example extracted employing spurrier algorithm infinitesimal spatial quantity denoted multiplicative rotation vector variation spin vector allows express variation exp alternatively variation expressed material spin vector via right translation exp expressing spin vectors associated frames respectively yields thus components spatial spin vector expressed local basis identical components material spin vector expressed global basis relation also holds pairs spatial material quantities considered work see based defining equation representation tensors means components determined later use components shall expressed previous equation follows relation infinitesimal additive multiplicative increments given tangent operator defined tan inversion previous equation expressing multiplicative means additive increments given sin sin details derivation transformations interested reader referred rotation vector parametrization presented far represents tool formulation geometrically exact beam elements type following section alternative parametrization large rotations proposed offers advantages description kirchhoff type beam elements parametrization via smallest rotation triads alternative parametrization considered section consists four degrees freedom context kirchhoff beam elements presented later work tangent vector aligned space curve representing beam centerline due kirchhoff constraint vanishing shear strains first base vector triad expressed tangent vector based tangent vector arbitrary given triad one determine triad following denoted intermediate medium triad index results triad rotated onto tangent vector via smallest rotation see resulting base vectors represented expressions order shorten notation abbreviation introduced mapping practical choices triad presented section see also discussion possible singularities mapping subsequently definition intermediate triad according triad defined based relative rotation intermediate triad respect tangent angle exp cos sin cos sin equations uniquely define triad parametrized four degrees freedom evidently one four degrees freedom namely norm tangent vector influence triad orientation however turn next sections tangent vector quantity directly results beam centerline description thus additional degree freedom introduced order describe triad orientation relative angle consequently proposed type triad parametrization redundant later use spatial spin vector shall expressed means additive increments four parameters therefore split component parallel component perpendicular follows throughout work indices vector denote components vector parallel perpendicular vector respectively taking advantage easily derived see also next step exploited order formulate tangential component spin vector variation basis vector defined tangential component determined inserting equations split relation yields following expression spatial spin vector far four degrees freedom applied order uniquely describe tangent vector defined orientation length well orientation triad aligned parallel tangent vector next tangent vector triad shall described alternative set four degrees freedom represents rotation vector associated triad via rodrigues formula norm tangent vector following transformations hold two sets exp exp based set calculated way round next also transformation rule variations variations associated set shall derived combining equations yields transformation rule two sets inverse transformation derived similar manner mappings transform multiplicative rotation increments additive increments chosen parametrization thus represent analogon transformations case rotation vector parametrization see section since kirchhoff constraint vanishing shear deformation solely influences component spin vector following sections often useful express component additive increments tangential spin vector component instead additive increment regarded independent primary variable transformation basically represents reformulation represent corresponding mappings since mappings solely transform component independent actual definition triad consequently index omitted transformation matrices beam theory section fundamentals geometrically exact beam theory based work reissner well simo briefly summarized results section provide essential basis subsequent derivation kirchhoff type beam formulations figure kinematic quantities defining initial deformed configuration geometrically exact beam basic kinematic assumptions throughout work prismatic beams anisotropic shape considered initial unstressed configuration beam centerline defined line connecting centroids described space curve following index quantity refers unstressed initial configuration furthermore parametrization curve beam length initial configuration description initial configuration completed field orthonormal triads also denoted material triads following attached beam assumed undeformable according bernoulli hypothesis context represents unit tangential vector initial centerline base vectors coincide principal axes inertia throughout work prime denotes derivative respect parameter rotation global frame onto initial local frame described via orthogonal transformation introduced leading following definition initial configuration correspondingly deformed configuration beam time given orthogonal transformation maps global frame onto current local frame base vector general tangential deformed centerline anymore due shear deformation according section represented three rotation parameters rotation vector leading pointwise six three translational three rotational degrees freedom basic kinematic assumption geometrically exact theory considered far easily summarized following constrained position vector field associated initial deformed configuration represent convective coordinates describing position arbitrary material point within rigid order simplify notation subsequent derivations convective coordinate vector given well redundant name introduced kinematic quantities defining initial deformed configuration introduced far illustrated figure order simplify notation time argument dropped following later section derivative base vectors required similar equation derivative formulated referred spatial material curvature vector derived similar manner spatial material angular velocity vectors defined according completeness also spin vectors introduced section repeated denotes derivative respect time applying young theorem throughout work dot making use following compatibility relations shown following sections stress resultants mechanical equilibrium proper constitutive relations presented stress resultants mechanical equilibrium objective deformation measures denoting distributed external forces moments per unit length representing force moment contributions due inertia effects strong form equilibrium reads see force moment stress resultants acting beam area material form equilibrium equations derived inserting material stress resultants balance equations following principle virtual work admissible variations infinitesimal small arbitrary additive multiplicative changes current configuration compatible employed boundary conditions introduced next step represents vector additive virtual displacements vector multiplicative virtual rotations also denoted spin vector multiplication integration parts spatial weak form derived denote external forces moments neumann boundary considered beam based principle virtual work following objective spatial deformation measures identified material counterpart equation chosen starting point see material deformation measures material stress resultants determined analogous manner objective variation arbitrary vector appearing defined see easy verify deformation measures vanish initial configuration relations valid following variations derived components represent axial tension shear components represent torsion well bending constitutive relations finally constitutive relations stress resultants deformation measures required simplest constitutive law type given hyperelastic stored energy function material constitutive tensors following diagonal structure diag git diag young modulus shear modulus two reduced two principal moments inertia torsional moment inertia similar manner kinetic energy beam formulated diag mass density material inertia tensor spatial angular velocity vector already introduced linear momentum angular momentum derived similar external forces also energies momenta furnished total counterparts fext mext obtained integration fext mext definitions required subsequently based inertia forces moments yield similar also angular accelerations related via operator since beam problem discretized time see section details vectors either directly employed time integration scheme alternatively expressed via additive rate primary variable similar following relations formulated finally problem setup presented sections completed boundary initial conditions order end initial boundary value problem based trial space functions first derivatives satisfying weighting space functions first derivatives satisfying weak form equivalent strong form supplemented boundary conditions following two sections intended supplement geometrically exact beam theory sdsd far however provided information necessarily required derivation problem statement beam theory subsequent discretization procedures thus reader may alternatively skip content proceed section concretely section alternative procedure deriving weak strong form balance equations presented section basis variational problem statement properly defined lagrangian undertaken section meant confirm constitutive relations presented based derivation continuum mechanics consistency continuum model sense fully induced beam theory verified case locally small strains remark two possible time integration schemes derived variants either employing directly expressing via additive rates given indices refer two successive time steps problem represents typical finite difference time integration scheme newmark scheme first variant see section considered flexible since require specific rotation parametrization directly applied reissner type beam formulations well kirchhoff type beam formulations strong weak kirchhoff constraint enforcement without need adaptions due flexibility simple compact time integrator resulting procedure method choice employed throughout work interlude variational problem formulation beam theory case external forces act beam strong weak forms shall equivalently formulated basis hamilton principle lagrangian occurring defined difference kinetic potential energy based kinetic energy energy variation reads first second line objective variations see inserted furthermore additional relation applied second third line partial integration inertia terms together boundary conditions applied since external forces moments considered partial integration yields arbitrariness directly yields strong form inverting last step partial integration weighted strong form time along beam length yields weak form balance equations terms occurring weak form unloaded beam already identified third line conservative external forces could also included lagrangian consideration external forces external moments known see starting point strong form mechanical balance equations interlude relation constitutive laws aim following considerations derive constitutive laws consistent manner continuum theory thereto deformation gradient position field subject kinematic constraints geometrically exact beam theory shall derived subsequently also deformation tensor required two objects formulated based following definitions covariant basis vectors determined also contravariant base vectors determined second line via definition git making use relation inserting first line well yields finally inserting individual components deformation tensor determined however order gain insight underlying structure deformation gradient ellegant procedure suggested geradin cardona initially straight beams well linn initially curved beams employed slightly reformulating expression basis relative rotation tensor applying auxiliary relation solving products deformation gradient eventually reformulated according components vector identified denoted material strain vector read based deformation gradient deformation tensor derived according result consistently derived basic kinematic assumptions without additional approximations however order finally end simple constitutive laws geometrically exact beam theory based quadratic form following assumption small local strains made assumptions state small local axial shear strains considered radii initial deformed centerline curvature small compared radius following firstorder approximation small quantities considered setting neglecting last quadratic term based assumptions approximated deformation tensor reads components approximated deformation tensor resulting procedure yield applying material constitutive tensor jkl piolakirchhoff stress tensor formulated based approximated tensor scaled young modulus shear modulus introduced according young modulus poisson ratio gets obvious standard relations known geometrically exact beam theory terms vanishing stress components constitutive parameter front normal stress holds special case consequence kinematic assumption rigid requires existence reaction forces general order resolve two putative contradictions general cases constraint rigid weakened allowing uniform lateral contraction strain components proper warping field see details alternatively approximation employed first two lines praxis slight inconsistency often taken account unusual field structural theories see last step material force moment stress resultants determined integration stress vector acting beam material normal vector according definition proposed original work gip defintions moments inertia area well applied expected yield constitutive law identical one posa tulated comparable derivations based similar assumptions found original works kirchhoff love context beam formulations well current contributions context geometrically exact type formulations mentioned presented derivation yields alternative material objects well pulledback curved initial reference configuration straight reference configuration case material objects considered far see also remark page however components alternative material objects expressed respect curved local basis identical components original material objects expressed respect straight global basis via also stress tensor cauchy stress tensor determined fsft detf starting stress tensor cauchy stress tensor spatial stress resultants derived similarly since terms small strains relevant sufficient approximate deformation gradient required according detf consequently already postulated sections relevant operator given rotation tensor derivations continuum formulation material strain stress measures based straight reference configuration applied resulting material objects would based global basis total rotation tensor could identified relevant operator beam theory configuration space reissner type beams described pointwise six degrees freedom namely three translational components three rotational degrees freedom parametrizing section assumption vanishing shear strains made assumed sensible approximation highly slender beams see thus spanned remain perpendicular principally kirchhoff constraint vanishing shear deformations enforced strong weak manner parametrization reissner case chosen additional fields lagrange multipliers necessary order integrate constrained variational problem weak sense see section following sections concept developing parametrization degrees freedom fulfills kirchhoff constraint strong manner already derived briefly repeated extended dynamic case already introduced section quantity describe relative rotation material frame intermediate frame respect tangent vector according means one example suitable intermediate frame already given smallest rotation intermediate frame nevertheless following derivations presented rather general manner allows insert arbitrary alternative intermediate frame definitions kinematics within following three sections kirchhoff constraint strongly enforced see also according exp intermediate triad base vectors completely defined centerline field specified possible example given kirchhoff constraint incorporated expressing current configuration via new set primary variables inserting definition curvature vectors yields see represents torsion arbitrary intermediate triad field curvature beam centerline components spatial well material curvature vector read intermediate torsion term depends specific choice intermediate triad besides curvature vectors also spin vector adapted kirchhoff constraint see also analogy reissner beam formulations first component spin vector representing multiplicative increment directly employed weak form expressed via additive increments according consequently admissible variations new set variational primary variables defining kirchhoff case compatibility conditions similar stated due kirchhoff constraint conditions required tangential vector components relations second line yields deformation measures stress resultants defined kinematics compatible kirchhoff constraint according deformation measures constitutive relations stress resultants presented section adapted case inserting constrained curvature vectors deformation measures according yields objective variation still given spin vector constrained according construction shear components vanish due kirchhoff constraint abbreviation introduced remaining component representing axial tension based force split also constitutive relations simply consequence kirchhoff constraint inertia forces well inertia moments identical like reissner case spatial material angular velocities well spatial material angular accelerations either directly used employed time integration scheme expressed via additive rate primary variables latter approach transformation matrices depending definition employed intermediate triad see case intermediate triad well required order formulate relations similar kirchhoff case third variant advantageous similar curvature vector see spin vector see also angular velocity split following two components determined specified expressed via done remark three possible time integration schemes result different variants given remark tangent transformations following similarities become obvious strong weak form section spatial representation mechanical equilibrium considered following notation simplified summarizing external forces moments well inertia forces moments according shear forces provide work contribution kirchhoff case eliminated strong form mechanical equilibrium yielding following set four differential equations set sufficient order solve four primary variables soon stress resultants expressed kinematic constitutive relations section multiplying admissible translational rotational variations integrating parts gives equivalent weak form equilibrium constrained spatial spin vector according identified already substituted symbol indicated curly brackets stress resultants represented objective variations underlining geometrical exactness proposed kirchhoff beam formulation finally problem setup completed proper boundary initial conditions prescribes orientation tangent vector orientation respect rotation around tangent vector conditions modeled practice shown appendix introducing trial space satisfying weighting space beam problem fully defined emphasized concrete analytic expressions depend specific choice intermediate triad definition remark special choice intermediate triad field proposed triad beam centerline argued work choice favorable numerical purposes consequence singularities occurring straight beam segments however frame beneficial analytical treatment beam problems applied order derive analytical reference solutions numerical examples section remark shown beam theory presented far provides ideal basis derivation reduced beam models beam models valid certain restrictions concerning initial geometry external loads eventually yield considerably simplified finite element formulations basis special beam element formulation could derived completely abstains rotational degrees freedom distinct property drastically simplifies many standard procedures numerical treatment geometrically exact beam element formulations weak enforcement kirchhoff constraint last sections set primary variable fields chosen way kirchhoff constraint vanishing shear strains strongly fulfilled construction however flexibility subsequent discretization process see section could gained formulating reissner type beam problem allows two independent interpolations centerline field well triad field weakly enforcing kirchhoff constraint vanishing shear strains means additional constraint equations gtj order integrate constraints variational framework additional lagrange multiplier potential required lagrange multiplier fields interpreted shear force components reaction forces enforce constraint vanishing shear strains variation lagrange multiplier potential leads contribution kirchhoff constraint weak form first term represents weak statement kirchhoff constraint second term interpreted work contribution shear reaction forces similar displacement primary fields proper trial space proper weighting space introduced uniquely define resulting mixed beam formulation discrete realization presented section temporal discretization primary fields next section beam problems presented far discretized spatial discretization discussed section based finite element method fem recently proposed extension method vector spaces lie groups directly applicable beam element formulations proposed work presented section time integrator represents implicit finite difference scheme inherits desirable properties standard variant context finite element methods solid mechanics often convenient perform time discretization problem setting resulting spatial discretization opposite succession initial time discretization followed subsequent spatial discretization chosen second variant often applied development geometrically exact beam finite element formulations lead simpler discrete expressions following considered total time interval subdivided equidistant subintervals constant time step size time step index consequently solution primary variable fields describing current configuration computed series discrete points time associated configurations order simplify notation required subsequent derivations weak form see split contributions gint internal forces gkin kinetic forces gext external forces next step basics lie group method originally proposed presented method applied time discretization reissner kirchhoff type beam element formulations presented subsequent sections whose configuration space defined position field rotation field emphasized following procedure independent rotation parametrization employed different beam element formulations order express material angular velocities accelerations end time interval terms known quantities time unknown rotation field vectors representing spatial material multiplicative rotation increment time steps introduced exp besides distinctions made vector space time integrators implicit explicit scheme onestep scheme employed methodology order guarantee stability conservation properties two classifications made time integration schemes applied rotational variables first depending type spatial rotation interpolation succession spatial temporal discretization cases influence resulting discrete solution secondly distinguished approaches apply time integration scheme directly vectors approaches express angular velocities accelerations means additive rates primary variables see also axial vectors associated elements lie algebra consequently time integration schemes former type commonly denoted lie group time integration schemes context distinguished lie group schemes based material vectors schemes based spatial counterparts former variant considered work arguably one first lie group time integration schemes least context geometrically exact beam formulations proposed simo represents lie group extension classical newmark scheme contrary scheme presented following lie group extension standard method also scheme based four parameters simplifies variant simo special choice distinctive feature lie group scheme lies fact terms weak form evaluated end point considered time interval next step update formulas translational quantities given standard newmark scheme update scheme slightly changed form multiplicative configuration update rotations exp amod amod modified acceleration vectors occuring related physical acceleration vectors according similar manner modified well real physical angular accelerations related according amod amod later use favorable express vectors terms primary unknown similar relation formulated material vectors angular velocity acceleration amod amod proven integration scheme given equations yields favorable properties standard method accuracy unconditional stability within linear regime controllable damping modes minimized damping lowfrequency modes remarkably parameter choice leading optimal behavior identical standard scheme furthermore shown scheme consistently treat mass matrix contributions term occurring geometrically exact reissner kirchhoff type beam formulations stated extended scheme guarantee exact conservation energy linear angular momentum field lie group time integration schemes large variety methods aiming guarantee conservation properties proposed however perhaps essential advantage extended scheme compared alternatives lies simplicity flexibility independent beam theory reissner kirchhoff type employed spatial interpolation schemes well chosen set nodal primary variables terms rotation parametrization time integration scheme directly applied without need adaptions review mentioned section large number scientific contributions considers development schemes type formulations however hold true beam element formulations example formulations anisotropic type see requirements table considered dynamics far scheme proposed temporal discretization beam centerline torsional problem associated rotational degree freedom considered static manner arbitrary dynamic problems containing also rigid body rotations respect beam axis accessible approach temporal discretization applied present work encompasses translational rotational fields verified numerically section numerical dissipation provided extended scheme enable time integration highly nonlinear problems concretely example section considerably improved energy stability observed compared extension standard newmark scheme employed nevertheless supplementation proposed beam elements energymomentum schemes temporal discretization seems interesting direction future research spatial discretization methods primary fields spatial discretization exclusively considered context finite element methods within work represents core topic development geometrically exact kirchhoff beam elements considered work discretization beam centerline first spatial discretization beam centerline conducted thereto elementwise parameter space introduced element jacobian mapping infinitesimal increments parameter space space according following two sections two different interpolation schemes based lagrange polynomials hermite polynomials respectively presented discretization beam centerline based lagrange polynomials highest derivative primary variable occurring weak form beam theory first derivative centerline curve consequently interpolation beam centerline sufficient case thus standard choice trial functions given lagrange polynomials yielding vectors represent nodal positions whereas standard lagrange polynomials satisfying interpolation property element node coordinates well proper completeness conditions represents kronecker delta symbol matrix vector represent assemblies shape functions position vectors given lnr well following approach interpolation trial functions reads discretization beam centerline based hermite polynomials highest derivative primary variable occurring weak form kirchhoff beam given second derivative consequently interpolation centerline shape functions required fulfill element boundaries besides requirement centerline representation beneficial problem class requiring smooth geometry representation contact problems see order guarantee hermite shape functions employed hdi hdi vectors represent nodal position nodal tangent vectors two boundary nodes resulting finite elements matrix vector represent proper assemblies shape functions hti hdi well nodal position tangent vectors explicit expressions well already shown hermite shape functions fulfill interpolation property nodal positon tangent vectors well proper completeness conditions polynomials optimal choice constant lele given element length lele discretized initial geometry determined iterative manner basis first second derivative determined jacobi factor parameter derivative appearing given following relations finally also variations discretized properly chosen test functions following bubnovgalerkin approach also interpolation based hermite polynomials given expression hdi analogous manner procedure presented also extended hermite polynomials higher order discretization rotation field section two parametrizations rotation tensors investigated parametrization via rotation vectors parametrization via mapping basis set following two sections two variants employed order parametrize rotation tensors element nodes subsequent sections also two possible approaches interpolation nodal triads elements interior one based rotation vectors one based mapping presented parametrization nodal triads via rotation vectors according section rotation vectors employed primary variables describing nodal triads time step update iteration nonlinear solution scheme might either based additive rotation increments multiplicative rotation increments given exp rotations magnitude smaller unique rotation vector extracted given triad applying spurrier algorithm see within work rotation vectors always extracted manner within range transformation matrix additive multiplicative rotation vector increments see inverse exist always parametrization nodal triads via smallest rotation mapping alternatively nodal triads defined via relative rotation nodal intermediate triads relative angle respect tangent see nodal intermediate triads defined smallest rotation mapping nodal intermediate triad last time step onto basis vector current step exp variant used kirchhoff type beam element formulations combination hermite centerline interpolation first base vector defined via tangent vector beam centerline nodal triad defined nodal relative angle tangent vector node however emphasized contrary vector necessarily nodal primary variable considered node coincides one two element boundary nodes employed hermite interpolation tangent vector indeed represents nodal primary variable otherwise tangent vector simply represents interpolated centerline derivative position based additive increments nodal relative angles configuration update iteration iteration scheme time step reads exp fully defined centerline base vectors based additive updates well remark within work intermediate triads based mapping used two different purposes firstly used definition nodal material triads based nodal relative angles associated nodal intermediate triads mapping time see secondly used definition interpolated material triad field based relative angle field associated intermediate triad field mapping space reference triad see order distinguish two applications additional index refers associated relative angle review appendix appendix shown rotation parametrization nodal triads according last section considerably simplify modeling complex dirichlet boundary conditions joints compared tangent variant considered section straight formulations literature see requirement table employ rotation triad parametrization supports simple prescription conditions section turn tangent parametrization nodal triads hand lead better nonlinear solver performance thus practice one might combine advantage approaches employing rotation parametrization nodes complex dirichlet constraint conditions modeled tangent variant remaining nodes triad interpolation based local rotation vectors section triad interpolation presentented originally proposed shoemake first time employed geomerically exact beam element formulations crisfield considered nodes triad defined primary degrees freedom either according section section interpolation strategy presented section independent specific choice nodal primary variables first reference triad based triads nodes defined exp exp nodes chosen two middle triads elements even number nodes one middle triad elements odd number nodes see also equation based sightly different node numbering based definition interpolated triad field defined follows exp exp represent standard lagrange polynomials order material rotation vectors associated relative rotation triad node reference triad interpolation represents orthonormal interpolation scheme thus interpolated triad field still element rotation group furthermore interpolation scheme preserves objectivity deformation measures see curvature vector see resulting reads exactly represent state constant curvature const thus variant identified geodesic interpolation scheme since connects two points nonlinear manifold via shortest distance consequently variant interpolation represents linear interpolation quantities contrast interpolations beam centerline rotation interpolation nonlinear nodal degrees freedom thus field rotation vectors nodal values employed triad parametrization rotation vector interpolation resulting written abstract manner form explicit interpolation rule rotation vectors needed practical purposes triad field already given discrete version spin vector field field multiplicative rotation vector increments required next sections spatially discretized weak form balance equations linearization discretized fields consistently derived triad interpolation leading derivatives interpolations follow straightforward manner given generalized shape function matrices well derivatives derived original work see also appendix assembly matrices vectors well introduced shape functions depend rotational primary variables nonlinear manner nodal rotation vectors according section employed consequently every new configuration dependency rotational primary variables would considered within consistent linearization procedure case spin vector interpolation given used weak form according procedure order avoid additional linearization sensible follow approach based interpolation via lagrange polynomials strategy also applied within work leads nevertheless interpolation still based order end consistent linearization emphasized generalized shape functions fulfill following interpolation completeness properties shape functions exactly represent constant rotation vector increment fields since properties also fulfilled lagrange polynomials interpolation well petrovgalerkin interpolation spin vector field exactly represent constant distribution const element property important respect conservation angular momentum see sections triad interpolation based smallest rotation mapping section triad interpolation nodes considered nodes triad defined primary degrees freedom either according section section similar last section interpolation strategy presented following independent specific choice nodal primary variables concretely novel interpolation scheme proposed defines orthonormal triad field based given tangent vector field nodal triads following tangent vector field defined hermite interpolation beam centerline according based two nodes element boundary six degrees freedom respectively emphasized number nodes triad interpolation general differ number nodes hermite centerline interpolation see figure node numbering applied elements node numbering applied elements figure node numbering translational rotational primary fields nodal triads oriented tangential beam centerline curve thus first base vectors yield similarly section one nodal triad initially chosen reference triad interpolation scheme based reference triad nodal triads interpolation procedure defined according exp exp general curvature vector interpolations fulfill kirchhoff constraint strong manner given total torsion resulting derived manner remark ihe nodal relative angles different nodal primary variables section parametrization nodal triads quantities symbol chosen since cases relative angle material triad intermediate triad case stemming mapping space intermediate triad case stemming mapping time measured difference becomes clear realizing intermediate triad resulting smallest rotation onto see general differ intermediate triad resulting smallest rotation onto see review argued mapping exhibits singularity proposed finite element formulations occur rotation increments per time step larger large element deformations exhibiting relative rotations element center boundary nodes larger see consequently singularities practically relevant reasonable spatial temporal discretizations also argued singularities occurring relative rotations represent optimum achieved mappings tangent vector approach employed anisotropic formulations literature sightly different see requirement table one spatially fixed reference triad beam endpoint used initial mapping space one temporarily fixed reference triad every centerline position mapping time consequently formulations singularities could occur practically relevant configurations case relative rotation beam end point arbitrary centerline position exceeds case total rotation initial current configuration exceeds centerline position review literature smallest rotation mapping defined section often denoted rotation without twist thus sometimes mistakenly assumed intermediate triad field employed would exhibit vanishing torsion see slightly different interpolation scheme employed however according torsion intermediate triad field constructed via mapping vanish general curved configurations easily shown torsion vanishes limit fine discretizations lele limit intermediate triad field becomes identical elementwise bishop frame relation lim lim lele lim holds true however verified numerically section neglecting range finite element lengths general lead decline spatial convergence rate contrary arclength derivatives variations derived alternative anisotropic formulation see requirement table contain required terms stemming mapping discrete version spin vector field determined following approach spin vector discretized follows interpolation well follows matrix well vector represent assemblies lagrange shape functions nodal twist components alternatively discretized manner based case reads via assemblies spin vector shall completely expressed via nodal variations nodal primary variables thereto expressed directly follows interpolation inserting relations spin vector yields git notion introduced term distinguishing variant making use abbreviations relations interpolation spin vector given equation finally formulated ltk comparing equations leads conclusion difference variant expressed one additional term involving derivative reads shall investigated variants represent constant distribution const case counterparts thereto nodal variations chosen inserting according gives desired result result inserting spin vector interpolations fulfill interpolation property tangential spin vector components shown least represented correctly element nodes grassmann identity well employed next investigated variants choice leads constant spin vector field along beam element inserting using grassmann identity yields thus variant represent constant spin vector field since solely differ see term investigated inserting yields thus adding term variant yields desired result const spin vector interpolation case nodal variations given alternatively result obtained considering represents consistent variation objective triad interpolation see also section since interpolation objective variation discrete internal energy vanish infinitesimal rigid body rotations arbitrary stress resultant possible consequently const displayed exactly see also weak form recapitulatory interpolation represent arbitrary constant spin vector distributions interpolation possible problems special case result important order investigate conservation properties resulting finite element formulations see sections finally field multiplicative rotation vector increments derivative required consistent linearization discretized weak form follow equations simply replacing variations increments spin vector considered expressed via multiplicative nodal increments nodal primary variables case rotation parametrization nodal triads via nodal rotation vectors according section employed nodal vector multiplicative iterative rotation increments given tgi directly used triad update shown however rotation parametrization nodal triads via mapping nodal relative angles according section employed rotation vector increments shall expressed means additive increments nodal primary variables shown relation easily derived basis equations element boundary nodes last term simplified review triad interpolation scheme presented section similar approach proposed authors earlier contributions see section intermediate triad field constructed manner similar choice essential properties two approaches comparable slight advantages procedure presented choosing material triad reference triad makes interpolation scheme independent choice nodal primary variables according section according section furthermore locating reference triad element middle node makes element formulation symmetric extends maximal orientation difference material triads element boundary nodes represented latter property results maximal orientation difference allowed two tangent vectors order yield unique mapping see also section besides authors earlier work also straight anisotropic formulations literature exhibit mentioned lack symmetry see requirement table review alternative triad interpolation see section investigated defines intermediate triad field directly via smallest rotation intermediate triad field last time step onto current tangent vector field according exp first glance interpolation seems straightforward since nodal triads required constructing intermediate triad field however shown interpolation kind neither objective interpolation applied anisotropic formulations literature see requirement table requirements spatial discretization methods section essential requirements spatial discretizations translational rotational fields stated subsequently different beam elements presented fulfillment requirements investigated differentiability discrete fields first requirement spatial discretization methods concerns differentiability one hand requirement related weak form balance equations highest derivative occurring weak form beam theory order one leading requirement least discrete centerline triad fields continuity element boundaries provided lagrange centerline interpolation well two discussed approaches triad interpolation according sections second derivative beam centerline weak form balance equations distinctive property beam theory requires interpolation centerline least guaranteed hand certain applications contact formulations see benefit considerably smooth geometry representation existence tangent vector field along entire beam centerline conveniently furnished hermite interpolation objectivity properties objectivity play central role development geometrically exact beam finite element formulations importance properties traced back nonlinear nature configuration space resulting occurrence large rotations complicates interpolation rotational quantities furthermore explained historic background none early geometrically exact beam formulations fulfilled properties see already explained employed discretizations directly concluded fact none interpolation schemes based history values interpolated quantities nodal primary variables depend history values however corresponding nodal displacements always arise way finite element solution independent actual load path case considered physical problem words arising nodal displacements yield solution discrete optimization problem based proberly defined lagrangian associated physical problem numerical investigations performed end chapter property verified however throughout section fundamental property objectivity invariance applied deformation measures rigid body motions investigated thereto rigid body translation rigid body rotation superimposed onto beam centerline curve triad field rigid body motion characterized constant fields along beam thus following subscript denotes quantities result superimposed rigid body motion thus formulation denoted objective rigid body motion affect material deformation measures straightforward show versions deformation measures objective see question interest objectivity preserved employed spatial discretization schemes shown fulfillment following requirement guarantees objectivity geometrically exact beam formulation based theory introduced section following investigations exclusively applied discretized quantities order shorten notation subscript omitted throughout section first shall shown validity also sufficient invariance deformation measures kirchhoff beam theory valid follows based axial tension curvature vector well total torsion read identities original rotated deformation measures direct consequence thus also kirchhoff beam elements requirements sufficient order ensure objectivity following validity investigated interpolations sections objectivity centerline interpolations due linear dependence centerline interpolations nodal vectors proof first part independent intermediate triad field objectivity triad interpolation based local rotation vectors fulfillment objectivity interpolation shown original work interested reader referred reference objectivity triad interpolation based smallest rotation mapping based relations strong fulfillment kirchhoff constraint base vector resulting rigid body rotation yields next step nodal primary variables chosen nodal triads also rigidly rotated using following relation vectors intermediate triad see counterparts resulting rigid body motion derived gri concluded intermediate triad field rigidly rotated transformation property together considered following result derived fourth equation triad interpolation scheme exp exp exp exp thus interpolation unchanged rigid body motion together equations desired result already stated derived exp exp exp exp reformulations made use made transformation property according triad interpolation proposed section fulfills requirement objectivity mentioned derived third line fulfillment guarantees objective deformation measures provided consistently derived triad interpolation order verify latter restriction two individual contributions appearing third line shall subject closer investigation besides relation already deduced also torsion intermediate system calculated configuration resulting rigid body motion expected affected rigid body motion torsion remains unchanged underlines objectivity interpolation consistency torsion measure avoidance locking effects purely finite elements prone locking locking effects particularly relevant geometrically exact beam formulations shear locking well membrane locking shear locking definition appear beam formulations type membrane locking already observed geometrically linear kirchhoff beams see general membrane locking refers inability elements exactly reproduce inextensibility viz vanishing axial strain curved structures shells beams behavior traced back coupling kinematic quantities describing axial tension mode curved geometry focus subsequent investigations lies membrane locking least remarks concerning shear locking made end section characterization locking one possible definition locking deterioration spatial convergence rate dependence certain key parameter subsequent investigations reveal element slenderness ratio lele plays role key parameter associated membrane locking effect slender beams one question interest liability finite element formulation locking assessed quantitative manner mathematical point view question answered investigating stability finite element formulation example mixed finite element formulations stability criterion given wellknown lbb condition also denoted condition see since direct general analysis conditions often intricate also numerical tests suggested literature see mechanical point view locking typically explained occurrence parasitic stresses viz occurrence modes discrete solution part analytical solution consequently question formulation prone locking also answered investigating proper representative test cases parasitic stresses besides mathematical mechanical interpretations locking third namely numerical perspective see helpful numerical point view locking seen consequence system equations introduced constraint ratio allows least heuristic evaluation locking behavior finite element formulation constraint ratio defined ratio total number equilibrium equations neq total number constraints neq neq neq order analyze locking behavior constraint ratio continuous problem constraint ratio discretized problem evaluated infinite number elements compared underlying proposition elements especially means constraints degrees freedom present tendency lock whereas values constraint ratio indicate enough constraint equations available order reproduce constraint accurate manner following hypothesis case regarded optimal constraint ratio throughout contribution relevant locking phenomena analyzed based mechanical well numerical perspective corresponding concepts applicable straightforward manner future work stability promising element formulations also investigated mathematically rigorous manner either based direct analysis numerical stability tests membrane locking effect membrane locking context geometrically exact kirchhoff beam elements based hermite centerline interpolation according section investigated detail main results shall recaptulated found state exactly vanishing axial strains represented straight configurations abritariy curved configurations furthermore amount parasitic axial strain energy occuring states pure pending shown increase quadratically beam element slenderness ratio leading progressively stiff system answer constraint ratios discrete beam problem determined indicating considered element formulation prone membrane locking different solution strategies exist order get rid locking effects approach assumed natural strains ans see reduced integration see alternative procedure proposed denoted minimally constrained strains mcs compared proposed mcs scheme characterized assumed approach see derived variationally consistent manner concretely contribution axial tension weak form mechanical equilibirum equations replaced ncp ncp denotes parameter coordinate ncp number collocation points cps original strains evaluated moreover lagrange shape functions polynomial order ncp linearizing undeformed straight configuration yields structure typical geometrically linear finite elements ncp last equation additional abbreviation introduced order shorten notation remark actually residual vector stated consists functional expressions integrated exactly gauss integration however numerical simulations deviations results number gauss points turned small compared discretization error therefore notion exact integration used whenever four gauss points applied remark last section shown strain field objective meaning strain field change consequence rigid body motion value certain configuration independent deformation path leading configuration since assumed strain field represents pure original strain field evaluated fixed collocation points former also fulfill objectivity number location cps chosen according ncp order motivate choice constraint ratio resulting mcs method different sets cps well alternative methods ans presented following case strain given sufficient constraint vanishing axial strains fulfilled collocation points order end vanishing axial strain energy since hermite interpolation provides centerline curve first derivative consequently also axial tension element boundaries thus exactly one constraint equation results interior element boundary remark chosen cps motivated axial strains element boundaries element boundary node would provide two constraint equations one previous element one subsequent element following table constraint ratios resulting three methods ans mcs compared different choices concerning number locations collocation gauss points shown ans approach based cps vanishing axial strain values pure bending state requires four cps neq nele table quantitative comparison different methods index table indicates collocation gauss points lie elements interior variants marked index also employ element boundary nodes according statements made variants represent constraints associated axial tension optimal manner whereas variants tendency lock consequently first glance variants seem equally suitable however derived variants lead underconstrained system equations allowing modes yielding singular system equations rank deficient stiffness matrix straight configuration variant exactly provides minimal number constraint equations required order avoid modes straight configuration reason method denoted method minimally constrained strains similarly optimal constraint ratio avoidance modes verified choice cps general case remark course locking would avoided beam elements could exactly represent internal energy associated pure bending according mcs method fulfills requirement straight configurations arbitrary curved configurations yield slightly system equations thus state constant curvature vanishing axial tension displayed exactly however shown numerically membrane locking avoided result reasonable since variant still fulfills optimal constraint ratio arbitrarily curved configurations remark alternative variant also reduced integration scheme integration points could applied axial tension term weak form yielding number constraint equations neq mcs approach however within contribution mcs method preferred due arguably consistent variational basis uniform integration scheme resulting individual work contributions weak form concluded choice element boundary nodes element midpoint cps mcs method leads minimal possible number constraint equations optimal constraint ratio sequently successful avoidance locking effects could confirmed approach similar effectiveness expected reduced integration axial tension terms integration points contrary minimal number three integration points possible reduced integration scheme leads increased constraint ratio consequently suboptimal locking behavior mcs methodologies aim reduction number constraint equations functional principle ans approach different parameter coordinates determined constraint already correctly fulfilled original element formulation applying latter representative test case parameter coordinates typically chosen cps ans approach procedure avoid locking effects manner independent number constraint equations drawback ans method positions points may change general deformation states geometrically nonlinear regime might considerably deteriorate effectiveness approach theoretical considerations recommend proposed mcs approach method choice order avoid locking effects kirchhoff beam elements considered contribution numerical results presented confirmed prediction review three categories isotropic straight anisotropic geometrically exact finite elements available literature see requirement table consequences membrane locking observed however rigorous treatment seems missing works example oscillations membrane forces observed cured means special force averaging procedure however procedure seems rather step invasion actual finite element formulation would also improve final displacement solution axial tension term weak form replaced averaged constant approximation order able exactly representing solution straight beam axial load however influence procedure membrane locking effects curved configurations configurations actually relevant membrane locking investigated finally proposes mixed finite element formulation combination multi patch approach order treat locking effects interestingly variant patches could identified hermite interpolation comparable turned favorable avoidance locking compared single patches due lower continuity enforced former approach however geometrical compatibility equations multi patch approach well considered numerical examples seem cover geometrically linear regime contrast numerical examples section work confirm gap could closed proposed mcs method successfully cures membrane locking problems shear locking phenomenon shear locking lie focus current work nevertheless cause locking effects shall least briefly compared situation already discussed membrane locking shear locking denotes inability finite element exactly represent state vanishing shear strains situation illustrated means pure bending example assumed beam centerline discretized either lagrange polynomials according hermite polynomials according triad interpolation given already stated triad interpolation exactly represent constant curvatures hand shown hermite centerline representation also counterpart based lagrange polynomials exactly display state constant curvature means state displayed exactly combination constant curvature words interpolation spaces applied translation beam centerline rotation field triad field optimally match sense state constant curvature vanishing shear deformation represented exactly similar membrane locking ratio shear stiffness bending stiffness increases quadratically beam element slenderness ratio thus element slenderness ratio represents key parameter locking effect investigations membrane shear locking effects geometrically linear nonlinear beam element formulations example found review end section also view considerations shall made concerning element patch tests requirement table patch test geometrically exact beams could example require exact representability state constant curvature bending torsion constant axial tension well constant case formulations beam centerline curve associated configuration given helix constant slope investigated numerical example section following distinguished strain energy associated state exactly represented additionally also beam geometry terms centerline triad field state exactly represented sections verified theoretically numerically element formulations based strong enforcement kirchhoff constraint proposed section general neither represent strain energy beam geometry associated state due identical measure centerline curvature comparable centerline discretizations employed isotropic straight anisotropic formulations literature directly concluded also none formulations fulfill patch test neither energetic geometrical sense see table hand verified theoretically numerically element formulations based weak enforcement kirchhoff constraint see section well formulation proposed crisfield denoted element section represent exact energy state patch test due employed geodesic triad interpolation appropriate collocation scheme mcs reduced integration coupling centerline triad field exact geometry exist recent beam element formulations exactly represent patch case energetic also geometrical sense unfortunately explicite analytic representation discretization underlying formulations form helicoidal interpolation applied triad centerline field possible interpolations emphasized helicoidal scheme identified one geodesic interpolations semidirect product manifold denoted special euclidean group formulation crisfield consists two individual geodesic interpolations linear one constant slope euclidean vector space well spherical interpolation constant curvature special orthogonal group together composing interpolation scheme direct product manifold extension helicoidal interpolations schemes given representation interpolation possible anymore triad centerline field generated strain field implicit manner via numerical integration contrast formulations element formulations typically order guarantee reasonable convergence orders see also section clear increased numerical effort required implicitly defining helicoidal fields via numerical integration could overcompensate possible advantages terms higher approximation quality developing comparable element formulations exactly fulfill patch test energetic geometrical sense least seems promising direction future research compare formulations kind elements proposed optimal convergence orders order compare convergence behavior different finite element formulations error measure required thereto following relative considered numerical examples section umax rre denotes numerical solution beam centerline certain discretization examples without analytic solution standard choice reference solution rre numerical solution via element see section employing spatial discretization factor four finer finest discretization shown convergence plot normalization element length makes error independent length considered beam second normalization leads convenient relative error measure relates maximal displacement umax load case examples also relative energy error rel considered represent stored energy functions see associated certain discretization reference solution convergence plots selected numerical examples considered optimal convergence rates norms expected different beam element formulations shall briefly discussed convergence energy error minimized finite element method dominated highest derivative primary variable fields occurring energy see consequently weak form since employed hyperelastic stored energy functions represent quadratic forms derivatives primary variable fields convergence rate energy error yields lele element length polynomial degree completely represented trial functions convergence rate ritz solution variational problem order shown second term represents dependence convergence rate energy error convergence reflecting variational basis finite element method first term represents pure polynomial approximation trial functions respect considered primary variable field cases solid elements considered first exponent smaller second one dominates overall discretization error reason first term considered many authors however following shown especially kirchhoff type beam elements also second term important thereto expected convergence rates reissner kirchhoff type beam elements shall briefly discussed element formulations type highest derivative primary variable fields occurring weak form associated beam problem thus reissner beam elements considered work convergence rate six expected energy error dominated first term leading corresponding optimal convergence rate four element formulations type subsequently proposed kirchhoff beam elements lead values polynomial degree triad functions highest derivative weak form consequently convergence energy error order four furthermore exponents terms take value four also leading expected convergence rate four thus also second term considered kirchhoff problems kind reason least polynomials order three chosen trial functions kirchhoff beam elements reducing polynomial degree would lead undesirable decline rate four two thus kirchhoff beam elements proposed work regarded approximations lowest order reasonable numerical point view however fact exponents take value also means cases usually examples involving complex deformation states strain distributions second term might determine overall thus order fully exploit approximation power employed discretization often sensible apply trial functions increased polynomial degree case first term determine overall error level sufficiently fine discretizations since second term converges faster consequently discretization error exclusively limited approximation power applied polynomial order extension proposed kirchhoff beam elements hermite interpolations order possible straightforward manner treated future research work numerical example section first proof concept given extension remark based considerations question arises trial function orders kkl formulations based theory chosen numerical examples order perform reasonable comparison convergence behavior answer question depends primary interest might either lie energy convergence kirchhoff elements compared reissner elements leading equal different energy error rates alternatively one could compare kirchhoff elements reissner elements leading equal energy error rates different rates review later section quadratic interpolation applied relative angle field interpolation scheme since orientation material triad field determined relative angle well tangent vector field latter polynomial order two interpolation sufficient triad field discretization even higher polynomial degree could improve exact polynomial representation rotational strains first order hand interpolated linearly constant term rotational strains could exactly represented second term would dominate leading decline expected optimal order two meier considerations confirmed numerically formulations straight type see requirement table also apply interpolation relative angle comparable intermediate triad definition thus also formulations decline optimal convergence order two expected conservation properties since finite element solution converges towards corresponding analytic solution limit fine spatial discretizations elementary properties analytic solution conservation linear momentum rather equilibrium forces statics conservation angular momentum rather equilibrium moments statics well conservation energy rather balance external internal work problems also fulfilled numerical solution lele however often desirable provide properties already arbitrarily rough spatial discretizations question properties problem inherited spatially discretized problem later investigated different beam element formulations proposed subsequent sections thereto use made fact discretized weak form balance equations fulfilled arbitrary values nodal primary variable variations choosing nodal primary variable variations associated virtual motion represents rigid body translation given allows investigate conservation linear momentum special choice leads vanishing contributions internal forces moments discretized weak forms associated beam theory inserting weak forms yields fext fext exact conservation linear momentum const unloaded system viz fext since reaction forces dirichlet supports also included fext equivalent equilibrium forces static case similarly choice nodal primary variable variations representing rigid body rotation given allows investigate angular momentum relation leads well vanishing contributions internal forces moments discrete versions weak forms associated beam theory inserting weak forms yields mext mext exact conservation angular momentum const unloaded system viz mext since possible reaction moments dirichlet supports also included mext relation equivalent equilibrium moments static case finally choice nodal primary variable variations according allows investigate mechanical power balance inserting discrete versions weak forms associated theory making use relations int see section well yields following relation pext pext consequently exact energy conservation const unloaded system viz pext far shown exact conservation linear momentum angular momentum energy see equations guaranteed spatially discretized problem provided special choices translational rotational variation fields contained discrete weighting space considered finite element formulation following sections question indeed represented discrete weighting functions investigated proposed beam element formulations course also time integration scheme influences conservation properties fully discrete system considerably however investigation factor lie within scope work general stated three conservation properties considered fulfilled finite element formulation formulation objective test functions consistently derived manner first ensures unique potential exists form discrete storedenergy function invariant rigid body motions second ensures corresponding contribution weak form represents exact increments potential fulfills third requirement per definition also first second requirement since infinitesimal rigid body translations rotations according lead vanishing increments objective potential however opposite conclusion obviously hold true also formulations least fulfill first second requirement chosen test functions represent rigid body translations rotations see element section element section review following argumentation expected objective pathindependent fem discretizations employed formulations straight type see requirement table general fulfill conservation properties long variation discretization process required derive test functions conducted consistent manner additional approximations small tension assumptions according discussed applied could spoil consistency situation slightly different anisotropic formulations character employed discrete deformation measures might general lead energy increments infinitesimal rigid body translations rotations according exact representation energy increments resulting consistently derived test functions may case exactly reason conservation properties spoiled eventually also existing formulations straight type table shall discussed standard polynomial interpolation based lagrange shape functions applied tangential spin vector component similar variant consequently shown equation also formulations represent constant spin vector distributions exact conservation angular momentum shown straight formulations indeed represent formulations turn also guarantee conservation energy verify statement following considerations made discrete spin vector field employed consistent underlying triad interpolation sense answered since explicit triad interpolation scheme given required considered isotropic formulations however least derivative discrete triad field defined employed torsion interpolation based lagrange polynomials consistent interpolation identified based second compatibility condition reads since interpolated via lagrange polynomials interpolated via hermite polynomials relation holds consistent interpolation tangential spin vector component fulfilling based pure polynomial interpolation done mentioned references consequently interpolations type beam element throughout work reissner type beam element formulation proposed crisfield following referred element serve reference formulation numerical comparisons next section main constituents required derive element residual vector presented subsequent sections element formulation investigated respect possible locking effects fulfillment mechanical conservation properties introduced sections element residual vector section element residual vector element derived general beam problem first trial weighting functions replaced discrete counterparts taken trial subspace weighting subspace following finite elements vectors nodal primary variables considered centerline interpolation based lagrange polynomials order according section furthermore rotation field interpolation follows equation section based nodes combination approach spin vector discretization given contrast original works modified scheme section employed time integration defining velocities accelerations required inertia forces moments stated inserting interpolation schemes presented previous sections weak form balance equations yields element residual vector contributions rcj rcj according rcj rcj subscripts distinguish residual vector contributions associated variations within work linearization kcj based multiplicative rotation increments according given employed dynamics element residual vector rcj stiffness matrix kcj slightly differ original work due applied time integration scheme avoidance locking effects authors proposed reduced gauss integration scheme order avoid shear locking membrane locking range high beam slenderness ratios thereto integration points employed integration internal force contribution element effectiveness procedure verified subsequent numerical examples shall briefly motivated following considerations beam problem based neq differential equations describing beam problem pointwise neq constraint equations order represent state vanishing axial strains vanishing shear strains prevalent pure bending problem consequently constraint ratio problem yields neq consequence reduced integration discrete number constraint equations takes value neq nele given total number equations neq nele application proper dirichlet conditions constraint ratio discrete problem results lim nele nele nele relation yields optimal constraint ratio element formulation consequently locking effects expected investigations made far refined realizing element exactly represent internal energy associated pure bending state order understand statement internal energy split contributions stemming torsion bending contributions stemming axial tension shear deformation pure bending state energy contribution vanish thus total internal energy pure bending state given turn uniquely defined curvature vector field const order represent desired constant distribution curvature vector field possible employed triad interpolation nele nele nodal rotation vectors arise properly one remaining nodal rotation vector describes rotational rigid body modes beam since curvature vector field defined via derivative rotation field one remaining nodal rotation vector also interpreted integration constant resulting integration curvature field additionally nodal position vectors arise way requirement fulfilled exactly employed triad trial function spaces reduced gauss integration scheme applied element yields finite number constraint equations order satisfy thus similar rotation field nele nodal position vectors arise properly order fulfill constraint equations one remaining nodal position vector describes translational rigid body modes beam nodal position rotation vectors always arise way pure bending case represented consists constant curvature vector field vanishing reduced integrated energy contribution axial shear strains well six superposed rigid body modes consequently torsion bending modes represented well axial tension shear values gauss points represented locking effects expected considerations easily extended arbitrary curvature fields representable employed triad interpolation arbitrary fields term occurring energy integral integrated exactly reduced gauss integration scheme sections expected result discrete hyperelastic energies associated pure bending states represented exactly beam element formulation verified means corresponding numerical test cases conservation properties following investigated beam element formulation proposed crisfield repeated section represent variational fields required order guarantee conservation linear momentum conservation angular momentum well conservation energy representation rigid body translation trivial given nodal primary variable variations similarly rigid body rotation displayed nodal primary variable variations follows conservation linear angular momentum guaranteed statement holds discretizations discretized centerline variation since variants fulfill proper completeness conditions exactly represent constant vector field nodal velocities angular velocities problem chosen primary variable variations variant leads per definition exact representation rates spatially discretized hyperelastic kinetic energy consequently exact energy conservation spatially discretized problem contrary variant employed variationally consistent triad interpolation occurring discrete internal kinetic energies consequently weak form represent exact energy rates spatially discretized problem motivation beam theories geometrically exact beam elements unify high computational efficiency accuracy fields application thick beams involved effect shear deformation important favorable compared kirchhoff type counterparts however increasing beam slenderness ratio shear contribution overall beam deformation decreases furthermore exactly avoidance high stiffness contributions shear modes makes theory thin beams applicable also favorable range high beam slenderness ratios brief section possible benefits applying kirchhoff type beam elements range high slenderness ratios illustrated least approximately quantified section effects also investigated verified means numerical examples improved stability properties time integration scheme dynamic equations motion highly slender beams typically result stiff partial differential equations pdes increasing beam slenderness ratio ratio high eigenfrequencies associated shear modes intermediate eigenfrequencies associated axial tension twisting modes low eigenfrequencies associated bending modes increases considerably consequence stability requirement explicit time integration schemes leads small critical time step sizes compared large oscillation periods bending modes contrary implicit time integration schemes provide unconditional stability linear regime small deformations however large deformation regime also performance considerably deteriorated contributions despite stability aspect modes strongly affected time discretization error avoided long analysis required specific application order illustrate relevant frequency spectrum following proportionalities eigenfrequencies resulting pure bending pure torsion pure axial tension pure shearing given linearized beam problem git according relations ratio axial torsional eigenfrequencies bending eigenfrequencies increases linearly increasing slenderness ratio ratio shear eigenfrequencies bending eigenfrequencies increases quadratically increasing slenderness ratio thus theoretical point view avoidance shear modes could already improve numerical behavior considerably since numerical examples section mainly focus static analysis present brief outlook possible dynamic investigations numerical verification theoretical considerations lie within scope work however numerical investigations question available literature confirm predicted trend lang arnold investigated geometrically nonlinear oscillations slender beam modeled means geometrically exact theory discretized via finite differences see also order measure influence modes time integration stability maximal possible time step sizes applied explicit time integration scheme determined three different mechanical beam models full extensible beam formulation beam formulation subject kirchhoff constraint vanishing shear deformation beam formulation subject kirchhoff constraint additional inextensibility constraint enforcing vanishing axial tension roughest discretization considered numerical experiment investigated slenderness ratio rather moderate slenderness ratio compared many applications mentioned section time step size could increased factor abstaining shear mode factor additionally abstaining axial tension mode results indicate potential kirchhoff type beam formulations furthermore suggest first step towards extensible kirchhoff beam formulation might already represent essential one respect numerical savings improved performance iterative linear solvers according previous argumentation high ratio highest lowest dynamical eigenfrequencies measured dynamic spectral radius deteriorates performance time integration schemes similar manner performance iterative linear solvers see decreases increasing ratio highest lowest eigenvalue tangent stiffness matrix measure condition number matrix furthermore even direct linear solvers applied high condition numbers might considerably limit achievable numerical accuracy especially dynamics errors tend accumulate effects undesirable following influence different deformation modes condition number investigated simplicity physical units considered beam problem chosen element length lies range lele since element length better element jacobian typically enters element formulation different exponents occurring different stiffness matrix entries element length lele seems reasonable choice respect conditioning case resulting contributions element stiffness matrix beam element formulation linearized respect straight configuration typically obeys following proportionalities denote stiffness contributions bending torsion axial tension shear modes long lele holds discretization kept fixed radius decreases linearly increasing slenderness ratio according expected ratio high stiffness contributions shear axial tension modes low stiffness contributions torsional bending modes also condition number increases quadratically beam slenderness ratio furthermore expected pure neglect shear modes sufficient order improve conditioning thus supplementation proposed formulations additional inextensibility constraint seems beneficial order get also rid axial stiffness contributions improve also performance linear solvers however formulation additional inextensibility constraint lie focus contribution see also remark end section improved performance nonlinear solvers relation performance nonlinear solver scheme conditioning considered problem measured via condition number tangent stiffness matrix clear case linear solvers nevertheless typically least expected performance nonlinear solvers also deteriorates problems showing slope differences target function several orders magnitude stepping different directions directions activate shear axial tension modes directions activate bending torsional modes interestingly numerical examples investigated work confirm trend nonlinear solver performance considered reissner type beam elements deteriorates drastically increasing slenderness ratio total number newton iterations required kirchhoff type formulations remains almost unchanged reduced system size kirchhoff type beam element formulations require degrees freedom representing mode shear deformation expected least long convergence deteriorating phenomena locking occur polynomial approximation discretization error level guaranteed fewer degrees freedom prediction confirmed numerical examples section smooth geometry representation proposed beam elements based centerline interpolations interpolations eventually result smooth contact kinematics property highly desirable order yield efficient robust contact algorithms see abstaining algorithmic treatment large rotations already mentioned earlier work proposed beam formulations provide ideal basis derivation reduced beam models valid certain restrictions concerning external loads initial geometry example beam element formulation could derived extended dynamic problems based pure centerline representation consequently avoid treatment large rotations associated degrees freedom thus many steps within nonlinear finite element algorithm typically complicated presence large rotations spatial interpolation time discretization tangent stiffness matrix mass matrix incremental iterative configuration updates comparable standard solid finite elements formulation details reader referred sources potential benefits motivation development element formulations based theory thin beams different realizations element formulations based weak strong enforcement kirchhoff constraint presented next sections influence aforementioned aspects resulting numerical behavior verified section via proper test cases remark discussed performance iterative linear solvers could improved supplementing proposed kirchhoff type element formulations additional inextensibility constraint unfortunately contrast kirchhoff constraint straightforward way enforce inextensibility constraint directly special choice primary variables collocation approach would allow lagrange multiplier elimination element level long interpolation property fulfilled position vector field element boundary nodes statement easily illustrated considering straight beam element arbitrary order order avoid modes inextensibility means case solution nodal position vectors two boundary nodes arise independently fulfill constraint ele beam element based strong constraint enforcement section finite element formulation based strong enforcement kirchhoff constraint presented section variant based nodal triads parametrized via tangent vectors according section investigated section transition rotation parametrization section conducted similar element presented previous section also element formulation avoidance possible membrane locking effects well fulfillment mechanical conservation properties verified sections appendix suitability rotation nodal triad parametrizations modeling practically relevant dirichlet boundary conditions joints evaluated residual vector parametrization similar case trial weighting functions replaced discrete counterparts taken trial subspace weighting subspace following elements nodal primary variables well set nodal primary variable variations considered see also figure centerline interpolation based hermite polynomials according section completely defined two element boundary nodes rotation field interpolation follows equation concretely quadratic rotation interpolation based three nodes thus also involving element center node considered since orientation material triad field determined relative angle well tangent vector field latter polynomial order two interpolation sufficient triad field discretization meier confirmed higher interpolation order improve approximation quality lower interpolation order lead decline convergence rate time integration section employed thus leading inertia forces moments given inserting discretizations equation taking advantage spin vector interpolation given eventually yields element residual vector variant element formulation formulation transformed variant based spin interpolation scheme simply omitting terms yields order avoid membrane locking range high element slenderness ratios following axial tension variation based mcs procedure see also applied thus finite element formulation obtained simply replacing axial force discrete axial tension variation operator discrete weak form discrete expression internal energy associated modified axial tension eventually reads element formulation based degrees freedom residual denoted element stands strong kirchhoff constraint enforcement combined nodal triad parametrization via nodal tangents correspondingly combination degrees freedom residual denoted element referring consistent spin vector interpolation underlying variant emphasized replacement original axial tension terms corresponding mcs terms according standard kirchhoff type beam element formulations considered work examples comparison reasons also variants without mcs method considered additional abbreviation employed element linearization based increment vector employed see appendix contrary multiplicative rotation variations occurring quantities represent additive rotation increments nodal relative angles residual vector rotation parametrization scenarios applications complex rotational dirichlet coupling conditions prescribed element boundary beneficial employ alternative parametrization triads element boundary nodes via rotation vectors according section case alternative set nodal primary variables given employed represent rotation vectors associated boundary triads corresponding spin vectors represent magnitudes nodal tangents introduced section case nodal tangents hermite interpolation primary variables anymore expressed well see also transformation rule exp transformation variations well given transformation matrices according leads following relation blank entries zero required transformation matrices well follow equation exp order simplify transformation sets degrees freedom considered sections residual according slightly reordered residual rrot introduced rtt rtt rtt rtt rtt rtt rtt rtt rrot rtrot rtrot rrot rtrot rrot rrot rot inserting relation virtual work contribution resulting one beam element yields rrot rrot according matrix inverse long physical interpretation beam centerline position node initial length would compressed length zero since scenario impossible physical point view requirements assumed fulfilled consequently transformation residual vector residual vector rrot based matrix statement holds transformation global residual vectors rrot via matrix represents assembly element matrices based considerations following relation established rrot rank rrot long unique solution exists solution rrot lead mechanical equilibrium configuration words pure performed section section change results discretized beam problem nevertheless transformation matrix depends primary degrees freedom nonlinear manner considered within consistent linearization procedure throughout work element formulation based degrees freedom residual transformed via together mcs approach denoted element stands strong kirchhoff constraint enforcement combined nodal triad parametrization via nodal rotation vectors correspondingly combination degrees freedom residual transformed via together mcs approach denoted element since sktan element yield finite element solution expressed via different nodal primary variables following theoretical investigations concerning locking behavior conservation properties performed sktan element element linearization based increment vector employed see appendix represent multiplicative increments identified additive increment avoidance locking effects order investigate locking behavior element investigations already made section extended general case kirchhoff beam problem described neq differential equations constrained neq constraint equation case pure bending state shall represented thus constraint ratio problem yields neq due employed mcs method discrete number constraint equations takes value neq given total number equations neq application proper dirichlet boundary conditions discrete constraint ratio yields lim nele relation yields optimal constraint ratio element formulation consequently locking effects expected furthermore shown section requirement representing straight beam configuration arbitrary distribution yields number independent equations equals number degrees freedom consequently state represented exactly modes associated state expected extension statement straightforward investigated moreover also stated section discrete hyperelastic energy associated pure bending case displayed exactly element course statement still holds sections expected result discrete hyperelastic energies associated pure bending states represented exact manner beam element formulation verified means corresponding numerical test cases also shown property leads slightly increased discretization error level compared subsequently derived element however observation independent element slenderness ratio attributed membrane locking conservation properties also proposed beam element strong enforcement kirchhoff constraint shall investigated variational fields required conservation linear momentum conservation angular momentum conservation energy represented corresponding discrete weighting subspace representation rigid body translation given nodal primary variable variations result verified inserting choices made hermite interpolation making use completeness conditions underlying hermite polynomials see yields const well inserting relations together either results required vanishing discrete spin vector field thus well bubnovgalerkin variant spin vector interpolation lead exact conservation linear momentum next rigid body rotation displayed following choice nodal primary variable variations inserting according gives desired result section see shown based nodal values interpolation exactly represent virtual rigid body rotation variant thus interpolation guarantee exact conservation angular momentum result confirmed subsequent numerical examples finally conservation energy investigated nodal velocities angular velocities problem chosen nodal primary variable variations variant leads per definition exact representation rates discrete internal kinetic energy exact energy conservation spatially discretized problem contrast variant variationally consistent triad interpolation underlying discrete energies guarantee energy conservation problem beam element based weak constraint enforcement alternative formulation presented last section beam element presented based weak fulfillment kirchhoff constraint thus basis intended element formulation provided beam theory first step finite element formulation type continuous centerline representation derived section afterwards kirchhoff constraint vanishing shear strains enforced order end finite element formulation kirchhoff type following derivations section weak statement kirchhoff constraint realized introducing spatial interpolations lagrange multipliers variations choosing proper discrete trial space proper discrete weighting space resulting nonlinear system discrete equilibrium equations contain discrete lagrange multipliers additional nodal primary variables exhibit saddle point type structure order avoid additional effort solving large system equations saddle point structure slightly different approach chosen next section modified reissner type beam element formulation presented based smooth hermite centerline interpolation mcs type strain axial strain also shear strains applying constraint vanishing shear strains consistent manner directly reinterpolated strain fields yields collocation point type approach constraint enforcement require additional lagrange multipliers see section throughout contribution variant preferred since neither yield additional lagrange multiplier degrees freedom saddle point type system equations also element formulation two variants concerning nodal rotation parametrization according sections presented following sections basic formulation hermitian element reissner type beam element formulated section represents intermediate step derivation corresponding kirchhoff type beam element formulation next section discrete beam centerline representation given hermite interpolation based position tangent vectors two element boundary nodes furthermore rotation interpolation given representation nodal triads finite element formulation considered basis strain similar mcs method kirchhoff case axial strain treated entire deformation measure order avoid membrane well shear locking basis also stored energy function given replaced one introduce following set degrees freedom well associated variation vector based weak form definitions element residual vector derived follows rhs rhs lts element formulation could applied problems thick beams higher continuity requirements beam contact however formulation solely represents intermediate step derivation kirchhoff beam elements weak enforcement kirchhoff constraint performed next two sections residual vector parametrization due general weak constraint enforcement section simplified according kirchhoff constraint exactly fulfilled three collocation points following parametrization chosen directly fulfills constraints without need additional lagrange multipliers thereto set nodal degrees freedom well set nodal primary variable variations section chosen case approach following discrete spin vector field results triad interpolation combination kirchhoff constraint according see also since kirchhoff constraint exactly fulfilled three element nodes constrained variant nodal spin vectors combined based alternative approach kirchhoff constraint given relations yields following expression spin vector field similar element latter version employed throughout contribution final finite element residual vector resulting discretized counterparts fields reads following formulation based degrees freedom residual denoted element stands weak kirchhoff constraint enforcement combined nodal triad parametrization via nodal tangents element linearization based increment vector employed see appendix remark actually collocation type approach applied order enforce kirchhoff constraint nevertheless notion weak constraint enforcement kept throughout work since procedure still represents basis formulation moreover difference formulation based strong constraint enforcement section shall emphasized naming residual vector rotation parametrization also element formulation section based weak kirchhoff constraint enforcement coordinate transformation alternative primary variables performed transformation rule element residual vector rrot identical section throughout work element formulation based degrees freedom element residual vector transformed via denoted element stands weak kirchhoff constraint enforcement combined nodal triad parametrization via nodal rotation vectors element linearization based increment vector employed see appendix avoidance locking effects investigation locking behavior proposed element many results already derived section since numbers neq neq well discrete problem identical element readily concluded also element formulation shows optimal constraint ratio membrane locking effects expected element similar section shall shown also elements exactly represent internal energy associated pure bending state time internal energy split contributions stemming torsion bending contributions stemming axial tension pure bending state energy contribution vanish thus total internal energy pure bending state given uniquely defined curvature vector field const order represent desired constant distribution curvature vector field possible employed triad interpolation nodal triads arise properly one remaining nodal triad describes rotational rigid body modes beam although nodal triads necessarily parametrized nodal rotation vectors still three conditions result nodal triads thus resulting total neq conditions additionally axial strains collocation points vanish order yield vanishing contribution requirement results neq additional conditions fulfilled collocation points six conditions considered order superpose arbitrary rigid body modes representing minimally required number dirichlet boundary conditions static problems total number neq equations equals total number nuk unknowns contained global vector considered elements thus case unique fem solution existent pure bending case represented exactly similar element torsion bending modes represented well axial tension values collocation points represented locking effects expected considerations easily extended arbitrary curvature fields representable employed triad interpolation arbitrary polynomials according sections expected result discrete hyperelastic energies associated pure bending states exactly represented elements verified means corresponding numerical test cases conservation properties since beam element proposed basically combines triad interpolation spin vector interpolation element see section centerline interpolation variation applied element section corresponding conservation properties directly concluded investigations sections consequently element exactly fulfill conservation linear angular momentum conservation energy guaranteed spatially discretized problem case spin vector interpolation replaced counterpart numerical examples section previously proposed beam element formulations investigated numerically means proper test cases simulations results presented following rely software implementation proposed finite element formulations numerical algorithms within finite element research code baci wall gee developed jointly institute computational mechanics technical university munich numerical examples see sections considered problems eventually section also dynamic test case investigated first step numerical examples aim verify principle applicability accuracy proposed general reduced beam elements range different beam slenderness ratios verification crucially relies detailed comparisons analytic reference solutions benchmark tests known form literature well numerical reference solutions generated means geometrically exact beam element formulations type specifically also essential requirements formulated section objectivity avoidance locking effects consistent spatial convergence behavior well fulfillment conservation properties verified different beam element formulations presented sections finally based arguments given section focus also lie detailed comparisons reissner kirchhoff type beam element formulations example respect resulting discretization error level performance scheme since kirchhoff type beam element formulations based triad parametrization formulations based rotation triad parametrization compare sections sections shown yield identical fem solutions former category investigated respect spatial discretization errors furthermore examples without analytic solution standard choice reference solution rre see also section numerical solution via element see section employing spatial discretization factor four finer finest discretization shown corresponding convergence plot order achieve good comparability among different geometries load cases standard set geometrical constitutive parameters applied simulations unless stated otherwise standard set consists beam initial length square side length parameters lead area moments inertia area different beam slenderness ratios generated varying value side length standard choice constitutive parameters thus leading numerical examples considered following sections scheme based consistent linearization applied order solve set nonlinear equations resulting temporally spatially discretized weak form balance equations convergence criteria euclidean norms displacement increment vector residual vector checked convergence norms fall prescribed tolerances typical convergence tolerances chosen subsequent examples range well slenderness ratios static problems presented following sections external loads applied basis incremental procedure shall denote number load steps load step size long nothing stated contrary following simple procedure applied order adapt load step size static simulation efficient manner initially comparatively small load step size chosen according scheme converged within prescribed number niter max iterations step size halved load step repeated procedure repeated convergence achieved four converging load steps low step size level step size doubled also procedure successively doubling step size four converging load steps current step size level repeated original step size reached procedure drastically increase overall computational efficiency also allows comparatively objective fair comparisons performance scheme different element formulations subsequent numerical examples comparisons made basis accumulated number newton iterations niter tot niter required solve entire problem niter number iterations required load step load step adaption scheme steps considered total number iterations niter niter max example verification objectivity objectivity kirchhoff beam element formulations proposed sections already proven theoretically order verify results numerically following test case investigated see figure clamped end initially curved beam slenderness ratio whose centerline configuration equals quarter circle dirichlet rotation respect global imposed presented example total rotation angle increasing linearly load steps prescribed order investigate objectivity normalized internal hyperelastic energy plotted total number rotations see figure element well variant element formulation consistent spin vector interpolation according comparison reasons also kirchhoff beam element formulation investigated meier considered see also last remark end section clearness internal energy normalized factor equal amount mechanical work required bend initially quarter circle straight beam means discrete external course internal energy vanish beam merely rotated initial configuration problem setup internal energy due imposed rigid body rotation figure objectivity test rigid body rotation initially quarter circle figure becomes obvious however internal energy formulation increases number rotations clear indication already theoretically predicted within rotations normalized energy reaches value almost results clearly visible deformation initial quarter circle contrary internal energy investigated element formulations results value zero machine precision finite element formulations based interpolation schemes investigated element might show reasonable results static test cases see however especially dynamic problems involving considerable rigid body motions results well drastic deterioration conservation properties investigated section follow application element formulations example pure bending examples shown section exclusively focus geometries load cases section two subsections section two load cases pure pending case well combined load case yielding geometrically nonlinear still moderate centerline deformations considered section aims investigation membrane locking effects comparison different tools especially mcs method proposed section section pure bending combined load case considered however due higher load factors resulting degree deformation increased compared examples section higher degree deformation reveals clear differences approximation quality kirchhoff beam element variants besides comparison two variants also first proof concept development hermitian kirchhoff elements given comparison different methods initially straight beam clamped one end two different load cases analyzed first load case identical example analyzed section solely consists discrete applied one load step moment exactly bends beam shaped arc second load case additional tip force global applied one load step initial deformed geometries load cases illustrated figure straight beam bent straight beam bent figure initial deformed configuration initially straight beam two load cases standard parameters slenderness ratios chosen load case highest therefore critical slenderness ratio combination external force investigated load case first step element formulation according applied combination full integration reduced integration classical assumed natural strain approach well mcs method according based integration points see section details variants figure relative load case different slenderness ratios plotted respect analytic reference solution spatial discretization variant based meshes elements applied accordingly convergence slowed dramatically increasing slenderness ratio beam discretized one finite element lele relative error increases almost two orders magnitude enhancing slenderness ratio however figure also reveals effect decreases decreasing element sizes almost completely disappears discretizations elements rel rel element length rel rel element length reference analytic different slenderness ratios order order reference analytic different slenderness ratios order order element length reference analytic element length reference figure straight beam subject load cases relative reason behavior lies fact element slenderness ratio lele observed locking effect latter also decreases decreasing element sizes however typical engineering applications relative error bounds range effect means negligible sufficiently fine discretizations expected convergence order four reached figure relative error plotted slenderness ratios variant supplemented mcs method according expected locking effect completely disappears investigated slenderness ratios however shown figure load case highest investigated slenderness ratio effect alternatively achieved applying simple reduced integration procedure variant classical ans approach variant contrary load case figure reveals distinctive improvement locking behavior obtained alternative methods load case ans approach well reduced integration scheme slightly alleviate locking effect range rather coarse discretizations compared variant mcs approach however completely eliminates error offset due membrane locking also load case explanation observation obvious agreement statements section similar working principle mcs method reduced integration scheme alleviate locking reducing number constraint equations yet shown section mcs method leads lower number constraint equations compared simple reduced integration scheme makes latter method less effective reduced integration scheme seems sufficient load case special case yielding symmetric curvature distributions within elements general deformed configurations resulting load case already demonstrate limits simple method hand working principle standard ans method aim reduction number constraint equations rather evaluation critical axial tension term selected collocation points vanishing parasitic strains geometrically nonlinear regime large deformations parameter space positions optimal collocation points obviously load case already leads change positions extent almost completely destroys working principle impact ans method summing say based two examples proposed mcs method seems superior standard methods reduced integration ans terms locking avoidance combined considered geometrically exact kirchhoff beam elements example section comparison different methodologies basis general problem setting involving deformation states initially curved geometry presented confirm result completeness figures also element based weak enforcement kirchhoff constraint according plotted load cases formulation yields comparable convergence behavior discretization error level element next shown behavior change increasing deformation following mcs method employed per default abbreviation omitted comparison different element formulations convergence plots investigated two load cases noteworthy differences element could observed order investigate difference two general approaches enforcing kirchhoff constraint strong weak manner detail perform first comparisons geometrically exact beam element formulations type two additional load cases considered see figure first load case considered section following denoted simply increases magnitude external moment factor eight compared previous load case thus leading deformed geometry represented double circle see figure since contribution simo load case established standard test case geometrically exact beam element formulations finally fourth load case denoted supplemented tip force global whose magnitude time exactly chosen initial geometry well final configuration last load case illustrated figure straight beam bent straight beam bent figure initial deformed configuration initially straight beam two load cases figure resulting load case plotted element well reissner type beam element formulation proposed crisfield jelinic presented section discretizations elements employed discretizations rel rel crisfield jelenic order degrees freedom reference analytic order degrees freedom reference analytic energy error figure straight beam load case convergence energy error comparable load case since similarly example also roughest discretization based one finite element per segment analytic solution order enable reasonable comparison different element formulations following discretization error plotted total number degrees freedom resulting respective finite element discretization since shear deformation present example reissner kirchhoff type elements converge towards analytic solution element formulations exhibit expected optimal convergence order four indicated black dashed line element shows expected result kirchhoff element formulations represent discretization error level less degrees freedom compared reissner type element formulation see section furthermore example even shown lines representing discretization error element element formulation would almost identical discretization error plotted solely degrees freedom associated centerline interpolation thus observable difference figure pure result additional rotational degrees freedom required reissner type element formulations order represent shear deformation behavior expected pure bending example since two considered element formulations exactly represent internal energy associated pure bending state see sections consequently discretization error contribution stemming second term vanishes finite element problem degenerates pure problem polynomial curve approximation represented first term thus discretization error plotted number centerline dofs yields similar results lagrange centerline interpolation reissner type element hermite centerline interpolation kirchhoff type element situation completely different sktan element exactly represent internal energy associated pure bending state closer investigation would confirm expected result element exhibits remaining error lengthspecific hyperelastic stored energy less constant along beam length based finding easily answered discretization error level element applied load case considerably increased compared first load case identical level observed element fem solely distribution second centerline derivative optimized order yield minimal energy error within beam domain centerline field constrained clamped end beam thus increasing distance clamped end discretization error centerline field resulting integration less constant error second derivative along increasing segment also increases consequently assuming comparable errors energy comparable discretizations number finite elements representing angle segment analytic solution higher discretization error expected load case compared load case figure confirms expected result energy error element vanish example exhibits convergence order four furthermore crisfield jelenic order rel order rel crisfield jelenic order order degrees freedom reference analytic elements degrees freedom reference analytic elements figure straight beam load case convergence elements shown energy error averaged along entire beam length identical energy error averaged along first eighth beam representing quarter circle consequence error accumulation described lower similar load case averaged along first eighth beam see figure rather mathematical point view increased discretization error level element explained high level second term dominates overall discretization error behavior turn pure consequence fact two exponents identical trial functions polynomial degree however polynomial degrees second term expected converge higher rate consequently sufficiently fine discretizations first term reflecting pure polynomial approximation power determine overall discretization error level range lower discretization error per dof expected kirchhoff type beam element formulations independently beam length complexity deformation state type boundary conditions first proof principle figure resulting element lagrange interpolation well element based hermite interpolation depicted rough discretizations error contribution higher convergence rate still seems dominated overall discretization error element expected optimal gap reissner discretization error higher level kirchhoff discretization error lower level observed comparison reasons figure results corresponding variants figure repeated since work focuses development kirchhoff beam elements details construction hermite polynomials either introducing additional nodes considering derivatives given point however expected comparable behavior illustrated figure also achieved test cases considered following elements employed detailed investigation general geometrically exact beam element formulations type polynomial degree considered future research work eventually also convergence behavior fourth load case shall investigated figures element plotted two beam slenderness ratios since analytic solution available example numerical reference solution based element formulation crisfield employed consequence shear deformation induced tip force result derived reissner kirchhoff type beam element formulations differ limit fine discretizations model error kirchhoff elements becomes visible form kink convergence diagram certain cutoff error level remains constant even arbitrarily fine discretizations expected model difference beam theory decreases increasing beam slenderness ratio property reflected lower cutoff error level higher slenderness ratio lower slenderness ratio relative error distinguishing rel rel crisfield jelenic crisfield jelenic order order degrees freedom reference crisfield degrees freedom reference crisfield figure load case different element formulations rel rel order degrees freedom reference order degrees freedom reference figure load case different element formulations kirchhoff reissner solution lies assumed reasonable approximation many engineering applications high slenderness ratio relative error two models smaller investigated cases cutoff error scales almost quadratically slenderness ratio would expected result solution geometrically linear theory result remarkable highly nonlinear example figures observed element formulations exhibit expected convergence rate four error level element lies slightly error level element lies error level element despite fact formulations yield general solutions also contain effects shear deformation kirchhoff type element formulation considered numerical reference solution remaining examples throughout work procedure seems sensible since within contribution convergence behavior kirchhoff type elements reissner type elements shall studied nevertheless model error kirchhoff type beam elements still observable time form kink remaining cutoff error level convergence plots reissner type formulation element able exactly represent internal energy load case load case energy crisfield jelenic newton iterations newton iterations crisfield jelenic number elements moderate slenderness ratio number elements high slenderness ratio figure load case total number newton iterations convergence formulations compared see figures two slenderness ratios element formulations exhibit expected convergence order four similar load case element yields better approximation internal energy element observable form lower energy error level figures reason lower level visible figures possible explanation better performance element may found considering interaction employed translational rotational interpolation schemes shown section number unknowns equals number equations required element energetically represent pure bending state means exact representation internal energy associated states constant axial tension bending curvature torsion possible property hold element see section corresponding system equations fulfilled representing pure bending state slightly difference elements level expected vanish trial functions effect general predicted energy error eventually mentioned well energy error plots load case difference slenderness ratios evident underlines successful avoidance membrane locking effectiveness mcs method finally also performance scheme shall investigated compared kirchhoff reissner type element formulations see figure since computationally expensive steps solving nonlinear system equations evaluating tangent stiffness matrix conducted every newton iteration reduction total number newton iterations niter tot defined would considerably increase overall efficiency numerical algorithm figures total number newton iterations niter tot load case combination slenderness ratios plotted element formulations different spatial discretizations final fem solutions shown independent choice nodal rotation parametrization number newton iterations required variants might differ considerably therefore also newton performance variants investigated solving highly nonlinear beam problem load step adaption scheme mentioned based initial number load steps employed comparing figures one realizes newton performance kirchhoff type element formulations rarely influenced considered slenderness ratio number newton iterations required reissner element increases drastically increasing slenderness ratio furthermore seems variants require fewer newton iterations variants trends confirmed even pronounced general examples presented subsequent sections summary following conclusions drawn examples considered section proposed kirchhoff elements yield accurate results acceptable model errors slenderness ratios model error decreases quadratically increasing beam slenderness ratio expected convergence orders four well energy error could confirmed investigated kirchhoff elements combination mcs method none considered element formulations exhibited influence element slenderness ratio resulting discretization error result confirms effectiveness mcs method avoidance membrane locking discretization error level element lies error level reissner type element also error level element increased error level element shown vanish higher polynomial degree trial functions total number newton iterations required reissner type element formulations considerably increases increasing beam slenderness ratio number iterations remains less constant kirchhoff type formulations conclusions drawn tests confirmed examples investigated following sections example pure bending section extension pure bending examples load cases last section considered focus lies initially straight clamped beam standard length investigated two slenderness ratios however time beam loaded contains additional moment component beam length direction inducing torsion initial deformed configuration illustrated figure rel cri cri order degrees freedom reative reference analytic initial deformed geometry figure straight beam loaded discrete argued analytic solution example given following space curve representation sin cos sin solution represents helix whose points viz direction applied external moment special parameter choice example leads radius enveloping cylinder identical slope helix figure relative resulting two investigated slenderness ratios plotted element formulations well spatial discretizations based elements element formulations exhibit expected convergence order four discretization error level element lower whereas discretization error element slightly higher element furthermore visible difference observed discretization error levels associated two different slenderness ratios due choice git easily verified example results analytic solution exhibiting vanishing axial tension shear crisfield jelenic simo quoc newton iterations newton iterations crisfield jelenic simo quoc number elements number elements moderate slenderness ratio high slenderness ratio figure load case straight helix total number newton iterations deformation well constant spatial material curvature vector along entire beam pointing direction external moment vector const thus already roughest discretizations elements exactly represent hyperelastic stored energy function pure bending case interpreted simple patch test geometrically exact beams finally also number newton iterations shall investigated see figure order enable general conclusions time second reissner type beam element formulation based completely different triad interpolation scheme additionally included comparison concretely element represents interpretation formulated crisfield see chapter original variant proposed simo following denoted element time load step adaption scheme presented based initial number load steps employed similarly case considered last section newton performance reissner type element formulations drastically deteriorates increasing slenderness ratio whereas performance kirchhoff type elements remains unchanged even slightly improved case variants concretely slenderness ratio investigated discretizations elements exhibit remarkably constant number niter tot iterations total number iterations required elements increases niter tot niter tot increasing number elements total number iterations required reissner type formulations almost two orders magnitude higher elements lies constantly value niter tot seemingly considerable difference elements reissner type element formulations attributed two different effects firstly parametrization nodal triads via tangent vectors seems parametrization based nodal rotation vectors effect already gets visible difference variants seems less independent beam slenderness ratio secondly high stiffness contributions resulting shear mode seem considerably deteriorate newton convergence range high slenderness ratios effect becomes obvious difference elements reissner type elements linearizations four elements based multiplicative updates nodal rotation vectors observation emphasized two elements types additionally exhibit triad interpolation two elements differ centerline interpolation based lagrange hermite polynomials expected influence newton convergence drastic manner fact element additionally enforces constraint vanishing shear strains consequently avoidance shear modes seems main reason considerably improved performance kirchhoff type element formulations finally observation total number newton iterations required variants increases increasing number elements secondary practical interest since discretizations relevant practical applications located range small element numbers left figure observations made far confirmed subsequent examples remark discretizations investigated figure solutions element could already found one load step however since cases convergence could achieved simulations based two three load steps solution problem means one load step rather regarded lucky shot representative convergence behavior order avoid biased comparison resulting effects initial number load steps increased means mentioned load step adaption scheme evaluation comparison process intended fair objective possible nevertheless example shows absolute statement concerning robustness nonlinear solution scheme based single example discretization deliberately chosen step size sometimes done literature questionable degree arbitrariness intended minimized employing automated scheme determining optimal load step size comparing results different test cases different discretizations different element types reissner kirchhoff type beam elements well different representatives element type order avoid biased results consequence incorrect linearizations results reissner type elements derived basis analytic representation consistent tangent stiffness matrix verified simulations basis consistent tangent stiffness matrix derived via automatic differentiation tool example verification section fundamental property objectivity already verified proposed kirchhoff beam elements section shown element formulations also beam problems whose analytic solution independent specific loading path beam element formulations also yield discrete solution independent specific loading path numerical test case path independence initially straight clamped beam initial length slenderness ratio thus considered loaded moment defined exactly bends beam additional comparison reasons also case increased slenderness ratio correspondingly adapted loads investigated problem setup well deformed configuration example shown figure following two different possibilities apply tip loads investigated first load case moment force applied simultaneously load case sim second load case moment force applied successively load case suc latter case external moment increased linearly zero within interval whereas external force increased linearly zero within interval figure problem setup initially straight beam bent discrete deformed shapes load cases plotted steps figure apparently two load cases lead different deformation paths identical final configuration contradicts intuition final deformed configuration lies completely half space although tip force points positive case small forces observation verified deriving analytical solution based linearization equilibrium equations respect configuration resulting see also furthermore observation step sim step sim step sim step sim step suc step suc step suc step suc figure deformed configurations simultaneous sim successive suc loading straight beam moment force agreement results obtained similar example based slightly modified parameter choice analyzed order investigate possible path dependence effects also quantitative manner relative calculated solution suc load case suc certain discretization solution sim load case sim centerline discretization thus basically relative definition equation applied suc rre sim results obtained two different slenderness ratios investigated element formulations wktan illustrated figure accordingly investigated element types discretizations slenderness ratios error vanishes machine precision verifies path independence formulations completeness figure represents discretization error resulting elements considered slenderness ratios observations already made earlier examples crisfield jelenic rel number elements crisfield jelenic rel moderate slenderness ratio number elements high slenderness ratio figure load cases simultaneously successively rel crisfield jelenic isotropic order degrees freedom reference reference figure path independence different element formulations respect convergence rate discretization error level cutoff error kirchhoff reissner type element formulations confirmed comparison reasons also resulting reduced isotropic beam element formulation proposed plotted result similar element reason discretization error level slightly lower isotropic element lies twist interpolation requires two dofs isotropic element three dofs sktan element section importance consistent torsion intermediate triad field emphasized figure discretization error variant plotted exactly torsion term neglected surprisingly resulting discretization error level identical correct element formulation contradiction explained order answer question realized actual triad orientation important isotropic examples order yield consistent centerline convergence easily verified mechanical torsion represented correctly also functional principle isotropic beam element torsion intermediate triad field neglected total torsion solely represented derivative relative angle field consequently relative angle arises way total torsion represented exactly turn results inconsistent triad orientation however since isotropic beams torsion triad orientation enters weak form final result beam centerline correct later section shown situation changes anisotropic beams beams initial curvature anisotropic shapes neglect intermediate triad torsion indeed lead inconsistent centerline solution resulting decreased spatial convergence rate furthermore investigation explains certain kirchhoff element formulations available literature accidentally neglect torsion term nevertheless produce correct results consistent convergence rates centerline solution long isotropic beam problems considered finally figure total number newton iterations required different finite element formulations plotted load case sim two investigated slenderness ratios load step adaption scheme based employed obtained results similar last section newton performance reissner type element formulations drastically deteriorates increasing slenderness ratio whereas performance kirchhoff type elements remains unchanged slenderness ratio investigated discretizations elements exhibit remarkably constant number niter tot iterations total number iterations required elements increases niter tot niter tot increasing number elements total number iterations required reissner type beam element formulations almost two orders magnitude higher elements lies constantly value niter tot element formulation niter tot element formulation crisfield jelenic simo quoc newton iterations newton iterations crisfield jelenic simo quoc number elements number elements moderate slenderness ratio high slenderness ratio figure load case simultaneously total number newton iterations example load besides objectivity test section examples investigated previous sections based isotropic geometries straight beams initially curved beam considered initial geometry represented circular curvature radius lies completely global clamped one end section constitutive parameters beam result quadratic shape side length young modulus well shear modulus initial geometry loaded force global magnitude example initially proposed bathe bolourchi meanwhile considered standard benchmark test geometrically exact beam element formulations investigated many authors see original definition slenderness ratio yields value example slightly modified definition slenderness ratio according employed following comparison reasons also second variant example increased slenderness ratio adapted figure force initial red final green configuration formulation crisfield crisfield simo elements table case tip displacement relative error different formulations formulation crisfield crisfield simo elements table case tip displacement relative error different formulations force investigated initial deformed geometry illustrated figure tables tip displacements resulting two slenderness ratios different discretizations kirchhoff type elements well reissner type elements crisfield simo vuquoc plotted due rough spatial discretizations cases also due additional model simplifications corresponding values derived literature case show comparatively large variation contrary deviation results displayed table smaller investigated formulations fact results derived representatives different beam theories theory indicates correctness reissner kirchhoff values resulting discretization elements coincide fourth significant digit case corresponding values identical seven significant digits displayed case high slenderness ratio observations described also confirmed convergence plots figure formulations yield expected convergence orders similar last example element exhibits identical discretization rel crisfield jelenic order degrees freedom reference reference figure force different formulations crisfield jelenic simo quoc newton iterations newton iterations crisfield jelenic simo quoc number elements number elements number newton iterations number newton iterations number load steps number load steps figure force number newton iterations load steps error level element since multiple centerline loops involved example furthermore similar example section also variant investigated torsion intermediate triad field omitted omission term influence convergence order observed section inconsistency yields decline convergence rate four two anisotropic example considered underlines importance consistently considering term see also section explanation also example performance scheme evaluated however order enable comparison values available literature time load step adaption scheme employed following alternative procedure order determine maximal constant load step size starting scheme based one load step number load steps increased one nnew nold range increased increments nnew nold range newton convergence achieved load steps order avoid lucky shots see remark end section also remark section maximal load step size associated minimal number load steps nmin accepted also next incrementation step load step size according procedure described leads newton convergence load steps figure total number newton iterations well minimal number load steps nmin resulting maximal constant load step size plotted two different slenderness ratios results similar observations made previous sections however smaller difference elements elements concretely reference elements nmin niter tot remark number iterations reported application standard newton scheme application accelerated newton scheme number iterations reported variant proposed objective variant proposed objective variant proposed interpretation formulation interpretation formulation kirchhoff type beam element formulation kirchhoff type beam element formulation kirchhoff type beam element formulation kirchhoff type beam element formulation table case number load steps newton iterations literature top work bottom beam problem slenderness ratio solved load step total iterations discretizations load steps total iterations discretizations load steps total iterations discretizations based reissner type elements slenderness ratio problem solved load step total iterations discretizations load steps total iterations discretizations load steps total iterations reissner discretizations table corresponding values reported literature slenderness ratio summarized already mentioned earlier direct comparison results difficult since clear procedure applied different authors order determine minimal number newton iterations required also subsequent refinement steps convergent contrary also singular occurrences convergence special loading paths accepted nevertheless numbers summarized table least give first impression behavior newtonraphon scheme resulting different finite element formulations accordingly formulations solve problem less iterations furthermore case reissner type formulation proposed investigated yields lower number newton iterations elements however shown beam element formulation moreover examples investigated far real advantage kirchhoff type formulations occurred especially high slenderness ratio investigated literature range moderate high slenderness ratios concluded proposed kirchhoff beam elements considered robust efficient formulations compared many reissner type alternatives literature remark maybe reader wondering element formulation interpretation element formulation proposed based identical discretization eight elements investigated required newton iterations reported reference actually also numerical tests performed nonlinear problem resulting discretization eight firstorder elements could solved three load steps however since subsequent simulation based four load steps convergent procedure avoidance lucky shots explained applied thus leading total load steps newton iterations example helix loaded axial force example generality initial geometry shall increased helix linearly increasing slope clamped one ends loaded see figure illustration figure helix varying slope loaded discrete force problem setup space curve representing initial geometry helix described via following analytic representation sin cos radius enveloping cylinder helix chosen helix exactly consists loops along standard length also example investigated two different slenderness ratios associated axial forces well ratio forces chosen identical ratio bending stiffnesses cases leading comparable values maximal figure resulting relative element plotted discretizations elements element formulations show expected convergence rate four discretization error level element slightly lower discretization error level element however discretization error levels lie error level reissner type element authors former contribution also different approaches already investigated case section compared helix example derived results drawn conclusion similar example details comparison interested reader referred reference finally helix problem shall even rel rel crisfield jelenic order order crisfield jelenic degrees freedom reference degrees freedom reference figure helix loaded axial force different formulations figure twisted helix axial force initial final shape visualization based generalized accounting initial curvatures also anisotropic shapes initial twist illustrated figure two slenderness ratios investigated square last example extended rectangular shape dimensions assumed torsional moment inertia case case defined assumed torsional moment inertia external forces chosen initial twist one twist rotation per helix loop resulting plotted figure shows consistent convergence behavior similar figure additionally figure also variant element consistent spin vector interpolation plotted accordingly visible difference compared sktan variant observed last step also balances forces moments investigated general example figure table reaction forces moments clamped end helix force moment contributions respect point resulting external load applied plotted discretizations eight elements easily verified balance forces moments exactly fulfilled variants variant fulfills balance forces balances moments confirms prediction made section rel rel crisfield jelenic order order crisfield jelenic degrees freedom reference degrees freedom reference figure twisted helix loaded axial force different formulations cri jel table slenderness reaction forces moments different formulations example free oscillations elbow cantilever final example represents dynamic test case example initially investigated subsequently considered several contributions field geometrically nonlinear beam element formulations see elbow cantilever beam consists two straight beam segments length rigidly connected one ends initial configuration first segment points global second segment global first segment cantilever clamped position original work beam described section parameters git well easily verified section parameters example represented quadratic radius thus resulting slenderness ratio two segments rotational inertia contributions additionally scaled factor artificial scaling applied order emphasize properly verify rotational inertia contributions would otherwise comparatively small chosen slenderness ratio cantilever beam loaded discrete force pointing global acting rigid corner elbow magnitude force linearly increased linearly decreased remaining simulation time cantilever executes geometrically nonlinear free oscillations space amplitudes range segment lengths initial deformed configurations different time steps illustrated figure time integration modified scheme step step step step step step step step step step figure free oscillations elbow cantilever initial deformed configurations section employed spatial discretizations element element well variant variant consistent spin vector interpolation according applied example yields complexity present previous test cases modeling rigid beam connection occurring corner elbow already mentioned earlier sections modeling kinks beam centerline easier realized variants nevertheless resulting solutions identical variants figure internal kinetic total system energy plotted different time step sizes spatial discretizations discretizations elements per elbow segment time step size illustrated figures visible oscillations total system energy visible differences energy contributions reissner kirchhoff type elements observed figures energy contributions resulting larger time step size rougher spatial discretization based one beam element per elbow segment plotted accordingly also rough discretizations overall system energy preserved well fact total system energy approximation resulting energetically consistent discretization rarely improved compared variants indicates chosen spatial temporal discretizations influence temporal discretization error might dominate error total system energy compared reference identical time step size comparable spatial discretization consisting one element per elbow segment applied oscillations total system energy could considerably decreased instability observed end considered time interval could completely avoided improvement attributed applied lie group extension scheme section whereas reference lie group extension newmark time integration scheme proposed considered based simplified reissner type beam element formulation see similar observations already made reference lie group extension scheme proposed elements elements cri elements elements elements elements figure free oscillations elbow cantilever energy conservation conclusion current work focused geometrically exact finite elements highly slender beams aimed proposal novel formulations type detailed review existing formulations simoreissner type careful evaluation comparison proposed existing formulations authors recent contribution first geometrically exact beam element formulation fulfills essential properties objectivity capable representing arbitrary initial curvatures anisotropic shapes proposed developed beam element formulation incorporates modes axial tension torsion anisotropic bending subsequent work also important question membrane locking successfully addressed current contribution extended methodologies providing considerable improvements terms accuracy practical applicability well generalization dynamic problems thereto two alternative interpolation schemes proposed first scheme based strong enforcement kirchhoff constraint enabled novel orthonormal rotation interpolation scheme theory second scheme based weak enforcement kirchhoff constraint discrete realization kirchhoff constraint relies properly chosen collocation strategy entirely abstain use additional lagrange multipliers second formulation allows arbitrary rotation interpolations investigated numerical realization employed orthonormal geodesic triad interpolation scheme proposed furthermore two interpolation schemes two different sets nodal rotation parametrizations proposed one based nodal rotation vectors rot one nodal tangent vectors tan different choices shown yield identical fem solutions differ resulting performance nonlinear solvers effort required prescribing essential boundary transition conditions rigid joints four finite element formulations resulting combination two interpolation schemes two choices nodal primary variables denoted elements respectively taking advantage hermite interpolation element formulations provide centerline representation order avoid membrane locking effects regime high beam slenderness ratios concept minimally constrained strains mcs recently proposed employed axial tension field eventually proposed beam elements supplemented implicit accurate time integration scheme recently proposed literature time discretization large rotations integration scheme identified lie group extension method comparable properties generality flexibility scheme allow straightforward combination different element formulations considered work review existing geometrically exact beam elements revealed approaches kind suitable general problems approaches categorized isotropic straight anisotropic formulations detailed evaluation formulations led result typically essential requirements summarized table fulfillled contrary finite elements proposed work fulfillment essential properties objectivity consistent spatial convergence behavior avoidance locking high slenderness regime conservation energy momentum spatial discretization scheme shown theoretically verified means representative numerical examples context conservation properties especially influence applying either discretizations focus concerning locking behavior recently proposed mcs method compared alternative methods known literature assumed natural strains ans reduced integration schemes see also contrast alternative methods mcs method could effectively avoid evidence membrane locking investigated load cases slenderness ratios contrast previously existing formulations formulations fulfilling requirements table found literature well however argued formulations provide considerable numerical advantages lower spatial discretization error level improved performance time integration schemes well linear nonlinear solvers smooth geometry representation compared formulations applied highly slender beams basis several numerical examples detailed systematic numerical comparisons resulting discretization error levels performance nonlinear solver performed four variants proposed geometrically exact beam elements two geometrically exact beam element formulations known literature examples investigated two different slenderness ratios low slenderness ratio general model difference theory theory beams measured form relative remaining limit arbitrarily fine spatial discretizations typically lay also quadratic decrease model difference increasing slenderness ratio could confirmed numerically investigated examples proposed elements shown lower discretization error level investigated beam element formulation results confirm theoretical prediction kirchhoff type formulations achieve discretization error level reissner type formulations less degrees freedom since shear deformation represented compared excellent results elements elements showed increased discretization error level examples even higher elements based underlying convergence theory phenomenon could attributed polynomial degree employed trial functions predicted vanish elements prediction confirmed means first numerical test case employing hermite polynomials order resulted expected optimal discretization error level lying error level reference formulation investigated examples conducted manner also one dynamic benchmark test literature conducted accuracy inertia contributions well energy stability employed time integration scheme could confirmed besides resulting discretization error level also total number iterations required solve considered test cases means different element formulations different slenderness ratios analyzed systematic manner investigated examples slenderness ratios proposed elements required less newton iterations solve problem compared two formulations chosen reference small slenderness range results four proposed variants two investigated formulations lay least order magnitude behavior formulations remained less unchanged number newton iterations required two different formulations increased considerably increasing slenderness ratio investigated examples slenderness ratio number two orders magnitude higher elements compared proposed elements also number iterations required elements based nodal rotation vectors triad parametrization independent considered slenderness ratio higher elements still considerably lower reissner type elements recapitulatory four proposed kirchhoff variants element based weak enforcement kirchhoff constraint triad parametrization via nodal tangent vectors recommended terms low discretization error level excellent performance scheme course factors could considered comprehensive comparison example elements based nodal rotation vectors simplify prescription dirichlet conditions flexibility proposed beam element variants allows combine advantages two different rotation parametrizations choosing element basic formulation provides excellent newton raphson performance replacing nodal tangents nodal rotation vectors nodes complex boundary coupling conditions prescribed realized simple transformation applied residual stiffness contributions relevant node abstaining stiff shear mode contributions underlying proposed element formulations may yield improved performance also highest eigenfrequency band slender beams associated shear modes avoided means theoretical considerations made work give hope considerably improved stability properties numerical time integration schemes combined developed elements future numerical investigation topic seems provide considerable scientific potential appendix definition rotational shape function matrices appendix shape functions required multiplicative rotation increments associated triad interpolation originally derived shall presented summation double indices applied vectors defined tan tan common abbreviation moreover derivative reads finally required derivative given see also cos sin cos sin sin cos abbreviations well applied mentioned original work limit derived small angles appendix modeling dirichlet boundary conditions joints many applications formulation proper dirichlet boundary conditions joints nodes different beam elements high practical relevance appendix represents brief summary possibility formulating basic constraint conditions investigated element appendix element since element simplifies formulation dirichlet boundary conditions kinematic constraints many practically relevant cases considered first dirichlet boundary conditions simple support element node realized via clamped end modeled also orientation fixed thus modeling dirichlet boundary conditions employed translational rotational degrees freedom similar standard finite elements purely based translational degrees freedom procedure also extended inhomogeneous conditions however determination requires special care case exp multiplicative procedure second line simplifies additive procedure according first line prescribed rotation additive holds rotations connections simple joint two nodes two connected elements reads thus degrees freedom eliminated global system equations standard manner simply assembling corresponding lines columns global residual vector stiffness matrix properly rigid joint two elements prescribed nodes additionally requires suppress relative rotation associated nodal triads assumed nodal triads differ fixed relative rotation exp exp following relations associated rotation increments derived consequently also rotational degrees freedom eliminated standard manner simply assembling corresponding lines columns global residual vector global stiffness matrix properly remark emphasized rigid joint according formulated via rotation tensor mandatory since rigid joint represents fixed orientation difference material quantities fixed relative rotation respect material axes via fixed relative rotation respect spatial axes different physical meaning remark additive increments rotation vectors instead multiplicative increments applied linearization process equation replaced case direct elimination degrees freedom via proper assembly global stiffness matrix possible instead corresponding columns scaled matrix remark physically reasonable boundary conditions completely defined orientation centroid position considered types boundary conditions degrees freedom measure nodal axial force part fem solution must prescribed appendix element treatment translational degrees freedom required subsequently considered boundary conditions identical last section therefore omitted dirichlet boundary conditions order model clamped end element simplest case tangent vector parallel global base vector considered supplemented representation tangent global frame exploited order prescribe boundary conditions arbitrary triad orientation tangent expressed basis prescribed triad gai consequently case equations linearized residual vector associated degrees freedom transformed rotation tensor dirichlet conditions formulated rotated coordinate system first component tangent vector expressed material frame represents magnitude must prescribed dirichlet conditions prescribed evolution relative angle adapted since intermediate frame might change time exp thus required value determined based prescribed current triad intermediate triad last step see also section remaining conditions remain unchanged compared connections based following relations stated combining two relations eventually yields following total transformation matrix trc similar relation also formulated iterative increments since multiplicative rotation increment components see expressed additive increments chosen linearization scheme additional transformation required compared trc equations allow transform corresponding lines columns global residual vector global tangent stiffness matrix properly eliminate degrees freedom global system equations magnitude tangent vector influenced rigid joint enters system equations new degree freedom last section motion rigid joint completely determined set section alternative set employed concluded realization clamped ends arbitrary orientation rigid joints beams simpler formulation based nodal rotation vectors elements conditions directly formulated global coordinate system tangent formulation requires additional transformation corresponding lines columns global residual vector stiffness matrix section properties tangent variant become apparent make type formulation favorable many applications certain element nodes require dirichlet conditions type considered still possible apply hybrid approach replace nodal tangents nodal rotation vectors primary variables specific nodes conditions required results derived far apply similar manner elements derived next section appendix linearization element deriving linearization element former definitions repeated quantities required later derivations linearization yields following linearization element derived completeness underlying residual vector inserted strain repeated linearization element residual vector obeys following general form order identify element stiffness matrix brought form vector already defined section terms yield ltk ltk many relations already derived section could field multiplicative rotation vector increments follows directly equation ltk similar manner associated derivative follows equation remaining linearizations required equation already derived contrast spin vector field increment field expressed via additive relative angle increments required relation given repeated git linearization element residual terms associated axial tension results based equation linearization inertia forces written time integration factor modified scheme according section slightly differs corresponding factor standard scheme linearization inertia moments yields clarity indices current previous time step explicitly noted quantities occurring quantities evaluated already introduced section fields material spatial multiplicative rotation increments relating current configuration converged configuration previous time step two vectors related transformation second step valid since eigenvector eigenvalue one rotation tensor configurations thus furthermore represent fields additive increments two successive newton iterations whereas given represents field multiplicative rotation increments two successive newton iterations appendix linearization element residual vector element given equation repeated linearization element residual vector obeys following general form order identify element stiffness matrix brought form linearization vectors originally defined follows linearization vectors already stated last section also linearization moment stress resultant form last section however fields originally defined section time given due kirchhoff constraint nodal increments expressed according git linearization inertia forces identical corresponding results last section statement also holds linearization inertia moments reads however element rotation increment field given equation appendix linearization elements nodal primary variable variations elements read similar manner set iterative nodal primary variable increments defined transformations primary variable variations increments given transformation matrices originally defined following form two different matrices required since primary variable variations elements based multiplicative quantities whereas corresponding iterative primary variable increments based additive quantities matrices tim evaluated element boundary nodes tim section already shown following residual transformation valid rrot similar manner also linearized element residual vector transformed matrix introduced order represent linearization calculating derivative result submatrices stated following transformation rule element stiffness matrix stated krot krot order apply transformation components element stiffness matrices krot arranged order components element residual vectors rtt rtt rtt rtt rrot rtrot rtrot 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feb exact consistent interpretation piecewise linear neural networks closed form solution lingyang chu xia juhua simon fraser university burnaby canada simon fraser university burnaby canada huxiah simon fraser university burnaby canada juhuah lanjun wang jian pei huawei technology ltd beijing china simon fraser university jpei abstract strong intelligent machines powered deep neural networks increasingly deployed black boxes make decisions risksensitive domains finance medical reduce potential risk build trust users critical interpret machines make decisions existing works interpret pretrained neural network analyzing hidden neurons mimicking models approximating local predictions however methods provide guarantee exactness consistency interpretation paper propose elegant closed form solution named openbox compute exact consistent interpretations family piecewise linear neural networks plnn major idea first transform plnn mathematically equivalent set linear classifiers interpret linear classifier features dominate prediction apply openbox demonstrate effectiveness nonnegative sparse constraints improving interpretability plnns extensive experiments synthetic real world data sets clearly demonstrate exactness consistency interpretation keywords deep neural network exact consistent interpretation closed form acm reference format lingyang chu xia juhua lanjun wang jian pei exact consistent interpretation piecewise linear neural networks closed form solution proceedings acm arxiv version acm new york usa pages https introduction machine learning systems making significant decisions routinely important domains medical practice autonomous driving criminal justice military decision making impact decisions increases demand clear interpretations machine learning systems growing ever stronger blind deployments decision machines accurately reliably interpreting machine learning model key many significant tasks identifying 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 must honored uses contact arxiv version canada copyright held acm isbn https failure models building trust human users discovering new knowledge avoiding unfairness issues interpretation problem machine learning models studied decades conventional models logistic regression support vector machine well interpreted practical theoretical perspectives powerful nonnegative sparse constraints also developed enhance interpretability conventional models sparse feature selection however due complex network structure deep neural network interpretation problem modern deep models yet challenging field awaits exploration reviewed section existing studies interpret deep neural network three major ways hidden neuron analysis methods analyze visualize features learned hidden neurons neural network model mimicking methods build transparent model imitate classification function deep neural network local explanation methods study predictions local perturbations input instance provide decision features interpretation methods gain useful insights mechanism deep models however guarantee compute interpretation truthfully exact behavior deep neural network demonstrated ghorbani existing interpretation methods inconsistent fragile two perceptively indistinguishable instances prediction result easily manipulated dramatically different interpretations compute exact consistent interpretation pretrained deep neural network paper provide affirmative answer well elegant closed form solution family piecewise linear neural networks piecewise linear neural network plnn neural network adopts piecewise linear activation function maxout family relu wide applications great practical successes plnns call exact consistent interpretations overall behaviour type neural networks make following technical contributions first prove plnn mathematically equivalent set local linear classifiers linear classifier classifies group instances within convex polytope input space second propose method named openbox provide exact interpretation plnn computing equivalent set local linear classifiers closed form third interpret classification result instance decision features local linear classifier since instances convex polytope share local linear classifier interpretations consistent per convex polytope fourth also apply openbox arxiv version canada study effect sparse constraints interpretability plnns find plnn trained constraints selects meaningful features dramatically improve interpretability last conduct extensive experiments synthetic data sets verify effectiveness method rest paper organized follows review related works section formulate problem section present openbox section report experimental results section conclude paper section related works interpret overall mechanism deep neural networks emergent challenging problem hidden neuron analysis methods hidden neuron analysis methods interpret deep neural network visualizing labeling features learned hidden neurons yosinski visualized live activations hidden neurons convnet proposed regularized optimization produce qualitatively better visualization erhan proposed activation maximization method unit sampling method visualize features learned hidden neurons cao visualized neural network attention target objects feedback loop infers activation status hidden neurons visualized compositionality clauses analyzing outputs hidden neurons neural model natural language processing understand features learned hidden neurons mahendran proposed general framework features learned image reconstruct image dosovitskiy performed task mahendran training neural network zhou interpreted cnn labeling hidden neuron best aligned semantic concept however hard get golden dataset accurate complete labels human semantic concepts hidden neuron analysis methods provide useful qualitative insights properties hidden neuron however qualitatively analyzing every neuron provide much actionable quantitative interpretation overall mechanism entire neural network model mimicking methods imitating classification function neural network model mimicking methods build transparent model easy interpret achieves high classification accuracy proposed model compression method train shallow mimic network using training instances labeled one deep neural networks hinton proposed distillation method distills knowledge large neural network training relatively smaller network mimic prediction probabilities original large network improve interpretability distilled knowledge frosst hinton extended distillation method training soft decision tree mimic prediction probabilities deep neural network che proposed mimic learning method learn interpretable phenotype features proposed tree regularization method uses binary decision tree mimic regularize classification function deep model lingyang chu xia juhua lanjun wang jian pei mimic models built model mimicking methods much simpler interpret deep neural networks however due reduced model complexity mimic model guarantee deep neural network large successfully imitated simpler shallow model thus always gap interpretation mimic model actual overall mechanism target deep neural network local interpretation methods local interpretation methods compute visualize important features input instance analyzing predictions local perturbations simonyan generated image map class images computing gradient class score respect input image ribeiro proposed lime interpret predictions classifier learning interpretable model local region around input instance zhou proposed cam identify discriminative image regions class images using global average pooling cnns selvaraju generalized cam identifies important regions image flowing classspecific gradients final convolutional layer cnn koh used influence functions trace model prediction identify training instances responsible prediction local interpretation methods generate insightful individual interpretation input instance however interpretations perspectively indistinguishable instances may consistent purposefully manipulated simple transformation input instance without affecting prediction result problem definition plnn contains layers neurons write layer hence input layer output layer layers hidden layers neuron hidden layer called hidden neuron let represent number neurons total number hidden neurons computed denote neuron bias output total weighted sum inputs neurons write biases vector bnl outputs vector anl inputs vector znl neurons successive layers connected weighted edges denote weight edge neuron neuron matrix compute denote piecewise linear activation function neuron hidden layers extend apply vectors fashion znl compute exact consistent interpretation piecewise linear neural networks closed form solution table frequently used notations arxiv version canada configuration plnn notation description neuron layer number neurons layer total number hidden neurons input neuron layer configuration neuron layer configuration plnn convex polytope determined linear classifier determined set linear inequalities define hidden neuron piecewise linear activation function following form input instance denoted input space also called instance short denote dimension input layer contains neurons output dimensional output space output layer adopts softmax function compute output softmax plnn works classification function maps input output widely known piecewise linear function however due complex network plnn overall behaviour hard understand thus plnn usually regarded black box interpret overall behavior plnn humanunderstandable manner interesting problem attracted much attention recent years following principled approach interpreting machine learning model regard interpretation plnn decision features define decision boundary call model interpretable explicitly provides interpretation decision features closed form definition given fixed plnn constant structure parameters task interpret overall behaviour computing interpretable model satisfies following requirements exactness mathematically equivalent interpretations provided truthfully describe exact behaviour consistency provides similar interpretations classification similar instances table summarizes list frequently used notations openbox method section describe openbox method produces exact consistent interpretation plnn computing interpretation model piecewise linear closed form first define configuration plnn specifies activation status hidden neuron illustrate interpret classification result fixed instance last illustrate interpret overall behavior computing interpretation model mathematically equivalent constant integer consists linear functions constant slopes constant intercepts collection constant real intervals partition given fixed plnn instance determines value determines linear function apply according linear function applied encode activation status hidden neuron states uniquely corresponds one linear functions denote state since inputs different neuron neuron states different hidden neurons may differ denote vector cnl states hidden neurons configuration vector denoted specifies states hidden neurons configuration fixed plnn uniquely determined instance write function maps instance configuration conf neuron denote variables slope intercept respectively linear function corresponds state uniquely determined hidden neurons write variables slopes intercepts rnl tnl respectively rewrite activation function neurons hidden layer hadamard product next interpret classification result fixed instance exact interpretation classification result fixed instance given fixed plnn interpret classification result fixed instance deriving closed form follows following equations plugging equation rewrite arxiv version canada lingyang chu xia juhua lanjun wang jian pei extended version hadamard product entry row column iteratively plugging equation write algorithm openbox train input fixed plnn train set training instances used train output set active llcs plugging equation rewrite coefficient matrix sum remaining terms superscript indicates equivalent plnn forward propagation layer layer since output input softmax closed form softmax fixed plnn fixed instance constant parameters uniquely determined fixed configuration conf therefore fixed input instance linear classifier whose decision boundary explicitly defined inspired interpretation method widely used conventional linear classifiers logistic regression linear svm interpret prediction fixed instance decision features specifically entries row decision features class instances equation provides straightforward way interpret classification result fixed instance however individually interpreting classification result every single instance far understanding overall behavior plnn next describe interpret overall behavior computing interpretation model mathematically equivalent initialization train compute configuration conf end end return exact interpretation plnn fixed plnn hidden neurons configurations represent configuration set configurations recall instance uniquely determines configuration conf since volume denoted number instances arbitrarily large clear least one configuration shared one instances denote conf set instances configuration prove theorem configuration convex polytope theorem given fixed plnn hidden neurons conf convex polytope proof prove showing conf equivalent finite set linear inequalities respect follows equation linear function constant parameters fixed summary given fixed linear function show convex polytope showing conf equivalent set linear inequalities respect recall denote bijective function maps configuration real interval conf equivalent set constraints denoted since linear function real interval constraint equivalent two linear inequalities respect therefore conf equivalent set linear inequalities means convex polytope according theorem instances sharing configuration form unique convex polytope explicitly defined linear inequalities since also determines linear classifier fixed instance equation instances convex polytope share linear classifier determined denote linear classifier shared instances interpret set local linear classifiers llcs llc linear classifier applies instances convex polytope denote tuple llc fixed plnn equivalent set llcs denoted use final interpretation model fixed plnn states hidden neurons independent plnn configurations means contains llcs however due hierarchical structure plnn states hidden neuron strongly correlate states neurons former layers therefore volume much less number local linear classifiers much less discuss phenomenon later table section practice need compute entire set llcs instead first compute active subset set llcs actually used classify available set instances update whenever new llc used classify newly coming instance exact consistent interpretation piecewise linear neural networks closed form solution coefficients also discover linear inequalities redundant whose hyperplanes intersect simplify interpretation polytope boundaries remove redundant inequalities caron method focus studying pbfs ones advantages openbox follows first interpretation exact set llcs mathematically equivalent classification function second interpretation consistent due reason instances convex polytope classified exactly llc thus interpretations consistent respect given convex polytope last interpretation easy compute since openbox computes forward propagation instance train data sets syn https neurons plnn table detailed description data sets training data positive negative data sets syn experiments section evaluate performance openbox compare method lime particular address following questions llcs look like interpretations produced lime openbox exact consistent decision features llcs easy understand improve interpretability features sparse constraints interpret pbfs llcs effective interpretations openbox hacking debugging plnn model table shows details six models used plnn use network structure described table adopt widely used activation function relu apply sparse constraints proposed chorowski train since goal comprehensively study interpretation effectiveness openbox rather achieving classification performance use relatively simple network structures plnn plnnns still powerful enough achieve significantly better classification performance logistic regression decision features used baselines compare decision features llcs python code lime published methods models implemented matlab plnn plnnns trained using deeplearntoolbox experiments conducted cpu ghz main memory rpm hard drive running windows use following data sets detailed information data sets shown table synthetic syn data set shown figure data set contains instances uniformly sampled quadrangle plnn table network structures number configurations plnn neurons successive layers initialized fully connected number linear functions relu number hidden neurons testing data positive negative training data syn prediction results plnn models flip equation linear function respect pbfs table models interpret logistic regression means sparse constraints flip means model trained instances flipped labels algorithm summarizes openbox method computes active set llcs actually used classify set training instances denoted train ready introduce interpret classification result instance first interpret classification result using decision features section second interpret contained using polytope boundary features pbfs decision features polytope boundaries specifically polytope boundary defined linear inequality arxiv version canada convex polytopes llcs figure llcs plnn trained syn euclidean space red blue points positive negative instances respectively use instances syn training data visualize llcs plnn data sets data sets contains two classes images fashion mnist data set consists images ankle boot bag consists images coat pullover images grayscale images represent image cascading pixel values feature vector fashion mnist data set available https arxiv version canada interpretations exact consistent exact consistent interpretations naturally favored human minds subsection systematically study exactness consistency interpretations lime openbox since lime slow process instances hours uniformly sample instances testing set conduct following experiments sampled instances first analyze exactness interpretation comparing predictions computed local interpretable model lime llcs openbox plnn respectively prediction instance probability classifying positive instance figure since lime guarantee zero approximation error local predictions plnn predictions lime exactly plnn dramatically different plnn difference predictions significant images difficult distinguish makes decision boundary plnn complicated harder approximate also see predictions lime exceed output interpretable model lime probability result arguable interpretations computed lime may truthfully describe exact behavior plnn contrast since set llcs computed openbox mathematically equivalent plnn predictions openbox exactly plnn instances therefore decision features llcs exactly describe overall behavior plnn next study interpretation consistency lime openbox analyzing similarity interpretations similar instances general consistent interpretation method provide similar interpretations similar instances instance denote nearest neighbor euclidean distance decision features classification respectively measure consistency interpretation cosine similarity larger cosine similarity indicates better interpretation consistency plnn openbox lime prediction lime prediction plnn openbox index instance index instance figure predictions lime openbox plnn sort results plnn predictions descending order lime openbox lime openbox llcs look like demonstrate claim theorem visualizing llcs plnn trained syn figures show training instances syn prediction results plnn respectively since instances used training prediction accuracy figure plot instances configuration colour clearly instances configuration contained convex polytope demonstrates claim theorem figure shows llcs whose convex polytopes cover decision boundary plnn contain positive negative instances shown solid lines show decision boundaries llcs capture difference positive negative instances form overall decision boundary plnn convex polytope cover boundary plnn contains single class instances llcs convex polytopes capture common features corresponding class instances analyzed following subsections set llcs produce exactly prediction plnn also capture meaningful decision features easy understand lingyang chu xia juhua lanjun wang jian pei index instance index instance figure cosine similarity decision features instance nearest neighbour results lime openbox separately sorted cosine similarity descending order shown figure cosine similarity openbox equal instances openbox consistently gives interpretation instances convex polytope since nearest neighbours may belong convex polytope cosine similarity openbox always equal instances constrast since lime computes individual interpretation based unique local perturbations every single instance cosine similarity lime significantly lower openbox instances demonstrates superior interpretation consistency openbox summary interpretations openbox exact much consistent interpretations lime decision features llcs effect sparse constraints besides exactness consistency good interpretation also strong semantical meaning thoughts intelligent machine easily understood human brain subsection first show meaning decision features llcs study effect sparse constraints improving interpretability decision features decision features plnn computed openbox decision features used baselines table shows accuracy models figure shows decision features models interestingly decision features plnn easy understand decision features features clearly highlight meaningful image parts ankle heel ankle boot upper left corner bag closer look average images suggests decision features describe difference ankle boot bag decision features plnn capture detailed difference ankle boot bag decision features llcs plnn capture difference subset instances within convex polytope however capture overall difference instances exact consistent interpretation piecewise linear neural networks closed form solution arxiv version canada table training testing accuracy models data set accuracy plnn avg image plnn train test train test avg image plnn figure decision features models show average image decision features models ankle boot bag respectively plnn show decision features llc whose convex polytope contains instances easier understand decision features plnn particular shown figure decision features clearly highlight collar breast coat shoulder pullover much easier understand cluttered features plnn results demonstrate effectiveness sparse constraints selecting meaningful features moreover decision features capture details thus achieves comparable accuracy plnn significantly outperforms accuracy summary decision features llcs easy understand sparse constraints highly effective improving interpretability decision features llcs avg image plnn avg image plnn figure decision features models show average image decision features models coat pullover respectively plnn show decision features llc whose convex polytope contains instances ankle boot bag accuracies plnn comparable instances ankle boot bag easy distinguish however shown figure instances hard distinguish plnn captures much detailed features achieves significantly better accuracy figure shows decision features models shown capture decision features strong semantical meaning collar breast coat shoulder pullover however features general accurately distinguish coat pullover therefore achieve high accuracy interestingly decision features plnn capture much details leads superior accuracy plnn superior accuracy plnn comes cost cluttered decision features may hard understand fortunately applying sparse constraints plnn effectively improves interpretability decision features without affecting classification accuracy shown figures decision features highlight similar image parts much pbfs llcs easy understand polytope boundary features pbfs polytope boundaries pbs interpret instance contained convex polytope llc subsection systematically study semantical meaning pbfs limited space use models trained target model interpret llcs computed openbox recall defined linear inequality pbfs coefficients since activation function relu either since values pbfs convex polytope images strongly correlate pbfs images strongly correlated pbfs analysis pbs pbfs demonstrated results tables figure take first convex polytope table example pbs whose pbfs figures show features ankle boot bag respectively therefore convex polytope contains images ankle boot bag careful study results suggests pbfs convex polytopes easy understand accurately describe images convex polytope also see pbfs figure look similar decision features figures shows strong correlation features learned different neurons probably caused hierarchy network structure due strong correlation neurons number configurations much less shown table surprisingly shown table convex polytope contains training instances instances training accuracy llc much higher training accuracies means arxiv version canada lingyang chu xia juhua lanjun wang jian pei table pbs convex polytopes containing instances indicates redundant linear inequality accuracy training accuracy llc ankle boot bag accuracy table pbs convex polytopes containing instances accuracy training accuracy llc figure show pbfs show pbfs lime avg cpp openbox hacked features lime openbox lime openbox hacked features hacked features figure hacking performance lime openbox show average avg cpp show nlci training instances convex polytope much easier linearly separated training instances perspective behavior like divide conquer strategy set aside small proportion instances hinder classification accuracy majority instances better separated llc shown convex polytopes table set aside instances grouped convex polytopes corresponding llcs also achieve high accuracy table shows similar phenomenon however since instances easy linearly separated training accuracy marginally outperforms coat pullover accuracy openbox hacked features nlci nlci avg cpp lime hack model using openbox knowing intelligent machine thinks provides privilege hack hack target model significantly change prediction instance modifying features possible general biggest change prediction achieved modifying important decision features precise interpretation target model reveals important decision features accurately thus requires modify less features achieve bigger change prediction following idea apply lime openbox hack compare quality interpretations comparing change prediction modifying number decision features instance denote decision features classification hack setting values decision features zero prediction changes significantly change prediction evaluated two measures follows first change prediction probability cpp absolute change probability classifying positive instance second number instance nlci number instances whose predicted label changes hacked due inefficiency lime use sampled data sets section evaluation figure average cpp nlci openbox always higher lime data sets demonstrates interpretations computed openbox effective lime applied hack target model interestingly advantage openbox significant shown figure prediction probabilities instances either provides little gradient information lime accurately approximate classification function case decision features computed lime describe exact behavior target model summary since openbox produces exact consistent interpretations target model achieves advanced hacking performance lime debug model using openbox intelligent machines perfect predictions fail occasionally failure occurs apply openbox interpret instance figure shows images high probability figures original image coat however since scattered mosaic pattern cloth hits features pullover coat original image classified pullover high probability figures original image pullover however coat white collar breast hit typical features coat dark shoulder sleeves miss significant features pullover similarly ankle boot figure highlights features upper left corner thus bag exact consistent interpretation piecewise linear neural networks closed form solution figure images coat pullover ankle boot bag show original images rest subfigures caption shows prediction probability corresponding class image shows decision features supporting prediction corresponding class bag figure ankle boot hits features ankle heel ankle boot however misses typical features bag upper left corner conclusion demonstrated examples figure openbox accurately interprets potentially useful debugging abnormal behaviors interpreted model conclusions future work paper tackle challenging problem interpreting plnns studying states hidden neurons configuration plnn prove plnn mathematically equivalent set llcs efficiently computed proposed openbox method extensive experiments show decision features polytope boundary features llcs provide exact consistent interpretations overall behavior plnn interpretations highly effective hacking debugging plnn models future work extend work interpret general neural networks adopt smooth activation functions sigmoid tanh references aishwarya agrawal dhruv batra devi parikh analyzing behavior visual question answering models jimmy rich caruana deep nets really need deep nips osbert bastani carolyn kim hamsa bastani interpreting blackbox models via model extraction bishop pattern recognition machine learning information science statistics springer new york cao liu yang wang wang huang wang huang look think twice capturing visual attention feedback convolutional neural networks iccv caron mcdonald ponic degenerate extreme point strategy classification linear constraints redundant necessary jota che purushotham khemani liu distilling knowledge deep networks applications healthcare domain jan chorowski jacek zurada learning understandable neural networks nonnegative weight constraints tnnls alexey dosovitskiy thomas brox inverting visual representations convolutional networks cvpr erhan yoshua bengio courville vincent visualizing higherlayer features deep network university montreal ruth fong andrea vedaldi interpretable explanations black boxes meaningful perturbation arxiv version canada nicholas frosst geoffrey hinton distilling neural network soft decision tree amirata ghorbani abubakar abid james zou interpretation neural networks fragile xavier glorot antoine bordes yoshua bengio deep sparse rectifier neural networks icais ian goodfellow yoshua bengio aaron courville deep learning mit press http ian goodfellow david mehdi mirza aaron courville yoshua bengio maxout networks goodman flaxman european union regulations algorithmic right explanation nick harvey chris liaw abbas mehrabian bounds piecewise linear neural networks zhang ren sun delving deep rectifiers surpassing performance imagenet classification iccv geoffrey hinton oriol vinyals jeff dean distilling knowledge neural network patrik hoyer sparse coding wnnsp kindermans sara hooker julius adebayo maximilian alber kristof sven dumitru erhan kim reliability saliency methods pang wei koh percy liang understanding predictions via influence functions pascal koiran eduardo sontag neural networks quadratic dimension nips alex krizhevsky ilya sutskever geoffrey hinton imagenet classification deep convolutional neural networks nips yann lecun yoshua bengio geoffrey hinton deep 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compatible discretization hexahedrals apr jasper kreeft marc gerritsma abstract derive compatible discretization method relies heavily underlying geometric structure obeys topological sequences commuting properties constructed sample problem consider formulation stokes problem motivate choice mixed variational formulation based geometric well physical arguments numerical tests confirm theoretical results obtain pointwise solution stokes problem method obtains optimal convergence rates introduction sample problem consider stokes flow problem formulation curl curl grad div article consider prescribed velocity boundary conditions method holds admissible types boundary conditions see despite simple appearance stokes flow model exists large number numerical methods simulate stokes flow reduce two classes jasper kreeft shell global solutions grasweg amsterdam shell com marc gerritsma delft university technology kluyverweg delft jasper kreeft marc gerritsma either circumventing lbb stability condition like stabilized methods satisfying condition compatible mixed methods last requires construction dedicated discrete vector spaces best known curl conforming divergence conforming spaces consider subclass compatible methods mimetic methods mimetic methods solely search appropriate vector spaces aim mimic structures symmetries continuous problem see consequence mimicking mimetic methods automatically preserve physical mathematical structures continuous formulation among others lbb condition important pointwise solution heart mimetic method integral theorems stokes gauss couple operators grad curl div action boundary operator manifold therefore obeying geometry orientation result satisfying exactly mentioned theorems consequently performing vector operators exactly finite dimensional setting distinguish four types points lines surfaces volumes two types orientation namely examples shown figure together action boundary operator outer orientation inner orientation fig four geometric objects possible point line surface volume outer orientation boundary operator maps objects objects creating quadrilateral hexahedral mesh divide physical domain large number geometric objects geometric object associate discrete unknown implies discrete unknowns integral quantities since three earlier mentioned theorems integral equations follows example taking divergence volume equivalent taking sum integral quantities associated surrounding surface elements fluxes using integral quantities degrees freedom perform grad compatible discretization hexahedrals curl div equivalent taking sum degrees freedom located boundary relations purely topological nature form topological sequence complex sequence fundamental direct connection complexes related physical domain computational domain physical problem discretization although original work presented terms differential geometry algebraic topology use vector calculus common mathematical language nevertheless put emphasis distinction topology metric complexes commuting diagrams drives former two languages make use spectral element interpolation functions basis functions past nodal spectral elements mostly used combination galerkin projection gsem gsem satisfies lbb condition lowering polynomial degree pressure two respect velocity results method weakly meaning divergence velocity field convergence zero mesh refinement present study uses mimetic spectral element interpolation basis functions mixed mimetic spectral element method mmsem satisfies lbb condition gives pointwise solution mesh sizes really discretize exactly since stokes flow model hold certain physical domain include geometry means integration case relate every physical quantity geometric object starting incompressibility constraint due gauss divergence theorem div using stokes circulation theorem relation written curl first relation follows div associated volumes association geometric object velocity less clear fact associated two different types geometric objects representation velocity compatible incompressibility constraint given terms velocity flux surface bounds volume circulation relation velocity represented along line bounds surface call velocity vector surface velocity along line segment inneroriented similar distinction made vorticity see jasper kreeft marc gerritsma last equation considered equation shows classical stokes circulation gauss divergence theorems tell half story perspective classical theorem gradient acting pressure relates line values corresponding end point stokes circulation theorem shows curl acting vorticity vector relates surface values line segment enclosing fit one equation fact geometric perspective exists two gradients two curls two divergence operators one related mentioned integral theorems explained others formal adjoint operators let grad curl div original differential operators associated mentioned integral theorems formal hilbert adjoint operators defined div curl grad geometric interpretation adjoint operators detours via opposite type orientation div relates vector quantity associated surfaces scalar quantity associated volume enclosed surfaces adjoint operator relates scalar quantity associated volume vector quantity associated surrounding surfaces illustrated figure following figure adjoint operator consists three consecutive steps first switch outer oriented scalar associated volumes inner oriented scalar associated points take derivative finally switch inner oriented vector associated lines outer oriented vector associated surfaces similar way describe derivatives outer orientation grad div curl div curl grad inner orientation fig geometric interpretation action boundary operators vector differential operators formal hilbert adjoint operators compatible discretization hexahedrals since horizontal relations purely topological vertical relations purely metric operators grad curl div purely topological operators metric makes much harder discretize could either associated line segment rewriting grad associated surface rewriting curl without geometric considerations could never make distinction grad curl div associated hilbert adjoints since focus obtaining pointwise discretization decide use expression equations associated geometric objects curl grad div first equation associated line segments second surfaces third volumes complexes figure reveals already number sequence complex structures starting geometry consider points lines surfaces volumes possess sequence combination boundary operator boundary volume surface boundary surface line boundary line two end points results following complex important property complex apply boundary operator twice always find empty set follows directly previously mentioned integral theorems follows consequence curl grad div curl derivatives also form complex hilbert setting becomes grad curl div curl div using hilbert adjoint relations also obtain adjoint complex properties jasper kreeft marc gerritsma hilbert setting variables stokes problem following spaces curl div hard even possible find discrete vector spaces subsets function spaces simultaneously satisfy complex properties instead stokes problem cast equivalent variational mixed formulation make use hilbert adjoint properties simplifies function spaces flow variables mixed formulation reads find curl div div given curl div curl curl div div formulation corresponding function spaces able construct compatible discrete vector spaces note completely avoid metric dependent derivatives corresponding complex discretization stokes problem degrees freedom many numerical methods especially finite difference finite element methods discrete coefficients point values proposed mimetic structure discrete unknowns represent integral values submanifolds ranging points volumes submanifolds oriented constitute computational domain span physical domain concept orientation shown figures gave rise boundary operator represented connectivities consisting see also space degrees freedom given spaces form duality pairing geometric spaces degrees freedom integral values definition degrees freedom spaces previously mentioned integral theorems define formal adjoint boundary operator coboundary operator coboundary operator discrete representation topological derivatives grad curl div since follows discrete stokes gauss theorem applying coboundary compatible discretization hexahedrals operator twice always zero see coboundary operator also matrix representations transpose connectivity matrices obtain following topological sequence matrices explicitly appear final matrix system illustration given figure details structure fig action twice coboundary operator vorticity zero net result surrounding volumes positive negative contribution neighboring velocity faces geometry orientation degrees freedom found gerritsma mimetic operators let curl div discretization flow variables involves projection operator complete vector spaces discrete vector spaces flow variables expressed terms defined corresponding interpolation functions also called projection operator actually consists two steps reduction operator integrates flow variables reconstruction operator interpolates using appropriate mimetic operators defined composition two operators gives projection operator reduction operator simply defined integration possesses following commutation relations rgrad rcurl rdiv treatment reconstruction operator leaves freedom long satisfies following properties right inverse reduction approximate left inverse reduction possess following commutation relations grad curl div completeness hilbert setting projection needs additional smoothing argument step ignored increase readibility see details jasper kreeft marc gerritsma reduction reconstruction operators commute continuous discrete differentiation also projection operator possesses commutation relation differentiation case divergence operator relevant obtain pointwise solution commutation relation given div div rdiv div commutation relations case divergence illustrated div div div div since property also holds grad curl obtain following complex discrete vector spaces grad curl div practice use computations relation implies among others satisfies discrete lbb condition div inf sup kqh kvh constant continuous problem whereas lbb condition measurement numerical stability commutation relation indicates physical correctness numerical method last much stronger statement includes also former conditions reconstruction operator led construction mimetic spectral element since use construction point line surface volume corresponding need nodal edge interpolation functions nodal interpolation functions lagrange polynomials edge polynomials derived lagrange polynomials based given conditions set lagrange polynomials edge polynomials given dlk lagrange edge polynomials possess condition compatible discretization hexahedrals kronecker delta interpolation function variable associated surface example given example divergence operator one interesting properties mimetic method presented paper within weak formulation constraint satisfied pointwise let velocity flux defined change mass equal divergence div compactly written note mass production zero model problem incompressibility constraint already satisfied discrete level interpolation using results solution velocity pointwise priori error estimates standard interpolation theory follows obtain following rates interpolation errors flow variables curl ukh div kdiv div due commuting property cases empty harmonic vector spaces discrete vector spaces conforming moreover due commuting property follows spaces compatible curlwh divvh finally possess decomposition grad jasper kreeft marc gerritsma curl terms vector spaces zwh zvh refers kernel nullspace orthogonal complement properties priori error estimates derived show optimal convergence rates admissible boundary conditions including boundary condition mixed finite element methods priori error estimates given inf inf inf inf inf inf constants differ case independent shows rate convergence approximation errors interpolation errors numerical results many years cavity flow considered one classical benchmark cases assessment numerical methods verification incompressible navier codes cavity test case deals flow unit box five solid boundaries moving lid top boundary moving constant velocity equal minus one especially two line singularities make cavity problem challenging test case left plot figure shows slices magnitude velocity field three dimensional cavity stokes problem obtained element mesh element contains mesh slices taken right plot figure shows slices divergence velocity field figure confirms mixed mimetic spectral element method leads accurate result solution second testcase shows optimal convergence behavior stokes problem boundary conditions testcase originates recent paper arnold convergence shown proven boundary conditions using elements since raviartthomas elements popular div conforming elements compare method results figure shows results stokes problem unit square velocity pressure fields given velocity methods show optimal convergence pressure difference noticed rate convergence vorticity difference rate convergence revealed compatible discretization hexahedrals div fig left slices magnitude velocity field three dimensional cavity stokes problem obtained element mesh right slices divergence velocity confirms velocity field div curl mimetic spectral mimetic spectral fig comparison mimetic spectral element projections stokes problem boundary conditions jasper kreeft marc gerritsma references arnold falk gopalakrishnan mixed finite element approximation vector laplacian dirichlet boundary conditions mathematical models methods applied sciences bochev hyman principles mimetic discretizations differential operators arnold bochev lehoucq nicolaides shashkov editors compatible discretizations volume ima volumes mathematics applications pages springer brezzi buffa innovative mimetic discretizations electromagnetic problems journal computational applied mathematics brezzi fortin mixed hybrid finite element methods volume springer series computational mathematics new york gerritsma edge functions spectral element methods hesthaven editors spectral high order methods partial differential equations pages springer gerritsma hiemstra kreeft palha rebelo toshniwal geometric basis mimetic spectral approximations issue hughes franca balestra new finite element formulation computational fluid dynamics circumventing condition stable formulation stokes problem accomodating interpolations computer methods applied mechanics engineering kreeft gerritsma priori error estimates compatible spectral discretization stokes problem admissible boundary conditions submitted kreeft gerritsma mixed mimetic spectral element method stokes flow pointwise solution journal computational physics kreeft palha gerritsma mimetic framework curvilinear quadrilaterals arbitrary order submitted perot discrete conservation properties unstructured mesh schemes annual review fluid mechanics
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replace line paper identification number edit novel scheme improve prediction sand fraction seismic attributes using neural networks soumi chaki student member ieee aurobinda routray member ieee william mohanty paper presents novel scheme improve prediction sand fraction multiple seismic attributes seismic impedance amplitude frequency using machine learning information filtering available well logs along seismic data used benchmark proposed stage using methodology primarily consists three steps preprocessing training artificial neural network ann learning algorithm used model sand fraction available sand fraction data high resolution well logs far information content low resolution seismic attributes therefore regularization schemes based fourier transform wavelet decomposition empirical mode decomposition emd proposed shape high resolution sand fraction data effective machine learning input data sets segregated training testing validation sets test results primarily used check different network structures activation function performances network passes testing phase acceptable performance terms selected evaluators validation phase follows validation stage prediction model tested unseen data network yielding satisfactory performance validation stage used predict lithological properties seismic attributes throughout given volume finally scheme using spatial filtering implemented smoothing sand fraction volume prediction lithological properties using framework helpful reservoir characterization index artificial neural network ann wavelets empirical mode decomposition emd entropy fourier transform normalized mutual information nmi preprocessing regularization reservoir characterization sand fraction median filtering introduction ydrocarbons migrate source rock porous medium reach reservoir rock temporary preservation finally mobile hydrocarbons get seized cap rocks identification characterization layer borehole enormous importance explorers soumi chaki aurobinda routray department electrical engineering indian institute technology kharagpur india aroutray recognition potential zones prospective oil exploration field carried using well logs categorize layers different sections dry water containing hydrocarbon bearing layers lithological properties neighborhood borehole estimated well logs whereas distributions become difficult predict away wells cases available seismic attributes used guidance predict lithological information traces area interest well logs seismic attributes integrated available well locations design reservoir model least uncertainty however mapping well logs seismic attributes governed nonlinear relationship characterized mismatch information content nonlinear problems approached using methods like hybrid systems multiple regression neural networks systems etc found literature ann widely used researchers engineers diverse backgrounds model single multiple target properties predictor variables different research problems accurate prediction generalization capability example ann used climatological studies ocean engineering telecommunications text recognition financial time series reservoir characterization etc diverse dataset containing information assembled multiple domains used learning validation ann however inherent limitations firstly performance ann dependent selection network structure associated parameters secondly training complex multilayered network time intensive process furthermore complex network trained relatively smaller number learning patterns may lead fitting equally important assess possibility modelling target property predictor variables using ann nonlinear modelling approach sometimes model performance improved applying suitable filtering techniques variables stage several studies contributed performance analysis ann along machine learning algorithms model target variable single multiple predictors respect william mohanty department geology geoscience indian institute technology kharagpur india wkmohanty replace line paper identification number edit study area workflow study area study working dataset acquired western onshore hydrocarbon field india hydrocarbon field located basin spread along western periphery central india surrounded aravalli range deccan craton saurashtra craton located direction along basin structurally field located broad nosing feature thus housing hydrocarbons two major synclines hydrocarbon present within series vertically stacked sandstone reservoirs individually separated intervening shale average thickness sand layer order due discrete sand depositions thin layers imaging seismic data obtaining mapping seismic borehole dataset challenging tasks spatial database containing seismic attributes well logs acquired study area format respectively file format one numerous standards developed society exploration geophysicists seg storing geophysical data log ascii standard las format developed canadian well logging society standardize organization digital log curves information workflow developed matlab platform data files converted format matlab platform compatible format matlab compatibility fig represents placements four wells across study area wells hereafter referred terms inlines crosslines xlines additionally well well well well locations terms universal transverse mercator utm coordinates depth well around meter ground whereas zone interest meter meter subsurface well zone interests meter meter well inline problem however following aspects still remain unexplored design appropriate stage effectiveness machine learning algorithms proper choice structure parameters associated selected machine learning algorithms ann model activation function type number hidden layers etc suitable methods predicted output paper proposes novel scheme demonstrates use said scheme appropriately designed framework consisting three modelling estimate sand fraction multiple seismic attributes target variable represents per unit sand volume within rock workflow starts stage uses three alternative approaches target variable regularization based fourier transform wavelet decomposition empirical mode decomposition emd next functional relationship regularized target sand fraction seismic attributes calibrated using ann study simple network structure consisting single hidden layer selected relatively complex networks network parameters training algorithm decided empirically results obtained evaluated terms four performance correlation coefficient root mean square error rmse absolute error mean aem scatter index inclusion stage workflow found produce remarkable improvement prediction results opposed use original sand fraction using ann structure learning validation post processing carried improve prediction result study area way complete workflow devised modelling lithological property sand fraction given seismic attributes paper organized follows section describes study area complete workflow adopted paper section iii explains steps section illustrates ann model building validation section described finally section concludes paper discussion achievements study possible future directions research xline fig location four wells study area terms inlines crosslines xlines borehole dataset contains basic logs gamma ray resistivity density along derived logs sand fraction permeability porosity water saturation etc well logs treated one dimensional signals processing study hand seismic dataset contains seismic impedance amplitude instantaneous frequency throughout volume workflow systematic workflow consisting three stages preprocessing model demonstrated fig workflow implemented matlab platform installed dell precision cpu ghz ram regularization approach stage highlighted using small dotted rectangle fig replace line paper identification number edit fig proposed workflow model sand fraction multiple seismic attributes three alternative regularization methods based emd implemented study model building stage training testing validation carried consecutively validation performance improve expected even implementing either three regularization techniques parameters associated selected regularization method modified carrying training testing validation tuning carried using loss high frequency information target attribute acceptable performance evaluators improvement ann performance regularization technique quantified using performance evaluators presented later first ann trained original sand fraction target variable without involving procedure repeated regularized sand fraction target variable ann helpful reassure improvement ann performance using regularized target variables validated network used obtain sand fraction study area three seismic attributes stage includes smoothing filtering estimated output remove unwanted predictions iii plays crucial role performance machine learning algorithm paper efficient preprocessing approach proposed part adopted methodology obtain functional relationship seismic attributes lithological properties signal reconstruction borehole data recorded specific well locations along depth high vertical resolution conversely seismic data collected spatially time domain sampling interval two milliseconds signal reconstruction important step discussed existing literature acoustic impedance log carried wavelet transform based method well geostatistical technique acoustic impedance log computed recorded logs pwave velocity density corresponding welllocation reconstructed acoustic impedance logs obtained raw impedance log wavelet transform geostatistical methods turned similar seismic attributes well logs integrated predict facies using methodology complexities regarding resolution individual datasets seismic well logs difference individual sampling intervals discussed detail present work well logs converted depth time domain milliseconds sampling interval using given velocity profile obtained noted sampling intervals two datasets seismic attributes well logs different hence order integrate two band limited seismic attributes reconstructed time instant corresponding well logs sinc interpolator adhering sampling theorem fig shows three seismic attributes sand fraction along well red dots seismic attributes represent original values time interval two milliseconds green curves represent corresponding reconstructed signals time instants marked well logs blue high frequency curve fig represents sand fraction along well time replace line paper identification number edit seismic seismic impedance amplitude reconstructed seismic attributes inst frequency sand fraction original seismic attributes fig seismic attributes sand fraction along well red dots seismic attributes original values green line reconstructed signal signal regularization observed fig frequencies present seismic signals much lower compared sand fraction words sand fraction carries much information compared seismic attributes according laws information theory higher informationcarrying signal modelled using single multiple lower predictor signals part target variable dependent predictor variables modelled present case spatially distributed seismic attributes low vertical resolution whereas high resolution borehole data recorded specific well locations carry high information content hence estimation subsurface lithological properties seismic attributes deals uncertainty due mismatch information content order circumvent disparity seismic attributes time instants corresponding well logs integrate lithological properties however contribute additional information seismic information would able successfully delineate rock properties completely key step new information added seismic attributes amount information filtered high information carrying signal order calibrate functional relationship thus necessity information filtering prediction regularization established paper three different signal processing approaches selected implemented order filter target signal parameters belonging stage tuned following changes entropy filtering along visual inspection output signal respect original target signal entropy computed power spectral density psd signal average amount information gained measurement specifies defined entropy system formally defined log probability value dataset known shannon entropy another random variable described dataset mutual information two expressed conditional entropy observed reservoir property present case represented seismic attribute seismic impedance represented statistical property interpreted reduction uncertainty reservoir property due observation attribute normalized mutual information nmi defined mutual information normalized minimum entropy variables nmi min study nmi computed predictor target signal used adjust parameters information filtering algorithms starting regularization seismic inputs sand fraction values normalized methods respectively using data four wells taken together paper modelling target three seismic attributes carried single hidden layer ann trained using back propagation algorithm network structure training procedure discussed detail following section target variable normalized within range output activation function keeping offset limiting value activation function otherwise back propagation algorithm tends drive free network parameters infinity result learning process may slow hence target variables normalized overlap saturation region function fourier transform first regularization approach based fourier transform algorithm spectrums target predictor variables compared higher frequency components target signal truncated target signal reconstructed using inverse fourier transform ift algorithm regularization based fourier transform task regularizing target sand fraction based fourier transform input predictor signal target signal sand fraction extracted proper depth time conversion using given velocity profile let seismic amplitude along compute fourier transform root unity similarly target computed root unity compare spectrums target predictor signals select bandwidth parameter max replace line paper identification number edit part target spectrum exceeding max truncated zero modified target ymod construct regularized target signal carrying ift truncated spectrum mod root unity calculate entropies predictors seismic attributes well original regularized target signals sand fraction entropy regularized target comparable predictor signal regularization result satisfactory regularization completed else step output regularized target signal comparison spectrums sand fraction fig seismic impedance fig reveals presence higher order frequencies former research find prediction capability given nonlinear mapping process original regularized signals presented fig blue red curves respectively based regularization method sophisticated bandpass filtering time domain order carry bandpass filtering two frequencies ideal bandpass filter defined beyond frequency components removed case ideal bandpass filters zero phase shift important requirement however ideal bandpass filter realizable working practical datasets finite number samples moreover case practical bandpass filtering transition takes place pass band stop band frequency component beyond frequencies diminish abruptly case based regularization frequency components spectrum beyond selected parameter range completely truncated reconstruction regularized log additionally phase shift resulting bandpass filtering yield relationships actual filtered logs hand based regularization incur phase shift actual regularized logs shown fig table entropy bit psd signals well variables entropy value seismic amplitude inst frequency original regularized seismic impedance emd table normalized mutual information nmi among predictors target sand fraction well seismic seismic inst predictor variable impedance amplitude frequency regularized original sand fraction fig regularization based reconstruct target signal along well spectrum seismic impedance spectrum superimposed plot actual regularized observed fig spectrum bandlimited seismic impedance diminishes beyond certain frequency range part sand fraction spectrum belonging slightly wider frequency range green curve fig reconstructed obtain regularized target wider range frequency chosen assumption ann nonlinear predictor capable mapping input signals lower frequencies output signals higher frequencies course needs entirely different emd shown table information content original sand fraction higher compared seismic predictor variables makes difficult model target sand fraction predictor attributes regularization process decreases information content sand fraction seen table dependency predictor target variables terms nmi table also improves result regularization wavelet decomposition wavelet decomposition extensively used different fields research example applied eeg signal artifact removal study geomagnetic replace line paper identification number edit signals etc representation one dimensional signal obtained using wavelet analysis wavelets oscillating functions localized time frequency finite energy time domain signal decomposed expressed terms scaled shifted versions mother wavelet corresponding scaling function decomposition scaled shifted form mother wavelet corresponding scaling function mathematically represented initial levels original target signal made zero regularized signal constructed performing inverse discrete wavelet transform idwt modified coefficients signal level original signal first decomposed high frequency low frequency components using high pass low pass filters filtering step output time series two low frequency part approximates signal high frequency part denotes residuals original approximate signal successive levels approximate component decomposed using set filters time domain signal expressed terms aforementioned mother wavelet corresponding scaling function level approximate detailed coefficients level coefficients computed using filter bank approach fig describes steps wavelet decomposition three levels signal decomposed approximate detailed coefficients using low pass high pass filters respectively decomposition coefficients modified case signal reconstruction modified approximate detailed coefficients sampled two convolved respective synthesis filters resulting pair summed finally modified signal acquired following level synthesis performance wavelet analysis dependent mother wavelet selection decomposition level daubechies family wavelets compact support relatively number vanishing moments therefore cases different variants daubechies family wavelets used signal analysis initial choice mother wavelet decomposition level six respectively kind study however initial choice wavelet decomposition level modified regularization results satisfactory paper choice mother wavelet decomposition level carried empirically based regularization result performance ann algorithm describes steps associated based regularization fig represent results regularization target sand fraction predefined wavelet type decomposition level first three detailed coefficients original sand fraction signal demonstrated fig decomposition detailed coefficients level level fig demonstration wavelet decomposition signal level algorithm regularization based task regularizing target based wavelet decomposition input predictor signal target signal select wavelet type number decomposition levels apply procedure fig target signal decide detailed coefficients truncated regularization selected detailed coefficients made zero regularized target signal reconstructed modified coefficients calculate entropies predictors well original regularized target signals entropy regularized target comparable predictor signal regularization result satisfactory regularization completed else step output regularized target signal fig regularization based reconstruct sand fraction along well decomposition result actual superimposed plot actual regularized well first five detailed coefficients truncated regularized target reconstructed approximate detailed coefficients sixth level idwt replace line paper identification number edit regularization result well presented fig blue red curves represent original regularized target sand fraction signals respectively table reveal changes information content original regularized sand fraction increase dependency target predictor variables result regularization empirical mode decomposition emd seismic well log signals signals reports suggest cases frequency analysis signals carried selected windows respect given orthogonal basis disadvantage basis decomposition techniques mismatch signal trend constant basis functions necessitate new decomposition method namely empirical mode decomposition emd emd algorithmic decomposition method decomposes input signal set intrinsic mode functions imfs residue signal two properties associated imfs numbers extrema present imfs imfs symmetric respect local mean words emd detects extracts highest frequency component signal step extracted imf contains lower frequency component compared extracted step moreover adaptive method emd decomposes input signal variable number components thus emd overcomes inherent limitation deciding priori number decomposition levels algorithm iii describes detailed steps associated emd based regularization target number imfs distribution imfs observed target predictor decide number imfs truncated emd regularization construct regularized target signal imf calculate entropies predictors well original regularized target signals entropy regularized target comparable predictor signal regularization result satisfactory regularization completed else step output regularized target signal fig first four imfs sand fraction log imfs residue decomposed seismic impedance along well plotted algorithm iii regularization based emd task regularizing target sand fraction based emd input predictor signal target signal algorithm initialize extract ith imf initialize extract local minima maxima iii create upper envelope emax lower envelope emin interpolating local maxima minima emin calculate mean envelope max imf imf else imfi least two extrema else imf decomposed numbers imfs residue signal emd target signal carried following steps imf fig regularization reconstruct sand fraction signal along well decomposed decomposed seismic impedance superimposed plot actual regularized comparison fig reveals number imfs obtained higher case target signal sand fraction predictor counterpart first imf component suppressed imfs used reconstruct regularized sand fraction superimposed plots actual blue curve regularized sand fraction red curve signals presented fig user decides regularization result satisfactory based visual inspection original regularized target variables regularized target smoother compared original signal replace line paper identification number edit nevertheless trend original signal preserved even information filtering based either three proposed regularization methods number selected less decomposed imfs original signal regularized usually selected one less number decomposed imfs original signal regularization result satisfactory terms entropy criterion superimposed plots actual regularized initial value retained algorithm iii otherwise value modified initial selection would one less number decomposed imfs original target signal working datasets table represents entropies predictor target attributes well improvement mutual dependency predictor target variables terms nmi evident table linear transform frequency spectrum signal obtained based comparison frequency spectrums predictor seismic impedance target regularization parameter selected result based regularization changes according value selected regularization parameter value selected regularization parameter small regularized signal would become smooth shedding finer trends original target signal regularization large parameter value would carry sufficient information filtering enable effective modelling target variable basic assumption stationarity signal however seismic signals therefore next two approaches based emd opted performance based regularization selection mother wavelet decomposition level determines granularity decomposed target signal regularized target signal reconstructed modified coefficients parameters wavelet type decomposition level detailed coefficients truncated finalized empirically based regularization result however case emd number parameters less number imfs used reconstruction target property needs decided three cases ensured regularized target signal retains trend original signal case based approach regularization parameter decided based results predictor well target signal number imfs seismic attributes less compared case emd cases emd based regularization number detailed coefficients imfs decided irrespective decomposition results predictor seismic signals instead parameters decided based regularization results entropy criterion mentioned algorithm algorithm iii respectively model building validation order establish efficacy proposed regularization method task model building validation stage carried using original regularized sand fraction target variable training dataset created aggregating sample patterns wells training patterns scrambled remove possible trend along depths remaining samples four wells combined divided two parts create testing validation datasets first network trained using training patterns initial parameter values network structure activation functions tuned using testing patterns testing phase important evaluating generalization capability trained network network performs satisfactorily terms rmse aem selected use validation stage statistical analysis errors involved model important proper understanding performance initial network structure decided intuitively depending nature problem amount available training patterns study several runs training testing validation neural network structures varying number neurons layers activation functions learning methods carried decide best structure well effective learning algorithm finally hidden layer hyperbolic tangent sigmoid transfer function used tangent sigmoid transfer function automatic choice researchers use hidden layer achieve swing activation function used output layer nonsymmetric nature facilitates faster learning rate number nodes input layer number predictor attributes used model ann example case predicting sand fraction three predictor namely seismic impendence instantaneous frequency amplitude number input output nodes three one respectively finally network single hidden layer trained using scaled conjugate gradient scg back propagation algorithm backpropagation learning using scg method given appendix scg proceeds conjugate direction previous step instead following gradient direction gradient descent based algorithms case conjugate gradient algorithms step size selected line search without evaluating hessian matrix however line search method computationally expensive due evaluation multiple error functions moreover line search deals multiple parameters crucial performance sometimes hessian matrix becomes non positive definite leads increase error weight updates due ill conditioning matrix inversion scg hessian matrix made positive definite computationally inefficient line search every iteration avoided using step size scaling mechanism scg another advantage scg involve user dependent parameters scg method used solve problems different research domains prediction groundwater level telemarketing success etc table reveals nmis instantaneous frequency relatively lower compared cases first attempt two input attributes seismic impedance amplitude used build prediction model corresponding results documented table iii terms performance evaluators replace line paper identification number edit though variation instantaneous frequency comparatively lower important attribute therefore three seismic attributes used inputs prediction model table second attempt performance comparison method averaging proposed regularization procedures carried original filtered moving average filter span nine samples passed ann training module table represents validation performance ann modelling averaged seismic impedance amplitude instantaneous frequency prediction results improve terms performance evaluators case filtered averaging compared original target observed table table performance trained ann superior working processed using proposed regularization target compared original averaged results reported table iii table table lead following important observations performance trained networks improved use regularized target signals quantified terms higher ccs lower error values cases inclusion instantaneous frequency third predictor improves prediction proposed regularization technique superior averaging method observed table iii table network performance superior case regularization based two predictor variables hand three predictors based regularization outperformed two regularization approaches terms performance evaluators however cases performance improved using regularized sand fraction target instead original log extent advantage scheme others may vary among different datasets therefore given scheme universally recommended user provided multiple choices among user select regularization technique best suits working dataset application machine learning algorithm literature study reveals important step prior modelling example normalization relevant attributes selection importance sampling etc part technique different fields environmental modelling chemistry biomedical reservoir engineering generally values rmse aem values value considered good fit studies prediction model attains values respective performance indicators beyond mentioned limits model retrained adjusting associated parameters baziar compared performance three machine learning techniques svm multilayer perceptron mlp inference system canfis predict permeability well logs normalization carried preprocessing technique modelling however integration seismic borehole dataset required since well logs used predictor variables performances three approaches svm mlp canfis quantified terms average absolute error aae equivalent aem mean squared error mse square rmse methodology used model original almost similar methodology adopted permeability prediction using multilayer perceptron comparison validation performance prediction using data part training reported achieved study revealed regularization stage improves prediction result terms aem rmse step involving single ann similar case present paper modelling original target comparison among ann performances without regularization step reveals results ann improved terms evaluators former case case unsatisfactory validation performance user may modify ann structures network parameters retrain modified ann validation performance still improve user modify regularization parameters carry modelling regularized target signal changing regularization parameters information retention changes choosing narrower bandwidth based method well logs become smoother reduced entropy well name table iii statistics validation performance two predictors original target sand fraction filtered fourier filtered emd rmse aem rmse aem rmse aem filtered wavelets rmse aem well name table statistics validation performance three predictors filtered fourier filtered emd rmse aem rmse aem filtered wavelets rmse aem original target signal rmse aem replace line paper identification number edit table statistics validation performance three predictors target averaged filtered sand fraction averaging well name rmse aem prediction entire volume along inline cross line carried using network selected acceptable performance validation step fig represents variation seismic amplitude specific inline inline containing well sand fraction variation inline shown fig obtained prediction sand fraction study area available seismic attributes using validated network parameters network used prediction calibrated using fraction target area contained selected window thus neighborhood values particular pixel play important role determination modified pixel value result order filtering case order statistics filtering window size selected define neighborhood around centered pixel selection window value crucial degree smoothing predicted sand fraction volume used input operation every element volume considered pixel smoothened using median filter respect neighborhood within window size missing values along boundaries ignored order carry median filtering values pixel neighbors within selected window first sorted centered pixel value replaced median value determined sorted pixel values example case median filtering window largest value neighborhood replaces pixel value center neighborhood logs area predicted seismic attributes variation predicted particular inline presented fig detailing variation revealed color code fig smoothened values attained median filtering may lose exactness predicted logs however volumetric representation filtered collectively offers better understanding variation fig represents result median filtering along inline effect localizing different levels sand fraction values observed comparing fig fig fig variation seismic amplitude inline fig result median filtering window predicted sand fraction variation inline fig variation predicted sand fraction inline observed fig variation seismic attributes study area smooth however sand fraction across study area shown fig changes abruptly transition sand fraction values smoother less agree patterns seismic data thereby rises need stage order obtain smoother sand fraction variation across volume incorporate rationale predicted values filtered median filter median filter order statistic filter spatial filter case order statistics filters filtered output dependent ordering ranking pixels image larger window sizes example higher order would smoothen predicted volume cost detailed variation subsequent research attempts adaptive post processing method potential improve prediction result worked would help subsurface characterization different types spatial filters would experimented observe changes filtered thus complete framework including learning validation finally successfully carries mapping seismic attributes sand fraction discussion conclusion paper brings complete workflow consisting elegant regularization step enhance replace line paper identification number edit learning capability ann carrying mapping seismic attributes lithological property sand fraction successfully improvement mapping introduction regularization step observed performance analysis comparison among results achieved work existing literatures reveals superiority proposed regularization step obtain improved performance terms performance evaluators present study synthetic logs generated study area available seismic information using validated network parameters obtaining volume postprocessing carried improve visualization means median filtering similarly important petrophysical properties porosity permeability modelled seismic attributes following proposed workflow prediction results petrophysical properties porosity permeability enables user identify zones high sand content porosity higher probability hydrocarbon presence thus study would help identify potential drilling locations new well study area present work differs work done terms stage division dataset adopted machine learning technique algorithm addition well tops horizon information used carry zone wise division overall dataset modular ann applied model multiple seismic attributes performance attained would improved use regularized log target instead original one selection initial parameters crucial achieving acceptable performance ann present work ann structure activation functions hidden output layers empirically adjusted also initial values weight bias matrices chosen randomly possible future extension domain metaheuristic algorithms explored order automate selections strengthen framework additionally postprocessing stage use spatial filtering provided significant improvement sand fraction variation study area adaptive method investigated future appendix learning ann using scaled onjugate gradient learning learning multilayer perceptron carried two phases synoptic weights network constant input signal propagated hidden layers output layer forward phase changes take place activation potentials output neurons contrast error signal computed difference network output actual desired target response error signal propagated layers backward direction output layer input layer backward phase training procedure described details follows assume set training samples used learning back propagation multilayer perceptron input vector applied input layer nodes desired output vector present output node compute error desired actual output forward signal propagated input layer output layer single multiple hidden layers induced local field neuron layer computed output neuron previous layer iteration yil synaptic weight wlji connected neuron layer expressed jil case output layer network depth output neuron written therefore error computed supervised learning multilayer ann viewed problem numerical optimization error surface multilayer ann nonlinear function weight vector assume error energy averaged training samples empirical risk avg using avg computed set contains neurons output layer second derivative cost function avg respect weight vector called hessian matrix denoted avg hessian matrix considered positive definite unless mentioned otherwise several algorithms train ann case conjugate gradient methods computational complexity memory usage large calculation storage hessian matrix stage indefiniteness handled scaling coefficient case scaled conjugate gradient scg parameters represent search direction steepest descent direction respectively different training algorithms experimented train single layer ann model multiple seismic attributes finally scg algorithm selected algorithms train ann due superiority terms obtained performances evaluators steps associated scg presented follows replace line paper identification number edit algorithm scaled conjugate gradient scg train network task obtain weight vector training input training dataset weight vector parameters associated training selection parameters weight vector assume success true success true compute information modify make hessian matrix positive definite evaluate step size compute comparison parameter error reduced success true mod else modify scale parameter else success false increase scale parameter check steepest descent direction desired minimum otherwise output trained weight vector references michelena lithologic characterization reservoir using transforms ieee trans geosci remote vol bosch mukerji gonzalez seismic inversion reservoir properties combining statistical rock physics geostatistics review geophysics vol ali ahmadi zendehboudi lohi elkamel chatzis reservoir permeability prediction neural networks combined hybrid genetic algorithm particle swarm optimization geophys vol may guo zhu method seismic reservoir fuzzy rules extraction expert syst vol mar hampson schuelke quirein use multiattribute transforms predict log properties seismic data geophysics vol wong bruce gedeon confidence bounds petrophysical predictions conventional neural networks ieee trans geosci remote vol sharma ali neural network approach improve vertical resolution atmospheric temperature profiles geostationary satellites ieee geosci remote sens vol ali jagadeesh lin hsu neural network approach estimate tropical cyclone heat potential 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neural network concept case study western onshore india pet sci vol jul barry cavers kneale report recommended standard digital tape formats geophysics vol victorine kansas geological survey http douillard schlumberger balz upscaling well logs seismic scale comparison based method geostatistical technique petrophysics meets geophysics november stright boucher mukherji derksen revisiting use seismic attributes soft data subseismic facies prediction proportions versus probabilities lead edge vol oppenheim schafer buck signal processing prentice hall cover thomas elements information theory new york usa john wiley sons mukerji avseth mavko takahashi statistical rock physics combining rock physics information theory geostatistics reduce uncertainty seismic reservoir characterization lead edge vol mar balasis donner potirakis runge papadimitriou daglis eftaxias kurths statistical mechanics informationtheoretic perspectives complexity earth system entropy vol tesmer perez zurada normalized 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anal vol smolinska hauschild fijten dallinga baumbach van schooten current breathomics review data techniques machine learning metabolomics breath breath vol jun rathi gupta approach predict breast cancer drug suggestion using machine learning techniques aceee int inf vol baziar tadayoni khalili prediction permeability tight gas reservoir using three soft computing approaches comparative study nat gas sci vol chaki verma routray mohanty jenamani classification framework using svdd application imbalanced geological dataset proc ieee students technology symp techsym gonzalez woods digital image processing second new jersey usa prentice hall huang yang tang fast median filtering algorithm ieee trans speech signal vol soumi chaki received degree iiest shibpur formerly besu shibpur india currently working toward degree electrical engineering indian institute technology kharagpur india research interests include signal processing machine learning algorithms aurobinda routray professor department electrical engineering iit kharagpur india research interests include nonlinear statistical signal processing embedded signal processing numerical lineal algebra diagnostics william mohanty professor department geology geophysics iit kharagpur india research interests include seismology reservoir characterization
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ideals coordinate sections determinantal varieties jan aldo conca volkmar welker abstract motivated questions algebra combinatorics study two ideals associated simple graph ideal defining orthogonal representations graph complementary determinantal ideal generic symmetric positions prescribed graph characteristic two ideals turn closely related algebraic properties radical prime complete intersection transfer ideal determinantal ideal ideals link properties combinatorial properties show always hold large enough specific classes graphs forests give complete picture classify radical prime complete intersection ideals introduction let field integer set simple graph vertex set edge set study following two classes ideals associated ideals integer consider polynomial ring every edge set ideal lkg called ideal short respect ideal lkg defines variety orthogonal representations graph complementary see coordinate sections generic symmetric determinantal ideals consider polynomial ring xij let generic symmetric matrix entry xij xji let sym matrix obtained replacing entries positions sym integer let idk xij ideal sym sym ideal defines coordinate hyperplane section generic symmetric determinantal variety similarly consider ideals defining coordinate hyperplane sections generic determinantal varieties generic pfaffian varieties sym observe section ideal ideal lkg closely related indeed characteristic classical results invariant theory employed show sym radical resp prime resp expected height provided lkg date january mathematics subject classification primary key words phrases work paper partly supported daad vigoni project indam conca welker radical resp prime resp complete intersection also exhibit similar relations variants lkg ideals defining coordinate sections determinantal pfaffian ideals facts turn focus algebraic properties lkg particular analyze questions lkg radical ideal complete intersection prime ideal show large enough three properties hold lkg among others able give necessary conditions lead full classification graphs lkg complete intersection prime case small characteristic deduce sym sufficient conditions radical prime expected height knowledge coordinate sections determinantal varieties systematically studied case maximal minors example results study properties roots work orthogonal graph embeddings see references motivation overview think interesting orthogonal embedding graph map edge edge set graph complementary thus definition real variety associated complementary graph coincides set orthogonal graph embeddings note variety includes degenerate embeddings injective send vertices zero vector since geometry variety orthogonal graph embeddings first studied ideals lkg carry name ideals indeed many algebraic results inspired results geometry real variety general position orthogonal embeddings well understood objects lkg called edge ideal graph squarefree monomial ideal clearly radical ideal respect every field prime empty complete intersection matching two edges empty intersection starting properties radical prime complete intersection become subtle results case see know general results beyond ones described section section generalize hypergraphs able state results section generality questions hypergraph remain unanswered nevertheless extending link graphs ideals minors hypergraph uniform hypergraphs related closure space symmetric tensors bounded rank prescribed expansion standard basis see proposition paper organized follows section introduce basic concepts notation graph theory bases theory section formulate main results lssideals sketch proofs section provide proofs results showing persistence properties complete intersection primeness particular follows fixed graphs minimal obstructions properties section exhibit obstructions prove necessity small give complete characterizations graphs prime complete intersection section define new combinatorial invariant graphs use prove lkg radical complete intersection prime large enough section define notation ideals minors pfaffians generic matrices recall classical results relation invariant theory section use connection invariant theory prove lkg radical prime property hold ideal minors generic matrices positions prescribed graph addition give obstructions preventing corresponding ideals minors prime finally section pose questions state open problems notations generalities graph hypergraph theory following introduce graph theory notation mostly follow conventions graph simple graph finite vertex set particular subset set subsets cases assume subgraph graph graph generally hypergraph pair consisting coordinate sections finite set vertices set subsets interested situation sets inclusionwise incomparable set subsets called clutter particular graph hypergraph clutter use following notations denotes complete graph vertices denotes complete bipartite graph bipartition denotes subgraph obtained removing edges denote subgraph edges denote path length subgraph edges denote graph complementary let write graph induced vertex set subgraph induced case simply write graph vertex set size called vertex connected every graph connected deg denote degree vertex deg maximal degree vertex clearly every vertex degree least finally denote clique number largest contains copy complete subgraph following well known fact follows directly definitions lemma given graph integer following conditions equivalent contain subgraph basics generalization hypergraphs let hypergraph define clutter call ideal lkh hypergraph sometimes useful consider lkh multigraded ideal equip multigrading induced deg unit vector clearly polynomial multigraded degree particular lkh following remark immediate consequence fact clutter two polynomials corresponding distinct edges incomparable multidegrees remark let hypergraph clutter generators lkh form minimal system generators particular lkh complete intersection polynomials form regular sequence following alternative description lkg graph turns helpful places remark let graph consider matrix lkg ideal generated entries matrix positions denotes transpose conca welker similarly bipartite graph say subgraph one considers two sets variables yij matrices yij lkg coincides ideal generated entries product matrix positions bases use following notations facts bases theory see example proofs details consider polynomial ring yij vector wij rnd polynomial set inw latter called initial term respect fixed term order set either following allows deduce properties ideals properties initial ideals proposition let homogeneous ideal polynomial ring term order rnd inw radical resp complete intersection prime moreover generators initial terms resp inw inw form regular sequence form regular sequence basis results counterexamples ideals first part section let graph start studying radicality lkg mentioned introduction lkg always radical trivial reasons following result gives complete answer theorem thm let graph char ideal lkg radical char lkg radical bipartite next examples show lkg need radical examples assume characteristic consider likely ideals radical field quick criterion implying ideal polynomial ring radical identify element call witness fact radical course potential witnesses must sought among elements somehow closely related alternatively one try compute radical even primary decomposition directly read whether radical direct computations extremely time consuming terminate computers examples nevertheless examples quickly identified witnesses example present three examples graphs lkg radical field characteristic first example vertices edges smallest example found terms edges vertices second example vertices edges complete intersection shows lkg complete intersection without radical third example bipartite subgraph edges smallest bipartite example found cases since lkg integral coefficients may assume exhibit witness polynomial lkg lkg latter inequality checked help cocoa macaulay let graph vertices edges depicted figure edges coordinate sections figure graphs lkg witness chosen follows denote yij generic matrix discussed remark ideal generated entries corresponding positions taken row indices let graph vertices edges depicted figure edges witness chosen follows denote yij generic matrix discussed remark ideal generated entries corresponding positions taken row indices fact complete intersection checked quickly cocoa macaulay let subgraph complete bipartite graph depicted figure edges denote xij generic matrix yij generic matrix explained remark ideal generated entries corresponding positions witness taken corresponding column indices following result shows properties complete intersection prime lkg closely linked persist occur theorem let graph lkg prime lkg complete intersection lkg complete intersection lkg prime lkg prime lkg prime lkg complete intersection lkg complete intersection lkg complete intersection complete intersection every subgraph lkg prime prime every subgraph proof theorem consists several steps first briefly sketch present full detail section sketch proof prove interpret lkg defining ideal symmetric algebra module quotient polynomial ring show statement follows induction employing result avramov prop characterizing complete intersection symmetric algebras result huneke thm characterizing symmetric algebras domains conca welker prove consider vector vij entries vij every observe inv therefore inv lkg lkg either lkg complete intersection assumption case assumption lkg prime implies inv lkg lkg assertions follow transfer properties inv recalled proposition assertion obvious one observes lkg also complete intersection general fact regular sequence homogeneous polynomials generates prime ideal every subsequence remark persistence result property radical indeed already seen lkg always radical lkg always radical case char hand example gives examples lkg simple examples also show radicality inherited subgraphs hand radicality inherited induced subgraphs follows fact every subset one lkgw lkg yij checked using multigraded structure see lkg complete intersection prime hence radical large enough prove fact section indeed generally show hypergraph clutter hypergraph lkh radical complete intersection vehicle define purely hyper graph theoretic invariant pmd called positive matching decomposition show following theorem let hypergraph clutter pmd ideal lkh radical complete intersection initial ideal pmd ideal lkh radical complete intersection graph pmd ideal lkg prime graph pmd min furthermore bipartite graph pmd min proof theorem consists several steps first briefly sketch present full details section sketch proof show every pmd exists vector vij rnd set inv consists pairwise coprime monomials follows inv lkg inv hence inv lkg radical complete intersection complete proof proposition implied therefore graph case theorem implies claim derived simple estimates using combinatorial structure graph complete graphs char provide asymptotic terms results radical complete intersection prime proposition corollary using sym transfer properties bounds derived using basis arguments corollary case complete bipartite graphs results concini strickland musili seshadri imply following theorem theorem let lkg radical every lkg complete intersection lkg prime pmd coordinate sections proof taking account remark assertions follow form general results variety complexes proved different techniques observed tchernev assertions refer hodge algebra structure variety complexes correct however assertions replaced statements concerning bases done example similar case hence still deduced arguments alternative proofs given also section assertion proved section seeing theorem one may wonder assertion theorem reversed next example shows general case example let even ideal lkg prime complete intersection assertions special cases subsequent theorem view theorem fixed graphs lkg complete intersection prime define downward monotone graph properties thus persistence graphs numbers lkg prime prime proper subgraph lkg prime pair considered minimal obstruction primeness similarly minimal obstructions complete intersection next results first steps towards classification minimal obstructions results partly inspired theorems book proposition let lkg prime contain subgraph isomorphic furthermore char contain subgraph isomorphic lkg complete intersection contain subgraph isomorphic furthermore char contain subgraph isomorphic obstructions derived proposition corollary example characteristic lkg prime contain small values implications proposition actually equivalences theorem let graph following equivalent lkg prime contain subgraph isomorphic following conditions equivalent lkg complete intersection contain subgraph graph contain even cycle graph contain even cycle forests graphs without cycles give quite complete picture theorem lkg lkg lkg let forest field radical complete intersection prime main tool proof theorem notion ideals developed inspired indeed turns forest ideal lkg ideal conca welker stabilization algebraic properties lkg section prove theorem state consequences embarking proofs need recall important results properties symmetric algebra module state results way suit needs best recall given ring presented cokernel map symmetric algebra symr isomorphic quotient symr ideal generated entries matrix representing vice versa every quotient ideal generated homogeneous elements degree symmetric algebra part following special case prop part special case thm rest paper denote matrix entries ring number ideal generated theorem let complete intersection symr complete intersection height symr domain domain height equivalent conditions imply remark let graph multigraded ideal lkg generated elements degree one block variables hence lkg seen ideal defining symmetric algebra various ways example set symmetric algebra cokernel map associated matrix yij remark order apply theorem case described remark important observe every minors matrix yij vanish modulo lkg lkg contained ideal generated monomials yik yjk terms minors yij belong obvious reasons formulate next results terms following algebraic invariants given algebraic property ideals graph set asymk inf lkg property course interested properties radical prime properties ideals defining normal ring ufd interesting well treated use new notation provide proof theorem proof theorem assume lkg complete intersection minimal generating set regular sequence remark form minimal generating set hence regular sequence particular subset regular sequence well follows consider vector assigning weight variables weight variables inw every hence initial forms generators form regular sequence follows form regular sequence hence lkg complete intersection terms asymk theorem yields following corollary coordinate sections corollary let graph asymk inf lkg position prove theorem proof theorem first show implies remark know form minimal system generators thus lkg complete intersection generators form regular sequence regular sequence generates prime ideal standard graded algebra local ring every subset sequence follows prove argue induction number vertices usual assume case trivial use notation remark note algebra retract therefore lkg prime induction follows complete intersection since degree vertex hence proposition virtue remark minors matrix particular hence theorem holds theorem holds well lkg complete intersection immediate corollary theorem obtain corollary let graph asymk asymk prime proceed proof theorem need another technical lemma lemma let matrix entries noetherian ring assume let polynomial ring let matrix entries obtained adding column height height height min height height proof set min height height let prime ideal containing prove height height height may assume corresponding first rows column hence height height contains since elements algebraically independent height height height turn proof theorem proof theorem first show implies lkg prime lkg complete intersection follows lkg prime completes proof proof argue induction number vertices assertion obvious assume set yij yij yij construction symmetric algebra presented cokernel map associated assumption lkg complete intersection hence complete intersection well follows induction prime hence domain proposition remark therefore theorem lkg prime heightit every conca welker equivalently prove height every height consider weight vector defined wij construction initial terms standard generators inw standard generators since standard generators coincide initial terms respect inw follows inw indeed equality holds need fact therefore inw enough prove height every equivalently heightit every variables appear generators hence yij let matrix column removed regarded symmetric algebra presented cokernel assumption complete intersection hence theorem know height every since obtained adding column variables lemma height min height height theorem together corollary directly imply following corollary let graph asymk prime inf lkg prime asymk asym prime asymk following proposition immediate consequence theorem theorem proposition let graph asymk radical asymk asymk prime bipartite asymk radical asymk asymk prime bounds tight general following example shows example using cocoa macaulay one check fields characteristic complete intersection hence theorem asymk similarly one check prime hence theorem implies asymk prime finally one checks radical hence asymk radical corollary able analyze asymptotic behavior asymk prime coordinate sections obstructions algebraic properties section prove theorem study necessary sufficient conditions lkg radical complete intersections prime first turn necessary conditions yield lower bounds asymk radical asymk prime asymk start proof proposition proof proposition first show implies lkg complete intersection theorem follows lkg prime hence contains case char violates conditions primeness lkg implies first prove lkg prime contain suppose contradiction lkg prime contains may decrease either assume right away particular assume subgraph renaming vertices complete bipartite graph edges set since subgraph assumption domain seen matrix identity field fractions hence rank rank follows rank rank implies zero ideals remark none minors lkg contradiction hence lkg prime remains shown char contain copy unfortunately resort proposition lemma iii section easily seen derivation independent results preceding sections theorem know lkg prime lkbd proposition gen implies prime generic matrix arbitrary size contradicts lemma iii next provide proof theorem proof theorem part lemma conditions theorem equivalent hence suffices prove equivalence statements obvious lkg prime edges set equivalent containing know proposition implies holds edges pairwise disjoint follows monomial ideal lkg complete intersection theorem assertion follows proposition condition implies prove implies may assume algebraically closed since tensor product domains domain see corollary proposition bourbaki algebra chapter prop may assume graph connected connected graph satisfying either isolated vertex path vertices cycle length hence prove lkg prime pmd indeed pmd seen easily form definition using lemma check maximal matching complement form positive matching decomposition hence theorem follows lkpn prime let set prove lkcn prime use symmetric algebra perspective observe set lkpm yij already proved conca welker prime complete intersection height prove symmetric algebra cokernel map domain since remark taking consideration remark may apply theorem therefore enough prove height height equivalently enough prove height height prove first since height obvious observe written path length vertices use disjoint set variables height proves note passant condition height holds domain hence deduce theorem lkcn complete intersection remains prove since height assertion obvious hence may assume use let prime ideal containing prove height contains height may assume contain say prove height since since mod either first case contains associated path length vertices hence height desired finally contains ideal associated vertices already observed ideal complete intersection since well follows height proof theorem part lemma conditions theorem equivalent hence suffices prove equivalence prove first implies proposition implies contain suppose contradiction contain hence complete intersection height generators sign among matrix ideal matrix height contradiction prove implies may assume algebraically closed since tensor product perfect field reduced reduced thm chapter may assume connected connected graph satisfying either isolated vertex path odd cycle already observed pmd theorem coordinate sections follows lkpn complete intersection remains prove complete intersection height note know already complete intersection height hence remains prove belong minimal prime generators sign adjacent matrix minimal primes described proof see also description given easy see minimal primes exception contained ideal yij clearly finally one since monomial divisible monomials support generators proceed proof theorem proof first formulate result special case general statement need introduce concept sturmfelscartwright ideals concept coined inspired earlier work developed applied various classes ideals consider polynomial ring yij multigrading degyij group gldn acts naturally group automorphism borel subgroup udn denotes subgroup gld upper triangular matrices ideal borel fixed every ideal called ideal exists radical borel fixed ideal multigraded theorem let yij polynomial ring multigrading induced degyij forest graph without cycles let polynomial degree ideal particular initial ideals radical proof first observe may assume generators form regular sequence end introduce new variables add monomial new variables degree coprime new polynomials form regular sequence proposition since initial terms respect appropriate term order pairwise coprime monomials ideal arises multigraded linear section ideal setting new variables thm family ideals closed multigraded linear section hence enough prove statement ideal equivalently may assume right away generators form regular sequences multigraded hilbert series multigraded written numerator laurent polynomial polynomial integral coefficients called since form regular sequence polynomial prove prove radical ideal taking consideration duality ideals ideals discussed enough exhibit monomial ideal whose generators polynomial ring conca welker equipped fine deg regarded claim assumption forest ideal desired property words prove tensor product truncated koszul complexes associated resolves ideal consider leaf set induction number edges resolves ideal homology since since leaf one two variables appear generators hence forms regular sequence hence resolves finally since concludes proof ideal ideal every initial ideal ideal ideal well property depends hilbert series particular every initial ideal ideal radical ready prove theorem proof theorem setting theorem lkg ideal hence radical lkg complete intersection proposition contain copy subgraph vice versa one proves lkg complete intersection using induction number vertices symmetric algebra point view forest may assume leaf forest hence induction complete intersection set may interpret lkg ideal defining symmetric algebra defined cokernel map associated matrix hence virtue theorem enough prove height equivalently height height true generators contained lkg prime proposition contain copy subgraph vice versa know lkg complete intersection hence theorem lkg prime coordinate sections hence forest complete picture asymptotic behaviour corollary let forest field asymk radical asymk asymk prime positive matching decompositions section introduce concepts positive matchings positive matching decomposition prove theorem definition given hypergraph positive matching subset pairwise disjoint sets matching exists weight function satisfying illustrate definition subgraphs edge set matching positive nevertheless matching positive graph edge set respect weight function next lemma summarizes elementary properties positive matchings lemma let hypergraph clutter positive matching positive matching induced hypergraph hvm positive matching matching positive matching iii bipartite graph bipartition positive matching matching directing edges yields acyclic orientation proof clearly weight function positive matching restricts making positive matching hvm conversely setting max max extends weight function shows positive matching let weight function positive set max define follow set set set max one easily checks positive matching respect iii change coordinates inequalities defining positive matchings simple reformulation get coordinates matching positive weight function equivalent existence region arrangement hyperplanes satisfying well known regions arrangement one one correspondence acyclic orientations see lemma position introduce key concept section conca welker definition let hypergraph whichse clutter positive matching decomposition partition pairwise disjoint subsets positive matching called parts smallest admits parts denoted pmd note definition one pmd obvious hand pmd smaller clutters next prove bound theorem proof theorem first consider case arbitrary graph set may assume complete graph induces subgraphs set example simplicity written edge clearly one since new edge inserted smallest index involved edge satisfies condition lemma respect current data example insert edge vertex satisfies condition lemma respect matching edges already used earlier step construction consider case bipartite graph let bipartition may assume numbers may assume show positive matching decomposition show positive matching assertion obvious since contains single edge assume lemma iii suffices show directing edges edges direction yields acyclic orientation assume resulting directed graph cycle cycle bipartite graph even length cycle must contain least two edges type equivalently assume chosen property maximal next edge directed cycle edge must following edge cycle must satisfy follows contradiction analogously consider edge preceding construction contradiction hence cycle positive matching bipartite case theorem shows pmd computer experiments show pmd holds small value next connect positive matching decompositions algebraic properties lemma let hypergraph clutter pmd positive matching decomposition exists term order every every proof define first define weight vectors purpose use weight functions associated matching weight vector defined follows yik coordinate sections definition weight monomial yik withqrespect hence definition weight positive negative follows yik define term order follows smallest arbitrary fixed term order simple induction using shows conclude section proofs theorem theorem simple corollary proof theorem let pmd hence lemma term order satisfying since matching implies initial monomials generators pairwise coprime hence lkh radical complete intersection prove follows proposition follows theorem also complete proof theorem proof theorem theorem know pmd theorem know lkkm prime pmd theorem know lkkm prime contain subgraph latter implies hence pmd therefore pmd using fact theorem primeness inherited subgraphs following immediate consequence theorem corollary let subgraph radical complete intersection min prime min let subgraph radical complete intersection min prime min determinantal rings point view invariant theory goal section recall classical results invariant theory see example treated modern terms concini procesi particular recall rings arise invariant ring group actions assume throughout base field characteristic generic determinantal rings rings invariants gen take matrix gen gen variables xij consider ideal gen xij gen generated consider two matrices variables size following action gld polynomial ring matrix acts automorphism sends entries product matrix clearly invariant action hence ring conca welker invariants contains subalgebra generated entries product first main theorem invariant theory action says surjective map gen sending xij clearly product matrix gen rank hence ker second main theorem invariant theory gen says ker hence gen generic symmetric determinantal rings rings invariants sym take symmetric matrix variables xnsym xnn consider ideal xnsym generated xnsym polynomial sym ring xij consider matrix variables size following action orthogonal group glt polynomial ring yij acts automorphism sends entries product matrix invariant action hence ring invariants contains subalgebra generated entries first main theorem invariant theory action asserts surjective presentation sym sending since product matrix rank ker second main theorem invariant theory says ker hence sym xnsym generic pfaffian rings rings invariants skew take matrix variables xnskew coordinate sections consider ideal generated pfaffians size xij xnskew polynomial ring skew xij consider matrix variables size let block matrix blocks diagonal positions sympletic group ajat acts polynomial ring yij follows acts automorphism sends entries product matrix invariant action hence ring invariants contains subalgebra generated entries first main theorem invariant theory current action says surjective presentation skew sending product matrix rank hence ker second main theorem invariant theory action says ker hence skew xnskew determinantal ideals matrices relation classical invariant theory point view described section shows generic determinantal pfaffian ideals prime kernels ring maps whose codomains integral domains height also well know see example references given gen matrix variables gen height ideal idk sym height ideal idk xnsym symmetric matrix variables skew height ideal pfaffians xnskew size degree matrix variables one replaces entries matrices general linear forms say variables bertini theorem combination fact generic rings implies ideals remain prime long radical case special linear sections determinantal ideals matrices case coordinate sections corresponding ideals prime radical describe coordinate sections employ following notation gen generic case take described introduction bipartite graph gen denote matrix obtained matrix variables xij replacing entries position sym generic symmetric case take subgraph denote sym matrix obtained generic symmetric matrix xnsym replacing entries row column row column skew generic skewsymmetric case take subgraph skew denote matrix obtained generic skewsymmetric matrix xnskew replacing entries row column row column gen sym gen sym terminology idk resp idk ideal resp gen sym skew skew skew resp write ideal pfaffians size gen sym skew ask conditions imply idk resp idk radical prime conca welker simple examples show special linear sections relatively small height generic determinantal ideals give ideals positive side maximal minors following results remark eisenbud proved ideal maximal minors matrix linear forms prime remains prime even modding set linear forms particular ideal maximal minors matrix linear forms prime provided ideal generated entries matrix least generators giusti merle studied ideal maximal minors coordinate sections generic case one main results characterizes combinatorial terms subgraph graphs variety associated gen gen irreducible radical prime gen boocher proved subgraph ideal radical combining result result giusti merle one obtains characgen prime terization graphs generalizing result boocher proved ideals maximal minors matrix linear forms either row column multigraded radical gen generic case every minor matrix type multiple factors gen might multidegree suggests determinantal ideals always radical following example shows case gen example let matrix associated graph example generic matrix set entries positions gen radical field characteristic likely field gen gen witness since contained one gen consider well matrix ideal turns radical similarly symmetric matrices sym example let generic symmetric matrix associated graph example generic symmetric matrix set entries positions sym well symmetric positions radical field characteristic sym witness since contained one consider well matrix ideal turns radical examples example example example indeed closely related explain let subgraph complete bipartite graph view isomorphism gen gen xij yij zij respectively matrices variables ideal generated lkg indeed equal classical result invariant theory derived fact linear groups reductive characteristic direct summand characteristic implies lkg next proposition immediate consequence coordinate sections proposition let field characteristic subgraph lkg radical resp complete intersection resp prime gen coordinate section generic determinantal ideal radical resp maximal height resp prime start subgraph may sym consider coordinate section xnsym using isomorphism obtain proposition let field characteristic subgraph lkg radical resp complete intersection resp prime coordinate sym section generic determinantal ideal radical resp maximal height resp prime back proposition let field characteristic let largest positive integer lkkn prime lkkn complete intersection proof set numbers chosen using formulas mentioned beginning section ideal generic symmetric matrix height consider graph vertices vertices appear edge sym contradiction ideal lkkn prime proposition ideal prime height one sym xhn contradiction check enough prove rank matrix sym mod xhn check rank block decomposition zero matrix size since latter obvious set numbers chosen ideal generic symmetric matrix height assume contradiction lkkn complete intersection proposition sym follows height sym xhn contradiction boils obvious statement rank matrix zero submatrix certain size using result analyze asymptotic behavior asymk asymk prime corollary let field characteristic asymk prime asymk lim lim conca welker proof corollary asymk prime proposition asymk asymk prime hence equalities follow fact lim using proposition theorem obtain obstructions corollary let graph vertices field characteristic lkg lkg prime complete intersection defined proposition get actual feeling obstruction list explicit example new obstructions derived corollary obstruction complete intersection obstruction primeness skew xnskew may consider coordinate section may well consider graph associated twisted defined follows every consider indeterminates set entry matrix row column call twisted associated twisted coincides binomial edge ideal defined studied remark assume bipartite bipartition coordinate transformation sends lkg particular bipartite graph radical resp prime lkg radical resp prime using isomorphism obtain proposition let field characteristic subgraph twisted radical resp resp prime coordinate section skew generic pfaffian ideal radical resp maximal height resp prime coordinate sections theorem let field characteristic gen gen gen subgraph ideals radical sym sym sym subgraph ideals radical skew skew subgraph ideal radical furthermore forest gen sym skew idk idk radical gen sym maximal height gen sym idk idk prime proof statements ideals cases gen sym follow proposition proposition using fact edge ideal graph radical indeed results hold field arbitrary characteristic corresponding ideals thm ideal lkg radical graphs using proposition gen sym proposition implies radical bipartite graphs radical arbitrary graphs cor ideal radical graphs using proposition implies skew radical arbitrary graphs finally forest results case minors derived proposition proposition theorem pfaffian case follows using theorem proposition following corollary immediate consequence assertion skew theorem corollary let coordinate ring subspaces equipped standard coordinates subset coordinates generates radical ideal note subsets generic matrix define ideals example ideal generated four generic radical statement analogous corollary higher order true indeed point set generic matrix generate radical ideal general example modulo class nilpotent gen sym next look necessary conditions idk idk prime condition tie proposition lemma let graph sym prime contain subgraph isomorphic gen assume bipartite bipartition prime deleting vertices vertices yields connected graph gen iii generic matrix prime assume vertices graph obtained deleting vertices disconnected implies sym selecting rows columns corresponding remaining vertices yields matrix reordering vertices least two blocks hence determinant reducible since determinant among sym minimal generating set follows prime one easily checks similar arguments proof first part verify assertion proof conca welker gen iii set xdd let polynomial ring whose indeterminates entries first prove every ideal expected height height ideal indeed prime height obvious generic matrix follows fact corresponding prime virtue proposition let prime containing contains height contain may assume inverting using standard localization trick determinantal ideals one sees contains change variables hence height height know height prove prime enough observe latter straightforward since mod submatrix consisting first rank questions open problems corollary corollary seen properties prime lkg persistence along parameter example shows persistence need hold property radical question patterns asymk radical lkg radical occur graphs expect erratic behavior occur example believe exists graph number lkg lkg lkg radical seen theorem cor thm certain fixed combinatorially classify graphs lkg radical complete intersection prime classifications based rather simple graph theoretic properties question fix number simple combinatorially defined classes gradical gprime say fields characteristic lkg radical gradical lkg complete intersection lkg prime gprime said expect pattern numbers lkg radical quite erratic therefore let concentrate properties prime complete intersection fact property inherited subgraphs supports hope classifications asked question theorem complete intersection primeness first parameters classification open even conjecture lkg complete intersection graph figure gives graph checked lkg complete coordinate sections intersection char still satisfies necessary conditions proposition subgraphs geometric results book indicate proposition still carries essential obstacles lkg prime question true lkg prime contain subgraph isomorphic subgraph isomorphic via fact primeness lkg implies primeness result giusti merle thm guides intuition behind following question question let subgraph graph assume true lkg prime contain subgraph isomorphic proposition proposition know lkg radical prime sym respectively general bounds asymk radical asymk prime corollary good enough make use implication indeed corollary shows properties complete intersection prime large enough graphs proposition prove primality interesting ideal hand use theorem theorem shows one take advantage connection cases would interesting exhibit classes different forests possible gen question interesting classes graphs asymk asymk prime despite fact proposition destroys hope using theorem general graphs would interesting replace asymptotic result actual value corollary large asymk prime numbers using notation proposition satisfy conjecture actual formula question value asymk prime radicality concrete conjecture case conjecture conjecture asymk radical least char words given matrix variables size conjecture ideal entries radical would also interesting study ideal generated entries note symplectic version problem investigated concini next turn open problems hypergraph know theorem hypergraph clutter ideal lkh radical complete intersection pmd prove theorem prime pmd case graph question true hypergraph clutter lkh prime pmd similarly persistence results theorem ask generalizations question let hypergraph clutter true lkh complete intersection resp prime lkh number call hypergraph graph every element cardinality particular clutter example graphs graphs say graph partition hypergraph hypergraph considered graph bipartite connect study ideal lkh graphs study coordinate sections space tensors given rank consider two mappings conca welker let standard basis vectors vectors consider map sends tensor take sums different tensors arising eir numbers permuting positions standard basis space symmetric tensors let natural numbers let knj standard basis vector knj vectors consider map sends vir eir knr take tensors eir numbers standard basis recall symmetric tensor symmetric rank written sum decomposable symmetric tensors details tensor rank geometry bounded rank tensors refer reader let hypergraph write lkh vanishing locus lkh definition maps immediately implies following proposition proposition let hypergraph algebraically closed field restriction map lkh parametrization space symmetric tensors rank expanded standard basis zero coefficient basis elements indexed particular image restriction irreducible lkh prime respect partition restriction map lkh parametrization space tensors knr rank expanded standard basis zero coefficient basis elements indexed particular image restriction irreducible lkh prime proposition gives motivation question indeed suggests strengthen question question let algebraically closed field one describe classes hypergraphs lkh prime bounded maximal symmetric rank symmetric sensor analogous question asked hypergraphs tensors bounded rank references system computations commutative algebra available http complete intersections symmetric algebras algebra determinantal rings lecture notes math berlin bases determinantal ideals commutative algebra singularities computer algebra sinaia nato sci ser math phys chem kluwer acad dordrecht coordinate sections free resolutions sparse determinantal ideals math res lett bourbaki algebra chapters berlin cartwright hilbert scheme diagonal product projective spaces int math res negri universal bases maximal minors int math res imrn negri universal bases ideals negri multigraded generic initial ideals determinantal ideals homological computational methods commutative algebra conca gubeladze joseph eds indam series heidelberg negri ideals associated graphs linear spaces concini symplectic standard tableaux adv math concini variety complexes adv math concini characteristic free approach invariant theory adv math lattice walks primary decomposition mathematical essays honor rota prog math boston boston graph theory graduate texts math heidelberg linear sections determinantal varieties amer math sections planes sections par les plans algebraic geometry lecture notes math berlin software system research algebraic geometry available http interpretation whitney numbers arrangements hyperplanes zonotopes partitions orientations graphs trans amer math soc binomial edge ideals conditional independence statements adv appl math macchia ideal orthogonal representations graph adv appl math ideals adjacent minors algebra symmetric algebra module algebra madani regularity binomial edge ideals combin theory ser landsberg tensors geometry applications graduate studies mathematics american mathematical society providence geometric representations graphs http orthogonal representations connectivity graphs linear alg appl regularity bounds binomial edge ideals commut algebra milne algebraic geometry version http schubert varieties variety complexes arithmetic geometry vol progr math boston boston graphs ideals generated comm alg universal complexes generic structure free resolutions michigan math classical groups invariants representations princeton univ press princeton dipartimento matematica genova via dodecaneso genova italy address conca marburg fachbereich mathematik und informatik marburg germany address welker
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link adaptation algorithms dual polarization mobile satellite systems anxo pol carlos ana mar universidade vigo anxotato mosquera centre telecomunicacions catalunya castelldefels barcelona universitat catalunya barcelona abstract use dual polarization mobile satellite systems promising means increasing transmission capacity paper study system uses simultaneously two orthogonal polarizations order communicate users application mimo signal processing techniques along adaptive coding modulation forward link provide remarkable throughput gains compared single polarization system gateway allowed vary mimo modulation coding schemes frame selection done means link adaptation algorithm uses tunable margin achieve predefined target frame error rate key words link adaptation adaptive coding modulation mimo satellite communications mobile satellite systems introduction recent years spectrum saturation increasing demand higher data rates ubiquitous way encourages engineers design new techniques order increase capacity communication systems without resorting expand occupied bandwidth two techniques leverage multiple antennas transmitter receiver means mimo multiple input multiple output signal processing techniques also adaptive coding modulation acm link adaptation part many current terrestrial wireless communication standards lte ieee cellular technologies ieee wireless local area networks paper propose apply mimo techniques mobile satellite communication systems exploiting polarization domain satellite links operating low frequency bands usually rely single circular polarization either rhcp right hand circular polarization lhcp anxo tato pol henarejos carlos mosquera ana left hand circular polarization avoid effects faraday rotation propose simultaneous use orthogonal polarizations within beam communicate users following preliminary studies presented within framework european satellite network experts algorithm switch among different mimo schemes presented particular siso orthogonal block codes optbc polarization modulation pmod vertical bell laboratories layer chosen function set effective signal noise ratio snr metrics hand use acm techniques allows system adapt instantaneous rate current channel capacity without designing system channel worst case paper following previous works propose use link adaptation algorithm adaptive margin selecting modulation coding scheme frame modcod mcs algorithm operates together mimo mode selection scheme increase spectral efficiency whilst trying maintain frame error rate fer level specified quality service qos parameter connection remainder article structured follows section provides description satellite communication system including link adaptation algorithms overview channel model channel series generated section describes link adaptation algorithms selecting mimo mode mcs section provides simulation results algorithms maritime mobile scenario lastly section collect main conclusions note future work topic system model consider satellite communications system serves mobile users operating link adaptation algorithm proposed forward link gateway towards mobile terminal typically bands one polarization used protect high leakage example service bgan broadband global area network standardized etsi employs rhcp higher frequency bands like high xpd cross polar discrimination would make possible use two independent streams polarizations system propose develop adaptive algorithms uses simultaneously polarizations yielding mimo system system analyse extent inspired bgan baseline single input single output siso system complemented additional mimo modes namely optbc polarization modulation pmod optbc orthogonal block code based alamouti coding used achieving transmit diversity introduced link adaptation algorithms dual polarization mobile satellite systems spatial components alamouti replaced two available polarizations optbc pmod analogous spatial modulation techniques one polarization per channel use transmitting symbols given constellation one extra bit information conveyed indicating polarization chosen qpsk constellation implies gain spectral efficiency two independent symbols transmitted per channel use receiver charge reducing interference perform detection three mimo schemes require channel state information csi transmitter therefore suitable satellite scenario csit outdated due long round trip time addition forward link algorithms operate assume feedback channel return link used mobile terminal inform gateway csi result frames decoding form optimum transmission mode calculated receiver information used gateway transmitter selection preferred mcs link adaptation algorithm propose signal model given time instant transmitted power channel matrix vector received signal component per polarization transmitted signal represents additive complex white gaussian noise awgn therefore resultant signal noise ratio snr transmitted symbols grouped blocks packets codewords span symbols transmitted baud rate gives frame length use frames one block length hereafter employ term frame referring codeword order speed simulations implement entire signal processing block chain thus use physical layer abstraction techniques particular effective snr comparison effective snr received frame threshold snr mcs used transmit frame allows decide frame decoded effective snr higher corresponding threshold effective snr frame snref given set snrs symbol obtained snref procedure called snr compression snr symbol period calculated different mimo modes anxo tato pol henarejos carlos mosquera ana siso optbc snr khn pmod mmse minimum mean square error receiver snr snr previous equations denotes coefficient channel matrix time instant indicates number stream identity matrix khn denotes frobenius norm matrix say case snr expression depends type receiver throughout work assume mmse receiver used reception given simplicity robustness noise table collects available transmission modes mcs including required effective snr correct decoding obtained curve snr mutual information corresponding spectral efficiency table coding rate options bearer qpsk constellation coding rate threshold snr snrth spectral efficiency spectral efficiency pmod spectral efficiency channel model simulation mobile satellite dual polarized channel done following work channel obtained sum three different components los specular reflected signal diffuse components produced scatterers near los specular components modelled rice random variables whereas diffuse component causes fast fading rayleigh distributed three components grouped following channel matrix link adaptation algorithms dual polarization mobile satellite systems dkd matrices collect rice factors polarization los specular diffuse components respectively two matrices related channel mixes two polarizations matrix entries complex gaussian random variables given covariance matrix parameters building matrices depend considered environment found channel generation fig shows block diagram channel time series generator one important aspect generated channel series time correlation doppler spread assuming clarke model coherence time approximated wavelength mobile terminal speed first generate independent realizations channel matrices uniformly distributed random phases complex gaussian random matrix make linear interpolation obtain channel matrices frequency equal doppler spread applied lastly channel matrices scaled yield given average snr simulation fig block diagram channel time series generator algorithm mode mcs selection paper link adaptation procedure select mcs mimo mode siso optbc pmod split tasks anxo tato pol henarejos carlos mosquera ana gateway one hand selects mimo mode achieves highest throughput given current snr receiver needs compute effective snr modes choose efficient task performed receiver reduce feedback load addition transmitter gateway informed preferred mimo mode effective snr mode selects consequently mcs following frame order formalize procedure let define following matrices matrix spectral efficiency combination mimo mode mcs example element sij spectral efficiency mimo mode mcs number also define vector snref represents csi whose elements effective snr one transmission modes lastly vector threshold snr mcs snrth note threshold mimo modes optimization problem choose optimum mode written rij max rij corresponding matrix rates given rij sij snref snrth mcs verifies required effective snr rij zero case tie mimo mode highest effective snr chosen reported back gateway choice mcs gateway follows previous work exploits adaptive margin lack space replicate derivation algorithm interested readers referred gateway selects mcs using lookup table lut represented means function maps snr intervals mcss shown graphically diagram fig value snr reported back plus margin introduced lut selecting mcs margin updated new feedback comes assume round trip time amounts duration frames recursive equation updating value margin recursion derived solve following optimization problem min min involved variables effective snr used lut decide mcs frame acknowledgement ack nack two constants take values respectively work suppose objective fer fixed value although studies like show benefits variable order maximize throughput link adaptation algorithms dual polarization mobile satellite systems fig block diagram mcs selection algorithm lut simulation results several simulations performed evaluate spectral efficiency gain use two polarizations simultaneously also understand potential robustness coming use adaptive margin link adaptation algorithm simulations results presented obtained maritime scenario vessel moving constant speed parameters channel generator taken carrier frequency simulations ghz typical frequency mobile satellite systems operating qpsk modulation frames symbols baud rate used physical layer similarly bearer bgan simulation comprises transmission reception frames specific average snr ranging steps average spectral efficiency defined rmi cumulative fer whole transmission also computed simulation previous expression rate jth mcs mcs selected frame corresponding ack lastly rtt set frames approximate feedback delay geo satellites contrary publications assume instantaneous feedback reasonable value proposed scheme robust even delay objective first set simulations show benefits using two orthogonal polarizations simultaneously serve mobile user low bands spectrum despite xpd lower bands bands provides significant throughput gain even available power split two polarizations simulations siso case transmit one polarization compared use two polarizations mimo based algorithm choosing dynamically mimo mode equation cases acm employed fixed margin lut see fig results collected table show use dual polarization provides gain higher cases interestingly optbc makes possible extend operation range low snr siso snr increases selected mode turns pmod instead optbc gains anxo tato pol henarejos carlos mosquera ana table results comparison single polarization siso adaptive dual polarization mimo average snr selected modes optbc efficiency siso efficiency mimo gain inf pmod pmod blast blast average snr pmod employed higher snrs mainly used mimo mode sends two simultaneous streams symbols offers gain terms throughput double capacity next let margin adapt effort match target fer figure shows results obtained terms average spectral efficiency fer acm scheme fixed margin compared link adaptation algorithm adaptive margin two different values target fer fer spectral efficiency fixed margin adaptive margin target fer adaptive margin target fer fixed margin adaptive margin target fer adaptive margin target fer average snr average snr fig average spectral efficiency left fer right fig observe behaviour algorithm terms spectral efficiency fer wide range los snrs spectral efficiency grows linearly snr reaches maximum value looking throughput significant differences using fixed adaptive margin furthermore observe target fer influences final spectral efficiency example simulations target fer slightly better efficiency target fer dependence spectral efficiency target error rate already studied authors show optimal fer maximizes throughput link adaptation algorithms dual polarization mobile satellite systems general margin converges value oscillates around except three higher snrs grows indefinitely observed value final margin depends operation point los snr target fer channel conditions mobile speed last fact exemplified show evolution margin intermediate tree shadow environment several speeds side fer behaves differently adapting margin respect fixed case shown right plot figure fixed margin guarantee prescribed fer different snr values opposed adaptive algorithm exposed earlier matches target fer example wide range los snrs therefore combination transmission mode selection adaptive margin achieves remarkable throughput gains compared siso case whilst guaranteeing physical layer fer suited prescribed qos conclusions future work simultaneous use dual orthogonal circular polarizations mobile satellite communication systems proposed paper application mimo techniques satellite scenario switching among several mimo modes generates significant increments spectral efficiency get close high values snr without using extra bandwidth power moreover proposed link adaptation algorithms select mimo mode mcs frame based adaptive margin helps system adapt channel conditions guarantee prefixed target frame error rate wide range snrs lastly future plan improve algorithm online calculation optimal fer maximizes spectral efficiency acknowledgements work partially funded agencia estatal spain european regional development fund project myrada also funded xunta galicia xeral universidades predoctoral scholarship european social fund consolidada galicia accreditation european regional development fund erdf part research done stay centre telecomunicacions catalunya cttc supported project cofunded european space agency esa work also received funding spanish ministry economy competitiveness ministerio competitividad project catalan government anxo tato pol henarejos carlos mosquera ana references lte evolved universal terrestrial radio access physical layer procedures etsi ieee standard air interface broadband wireless access systems ieee std revision ieee std ieee standard information information exchange systems local metropolitan area requirements part wireless lan medium access control mac physical layer phy specifications ieee std revision ieee std richharia mobile satellite communications principles trends wiley henarejos mazzali mosquera advanced signal processing techniques fixed mobile satellite communications advanced satellite multimedia systems conference signal processing space communications workshop sept arnau mosquera link adaptation mobile satellite links schemes different degrees csi knowledge international journal satellite communications networking vol online available http tato mosquera robust adaptive coding modulation scheme mobile satellite forward link signal processing advances wireless communications spawc ieee international workshop june tato mosquera gomez link adaptation mobile satellite links field trials results advanced satellite multimedia systems conference signal processing space communications workshop sept satellite component umts family satellite radio interface etsi alamouti simple transmit diversity technique wireless communications ieee journal selected areas communications vol oct henarejos dual polarized modulation reception next generation mobile satellite communications ieee transactions communications vol oct wolniansky foschini golden valenzuela architecture realizing high data rates wireless channel ursi international symposium signals systems electronics conference proceedings cat sep kaltenberger latif knopp scalability robustness accuracy physical layer abstraction evaluations lte networks asilomar conference signals systems computers nov sellathurai guinand lodge coding mobile satellite communications using channels ieee transactions vehicular technology vol jan park daniels heath optimizing target error rate link adaptation ieee global communications conference globecom dec
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aug canonical modules noetherian algebra mitsuyasu hashimoto department mathematics okayama university okayama japan dedicated professor shiro goto occasion birthday abstract define canonical modules algebra noether commutative ring study basic properties using modules generalize theorem araya iima generalize theorem syzygies evans griffith among others prove version aoyama theorem states canonical module descends respect flat local homomorphism also prove codimension modules coherent sheaf algebras module generalizing result author contents introduction preliminaries mathematics subject classification primary secondary key words phrases canonical module module approximation property canonical module module aoyama theorem theorem modules symmetric frobenius algebras argument introduction evg evans griffith proved criterion finite module noetherian commutative ring nth syzygy generalized theorem semidualizing module araya iima ari main purpose paper prove generalization results following settings ring finite may commutative module notion module introduced algebrogeometric situation criterion module nth syzygy generalized using modules standard argument see also generalized theorem schemes modules also generalize result theorem modules noncommutative sheaves algebras proposition let complete semilocal noetherian ring let dualizing complex rhomr dualizing complex lowest cohomology denoted called canonical module semilocal complete called canonical module canonical module completion module defined using canonical module finite right resp left module said ncanonical satisfies serre condition spec depthrp min dim suppr isomorphic right left module dim completion order study modules study analogue theory canonical modules developed aoyama aoy aoyg ogoma ogo commutative algebra among prove analogue aoyama theorem aoy states canonical module descends respect flat homomorphisms theorem main theorem following theorem evg ari let noetherian commutative ring may commutative let right set let mod following equivalent syz means suppr suppr spec depth min dim modified serre condition syz means means existence exact sequence still exact applying condition modified version takahashi condition free tak assumptions theorem let say resp canonical map injective resp bijective say see definition even commutative ring ring appears natural way even case definition slightly different takahashi original one prove general lemma modified version takahashi result tak application main theorem formulate prove different form existence approximations takahashi tak using modules see corollary corollary results strong enough deduce tak corollary commutative case related categorical results see section preliminaries depth serre conditions modules section discuss categorical abstraction approximations modules appeared tak everything done categorically theorem abstraction tak view fact general lemma section discuss prove lemma related lemmas section define canonical module algebra noetherian commutative ring prove basic properties section define module prove basic properties generalizing constructions results section section prove version aoayama theorem says canonical module descends respect flat local homomorphisms theorem corollary commutative case immediately localization canonical module canonical module important section section prove theorem related results approximations corollary corollary corollaries prove analogues theorems schenzel aoyg canonical module proposition corollary section define discuss symmetric frobenius algebras versions commutative algebra version gorenstein ring known rings discuss version rings scheja storch discussed relative notion definition absolute sense independent choice local property agrees gorensteinness discussed goto nishida see proposition corollary section show argument using existence modules still valid settings acknowledgments special thanks due professor osamu iyama valuable advice discussion special thanks also due professor tokuji araya work motivated advice proposition outcome discussion author also grateful professor iima professor takesi kawasaki professor ryo takahashi professor kohji yanagawa professor yuji yoshino valuable advice preliminaries unless otherwise specified module means left module let ring homb extb mean hom ext left denotes opposite ring nothing right let mod denote category mod also denoted mod left resp right noetherian ring mod resp mod denotes full subcategory mod resp mod consisting finitely generated left resp right derived categories employ standard notation found hart abelian category denotes unbounded derived category plump subcategory full subcategory closed kernels cokernels extensions denotes triangulated subcategory consisting objects ring denote mod mod mod dfg left noetherian throughout paper let denote commutative noetherian ring semilocal resp local jacobson radical say semilocal resp local say semilocal resp local semilocal resp local set consider convention subset inf means inf exists uniquely element similarly sup ideal mod define depthr inf extir call mat section also called semilocal denote depth depthr depth call depth lemma following functions valued equal ideal depthr inf depthrp spec inf hii otherwise length maximal function homr nonzerodivisor proof omit proof refer reader mat section subset spec define codim codimx codimension inf codim ideal mod define codim codim suppr ann ann denotes annihilator denote set spec rhni subset means rhni moreover use notation stands mod set minimal primes denoted min define spec depth similarly use notation spec depth let mod say satisfies snn snn spec depthrp min dimrp snr snr simply denoted say satisfies satisfies snm condition resp equivalent say resp maximal depth dim resp depth dim consider snn class modules also write snn snn lemma let exact sequence mod proof follows depth lemma depthrp min depthrp depthrp fact maximal modules closed extensions similar corollary let exact sequence mod assume proof proved using repeated use lemma lemma acyclicity lemma let noetherian local ring complex mod depth implies depth acyclic lemma let complex mod codim acyclic proof using induction may assume assume acyclic take assr assumption localize considering complex get contradiction lemma example let map mod injective injective indeed consider complex apply lemma lew bijective bijective consider complex time lemma let noetherian local ring min dim dim depth dim moreover depth dim proof ischebeck proved mod depth dim extir mat depthr dim rest easy corollary let min min proof let depth dim take min dim dim min min qrp min lemma dim dim depth hence corollary let mod proof obviously converse apply corollary let mod say satisfies suppr suppr lemma let mod following equivalent min min proof min min assumption min supp min min supp min corollary supp supp follows min min another case implies said full suppr spec finitely generated faithful full lemma let mod full satisfies condition satisfies snn condition annr annr satisfies snn condition satisfies annr condition proof first assertion dim dim spec second assertion follows first annr snn snn annr thing lemma let ideal commutative mod depthr depths particular semilocal depthr depths proof note hii lemma get lemma immediately lemma let finite homomorphism rings mod assume min min proof let depthrp depthsp depthsq lemma lemma maximal hence depthrp depthsq hence maximal let spec depthrp lemma lemma exists spec depthsq inf depthsp depthrp depth suffices show assumption universally catenary mat desired say satisfies resp regular resp gorenstein lemma let flat morphism noetherian rings mod ring satisfies spec faithfully flat satisfies resp satisfies resp spec satisfies resp proof left reader see mat let abelian category additive subcategory closed direct summands let define extia let sequence said exact addition exact homa say universal say write syz universal say upa obviously upa syza write define also cdim inf resolution define cdim define cdim sequence said exact also exact letting sequence exact sequence exact category denote order distinguish abelian category usual exact sequences let subset dim mean cdim respectively add smallest additive subcategory containing closed direct summands mean add add sequence means add sequence sequence exact definition object injective object let exact category additive subcategory define pushe exists exact sequence note pushe whole thus pushac upa direct summand object push lemma let exact category let additive subcategory consisting injective objects let exact sequence push push push push push push push push push proof let embedding consider full subcategory closed extensions sequence exact prove use induction case trivial assume let exact sequence push let exact sequence push surjective form commutative diagram exact rows columns closed extensions diagram diagram induction assumption push hence push prove let exact sequence push commutative diagram exact rows columns applying already proved push since lie push push desired prove let exact sequence push taking diagram push already proved since middle row splits exact sequence push desired corollary let lemma let push push proof part obvious lemma considering exact sequence prove part induction nothing prove let induction assumption push applying lemma exact sequence push push proved similarly corollary let sequence define obviously lemma exists sequence proof let take sequence corollary continue infinitely define also define let say lies short exact sequence define lemma let assume let short exact sequence proof assumption exact sequence cdim sequence commutative diagram exact rows columns top row exact assumption middle row corollary right column shows desired lemma let assume let short exact sequence proof take exact sequence taking get commutative diagram exact rows columns induction prove easily particular hence middle column splits replace definition exact sequence adding sequence get exact pulling back exact sequence get commutative diagram exact rows columns middle column top row shows desired theorem let assume following equivalent exact sequence moreover surjection conditions equivalent following exact sequence proof exact sequence exact sequence done let descending induction prove using lemma easily also proved easily using lemma trivial property rest paper let may commutative means let mod fixed set note also modulefinite denote mod mod mod mod denote syzmod upmod cdimmod respectively note mod mod standard isomorphisms first isomorphism sends map inverse given shows mod mod right adjoint hence right adjoint denote unit adjunction note mod map given denote unit adjunction mod view morphism opposite category mod counit adjunction lemma give contravariant equivalence add mod add mod proof suffices show isomorphism add isomorphism add verify may assume case trivial definition tak let mod say injective say bijective let say convention define mod lemma let sequence mod following proof commutative diagram exact rows prove may assume isomorphism injective five lemma injective case done also isomorphism isomorphism moreover vanish hence also proved similarly lemma tak proposition proof mod may assume let injection add commutative diagram injective shows prove first prove use induction case already done let definition sequence add induction hypothesis hence lemma proved next show use induction let let surjective map add map corresponds adjunction injective assumption composite surjective map assumption also surjective hence gives pushforward let proved pushforward let coker sequence add lemma induction assumption definition desired lemma mod syz proof let exact sequence mod add exact add shows syz denote class mod viewed lies see lemma assume satisfies syz proof follows easily corollary additive category additive subcategory denote quotient divided ideal consisting morphisms factor objects mod take presentation add denote coker coker trc transpose see ass call trc additive functor mod mod add mod mod add proposition let assume mod following trc additive functor mod mod exact sequence trc trc injective homomorphism ker trc exact sequence trc trc isomorphisms trc iii injective map trc proof obvious assumption consider complex degree zero consider degree zero complex trc spectral sequence trc general ker moreover coker follows assumption follows note moreover inclusion isomorphism iii follow corollary let trc trc canonical module let semilocal jacobson radical say dualizing complex normalized maximal ideal follow definition left right dim denotes dimension dimr independent choice call depthr also independent global depth depth denote depth called globally gcm short dim depth gcm maximal ideals height notion independent depends called globally maximal gmcm short dim depth say algebra gcm gcm however follows happens local gcm resp gmsm maximal thing used interchangeably assume complete semilocal let normalized dualizing complex lowest cohomology group extr denoted called canonical module note hence also sense define let center rhomr normalized dualizing complex shows rhomr rhoms hence definition also independent lemma number nothing dim moreover assr asshr minr dim dim proof may replace annr may assume faithful module may assume fundamental dualizing complex spec injective hull appears exactly dimension dim exists spec extrp suppr dim hand length nonzero asshr argument shows asshr assh supports asshr minr hand complex starts degree ass ass assh asshr lemma let complete semilocal satisfies condition proof easy see either zero maximal ideal hence may assume local replacing annr may assume faithful prove satisfies lemma replacing noether normalization may assume regular lemma homr syz lemma consider also assume semilocal may complete say finitely generated canonical module isomorphic canonical module unique isomorphisms denoted say mod right canonical module isomorphic mod completion exists right canonical module mod definitions independent sense right canonical module center thing right canonical module called left canonical module said weakly canonical bimodule left canonical right canonical canonical module canonically identified normalized dualizing complex normalized dualizing complex easy see exists agrees dim dimr case spec dualizing complex dualizing complex lowest nonzero cohomology group rhomrp module see also theorem lemma let local assume exists following assr asshr ann hence universally catenary proof assertions proved easily using case complete said suppr suppr lemma let local exists satisfies condition ann equidimensional proof proof ogo lemma use lemma let local normalized dualizing complex local duality hmd injective hull matlis dual homr let semilocal normalized dualizing complex note rhomr induces contravariant equivalence dfg dfg let dfg rhomr rhomr dfg dfg identified rhomr similarly rhomr dfg dfg identified note left right maximal rhomr concentrated degree dim dualizing complex sense yekutieli yek gcm isomorphism mod gmcm rhomr concentrated degree also case dualizing complex finite injective dimension left right prove may take completion may assume complete assertions independent taking noether normalization may assume local assertions follow mod gmcm rhomr rhomr rhomr hence also gmcm hence rhomr isomorphism words canonical map isomorphism isomorphism true without assuming dualizing complex assuming existence canonical module passing completion note exists dualizing complex similarly mod gmcm isomorphism particular letting gcm gmcm moreover isomorphism goes left multiplication similarly isomorphism let complete local ring dim local duality hmd homr homr module say right resp left weakly spec right canonical module resp left canonical module weakly canonical module canonical module suppr mention may mean center right resp left weakly rsemicanonical nothing left resp right weakly let mod resp mod mod mod say right resp left weakly right module resp left weakly mention may mean center example zero module dualizing complex lowest cohomology group lemma left right module satisfies thus right semicanonical module annr right semicanonical resp projective rank one right semicanonical resp normal domain reflexive module gorenstein equivalent say satisfies section let mod set moreover set map induced right action lemma let mod let mod following equivalent syz proof lemma trivial follows lemma immediately prove want prove injective example localizing may assume local may assume nonzero assumption nonzero hence assumption gcm hence isomorphism gcm gmcm isomorphism result follows lemma let right mod similarly mod proof note split monomorphism indeed left inverse assume coker nonzero let assr submodule assr assr min right canonical module isomorphism shows contradiction second assertion proved similarly lemma let local assume exists let gcm isomorphism proof possesses bimodule structure canonical map isomorphism gcm identified isomorphism lemma satisfies condition moreover ass ass ass min proof first assertion prove second assertion assr assr endr assr assr assr endr assr minr remains show suppr suppr suppr let spec hand identity map zero hence let right define minr hence isomorphism lemma also isomorphism assr minr lemma let right commutative proof commutative commutative ring prove commutative assr minr subring isomorphism hence also commutative lemma let right let left resp right modules assume let left resp right modules left resp right modules proof let commutative diagram clearly since injective easy see lemma let right restriction full faithful functor simiop larly gives full faithful functors proof consider case left modules mod homomorphic image hence suppr suppr suppr functor obviously faithful lemma also full done let left action induces map let get lemma let let left resp right modules assume let left resp right modules left resp right modules proof similar lemma left reader corollary let canonically isomorphic proof immediate lemma lemma let induces full faithful functor similarly also induced proof similar lemma left reader corollary let set canonical map induces equality similarly proof first assertion follows lemma second assertion proved symmetry lemma let right set let canonical map induced right action injective satisfies condition proof nothing result follows lemma immediately lemma let following equivalent canonical map injective map induced right action satisfies condition canonical map injective map induced left action proof corollary consequence lemma reversing roles left right get immediately lemma let right canonical map induced canonical map isomorphism modules proof composite map identity map also lemma local map isomorphism indeed verify may assume complete regular local annr hence homr see lemma apply hence corollary let local ring canonical module isomorphic lemma let right right left proof holds suffices prove right localization replacing may assume local right isomorphic mod isomorphic mod lemma similarly assuming local suffices show left identifying using symmetry proof lemma let mod right let mod following equivalent syz proof may assume faithful easy show example localizing may assume noetherian local ring dimension one formal denotes fibers hence completion may assume complete local may assume hence case dim similar proof lemma prove case dim note ideal module annihilated replacing may assume maximal isomorphism isomorphism identified isomorphism desired corollary let right canonical map isomorphism satisfies full proof follows immediately lemma applied let let left multiplication map induced right multiplication map induced let identified commutative diagram lemma lemma hence lemma isomorphism hence symmetry identification acts left also right actions extend indeed left multiplication right left hence left theorem let right restriction gives equivalence proof functor obviously full faithful lemma hand given isomorphism structure extends module structure also dense hence equivalence corollary let restriction gives equivalence proof obviously faithful morphism objects theorem note hence full let left resp right structure extendable left resp right structure theorem remains show structures make let given right hence right given left hence left desired proposition let opright give contravariant equivalence proof know contravariant adjoint suffices show unit unit isomorphisms isomorphism lemma note left lemma isomorphism lemma applied right corollary let give contravariant equivalence also give duality proof first assertion immediate proposition theorem second assertion follows easily first corollary aoyama theorem lemma let flat local homomorphism noetherian local rings let isomorphic let right module isomorphic right right proof taking completion may assume complete let decomposition mutually orthogonal primitive idempotents center replacing sei local ring sei maximal ideal may assume equivalent say isomorphic also isomorphic hence center let isomorphism write also write local exists automorphism also isomorphism faithful flatness isomorphism easy see mod projective replacing radical changing may assume field central simple one simple right direct sums copies dimension counting number copies equal hence isomorphic lemma let flat local homomorphism noetherian local rings let bimodule let isomorphic bimodule let right let right isomorphic right right proof easy see hence see corollary note lemma hence lemma hence proposition let flat local homomorphism noetherian local rings assume right canonical module gorenstein proof may assume complete replacing annr annr may assume faithful let dim dim hmd hmd hmd injective hull residue field isomorphisms last isomorphism hmd mod injective considering spectral sequence hmd mod hmd injectivity hmd follows hmd homr hmd injective however free module also module must injective hence must gorenstein lemma let flat local homomorphism noetherian local rings gorenstein assume canonical module exists canonical module proof may assume complete let normalized dualizing complex normalized dualizing complex dim dim since flat local homomorphism gorenstein closed fiber see avf definition normalized dualizing complex avf different follow one chapter extr theorem aoyama theorem aoy theorem let flat local homomorphism noetherian local rings canonical module canonical module right right canonical module right canonical module proof may assume complete canonical module exists localization canonical module canonical module hence may localize minimal element spec take completion may assume fiber ring gorenstein proposition lemma lemma isomorphisms bimodules right modules proofs complete corollary let noetherian local ring assume canonical resp right canonical module suppr localization canonical resp right canonical module particular semicanonical bimodule resp right module hence annr proof let prime ideal lying nonzero assumption hence canonical resp right canonical module using theorem canonical resp right canonical module last assertion follows let local assume exists assume faithful corollary letting corollary exists structure left structure extends original left structure unique obvious action similarly right action obvious action see theorem modules lemma aoy proposition ogo proposition aoyg proposition let local assume canonical module injective satisfies condition suppr equidimensional bijective ony satisfies condition proof replacing annr may assume faithful rmodule corollary full suppr equidimensional lemma consequence lemma follows corollary lemma proposition aoyg let local ring assume module assume moreover maximal proof second assertion follows first prove first assertion replacing annr may assume faithful let dim satisfies maximal lowest cohomology rhomr natural map induces isomorphism cohomology groups diagram commutative top horizontal arrow isomorphism lemma note rhomr rhomr rhomr left vertical arrow isomorphism maximal macaulay concentrated degree zero right vertical arrow isomorphism thus bottom horizontal arrow isomorphism applying map isomorphism macaulay desired corollary aoyg let local ring assume module resp maximal proof support macaulay one maximal also suffices prove assertion property verify may assume faithful note satisfies corollary conversely proposition theorem evg ari let noetherian commutative ring may commutative let right set let mod following equivalent syz proof easy prove lemma may assume lemma let resolution mod add suffices prove dual acyclic lemma may localize may assume dim split exact also exact may assume assumption mod maximal hence corollary lemma done corollary let assumptions notation theorem let assume satisfies condition contravariantly finite mod proof mod nth syzygy module satisfies condition theorem theorem short exact sequence right hence contravariantly finite corollary let assumptions notation theorem let assume satisfies condition contravariantly finite mod proof corollary suffices show let canonical map isomorphism lemma since left applying theorem isomorphism desired symmetric frobenius algebras let noetherian semilocal ring ralgebra say canonical module called symmetric gcm note resp symmetric denotes completion note also symmetric absolute notion independent choice sense definition change replace center noetherian ring say locally quasisymmetric resp locally symmetric spec resp symmetric equivalent say maximal ideal resp symmetric case semilocal locally resp locally symmetric resp symmetric converse true general lemma let noetherian semilocal ring following equivalent right canonical module left canonical module proof may assume complete replacing noether normalization annr may assume regular faithful prove lemma satisfies regular dim dim homr get map induced map given isomorphism mod induces homomorphism mod given verify isomorphism reflexive may localize take completion hence may assume dim finite free matrices transpose matrix invertible isomorphism follows considering opposite ring definition let semilocal say equivalent conditions lemma satisfied gcm addition called frobenius note definitions independent choice moreover resp frobenius completion general say locally resp locally frobenius resp frobenius spec lemma let semilocal following equivalent projective mod projective mod denotes completion proof may assume complete regular local faithful let denote functor finite dimensional mod mod number simple modules say indecomposable projective module mod nothing projective cover simple module mod mod mod indecomposable projectives homr equivalence add add also equivalence add add add add also indecomposables equivalent add add equivalent add add proved simply applying duality homr let semilocal equivalent conditions lemma satisfied say gcm addition say definitions independent choice note resp proposition let semilocal following equivalent gcm dim idim idim denotes injective dimension gcm dim idim proof definition gcm prove dim idim may assume local may assume complete replacing noetherian normalization annr may assume complete regular local ring dimension maximal module add add proof lemma suffices prove idim let minimal injective resolution homr injective resolution homr length extd extd idim may assume complete regular local maximal may assume field injective homr projective quasifrobenius see sky proved similarly corollary let arbitrary following equivalent spec maximal ideal gorenstein sense dim spec proof trivial proposition idim dim gorenstein follows proposition let arbitrary say spec definition let general say symmetric resp frobenius relative homr isomorphic resp right called relative right projective lemma let local dim dim resp right projective right resp gorenstein symmetric resp frobenius relative symmetric resp frobenius nonzero resp pseudofrobenius symmetric resp frobenius relative nonzero symmetric resp frobenius gorenstein symmetric resp frobenius relative proof take completion may assume complete local let dim dim let normalized dualizing complex hom hom homr result follows may assume nonzero finite projective maximal result follows part follows prove part nonzero dim dim free homr nonzero isomorphic direct sum copies hence hence result follows follows easily let semilocal let finite group act automorphisms let twisted group algebra product given makes modulefinite simply call say structures coming structure structure agree one thing action natural way action also right action action trivial actions make given homr right easy see standard isomorphism homr homr homr isomorphism right left consider case pairing given kronecker delta induces isomorphism finite free symmetric relative lemma resp symmetric action trivial resp symmetric proof taking completion may assume complete replacing noether normalization annr may assume regular local ring faithful action trivial resp symmetric seen easily particular commutative resp gorenstein action trivial resp symmetric general lemma let right let homomorphism right given particular split monomorphism mod split monomorphism mod proof straightforward proposition let finite group acting set action trivial resp symmetric resp frobenius resp proof lemma prove may assume complete regular local faithful module homr homr right isomorphic lemma since mod hence addition also hence frobenius proved similarly using lemma note assertions frobenius properties also follow easily lemma argument let locally noetherian scheme open subscheme coherent assume condition let inclusion follows use notation rings modules schemes coherent algebras modules obvious manner let mod coherent right restriction mod action get functor mod mod mod resp mod denote category resp lemma let notation assume large codimx canonical map isomorphism proof follows immediately proposition let notation let large assume right following proof question local may assume affine coherent subsheaf exercise let generated image composite note coherent let right let let double dual hence lies follows lemma immediately references aoy aoyama basic results canonical modules math kyoto univ aoyg aoyama goto endomorphism ring canonical module math kyoto univ ari araya iima locally gorensteinness macaulay rings ass assem simson elements representation theory associative algebras cambridge avf avramov foxby locally gorenstein homomorphisms amer math brodmann sharp local cohomology algebraic introduction geometric applications cambridge evg evans griffith syzygies london math soc lecture note series cambridge goto nishida towards theory bass numbers applications gorenstein algebras colloq math grothendieck partie ihes publ math hart hartshorne residues duality lecture notes math springer hartshorne algebraic geometry graduate texts math springer verlag hartshorne generalized divisors gorenstein schemes ktheory hashimoto equivariant class group iii almost principal fiber bundles iyama wemyss noncommutative conjecture reine angew math lew leuschke wiegand representations ams mat matsumura commutative ring theory first paperback edition cambridge ogo ogoma existence dualizing complexes math kyoto univ peskine szpiro dimension projective finie cohomologie locale publ ihes scheja storch und lokal durchschnitte manuscripta math addendum sky yamagata frobenius algebras basic representation theory european mathematical society tak takahashi new approximation theory unifies spherical approximations pure appl algebra thomason trobaugh higher algebraic schemes derived categories grothendieck festschrift iii yek yekutieli dualizing complexes noncommutative graded algebras algebra
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shape determination theory planar graphs application formation stabilization gangshan jinga guofeng zhangb heung wing joseph leeb long wangc mar center complex systems school engineering xidian university china department applied mathematics hong kong polytechnic university hong kong china center systems control college engineering peking university beijing china abstract paper presents approach distributed formation shape stabilization systems plane develop angle rigidity theory study whether planar framework determined angles segments uniquely translations rotations scalings reflections proposed angle rigidity theory applied formation stabilization problem multiple modeled agents cooperatively achieve formation formation process global coordinate system unknown agent wireless communications agents required moreover utilizing advantage high degrees freedom propose distributed control law agents stabilize desired formation shape desired orientation scale two simulation examples performed illustrating effectiveness proposed control strategies introduction formation stabilization problem design decentralized control law group mobile agents stabilize prescribed formation shape associated fundamental problem determine geometric shape graph embedded space based local constraints displacements distances bearings straightforward approach determining shape constraining location vertex graph formation strategy usually takes large costs unnecessary position agent corresponding author long wang email addresses nameisjing gangshan jing guofeng zhang heung wing joseph lee longwang long wang strictly required reduction information exchange improvement robustness control strategy formation method determines target formation shape relative positions agents extensively studied method also called formation since formation problem often transformed consensus problem hot topic widely studied investigations formation show shape graph determined displacements uniquely graph connected disadvantage formation control requirement global coordinate system last decade shape control gained lot attention since requirement global coordinate system agent different approach noncomplete graph embedded space straightforward answer whether shape determined edge lengths uniquely tool great utility deal problem traditional graph rigidity theory refer theory distance rigidity theory paper studied intensively area mathematics recent years formation control attracted many interests due low costs bearing measurements issue formation shape constrained bearings distinguish kind shapes uniquely determined bearings authors proposed bearing rigidity theory compared formation control advantage formation strategy fact restriction scale target formation imposed result simpler control scale formation benefit obstacle avoidance see unfortunately similar approach formation requires either global coordinate system agent developing observers based communications besides investigations issues associated formation control formation strategies details refer readers paper studies formation problem plane target formation shape shape planar graph encoded angles pair edges joining common vertex similar issues reported literature authors discussed possibility formation approach presented initial results another relevant reference authors solved cyclic formation problem constraining angle subtended vertex two neighbors case cyclic formation stretched preserving invariance angle thus target formation accurately stabilized contrast study stabilize formation shape via angle constraints stabilized formation congruent target formation translations rotations scalings reflections contributions summarized follows enlightened tance rigidity theory bearing rigidity theory propose angle rigidity theory study whether shape planar graph uniquely determined angles see section prove planar framework infinitesimal angle rigidity equivalent infinitesimal bearing rigidity theorem infinitesimal angle rigidity also generic property graph iii show framework embedded triangulated laman graph strongly rigid always determined angles uniquely translations rotations scalings reflections see theorem propose distributed control law achieving formation shape stabilization based angle rigidity theory shown control strategy locally exponentially stabilize multiple agents form infinitesimally angle rigid formation plane see theorem design distributed control law steer agents form target formation shape prescribed orientation scale see theorem note formation maneuver control literature simultaneously controlling orientation scale formation usually achieved advantages formation approach threefold agent measure relative displacements neighbors respect local coordinate system wireless communications agents required iii compared approaches shape higher degrees freedom precisely angles invariant motions including translations rotations scalings displacements distances bearings invariant subset motions result convenient achieve formation maneuver control using angle constraints paper structured follows section introduces preliminaries bearing rigidity theory section presents angle rigidity theory section firstly proposes distributed control law achieving formation stabilization based angle rigidity theory proposes distributed maneuver control law stabilizing formation shape prescribed orientation scale section presents application example verify validity formation strategy section concludes whole paper notations throughout paper denotes set real numbers euclidean space stands euclidean norm means transpose matrix kronecker product range null rank denote image space null space rank matrix represents identity matrix set elements belonging vector pts said degenerate collinear orthogonal group cos sin rotation matrix associated sin cos reflection matrix associated denote diag blockdiag undirected graph vertices edges denoted denote vertex set edge set respectively distinguish incidence matrix represented hij matrix rows columns indexed edges vertices orientation hij ith edge sinks vertex hij ith edge leaves vertex hij otherwise rank graph connected let denote complete graph vertices preliminaries graph rigidity theory section introduce preliminaries distance bearing rigidity theory plane taken distance rigidity theory answer whether uniquely determined translations rotations reflections partial length constraints edges bearing rigidity theory answer whether uniquely determined translations scalings partial bearing constraints edges following introduce two theories unified approach refer pair framework graph ptn called configuration coordinate vertex define rigidity framework smooth rigidity function first given positive integer given rigidity function several definitions associated rigidity induced follows framework said rigid exists neighborhood globally rigid infinitesimal motion assignment velocities guarantees invariance vnt velocity vertex say motion trivial satisfies equation framework vertices framework infinitesimally rigid every infinitesimal motion trivial denote rigidity matrix equation equivalent let dimension space formed trivial motions framework infinitesimally rigid rank traditional graph rigidity theory rigidity function commonly set following distance rigidity function eij recently authors developed bearing rigidity theory using following bearing rigidity function gij gij framework plane totally independent translations independent rotation independent scaling trivial motions framework determined distances translations rotations thus dimension trivial motion space trivial motions framework determined bearings translations scalings accordingly dimension trivial motion space following two lemmas used paper lemma framework infinitesimally bearing rigid infinitesimally distance rigid lemma framework infinitesimally distance rigid vertex relative position vectors collinear worth noting infinitesimal bearing rigidity implies global bearing rigidity whereas infinitesimal distance rigidity induce global distance rigidity angle rigidity section develop angle rigidity theory investigate encode geometric shapes graphs embedded plane angles framework employ gij gik constraint angle edges eij eik actually cosine angle let gij gik set constraints angles note framework often redundant angle information shape determination example fig available holds cos arccos arccos information partial angles graph often sufficient recognize framework therefore employing subset try study whether uniquely determined gij gik based angle rigidity theory developed paper note although subset elements involve vertices otherwise shape never determined framework angle rigidity function corresponding written gij gik sake notational simplicity denote ftg figure globally infinitesimally angle rigid framework framework angle rigid globally infinitesimally angle rigid framework globally angle rigid framework easy see whether determine unique shape congruent determined choice result definitions angle rigidity must associated present following definitions definition framework angle rigid exists neighborhood definition framework globally angle rigid holds definition framework minimally angle rigid angle rigid deletion edge make angle rigid definitions frameworks fig globally angle rigid framework moving vertices along blue arrows invariant shape deformed thus angle rigid framework since graph complete obviously holds thus globally angle rigid note shape still determined angles uniquely similar distance bearing rigidity theory define infinitesimal angle motion motion preserving invariance velocity corresponding infinitesimal motion satisfy equivalent following equation gik gij pij pij gij projection matrix defined xxt unit vector pij let gij follows chain rule pij diag diag termed angle rigidity matrix actually bearing rigidity matrix therefore equation equivalent next define infinitesimal angle rigidity distinguish trivial motions geometric shape intuitive observation motions always preserving invariance angles framework translations rotations scalings therefore dimension trivial motion space note trivial motion space always subspace null implying dim null present following definition definition framework infinitesimally angle rigid exists every possible motion satisfying trivial equivalently dim null definition frameworks fig infinitesimally angle rigid frameworks infinitesimally angle rigid since nontrivial infinitesimal angle motions interpreted arrows blue following lemma gives specific form trivial motions preserving invariance angles lemma trivial motion space angle rigidity space formed rotations span space formed scalings null space formed translations proof authors showed scaling translational motion spaces respectively always belong null since straightforward null next show null let gik arbitrary row suffices show note gik gij follows diag gik pij gij pik note also etij order eij vector one gij vector follows pij diag gik eij gij eik gik gij gij gij gij gik gik gik gik gij gij gik gik gij completes proof direct consequence lemma following result lemma framework infinitesimally angle rigid null rtg authors showed set includes configurations congruent always manifold dimension fact since shape least degrees freedom manifold dimension infinitesimally angle rigid regular point see following theorem theorem let infinitesimally angle rigid manifold proof presented later subsections aid theorem derive relationship infinitesimal angle rigidity angle rigidity given follows theorem infinitesimally angle rigid angle rigid proof proposition neighborhood manifold dimension theorem also manifold result coincide implying angle rigid converse theorem true typical framework degenerate configuration case globally angle rigid infinitesimally angle rigid relation bearing rigidity subsection establish connections angle rigidity bearing rigidity following theorem shows equivalence infinitesimal angle rigidity infinitesimal bearing rigidity plane also implies feasibility approach determining planar framework theorem framework infinitesimally angle rigid infinitesimally bearing rigid proof necessity since null null dim null reaches minimum dim null minimal recall always holds null infinitesimally angle rigid must hold null infinitesimally bearing rigid sufficiency note infinitesimal bearing rigidity implies null show null suffices show null range suppose vnt let gij gik component gij gik collinear implies gik diag equivalent etik pij etij pik note nonzero vectors perpendicular therefore always exist cij cik pij cij gij pik cik gik follows cij gij gij cik gik gik substituting cij etik gij cik etij gik note also rot cij cik gij gik since gij gik collinear gij gik follows cij cik cij cijk cik cijk cijk together cijk eij eij cijk eik eik far proved gij collinear gik holds cijk following constructing show exists common constant cro eij eij construct set gij gik collinear since infinitesimally bearing rigid lemma lemma vertex exist least two neighbors gij gik collinear result divide two sets gij gik collinear construct set following two steps step select vertex randomly let element step select vertex randomly let element example constructing shown fig figure example illustrate construct subgraph composed vertex neighbors note collinear collinear connected red line implies angle edge edge selected angle constraint also implies element let obvious gij gik collinear regard edge vertex adjacent belongs approach construction easy see either adjacent neighbors therefore graph corresponding connected regard cij state corresponding cij eij eij note implies adjacent share common state cijk since connected edges consensus state cro eij eij implies diag pij pij pij diag diag pij pij diag diag diag eij diag since proof completed remark authors proved infinitesimal bearing rigidity generic property graph infinitesimally bearing rigid infinitesimally bearing rigid almost configuration theorem easy obtain infinitesimal angle rigidity also generic property graph thus primarily determined graph rather configuration figure infinitesimally angle rigid globally angle rigid remark definition conclude minimal number angle constraints achieving infinitesimal angle rigidity fact also shown hand shown minimal number edges framework infinitesimally bearing rigid theorem true infinitesimal angle rigidity remark proof sufficiency theorem constructing appropriate show null range must correspond rotational motion nevertheless constructed may suitable determine unique shape shown fig constructed approach theorem although infinitesimally angle rigid may determine incorrect shape fig however let always determine correct shape implies fig infinitesimally globally angle rigid moreover conclude given infinitesimal angle rigidity induce global angle rigidity directly consider framework plane distance rigidity theory obvious shape uniquely determined bearing rigidity theory authors showed uniquely determines shape infinitesimally bearing rigid however angle rigidity theory immediately answered whether shape uniquely determined angles edges one hand angles constraints relationships edges joining common vertex even complete graph always exist disjoint edges angle pair disjoint edges constrained directly hand shown remark infinitesimal angle rigid framework specified determine unique shape particular cases following theorem connection established theorem given configurations figure exist angles red constrained angles determined remark easy see complete graph theorem replaced provided globally angle rigid globally bearing rigid note replaced general graph theorem hold shown fig although exist next present lemmas required prove theorem authors showed positive matrix rank specified permutation matrix cholesky decomposition unique uniqueness cholesky decomposition implies straightforward obtain following lemma lemma matrix rank let householder transformation unit vector geometrically reflection vector perpendicular list easily checked properties following lemma lemma given unit vectors following properties hxt iii exists unit vector eigenspace associated eigenvalue span aid lemma establish following result lemma unit vector proof note orthogonal matrix either rotation matrix reflection matrix without loss generality discuss problem three cases case implying hence case following procedure case one also obtain case follows lemma iii exists using lemma span implying result lemma let denote graph vertices edges following lemma holds lemma infinitesimally bearing rigid nondegenerate necessity lemma obvious sufficiency lemma obtained lemma given next subsection lemmas hand give proof theorem proof theorem note equivalent also equivalent therefore suffices show sufficiency straightforward gij gik gij rgik gij gik prove necessity consider following two cases case configuration degenerate let unit vector collinear gij gij rgij gij gij let rij gij rij gij prove necessity suffices show distinct vertices gij rij gij gik rik gik always holds rij rik rij rik without loss generality suppose gij gik gij gik gij gik holds gij gik rij gij rik gik rik gij lemma rij rik rij rik hgij since proof completed case configuration nondegenerate note complete hence vertex least two neighbors gij gik collinear result divide two sets gij gik collinear first show given always holds gijkl rijkl gijkl rijkl gijkl gij gik gil gjk gjl gkl since complete without loss generality consider triangle composed let gijk gij gik gjk since gtijk gijk gtijk gijk note gij gik collinear thus rank gijk virtue lemma cholesky decomposition gtijk gijk determines gijk orthogonal matrix rijk gijk rijk gijk similarly gijl rijl gijl rijl vertices follows case gjkl rjkl gjkl rjkl matter collinear since rijk gij rijl gij gij according lemma rijk rijl rijk rijl hgij suppose rijk rijl gjk gjl gjk gjl gjk rijk rijl gjl gjk hgij gjl gjk gjl gij gij gjl gij gij gjl since gij gik implies gjk collinear gij gjk also collinear similarly gij gjl collinear thus contradiction arises rijk rijl rijkl implies rijkl gij gik gil gjk gjl consider framework graph vertex set edge set since four vertices collinear according lemma infinitesimally bearing rigid thus globally bearing rigid implies gkl uniquely determined result gijkl rijkl gijkl proof implies given holds rijk rijl rikl rjkl note edge graph involved triangle including vertex therefore gij rgij important note theorem induce equivalence global angle rigidity global bearing rigidity examples show equivalence holds still idea prove nonetheless able establish following result theorem framework globally angle rigid globally bearing rigid proof suppose angle rigid exists neighborhood consider follows therefore theorem recall result bearing rigid bearing rigidity equivalent global bearing rigidity since global angle rigidity obviously leads angle rigidity also induce global bearing rigidity prove theorem introduce following theorem theorem level set theorem let smooth manifolds let smooth map jacobian matrix constant rank level set properly embedded submanifold codimension proof theorem theorem infinitesimally bearing rigid shows together theorem must hold next show manifold obvious scalar vector chain rule rank rank rank note smooth map according theorem properly embedded submanifold dimension fact even complete graph possible geometric shape determined information typical example degenerate configuration shown fig generally hope determine framework angles uniquely translations rotations scalings reflections plane next subsection introduce specific class frameworks satisfying condition class frameworks uniquely determined angles authors introduced particular class laman graphs termed triangulated laman graphs constructed modified henneberg insertion procedure follows show shape frameworks always determined angles uniquely let triangulated laman graph definition follows definition let graph vertex set edge set graph obtained adding vertex two edges graph satisfying note triangulated laman graph considered always undirected graph holds frameworks associated triangulated laman graphs also presents following result lemma strongly rigid collinear three vertices satisfying infinitesimally distance rigid easy see strongly rigid nondegenerate therefore infinitesimally distance rigid follows lemma infinitesimally bearing rigid note also graph vertices edges always triangulated laman graph sufficiency lemma follows following theorem shows shape strongly rigid framework plane always uniquely determined angles theorem strongly rigid minimally infinitesimally angle rigid globally angle rigid min otherwise iii fln proof lemma infinitesimally bearing rigid null suffices show null range always exists suppose vnt proof theorem shown gij collinear gik holds cijk recall strongly rigid holds cijk without loss generality suppose due definition triangle formed vertices regard vertex two vertices adjacent belong triangle let cij state cij eij eij easy see common state implying adjacent vertices must common state note every step generation graph new triangle generated based existing edge therefore must connected result exists constant cro eij eij similar analysis proof theorem obtain implies infinitesimally angle rigid moreover observe conclude minimally infinitesimally angle rigid prove statement induction obvious globally angle rigid suppose globally angle rigid next show globally angle rigid without loss generality let neighbors note must least one common neighbor vertex let minimum index among easy see suffices show always holds fkn fkn since globally angle rigid theorem exists matrix gij gin gij gin gji gjn gji gjn gni gnj gni gnj gijn gijn gtijn gijn gijn gij gnj gni using strong rigidity rank gijn lemma gijn rijn gijn rijn follows gij gij rijn gij according lemma rijn rijn hgij suppose rijn gik gik rijn hgij gik gin rijn gin follows gik gin gik hgij rijn rijn gin gik gin recall gik gin gik gin together follows gik hgij holds gin gij collinear either gik gin either qit qjt qkt qit qjt qnt degenerate implying either pti ptj ptk pti ptj ptn degenerate conflicts strong rigidity therefore rijn follows hence globally angle rigid iii theorem ftln example strongly rigid framework embedded triangulated laman graph shown fig angles red constrained angles determined graph fig contains triangulated laman graph triangulated laman graph application formation control section apply approach framework recognition distributed formation control plane target formation characterized constraints angles order form desired shape multiple mobile agents required meet constraints via distributed controller figure framework embedded triangulated laman graph infinitesimally angle rigid globally angle rigid angles red constrained angles termined framework embedded graph containing framework globally infinitesimally angle rigid tgf tgf form angles red constrained angles determined tgf formation stabilization problem consider agents moving plane agent simple kinematic point dynamics position control input agent respectively global coordinate frame consider global coordinate system absent agents agent local coordinate system let pij coordinate agent position respect agent local coordinate system agent measure relative position state pii pik paper employ infinitesimally angle rigid framework describe target formation shape agent viewed vertex framework interaction link two agents regarded edge graph also sensing graph interpreting interaction relationship agents virtue theorem target formation shape defined following manifold target formation make following assumption assumption contains triangulated laman graph subgraph strongly rigid set determining angle constraints given tgf remark assumption graphical condition condition achieving stability target formation assumption holds easy see tgf form since shown theorem infinitesimally angle rigid together follows infinitesimally angle rigid tgf also worth noting strongly rigid configurations form dense subset shown framework said realizable exists throughout paper always assume target formation shape realizable formation stabilization problem summarized follows problem given realizable formation satisfying assumption design distributed control law agent based relative position measurements pii pij asymptotically stable steepest descent formation controller denote gij gij gij gij gij gik gij gik form desired geometric shape system minimize following cost function gij gik gij gik basis function control strategy derived eij eik eji ejk nti tgf gik gij gik gik tgf eij eik gij gij eji ejk gji gjk gji gjk gkj observe tgf control input agent includes term associated ejk obtained simple subtraction eik eij form tgf therefore reasonable agent use ejk define desired equilibrium formation system gij gik gij gik tgf easy see subset globally angle rigid tgf example fig framework globally infinitesimally angle rigid tgf thus holds however framework fig infinitesimally angle rigid globally angle rigid scenario even angle constraints determined tgf satisfied possible target formation shape formed nonetheless definition infinitesimal angle rigidity exists neighborhood hence stability still sufficient local stability let ftgf ftgf written chain rule system control input written following compact form note requires agent sense relative displacements neighbors therefore distributed control strategy moreover implementing formation system following properties lemma control law following statements hold global coordinate system required agent degenerate iii centroid scale invariant proof straightforward similar approach validity also easy verify thus proofs omitted iii observe according rtgf show first note rtgf diag follows pij diag stability analysis theorem group agents dynamics controller moving plane assumption neighborhood proof let expanding taylor series equivalent rtt rtgf lemma validity assumption implies infinitesimally angle rigid therefore zero eigenvalues rest negative real numbers must exist orthonormal transformation hurwitz equivalent qjf diag note equilibrium point hence since follows observe manifold next show center manifold note invariant since equilibria must satisfy implicit function theorem neighborhood origin smooth since manifold must exist open set diffeomorphic neighborhood origin represented conclude center manifold flow manifold governed system sufficiently small recall manifold equilibria center manifold theory sufficiently small implies follows proof completed observe equilibrium set system rtt rtgf full row rank implies follows however rtgf varies formation system evolves difficult determine rank moreover rtgf never full row rank result undesired equilibria often exist system hence obtain local stability implies local stability theorem actually means implementing control law agents cooperatively restore desired formation shape small perturbation convergence rate fast dependent however uncertain whether exists uniform exponent compact exist finite subcover including orientation scale control shown formation degrees freedom higher formations ensures one advantage formation approach convenience orientation scale control subsection propose control scheme steer agents form target formation shape prescribed orientation scale given target formation shape satisfying assumption configuration forming target formation desired orientation scale written constant arbitrary translational vector worth noting denotes position agent global coordinate frame let target equilibrium described control orientation formation obviously necessary agents access global coordinate system keep target shape precise orientation try constrain displacement two adjacent agents similar since orientation scale ultimate formation determined two agents call leaders noteworthy two adjacent agents selected leaders controlling relative position sufficient control orientation scale formation fact shown later moreover different using approach target displacement leaders artificially specified satisfy fixed length constraint orientation also scale target formation controlled suppose agents leaders displacement formation target orientation scale summarize problem deal subsection problem given realizable target formation satisfying assumption target displacement known agents design distributed control law agent based relative position measurements pii pij asymptotically stable solve problem consider following set containing target equilibrium following lemma shows infinitesimally angle rigid coincide near point lemma infinitesimally angle rigid exists neighborhood proof let follows since must connected rank rank according theorem manifold next show also manifold near without loss generality suppose consisted row row let sij matrix sij define rtt easy obtain null null rtgf null first notice must infinitesimally angle rigid implying null rtgf trivial motion space shown lemma also note null null range null verified null null null span obtain rank max rank regular point proposition exists neighborhood manifold together follows virtue lemma initial positions agents close drive agents suffices constrain steering agents meet angle constraints determined tgf therefore wish agents cooperatively minimize following cost function form result control law induced form driving agents maintain target shape controlling formation orientation scale easy see control law distributed property lemma also holds formation system lemma becomes invalid moreover evolution centroid still invariant scale may changed define graph distinguish let incidence matrix hlt laplacian matrix corresponding graph denote using control law formation system written following compact form jacobian matrix desired equilibrium jhm rtgf rtt rtgf following theorem shows effectiveness control strategy theorem group agents dynamics controller moving plane assumption locally exponentially stable proof lemma show local exponential stability note system similar form equation moreover null null span matrix defined proof lemma process similar proof theorem shown locally exponentially stable lemma also locally exponentially stable remark besides orientation scale also control translation formation suffices add extra control term control law agent scenario desired equilibrium point let lemma neighborhood follows linearization approach theorem easy prove local exponential stability therefore also locally exponentially stable methods formation control literature apply problem subsection specifically approach neither orientation scale controlled approach formation scale controlled approach formation orientation controlled simulations section present two numerical examples illustrate effectiveness theoretical findings first example target formation shape formed exponentially fast formation control law second example implementing control law formation transformed another one desired orientation scale example consider group autonomous agents moving plane target formation shape regular pentagon described framework fig set desired angle information note triangulated laman graph strongly rigid assumption holds without loss generality choose cos sin set initial position vector agents perturbation component pseudorandom value drawn uniform distribution implementing control law fig initial positions initial positions final positions final positions time figure control law agents asymptotically form regular pentagon vanishes zero exponential speed obtained shows desired formation shape formed formation strategy fig describes evolution form observed implying exponential convergence formation system conclusion simulation result illustrates theorem fact repeat simulation choosing values way always obtained vanishes zero exponentially target formation shape eventually formed moreover select component uniform distribution target formation shape still formed cases cases angle constraints usually satisfied exponentially fast speed vanishes zero exponentially whereas target formation shape eventually formed tgf globally angle rigid note edge length pentagon formed therefore attraction region sizable example example control orientation scale formation formed example implementing control input let agents two leaders aim drive direction horizontal respect global coordinate system setting length edge suffices set target displacement two leaders fig shows trajectories agents evolution observe validity theorem conclusion paper developed angle rigidity theory study planar frameworks determined angles uniquely translations rotations scalings reflections also proved shape initial positions followers final positions followers initial positions leaders final positions leaders time figure control law regular pentagon formed agents asymptotically transformed another regular pentagon desired orientation scale vanishes zero exponentially triangulated framework always uniquely determined angles basis proposed angle rigidity theory distributed formation controller designed formation shape stabilization proved implementing control strategy formation containing strongly rigid triangulated framework locally exponentially stable taking advantage high degrees freedom proposed distributed control strategy drive agents stabilize target formation shape prescribed orientation scale two simulations given show effectiveness formation strategies future work includes global stability formation maneuver control references fax murray information flow cooperative control vehicle formations ieee transactions automatic control ren atkins distributed coordinated control via local information exchange international journal robust nonlinear control xiao wang chen gao formation control systems automatica coogan arcak scaling size formation using relative position feedback automatica jing zheng 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journal robotics research mou belabbas morse sun anderson undirected rigid formations problematic ieee transactions automatic control sun park anderson ahn distributed stabilization control rigid formations prescribed orientation automatica chen belabbas global stabilization triangulated formations siam journal control optimization asimow roth rigidity graphs transactions american mathematical society hendrickson conditions unique graph realizations siam journal computing liberti lavor maculan mucherino euclidean distance geometry applications siam review eren whiteley morse belhumeur anderson sensor network topologies formations direction bearing angle information agents proceedings ieee conference decision control zelazo franchi giordano rigidity theory unscaled relative position estimation using bearing measurements proceedings european control conference bishop shames anderson stabilization rigid formations constraints proceedings ieee conference decision control european control conference zhao zelazo bearing rigidity almost global bearingonly formation stabilization ieee transactionson automatic control zhao zelazo translational scaling formation maneuver control via approach ieee transactions control network systems wang shi chu zhang zhang aggregation foraging swarms lecture notes artificial intelligence vol springer park ahn survey formation control automatica lin wang chen han necessary sufficient graphical conditions affine formation control ieee transactions automatic control aranda zavlanos distributed formation stabilization using relative position measurements local coordinates ieee transactions automatic control zhao lin peng chen lee distributed control cyclic formations using measurements systems control letters zhao sun zelazo trinh ahn laman graphs generically bearing rigid arbitrary dimensions arxiv preprint dongarra moler bunch stewart linpack users guide society industrial applied mathematics lee introduction smooth manifolds vol springer 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type targeted testing eric seidel niki vazou ranjit jhala jan san diego abstract present new technique called type targeted testing translates precise refinement types comprehensive key insight behind approach lens smt solvers refinement types also viewed declarative test generation technique wherein types converted smt queries whose models decoded concrete program inputs approach enables systematic exhaustive testing implementations declarative specifications furthermore provides gradual path testing full verification implemented approach haskell testing tool called target present evaluation shows target used test wide variety properties compares testing approaches introduction programmer spend time writing better types thorough tests types long pervasive means describing intended behavior code however type signature often coarse description actual inputs outputs may subset values described types example set ordered integer lists sparse subset set integer lists thus validate functions produce consume values programmer must painstakingly enumerate values hand via generators unit tests present new technique called type targeted testing abbreviated target enables generation unit tests precise refinement types last decade various groups shown refinement types compose usual types logical refinement predicates characterize subset actual type inhabitants used specify formally verify wide variety correctness properties programs insight lens smt solvers refinement types viewed declarative test generation technique target tests implementation function refinement type specification using loop first target translates argument types logical query obtain satisfying assignment model smt solver next target decodes smt solver model obtain concrete input values function finally target executes function inputs get corresponding output check belongs specified result type check fails inputs returned counterexample otherwise target refutes given model force smt solver return different set inputs process repeated given number iterations inputs certain size tested target offers several benefits testing techniques refinement types provide succinct description input output requirements eliminating need enumerate individual test cases hand write custom generators furthermore target generates values given size inhabit type thus skip corner cases generator might miss finally advantages recovered approach discards inputs meet predicate show method significantly efficient enumerating valid inputs space target paves gradual path testing verification affords several advantages verification first programmer incentive write formal specifications using refinement types target provides immediate gratification automatically generated exhaustive suite unit tests expose errors thus programmer rewarded without paying front extra price annotations hints strengthened inductive invariants tactics needed formally verifying specification second approach makes possible use refinement types formally verify parts program using tests validate parts may difficult verify target integrates two modes using refinement types uniform specification mechanism functions verified half formally checked assuming functions tested half adhere specifications could even use refinements generate dynamic contracts around tested half desired third even formally verifying type specifications generated tests act valuable counterexamples help debug specification implementation event program rejected verifier finally target offers several concrete advantages previous testing techniques also potential gradual verification first instead specifying properties arbitrary code complicates task subsequent formal verification target properties specified via refinement types already several existing formal verification algorithms second symbolic execution tools generate tests arbitrary code contracts assertions find highly constrained inputs trigger path explosion precludes use tools gradual verification rest paper start overview target used loop implemented next formalize general framework testing show instantiated generating tests lists automatically generalized types benefits target come price limited properties specified refinement types present empirical evaluation shows tar get efficient expressive enough capture variety sophisticated properties demonstrating testing sweet spot automatic testing verification overview start series examples pertaining small grading library called scores examples provide bird eye view user interacts target target implemented advantages testing refinement types refinement type one basic types decorated logical predicates drawn efficiently decidable theory example type nat int type pos int type rng int refinement types describing set integers strictly positive interval respectively also build function collection types base refinement types like paper address issue checking refinement type signatures assume code typechecked ghc standard type signatures obtained erasing refinements instead focus using refinements synthesize tests execute function find counterexamples violate given specification testing types base types let write function rescale takes source range target range score source range returns linearly scaled score target range example rescale return first attempt rescale rescale nat nat rng rng rescale div run target immediately reports found indeed rescale results target rng latter empty could fix various ways requiring ranges rescale pos pos rng rng target accepts function reports passed tests thus using refinement type specification rescale target systematically tests implementation generating valid inputs given size bound respect running function checking output satisfies testing random unconstrained inputs would limited value function designed work int values case could filter invalid inputs shall show target effective containers let suppose normalized scores type score rng next let write function compute weighted average list scores average average average wxs total int score score total div sum wxs sum wxs tricky verify function requires reasoning unbounded collection however gain great degree confidence systematically testing using type specification indeed target responds found clearly unfortunate choice weights trigger fix requiring weights average int score score target responds found also triggers play safe require positive weights average pos score score point target reports tests pass ordered containers nature business requires end day order students scores represent ordered lists requiring elements tail greater head data ordlist ordlist note erasing refinement predicates gives plain old haskell lists write function insert score ordered list insert ord ordlist ordlist target automatically generates ordered lists given size executes insert check errors unlike randomized testers target thwarted ordering constraint require custom generator user structured containers everyone bad days let write function takes best scores particular student output must satisfy structural constraint size equals encode size list logical measure function measure len nat len len len stipulate output indeed scores best nat score score len best take reverse sort target quickly finds counterexample found course need least scores start best nat score len score len target assuaged reports counterexamples randomized testing would suffice best see sophisticated structural properties height balancedness stymie random testers easily handled target functions perhaps instead taking best grades would like pad individual grade furthermore want able experiment different padding functions let rewrite average take functional argument stipulate increase score padaverage score score pos score score padaverage padaverage wxs total div total sum wxs sum wxs target automatically checks padaverage safe generalization average randomized testing tools also generate functions functions unlikely satisfy constraints thereby burdening user custom generators synthesizing tests next let look hood get idea target synthesizes tests types strategy query smt solver satisfying assigments set logical constraints derived refinement type decode model haskell values suitable inputs execute function decoded values obtain output check output satisfies output type refute model generate different test repeat steps tests certain size executed focus steps query decode check others standard require little explanation base types recall initial buggy specification rescale nat nat rng rng target encodes input requirements base types directly corresponding refinements constraints multiple related inputs conjunction constraints input hence constraint rescale practice also contain conjuncts form restrict intvalued variables within size bound supplied user omit throughout paper clarity note easy capture dependencies inputs score range defined querying smt solver get model target decodes model executes rescale obtain value target validates checking validity output type constraint valid target moves generate another test conjoining constraint refutes previous model time smt solver returns model decoded executed yields result inhabit output type reported counterexample fix specification allow pos ranges test produces valid output target reports tests pass containers next use target test implementation average target needs generate haskell lists appropriate constraints since list recursively either nil cons target generates constraints symbolically represent possible lists given depth using propositional choice variables symbolically pick two alternatives every satisfying assignment choices returned smt solver gives target concrete data constructors used level allowing decode assignment haskell value example target represents valid pos score inputs depth required test average conjunction clist cdata clist cdata first set constraints clist describes lists size level choice variables determine whether level constructed list nil cons constraints uninterpreted functions represent nil cons respectively functions obey congruence axiom hence efficiently analyzed smt solvers data level constrained pair positive weight valid score choice variables level used guard constraints next levels first generating cons given level exactly one choice variables next level must selected second constraints data given level hold generating values level used guard constraints essential avoid system would cause target miss certain tests decode model haskell value type int int traverse constraints use valuations choice variables build list appropriately level true list level otherwise true decode cons results iteratively generate multiple inputs adding constraint refutes prior model important optimization refute relevant parts model needed construct list ordered containers next let see target enables automatic testing highly constrained inputs increasingly ordered ordlist values required insert type definition apparent ordered lists usual lists described clist except unfolded tail must contain values greater corresponding head unfold ordlist level ordlist true level ordlist level ordlist thus encode ordlist score depth conjoining clist cscore cord capture valid score ordering requirements respectively cord cscore structured containers recall best requires inputs whose structure constrained size list less specify size using special measure functions let relate size list unfolding hence let encode notion size inside constraints csize len len len len len len len len len len unfolding instantiate definition measure alternative datatype constraints len uninterpreted function derived measure definition relevant properties function spelled unfolded constraints csize hence use smt search models constraint hence target constrains input type best clist cscore csize len final conjunct comes refinement stipulates input least scores thus target generates lists large enough example model generate empty singleton list cases len would resp violating final conjunct manipulating refinements refinement reftype refinement subst reftype var var reftype manipulating types unfold ctor reftype var reftype binder reftype var proxy reftype proxy fig refinement type api functions finally target testing scales higherorder functions using insight quickcheck namely generate function suffices able generate output function tasked generation functional argument target returns haskell function executed checks whether inputs satisfy uses target dynamically query smt solver output satisfies constraints imposed concrete inputs otherwise specifications violated target reports counterexample concludes tour benefits implementation target notice property specification mechanism refinement types allowed get immediate feedback helped debug code also specification additionally specifications gave documentation behavior functions large unit test suite automatically validate implementation finally though focus specifications amenable formal verification programmer desire framework type targeted testing next describe framework type targeted testing formalizing abstract representation refinement types describing operations needed generate tests types using implement target via loop subsequently instantiate framework obtain tests refined primitive types lists algebraic datatypes functions refinement types refinement type type component decorated predicate refinement logic clarity describe refinement types refinements abstractly reftype refinement respectively write var alias refinement typically used represent logical variables appearing within refinement notation sequel use double brackets represent various entities used describe target example len score len var refinement reftype representing corresponding entities written brackets next describe various operations needed implement target operations summarized figure fall two categories manipulate refinements manipulate types operating refinements generate constraints check inhabitation use function refinement returns refinement decorates given refinement type generate fresh vars name values components use subst replace free occurrences variables given reftype suppose reftype represented score len refinement evaluates len subst evaluates score len operating types build compound values lists components integer list unfold breaks reftype list integers constituents integer list integers given constructor cons binder simply extracts var representing value refined reftype write generic functions reftypes use haskell type class machinery query decode components types associate refinement type proxy representing corresponding haskell type practice must passed around separate argument example score len unfold evaluates score score binder evaluates proxy evaluates value type proxy int targetable type class following quickcheck encapsulate key operations needed testing type class targetable figure class characterizes set types class targetable query proxy int reftype smt var decode var smt check reftype smt bool var toreft refinement fig class types tested target tested target operations interact external smt solver return values smt monad query takes proxy haskell type generating values integer depth bound refinement type describing desired constraints generates set logical constraints var represents constrained value decode takes var generated via previous query queries model returned smt solver construct haskell value type check takes value type translates back logical form verifies inhabits output type toreft takes value type translates back logical form specialization check loop figure summarizes overall implementation target takes input function refinement type specification proceeds test function specification via loop first translate refined inputtypes logical query next decode model satisfying assignment query returned smt solver obtain concrete inputs finally execute function inputs get corresponding output check belongs specified outputtype check fails return inputs counterexample test target refutes given test force smt solver return different set inputs process repeated user specified number iterations checksmt call may fail find model meaning exhaustively tested inputs upto given testdepth bound iterations succeed counterexamples found target returns indicating satisfies given depth bound target let txs inputtypes vars form txs query proxy testdepth query form testnum hasmodel checksmt hasmodel inputs form vars decode decode output execute inputs let zip map binder txs map toreft inputs let outputtype subst check output check refutesmt else throw counterexample inputs return fig implementing target via loop instantiating target framework next describe concrete instantiation target lists start constraint generation api use api implement key operations query decode check refutesmt thereby enabling target automatically test functions lists omit definition toreft follows directly definition check finally show list instance generalized algebraic datatypes functions smt solver interface figure describes interface smt solvers target uses constraint generation model decoding interface functions generate logical variables type var constrain values using refinement predicates determine values assigned variables satisfying models fresh smt var guard var smt smt constrain var refinement smt apply unapply ctor var smt var var smt ctor var oneof whichof var var var smt var smt var eval refinement smt bool fig smt solver api fresh allocates new logical variable guard act ensures constraints generated act guarded choice variable act generates constraint guard act generates implication constraint constrain generates constraint satisfies refinement predicate apply generates new var folded value obtained applying constructor fields also generating constraints measures example apply returns generates constraint len len unapply returns ctor vars input constructed oneof cxs generates constraint equals exactly one elements cxs example oneof yields query let ctors form fresh zipwithm queryctor fresh oneof zip constrain refinement return queryctor let fts return guard unfold scanm queryfield fts apply queryfield query proxy subst return ctors otherwise fig generating query whichof returns particular alternative assigned current model returned smt solver continuing previous example model sets resp true whichof returns resp eval checks validity refinement free variables example eval len would return true query figure shows procedure constructing query refined list type one required input best insert functions lists query returns var represent lists depth satisfy logical constraints associated refined list type end invokes ctors obtain suitable constructors depth lists depth use constructor otherwise use either ensures query terminates encoding possible lists given depth next uses fresh generate distinct choice variable constructor calls queryctor generate constraints corresponding symbolic var constructor choice variable constructor supplied queryctor ensure constraints guarded required hold corresponding choice variable selected model otherwise finally fresh represents value depth constrained oneof alternatives represented constructors satisfy refinement decode whichof unapply decodector decodector return decodector decode decode return fig decoding models haskell values constructors queryctor takes input refined list type depth particular constructor list type generates query describing unfolding constructor guarded choice variable determines whether alternative indeed part value constraints conjunction describing values individual fields combined via obtain value queryctor first unfolds type obtaining list constituent fields respective refinement types fts next uses scanm monad traverse fields left right building representations values fields unfolded refinement types finally invoke apply fields return symbolic representation constructed value constrained satisfy measure properties fields queryfield generates actual constraints single field refinement type invoking query proxy enables resolve appropriate typeclass instance generating query field value field described new symbolic name substituted formal name field refinements subsequent fields thereby tracking dependencies fields example substitutions ensure values tail greater head needed ordlist decode generated constraints query smt solver model one found must decode concrete haskell value test given function figure shows decode smt model lists lists decode takes input symbolic representation queries model determine alternative assigned solver nil cons alternative determined use unapply destruct constructor fields recursively decoded decodector constructors decodector takes constructor list symbolic representations fields decodes field value applies constructor obtain haskell value example case constructor fields return empty list case constructor decode head tail cons return decoded value decodector type targetable ctor var smt check let splitctor let fts unfold fmap unzip scanm checkfield zip fts apply let subst binder eval refinement return checkfield check subst return splitctor splitctor fig checking outputs decodable type decodector suffices decode lists primitives like integers directly encoded refinement logic base case value model directly translated corresponding haskell value check third step loop verify output produced function test indeed satisfies output refinement type function accomplish encoding output value logical expression evaluating output refinement applied logical representation output value check shown figure takes haskell output value output refinement type recursively verifies component output type converts component logical representation substitutes logical expression symbolic value evaluates resulting refinement refuting models finally target invokes refutesmt refute given model order force smt solver produce different model yield different test input implementation refutation follows let set variables appearing constraints suppose current model variable assigned value refute model add refutation constraint stipulate variable assigned different value implementation extremely inefficient smt solver free pick different value irrelevant variable even used decoding result next model decoding yield haskell value thereby blowing number iterations needed generate tests given size target solves problem forcing smt solver return models yield different decoded tests iteration end target restricts refutation constraint set variables actually used decode haskell value track set instrumenting smt monad log set variables choicevariables transitively queried via recursive calls decode call decode logs argument call whichof logs choice variable corresponding alternative returned let resulting set decoderelevant variables target refutes model using relevant refutation constraint ensures next model decodes different value generalizing target types implementation list types ctors decodector splitctor functions thus easily generalize implementation primitive datatypes integers returning empty list constructors algebraic datatypes implementing ctors decodector splitctor type functions lifting instances functions returning algebraic datatypes list implementation three pieces logic ctors returns list constructors unfold decodector decodes specific ctor splitctor splits haskell value pair ctor fields thus instantiate target new data type need implement three operations type implementation essentially follows concrete template lists fact observe recipe entirely mechanical boilerplate fully automated algebraic data types using generics library algebraic datatype adt represented component types generics library provides univeral type functions automatically convert adt universal representation thus obtain targetable instances adt suffices define targetable instance universal type universal type targetable automatically get instance new adt instance generic follows generate query simply create query universal representation refined type decode results smt solver decode universal representation use map back userdefined type check given value inhabits refinement type check universal representation value inhabits type universal counterpart targetable instance universal representation generalized version list instance relies various technical details higher order functions approach specification makes easy extend target functions concretely suffices implement typeclass instance instance targetable input targetable output targetable input output essence instance uses targetable instances input output create instance functions input output haskell type class machinery suffices generate concrete function values create instances use insight quickcheck generate constrained functions need generate output values function following route generate functions creating new lambdas take inputs calling context use values create queries output call smt solver decode results get concrete outputs returned lambda completing function definition note require input also targetable encode haskell value refinement logic order constrain output values suitably additionally memoize generated function preserve illusion purity also possible future extend implementation refute functions asserting output value given input distinct previous outputs input evaluation built prototype implementation next describe evaluation series benchmarks ranging textbook examples algorithms data structures widely used haskell libraries like containers xmonad goal evaluation first describe functions quantitatively compare target existing testing tools haskell namely smallcheck quickcheck determine whether target indeed able generate highly constrained inputs effectively second describe modules evaluate amount code coverage get testing comparison quickcheck smallcheck compare target quickcheck smallcheck using set benchmarks highly constrained inputs benchmark compared target smallcheck quickcheck latter two using approach wherein value generated subsequently discarded meet desired constraint one could possibly write custom operational generators property point evaluation compare different approaches ability enable declarative specification driven testing next describe benchmarks summarize results comparison figure http ist insert rbt ree add onad focus left time sec depth delete difference time sec target smallcheck depth lazy smallcheck lazy smallcheck slow fig results comparing target quickcheck smallcheck lazy smallcheck series functions target smallcheck lazy smallcheck configured check first inputs satisfied precondition increasing depth parameters minute timeout per depth quickcheck run default settings produce test cases target smallcheck lazy smallcheck configured use notion depth order ensure would generate number valid inputs depth level quickcheck unable successfully complete run due low probability generating valid inputs random inserting sorted list first benchmark insert function homonymous sorting routine use specification given element sorted list insert evaluate sorted list express type type sorted list insert sorted sorted ordering constraint captured abstract refinement states list head less every element tail inserting tree next consider insertion tree data rbt leaf node col rbt rbt data col black red trees must satisfy three invariants red nodes always black children black height paths root leaf elements tree ordered capture via measure recursively checks red node black children measure isrb rbt prop isrb leaf true isrb node isrb isrb red isblack isblack specify defining black height measure rbt int leaf node red else checking black height subtrees measure isbh rbt prop isbh leaf true isbh node isbh isbh finally specify ordering invariant type ordrbt rbt two abstract refinements left right subtrees respectively state root greater resp less element subtrees finally valid tree type okrbt ordrbt isrb isbh note specification internal invariants trees tricky specification public api add function straightforward add okrbt okrbt deleting third benchmark delete function module haskell standard libraries map structure balanced binary search tree implements purely functional dictionaries data map tip bin int map map valid must satisfy two properties size left right subtrees must within factor three keys must obey binary search ordering specify balancedness invariant measure measure isbal map prop isbal tip true isbal bin isbal isbal combine ordering invariant like ordrbt specify valid trees type okmap ordmap isbal check delete preserves invariants checking output okmap however also one step check functional correctness property delete removes given key type delete ord okmap okmap minuskey predicate minuskey defined predicate minuskey keys difference keys singleton using measure keys describing contents map measure keys map set keys tip empty keys bin union singleton union keys keys refocusing xmonad stacksets last benchmark comes tiling window manager xmonad key invariant xmonad internal stackset data structure elements windows must unique contain duplicates xmonad comes quickcheck properties select one states moving focus windows stackset affect order windows index foldr const focusup index quickcheck user writes custom generator valid stacksets runs function test inputs created generator check case result true target possible test properties without requiring custom generators instead user writes declarative specification type okstackset stackset noduplicates refer reader full discussion specify noduplicates next define refinement type type ttrue bool prop inhabited true use type quickcheck property nat okstackset ttrue property particularly difficult verify however target able automatically generate valid inputs test always returns true results figure summarizes results comparison quickcheck unable successfully complete benchmark low probability generating properly constrained values random list insert target able test insert way depth whereas lazy smallcheck times depth tree insert target able test add depth lazy smallcheck times depth map delete target able check delete depth whereas lazy smallcheck times depth checks ordering first depth checks balancedness first stackset refocus target able check property depth lazy smallcheck times depth target sees performance hit properties require reasoning theory sets invariant stackset lazy smallcheck times higher depths completes depth versus tar get minutes suspect theory sets relatively recent addition smt solvers improvements smt technology numbers get significantly better overall found small inputs lazy smallcheck substantially faster exhaustive enumeration tractable incur overhead communicating external solver additionally lazy smallcheck benefits pruning predicates exploit laziness force small portion structure ordering however found constraints force entire structure balancedness composing predicates wrong order force lazy smallcheck enumerate entire exponentially growing search space target hand scales nicely larger input sizes allowing systematic exhaustive testing larger complex inputs target eschews explicit results searching fewer needles larger haystacks sizes increas favor symbolically searching valid models via smt making target robust strictness ordering constraints measuring code coverage second question seek answer whether target suitable testing entire libraries much program automatically exercised using system keeping mind issues treating code coverage indication quality consider experiment negative filter end ran target entire api rbtree library using constrained refined types okmap okrbt okstackset specification exposed types measured expression branch coverage reported hpc used increasing timeout ranging one thirty minutes per exported function results results experiments shown figure across three libraries target achieved least expression alternative coverage shortest timeout one minute per function interestingly coverage metrics rbtree remain relatively constant increase timeouts small jump expression coverage minutes xmonad hand jumps expression alternative coverage one minute timeout expression alternative ten minute timeout data onad tack rbt ree coverage expressions timeout min booleans alternatives fig rbtree using tar get exported function tested increasing depth limits single run hit timeout ranging one thirty minutes lower better higher better everything else three things consider examining results first expressions evaluated due haskell laziness values contained map second expressions evaluated branches taken happen unexpected error condition triggered expressions dead code target considers inputs trigger uncaught exception valid counterexample rule inputs expect cover expressions target last remark intrinsically related target rather means collecting coverage data hpc includes otherwise guards category even though evaluate anything else contained guards marked manually counted otherwise guards remaining guards compared size subtrees rebalancing determine whether single double rotation needed unable trigger double rotation cases xmonad contained guards otherwise guards remaining guard dynamically checked function check failed error would thrown next case consider success target error branch triggered discussion sum experiments demonstrate target generates valid inputs quickcheck fails outright due low probability generating random values satisfying property efficiently lazy smallcheck relies lazy pruning predicates providing high code coverage libraries test cases course approach without drawbacks highlight five classes pitfalls user may encounter laziness function output refinement cause exceptions unthrown output value fully demanded example target would decide result undefined inhabits int score latter would evaluate undefined limitation specific system rather fundamental tool exercises lazy programs furthermore target generates values generate infinite cyclic structures generated values ever contain polymorphism like tool actually runs function scrutiny tar get test monomorphic instantiations polymorphic functions example testing xmonad instantiated window parameter char type parameters properties testing examined window helped drastically reduce search space target smallcheck advanced features gadts existential types may prevent ghc deriving generic instance would force programmer write targetable instance though tedious single instance allows target automatically generate values satisfying disparate constraints still improvement approach refinement types less expressive properties written host language expressible target logic user use approach losing benefits symbolic enumeration input explosion target excels space valid inputs sparse subset space inputs input space sufficiently constrained target may spend lose competitive advantage tools due overhead using solver related work target closely related number lines work connecting formal specifications execution automated testing next describe closest lines work situate respect approach testing testing encompasses broad range testing tools facilitate generating concrete abstract model system test systems generally though necessarily model system holistic level using state machines describe desired behavior may may provide fully automatic generation addition generating many testing tools spec explorer produce extra artifacts like visualizations help programmer understand model one could view testing including system subset testing focusing properties individual functions using functions scrutiny provide fully automatic generation testing many testing tools developed automatically generate testsuites quickcheck randomly generates inputs based property scrutiny requires custom generators consistently generate constrained inputs extends quickcheck randomly generate constrained values uniform distribution contrast smallcheck enumerates possible inputs depth allows check existential properties addition universal properties however difficulty generating inputs properties complex lazy smallcheck addresses issue generating constrained inputs taking advantage inherent laziness property generating values values containing filling holes demanded korat instruments repok method checks class invariants method monitor object fields accessed authors observe unaccessed fields effect return value repok thereby able exclude search space objects differ values unaccessed fields lazy smallcheck korat reliance functions source language specifying properties convenient programmer specification implementation language makes method less amenable formal verification properties would need another language restricted enough facilitate verification symbolic execution another popular technique automatically generating analyze source code attempt construct inputs trigger different paths program dart cute pex use combination symbolic dynamic execution explore different paths program executing program collect path predicates conditions characterize path program end run negate path predicates query constraint solver another assignment values program variables enables tools efficiently explore many different paths program technique relies path predicates expressible symbolically predicates expressible logic constraint solver fall back values produced concrete execution severe loss precision instead trying trigger paths program one might simply try trigger erroneous behavior check crash uses analyzer discover potential bugs constructs concrete designed trigger bugs exist similarly uses blast construct bring program state satisfying predicate contrast approaches target generally testing treats program requires expressible solver logic course expressing specifications source language contracts pex one use symbolic execution generate tests directly specifications one concrete advantage approach symbolic execution based method pex latter generates tests explicitly enumerating paths contract code suffers similar combinatorial problem smallcheck quickcheck contrast target performs search symbolically within smt engine performs better larger input sizes integrating execution target one many tools makes specifications executable via constraint solving early example approach testera uses specifications written alloy modeling language generate java objects satisfy method class invariants specifications written alloy one use alloy based model finding symbolically enumerate candidate inputs check crash uses similar idea smt solvers generate inputs satisfy given jml specification recent systems sbv kaplan offer monadic api writing smt constraints within program use synthesize program values sbv provides thin dsl logics understood smt solvers whereas kaplan integrates deeply scala allowing use recursive types functions test generation viewed special case indeed kaplan used generate testsuites preconditions similar manner target however also symbolic execution based methods like pex jcrasher specifications assertions sense consequently techniques limited testing functions monomorphic data types contrast target shows view types executable specifications yields several advantages first use types compositionally lift specifications flat values score collections score without requiring special recursive predicates describe collection invariants second compositional nature types yields compositional method generating tests allowing use machinery generate tests richer structures tests third refinement types proven effective verifying correctness properties modern modern languages make ubiquitous use parametric polymorphism higher order functions thus believe target approach making refinement types executable crucial step towards goal enabling gradual verification modern languages acknowledgements work supported nsf grants generous gift microsoft research thank lee pike reviewers excellent feedback draft paper references beyer chlipala henzinger jhala majumdar generating tests counterexamples icse software engineering boyapati khurshid marinov korat automated testing based java predicates issta software testing analysis acm claessen palka generating constrained random data uniform distribution flops claessen hughes quickcheck lightweight tool random testing haskell programs icfp acm csallner smaragdakis check crash combining static checking testing icse dias neto subramanyan vieira travassos survey testing approaches systematic review weaseltech acm dunfield refined typechecking stardust plpv sbv smt based verification haskell http findler felleisen contract soundness languages oopsla flanagan leino lillibridge nelson saxe stata extended static checking java pldi gill runciman haskell program coverage haskell acm godefroid klarlund sen dart directed automated random testing pldi jackson alloy lightweight object modelling notation acm transactions software engineering methodology tosem kuncak suter constraints control popl acm new york usa dijkstra jeuring generic deriving mechanism haskell haskell symposium acm marick misuse code coverage proceedings interational conference testing computer software marinov khurshid testera novel framework automated testing java programs ase ieee computer society washington usa moura generalized efficient array decision procedures fmcad nelson techniques program verification tech xerox palo alto research center nystrom saraswat palsberg grothoff constrained types languages oopsla runciman naylor lindblad smallcheck lazy smallcheck automatic exhaustive testing small values haskell symposium acm sen marinov agha cute concolic unit testing engine acm swamy chen fournet strub bhargavan yang secure distributed programming types icfp tillmann halleux box test generation tests proofs vazou rondon jhala abstract refinement types esop vazou seidel jhala liquidhaskell experience refinement types real world haskell symposium vazou seidel jhala vytiniotis refinement types haskell icfp veanes campbell grieskamp schulte tillmann nachmanson testing reactive systems spec explorer formal methods testing pfenning eliminating array bound checking dependent types pldi
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freeness random fundamental group oct andrew newman october abstract let denote probability space random simplicial complexes model let denote random complex chosen according distribution paper cohen costa farber kappeler shown high probability free following paper costa farber shows values satisfy high probability free improve results show explicit constants high probability free fundamental group high probability fundamental group either free trivial introduction positive integers space random ddimensional simplicial complexes introduced denoted probability space simplicial complexes vertices complete possible faces included independently probability primarily interested case suppress dimension parameter write question fundamental group nontrivial studied additionally series papers study regime describe results introduce two constants introduced needed state main theorem log let unique nonzero solution exp let unique root log build work prove following result fundamental group random note theorems stated asymptotic results use phrase high probability abbreviated mean property holds probability tending tends following theorem main result paper theorem high probability free group high probability free group approximations computed ohio state university lower bound section prove part theorem one free group high probability result follow adapting argument used prove following result theorem case theorem let contains subcomplex means simplicial complex simplicial complex say face free contained exactly one face free elementary collapse simplicial complex obtained removing unique containing sequence elementary collapses removes faces say observe elementary collapses homotopy equivalences graph particular free fundamental group therefore theorem almost proves lower bound except problem tetrahedron boundaries note impossible rule appearing subcomplex since expected number copies approaches poisson distribution mean additionally state result partial collapsibility presence copies indeed clear partial collapsibility result would hold however result needed imply fundamental group free see following convention core simplicial complex every edge contained least two faces also let denote simplicial complex obtained collapsing free edges let denote simplicial complex obtained repeatedly collapsing free edges free edges remain two key results use following theorem case theorem every exists constant every core subcomplex must contain boundary tetrahedron theorem case theorem let fixed suppose bound probability free group bound probability core contains tetrahedron boundary pair tetrahedron boundaries face disjoint upper bound probability free following proposition proposition let simplicial complex every core contains tetrahedron boundary tetrahedron boundaries free proof let simplicial complex obtained adding inside tetrahedron boundaries let obtained collapsing free face every face collapse remove tetrahedra tetrahedron boundaries face disjoint every tetrahedron faces free equivalently obtained deleting one face every tetrahedron boundary collapsing free faces homotopy equivalence furthermore indeed cores core would core well since obtained removing faces every core contains tetrahedron boundary tetrahedron boundaries since core must otherwise deleting isolated edges would give subcomplex faces degree zero one subcomplex would core thus particular homotopy equivalent graph free group ready prove part theorem free group high probability proof lower bound theorem let suppose proposition probability free bounded sum probability contains tetrahedron boundaries share face probability core tetrahedron boundary first easy bound probability contains tetrahedron boundaries share face two tetrahedron boundaries simplicial complex sharing face must meet exactly one face two tetrahedron boundaries meeting one face simplicial complex vertices faces expected number subcomplexes case markov inequality probability tetrahedron boundaries face disjoint use two theorems show probability core tetrahedron boundary let given theorem let denote collection vertices containing core tetrahedron boundary let denote collection vertices cores size note since cores elementary collapses theorem bound core faces choice theorem know probability core faces tetrahedron boundary thus completes proof upper bound turn attention proving high probability free group fact relevant results costa farber prove cohomological dimension refer reader background group cohomology theory main result following theorem theorem assume random asphericable complex obtained removing one face tetrahedron aspherical universal cover contractible costa farber prove following result theorem theorem constants satisfying cohomological dimension equals high probability prove upper bound theorem use following result linial peled reduce constant theorem argument follow exactly argument costa farber proof theorem current bestpossible also threshold emergence homology degree random theorem special case theorem suppose dim constant implicit given explicitly need ready prove second part theorem proof upper bound theorem fix suppose simplicial complex drawn let obtained removing one face every tetrahedron boundary high probability theorem aspherical therefore showing would imply cohomological dimension least two theorem know high probability also moment argument expected number tetrahedron boundaries bounded therefore markov inequality high probability say tetrahedron boundaries given simplicial complex removing face drop one therefore remove one face tetrahedron boundary obtain drop thus cohomological dimension least two actually equality holds theorem particular free group concluding remarks statement theorem perhaps implicitly suggests sharp threshold property random fundamental group free however worth mentioning property fundamental group simplicial complex free monotone property obvious sharp threshold exist however theorem theorem log high probability free combining result log property high probability fundamental group high probability free trivial free group property trivial group hand proves collapses graph high probability thus least coarse threshold fundamental group random either free trivial remains discover fundamental group right seem enough evidence establish conjecture following three possibilites happens fundamental group intermediate regime sharp threshold fundamental group free group group sharp threshold fundamental group free group group neither sharp threshold fundamental group free group group three would interesting way holds regime cohomological dimension equal asphericable thus remove face every tetrahedron group cohomological dimension however reason cohomological dimension must reason cohomological dimension regime indeed regime enough imply cohomological dimension fundamental group least second homology group trivial removal face tetrahedron boundary moreover apparent lack torsion proved extensive experiments conducted provide evidence support state following conjecture regarding torsion homology conjecture case conjecture every bounded away high probability torsion homology observed experimentally close torsion see say likely one able prove cohomological dimension proving prime hand holds regime homotopy equivalent wedge circles removal one face tetrahedron boundary follows fact asphericable aspherical space unique homotopy equivalence show following results homotopy wedge circles removal face tetrahedron boundary however homotopy equivalence given sequence elementary collapses reduces complex graph proved series elementary collapses possible furthermore points regime far sense constant fraction faces must deleted arrive complex thus regime holds would homotopy equivalent wedge circles via type homotopy equivalence exists smaller values summary regardless whether truth new techniques almost certainly required prove correct course possibility well indeed possible sharp threshold exists property discuss could also sharp threshold within intermediate regime positive probability free positive probability references aronshtam linial top homology random simplicial complex vanish random structures algorithms threshold random complexes random structures algorithms aronshtam linial luczak meshulam collapsibility vanishing top homology random simplicial discrete computational geometry babson kahle fundamental group random journal american mathematical society brown cohomology groups new york cohen costa farber kappeler topology random discrete computational geometry costa farber asphericity random complexes random structures algorithms kahle paquette spectral gaps random graphs applications random topology arxiv kahle lutz newman parsons heuristics torsion homology random complexes preparation linial meshulam homological connectivity random combinatorica linial peled random simplicial complexes around phase transition arxiv phase transition random simplicial complexes annals mathematics meshulam wallach homological connectivity random complexes random structures algorithms luczak peled integral homology random simplicial complexes arxiv
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reducing bias nonparametric density estimation via bandwidth dependent kernels nov kairat mynbaev international school economics technical university tolebi almaty kazakhstan email kairat mynbayev carlos department economics university colorado boulder usa email ifpri street washington usa email november abstract define new kernel density estimator improves existing convergence rates bias preserves variation error measured additional assumptions imposed extant literature keywords phrases kernel density estimation higher order kernels bias reduction classification thank anonymous referee associate editor excellent comments improved note significantly introduction given sequence independent realizations random variable density kernel estimator rosenblatt parzen given shn shn operator defined shn kernel function bandwidth one natural mathematically sound devroye devroye criteria measure performance estimator distance particular given distance random variable measurable function convenient focus denotes expectation taken using criterion simple bound devroye shn arbitrary shn called bias shn convolution term shn called variation exists large literature devoted establishing conditions assure suitable rates convergence bias zero see inter alia silverman devroye tsybakov particular order jth moment integrable derivative shn order hsn order improved see devroye theorem note show kernel allowed depend order hsn replaced order hsn without increasing order kernel smoothness density addition another result devroye throughout note integrals unless otherwise specified states kernel order greater derivative bias order achieve rate convergence kernels order main results let denote spaces integrable bounded continuous functions norms sup let sequence kernels define shn following theorem density degree smoothness kernels order devroye theorem bias order hsn instead hsn results kernels depend disappearing moments order theorem let sequence kernels order uniformly integrable absolutely continuous shn hsn proof note since kernel shn since differentiable taylor theorem furthermore given order shn dukn letting substituting obtain shn since adding subtracting side gives shn since write continuity modulus known see properties zhuk natanson nondecreasing lim shn hsn hsn hsn hsn given uniformly integrable implies sup using shn hsn remark kernel sequences satisfy restrictions imposed theorem easily constructed end denote space functions bounded norm kkkbs take functions define kernel note kernel order tends zero clear kernel order written conventional kernels obtain furthermore follows uniformly integrable obtain assume nonnegative kernel associate symmetric matrix det see mynbaev arbitrary vector let define polynomial transformation put satisfy thus following corollary theorem corollary let defined remark absolutely continuous shn hsn remark supported instead splitting use instead get shn hsn hence selecting way sup using fact get result precise shn hsn see property zhuk natanson remark young inequality variation using letting shn shn shn shn estimator shn shn hence shn shn since variation asymptotically bounded variation conventional estimator using shn assumptions variance devroye theorem showed shn nhn thus nhn shn last equality follows provide analog theorem devroye bias order achieved kernels orders greater following theorems obtain order bias kernels order theorem let sequence kernels order han sup absolutely continuous assume shn proof proof theorem shn hsn shn chan hence conditions statement theorem shn remark practitioners may find condition general preferring primitive conditions end say function defined satisfies global lipschitz condition order exist positive functions function called lipschitz constant function called lipschitz radius class lip defined set functions satisfy next lemma give two sufficient sets conditions lip first case compactly supported second lemma suppose compact support supp satisfies usual lipschitz condition set lip suppose let exp lip proof implies let sup sup implies gives exp let exp condition obviously satisfied cases part lemma compactly supported densities derivative satisfies usual alipschitz condition lip corresponds case treated theorem devroye part shows densities unbounded domains covered theorem derivative decays exponentially lip next provide version theorem densities derivative lip theorem suppose density derivative belongs respective norms finite lip let sequence kernels order han sup max shn proof proof theorem note since lip let han han letting noting given obtain consequently han max since hsn given han max thus using conditions statement theorem shn remark case theorem theorems address construction kernel sequence following corollary theorem shows han suitable kernel sequence defined corollary suppose density derivative belongs lip let satisfy belong intersection put kernel order condition definition han replaced respectively without affecting conclusion references devroye course density estimation boston devroye nonparametric density estimation view john wiley sons new york mynbaev nadarajah withers aipenova improving bias kernel density estimation statistics probability letters parzen estimation probability density mode annals mathematical statistics rosenblatt remarks nonparametric estimates density function annals mathematical statistics silverman density estimation statistics data analysis chapman hall london tsybakov introduction nonparametric estimation new york zhuk natanson seminorms continuity modules functions defined segment journal mathematical sciences
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ext tor cyclic quotient singularities may lars kastner freie berlin given two torus invariant weil divisors cyclic quotient singularity groups extix naturally interpret groups via certain combinatorial objects using methods toric geometry particular enough give combinatorial description polyhedra global sections weil divisors involved higher exti reduced case via quiver use description show denotes canonical divisor furthermore show matlis dual tori preliminaries let start recalling basic definitions notation toric geometry follow closely given two coprime positive integers define cyclic quotient singularity taking quotient action cyclic subgroup generated denotes root unity cyclic quotient singularity arises manner equivalent definition terms toric geometry follows let twodimensional lattice let homz dual lattice let associated spaces identify choosing usual scalar product pairing definition given two integers gcd define cyclic quotient singularity spec author supported dfg german research foundation priority program spp denotes dual cone cone denote coordinate ring let introduce running example example let hilbert basis dual cone four elements indicated dots picture thus label axes remark close relationship hilbert basis continued fraction expansion discovered riemenschneider already lead fruitful discussion deformation theory terms chains representing zero connection ext functor continued fraction expansion part torus invariant weil divisor integer linear combination orbits corresponding rays denoted write notation throughout article omit example write extix extix order study modules extix may instead study modules extir since affine toric language global sections torus invariant weil divisor given section polyhedron definition torus invariant weil divisor section polyhedron given lattice points section polyhedron correspond homogeneous global sections use describe divisorial ideal keep mind necessarily cartier hence summation divisors translate multiplication ideals general simplicial every minkowski sum rational vector write minimal homogeneous generators correspond exactly lattice points compact edges conv denote lattice points lies compact edge conv remark sort lattice points left right hur hur notation throughout paper use following shorthand notation let subset denote multiplication else note multiplication becomes furthermore every subset rather one needs convexity respect thus require actually little stronger needed example note class group denote torus invariant divisors divisors form system representatives class group furthermore let canonical divisor one calculate vertex canonical divisor example consider divisors first draw corresponding polyhedra global sections compute generators depicted dots respective colors approach resolve divisorial ideal projectively fashion choosing minimal generating set syzygies arrive short exact sequence may seen slight generalization cellular resolutions coordinate rings cyclic quotient singularities resolving torus invariant divisors cqs major obstacle stemming infiniteness complexes turns syzygies generators isomorphic direct sum divisorial ideals global sections torus invariant divisors finiteness class group yields encode information needed freely resolve finite quiver edges labelled elements effectively overcoming previously mentioned obstacle results recursive formula exti tori terms respectively denote generators take canonical surjection every pair consecutive generators build injective map kernel following way denote map direct sum ideals following proposition may seen generalization prop states every monomial ideal polynomial ring two variables resolved freely short exact sequence singular case proposition sequence exact furthermore fractional ideals let illustrate proposition example divisorial example example ideal generated obtain following sequence one immediately recognizes summands first term divisorial ideals begin proof proposition showing homogeneous elements ker image lemma let ker homogeneous element degree whenever particular proof cases nothing prove prove lemma explicitly case methods used case induction step general case since show split last summand homogeneous every entry multiple monomial hence take furthermore assume otherwise done implies together inequalities remark get therefore split following second summand clearly element ker conclude hence done proof proposition already clear surjective already know injective direct summands hence homogeneous element degree kernel gives rise equation factor consider equation vectors linearly independent hence factoring change thus must injective modules maps homogeneous degree therefore ker submodule particular ker generated homogeneous elements thus lemma implies ker final claim divisorial take polyhedron lattice points correspond exactly monomials furthermore polyhedron divisor hui remark alternative proof proposition given using modification criterion exactness lemma one needs replace expression least common multiple two monomials intersection corresponding principal ideals using approach sequence subcomplex taylor resolution divisorial ideal applying proposition recursively construct free resolutions desired length taking account finiteness class group encode information proposition different divisors quiver proposition gives sequence let introduce labelled quiver consists ordinary quiver set vertices set arrows two functions returning source target arrow additionally function equipping arrow label definition take quiver vertices every direct summand sequence add arrow exact sequences furthermore arbitrary weil divisor define sources incoming arrows definition takes care shifting divisors right way definition let divisor linearly equivalent define inq sequence proposition becomes grading imposed generators middle module example running example quiver looks follows one immediately recognizes first term sequence given example via inq one use construction define free resolution recursively resolution even minimal however quiver elegant way dealing infiniteness resolutions case gorenstein recover result eisenbud remark building quiver case one notices consists disjoint cycles length even one cycles length one loop particular every divisor direct sum beginning exact sequence proposition exactly one summand means resolution divisorial ideal already observed eisenbud due gorenstein choice mcm final remark use quiver construct higher exti tori recursively remark ext tor obtain following formulas using proposition applying hom taking long exact sequence cohomology extn torn thus need know compute applying hom short exact sequence proposition may consider long exact sequence cohomology first part gives formula section rephrase formula combinatorial terms weil divisor define following set definition int int example draw sets running example remember leftmost bottom boundary belong sets indicated dashed lines since two generators set looks like parallelepiped hand generated three elements hence set one dent let establish link proposition two weil divisors define ext ext start proving lemma support module hom lemma denote hom hom hom particular module hom divisorial ideal proof know hom thus generated homogeneous elements furthermore note map must injective implying image element completely determines map hence homogeneous maps given multiplication monomials leaves determining lattice points exactly lattice points hom lemma denote generators furthermore write elements inq ext proof using assumption write int int furthermore note use equations giving obtain multiplying adding sides yields desired formula proof proposition take sequence proposition apply hom considering first part long exact sequence cohomology hom hom hom hence need understand quotient hom image hom use lemma hom hom hom next determine image hom direct sum take generator boundary summands mapping via following map applying hom map gives hom hom hom hence dividing image hom means elements hom hom get identified explains take union consider map yields hom hom thus hom set zero one proceeds analogously explains part cut done let briefly remark mcm ness smcm ness mentioned introduction remark claim exti recursion formula enough show recall vertex proceed compute ext two steps first subtract second subtract one show argue otherwise would generate hence closure lattice points top rightmost edges particular lattice points lie valleys since valleys subtracting yields desired result remark assume simplest case second argument see dimc thus vanishes whenever generated exactly two elements thereby relating construction result wunram using strategy ext derive combinatorial description well depended polyhedra global sections involved divisors exclusively needs quiver additional datum definition proposition two weil divisors let tor tor proof weil divisor denote module middle sequence proposition build free resolution follows resulting repeatedly applying proposition tensorizing sequence taking cohomology yields ker tensorizing yields inserting see kernel exactly ker construction enough consider single summands remove image assume generated image exactly thus support equals tor hence finishing proof matlis dual definition define matlis dual homr one check setting exactly injective hull remark stated means module homc transpose particular means proposition matlis dual tor order prove proposition first closer look sets construction differ boundaries introduce set link common core meaning links sets intersecting lattice set behaviour shifts vertices key understanding theorem matlis duality ext tor definition weil divisor define link link example let construct link dashed lower left edge shift take closure intersection link indicated green alternatively one define link using set one close first set intersection first statement following proposition proposition given two weil divisors link link link proof already clear closures equal complementary boundary containments yield first formula proofs second third claim similar hence prove second claim containment link trivial construction contained even interior let similarly let hence want lattice points interior right hand side taking int evaluates integer since boundary corresponds exactly adding vertex canonical divisor obtain intersecting right hand side preserves relation final step note replace one recognizes intersection link proof proposition inserting formula tor ext terms use formula proposition obtain tor link ext inserting remark finishes proof main theorems theorem let two weil divisors cyclic quotient singularity proof want prove equality lattice points respective supports ext ext trick show lattice points respective complements since symmetric show ext ext reverse sign insert definition get ext int int ext ext result int cutting certain pieces top inserting equation ext rephrase intersection ext ext ext int set ext shift thus replace int intersection hence ext ext ext since hence mcm know ext stays true thus ext done finally show duality ext tor theorem let two weil divisors cyclic quotient singularity tori proof using recursion remark sides consequence lemma lemma given two weil divisors following equality proof first use description developed proposition insert formula matlis dual proposition tor applying recursion formula remark obtain desired result note although theorem shows symmetry theorem generalize theorem imply case particular case remains open references klaus altmann cyclic quotients toric english singularities brieskorn anniversary volume proceedings conference dedicated egbert brieskorn birthday oberwolfach germany july basel dave bayer irena peeva bernd sturmfels monomial english math res lett jan arthur christophersen components discriminant versal base space cyclic quotient english symmetric lagrangian singularities gauss maps theta divisors david cox john little henry schenck toric varieties english providence american mathematical society ams xxiv david eisenbud homological algebra complete intersection application group representations transactions american mathematical society lars kastner ext affine toric varieties available http phd thesis freie berlin ezra miller bernd sturmfels combinatorial commutative algebra english new york springer xiv oswald riemenschneider zweidimensionale quotientensingularitaeten gleichungen und german arch math jan stevens versal deformation cyclic quotient english symmetric lagrangian singularities gauss maps theta divisors diana kahn taylor ideals generated monomials phd thesis university chicago department mathematics wunram reflexive modules cyclic quotient surface singularities singularities representation algebras vector bundles springer
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seeding initial population evolutionary algorithms computational study tobias friedricha markus wagner nov jena germany university adelaide australia abstract experimental studies initialize population evolutionary algorithms random genotypes practice however optimizers typically seeded good candidate solutions either previously known created according method seeding studied extensively problems problems however little literature available approaches seeding individual benefits disadvantages article trying narrow gap via comprehensive computational study common test functions investigate effect two seeding techniques five algorithms optimization problems objectives observe functions family benefit significantly seeding others wfg profit less advantage seeding also depends examined algorithm keywords optimization approximation comparative study limited evaluations introduction many applications conflicting objectives play crucial role example consider route planning one objective might travel time another fuel consumption problems need specialized optimizers determine pareto front mutually corresponding author phone fax email preprint submitted elsevier december solutions several established evolutionary multiobjective evolutionary algorithms moea many comparisons various test functions however start random initial solutions practice however good initial seeding make problem solving approaches competitive would otherwise inferior prior knowledge exists generated low computational cost good initial estimates may generate better solutions faster convergence evolutionary algorithms methods seeding studied two decades see studies examples see recent categorization example effects seeding traveling salesman problem tsp jobshop scheduling problem jssp investigated algorithms seeded known good solutions initial population found results significantly improved tsp jssp investigate influence seeding optimisation varying percentage seeding used ranging interestingly also pointed seed necessarily successful either problems one reported seeding cases beneficial optimisation process necessarily always seeding technique dynamic environments investigated population seeded change objective landscape arrived aiming faster convergence new global optimum investigated seeding approaches successful others one studies found seeding techniques moeas one performed seeds created using information fed quality assessed benchmark family zdt results indicate proposed approach produce significant reduction computational cost approach general seeding well documented problems even problems seeding done typically approach outlined used comment worked preliminary experiments reader left dark design process behind used seeding approach quite striking one expects humans construct solutions hand even represent ranges objectives well least one able reuse existing designs modify iteratively towards extremes nevertheless even manual seeding rarely reported paper going investigate effects two structurally different seeding techniques five algorithms optimization moo problems seeding seeding use method preferences specified weights objective solutions objectives found arbitrary classical evolutionary algorithm experiments use details two studied weighting schemes presented section quality measure different ways measure quality solutions recently popular measure hypervolume indicator measures volume objective space dominated set solutions relative reference point disadvantage high computational complexity arbitrary choice reference point instead consider mathematically well founded approximation constant fact known approximation obtained optimal hypervolume distributions asymptotically equivalent best additive approximation constant achievable sets size rigorous definition see section notion approximation introduced several authors theoretical properties extensively studied algorithms use jmetal framework implementation ibea additionally classical moeas also study age aims directly minimizing approximation constant shown perform well larger dimensions algorithms compare regular behavior certain number iterations performance initialized certain seeding benchmark families compare aforementioned algorithms four common families benchmark functions dtlz wfg zdt last three families contain threedimensional problems dtlz scaled arbitrary number dimensions preliminaries consider minimization problems objective functions holds objective function maps considered search space real values order simplify presentation work dominance relation objective space mention relation transfers corresponding elements two points define following dominance relation assess seeding schemes algorithms achieved additive approximation known pareto front use following definition definition finite sets additive approximation respect defined max min max measure approximation constant respect known pareto front test functions better algorithm approximates pareto front smaller additive approximation value perfect approximation achieved additive approximation constant becomes however approximation constant achievable finite population respect continuous pareto front consisting infinite number points always strictly larger depends fitness function smallest possible approximation constant achievable population bounded size seeding task computing seeds employ evolutionary strategy extent perturbs decision variables generating new solutions based previous ones covariance matrix adaptation based evolutionary strategy covariance matrix multivariate normal distribution normal distribution used sample multidimensional search space variate search variable matrix allows algorithm respect correlations variables making powerful evolutionary search algorithm compute seed minimizes objective values solution preliminary testing noticed larger population values tended result seeds better objective values came cost significantly increased evaluation budgets learning correlations takes longer choice necessarily represent optimal choice across benchmark functions however take striking balance investing evaluations seeding investing evaluations regular optimization note large computational budgets seeding potential put unseeded approaches disadvantage final performance assessment done carefully number seeds coefficients used budget evaluations determined seeding approaches describe following cornersandcentre total evaluations equally distributed generation seeds rest population generated randomly seed coefficients set following way otherwise thus prevent seeding mechanism treating optimization problem purely way entirely neglecting relationships lastly weight vector uses equal weights per objective way aim getting seed relatively central respect others linearcombinations total seeds generated seed result running evaluations ranges objective values differ significantly coefficients adjusted accordingly cients linear combinations integer values construct following way first consider permutations coefficients one coefficient others consider permutations two coefficients value three coefficients value permutations based considered consider permutations based based based consequently achieve better distribution points objective space comes however increased initial computational cost furthermore budget per seed lower cornersandcentre approach typically results less optimized seeds noseed solutions initial population generated randomly approach typically used generation initial population evolutionary optimization algorithms following outline five optimization algorithms investigate benefits seeding initial populations many approaches try produce good approximations true pareto front incorporating different preferences example environmental selection first ranks individuals using sorting order distinguish individuals rank crowding distance metric used prefers individuals less crowded sections objective space metric value solution computed adding edge lengths cuboids solutions reside bounded nearest neighbors works similarly raw fitness individuals according pareto dominance relations calculated density measure break ties used individuals reside close together objective space less likely enter archive best solutions contrast two algorithms ibea general framework uses explicit diversity preserving mechanism fitness individuals determined solely based value predefined indicator typically implementations ibea come epsilon indicator hypervolume indicator latter measures volume dominated portion objective space frequently used ibea uses hypervolume indicator directly search process algorithm uses sorting ranking criterion hypervolume selection criterion discard individual contributes least hypervolume front often outperforms competition runtime unfortunately increases exponentially number objectives nevertheless use fast approximation algorithms algorithm applied solve problems many objectives well recently evolution age introduced allows incorporate formal notion definition approximation algorithm approach motivated studies theoretical computer science studying multiplicative additive approximations given optimization problems algorithm complete knowledge true pareto front uses best knowledge obtained far optimization process stores archive consisting objectives vectors found far aim minimize additive approximation population respect archive experimental results presented show given fixed time budget outperforms current algorithms terms desired additive approximation well covered hypervolume standard benchmark functions experimental setup use jmetal framework code seeding well used seeds available test problems used benchmark families dtlz zdt wfg used functions dtlz function variables objective order investigate benefits seeding even long run limit calculations algorithms maximum fitness evaluations maximum computation time four hours per run note time restriction used runtime algorithms increases exponentially respect size objective space age uses random parent selection algorithms parents selected via binary tournament variation operators polynomial http mutation simulated binary crossover applied used widely moeas distribution parameters associated operators crossover operator biased towards creation offspring close parents applied mutation operator specialized explorative effect moo problems applied number decision variables population size set setup repeated times note parameter settings default settings jmetal framework often found literature makes easier best knowledge parameter setting favor particular algorithm put one disadvantage even though individual algorithms differing optimal settings individual problems scenario algorithm run several times restarts seeding might calculated case might make sense compare unseeded seeded variant algorithm number fitness evaluations however observed expected outcome case seeding almost always beneficial therefore consider difficult scenario optimization run number fitness function evaluations used seeding deduced number fitness evaluations available moea pointed earlier assess seeding schemes algorithms using additive approximation pareto front however difficult compute exact achieved approximation constant known pareto front approximate quality assessment wfg zdt functions compute achieved additive approximations respect pareto fronts given jmetal package dtlz functions draw one million points front uniformly random compute additive approximation achieved set also measure hypervolume experiments behaviors five algorithms differ significantly single reference point allows meaningful comparison functions however observe qualitative comparison hypervolume additive approximation therefore omit hypervolume values paper additive approximation constant gives much better way compare results benchmark functions pareto fronts known advance addition calculating average ratio achieved approximation constant without seeding also perform test significance observed behavior compare final approximation runs without seeding runs seeding using test confidence level experimental results results summarized tables compare approximation constant achieved cornersandcentre seeding table linearcombinations seeding table number iterations without seeding seeding requires number fitness function evaluations cornersandcentre linearcombinations allocate seeded algorithms fewer fitness function evaluations makes harder seeded algorithms outperform unseeded variant discussed figures show representative charts approximation constant behaves runtime algorithms first note approximation constant mostly monotonically decreasing smaller approximation constant corresponds better approximation pareto front means algorithms achieve better approximation time exceptions unable handle six dimensional variants dtlz sometimes gets worse certain time problems algorithms total maximal number fitness function evaluations enough algorithms converged small black circles figures indicate average approximation constant initial seeding number fitness function evaluations needed calculate note specific selection schemes algorithms like sometimes increase approximation constant initial seeding another surprising effect observed seeding disadvantageous advantageous considered test problems ones shown figures either age reach best approximation constant however test problems two three dimensions figure fails due high computational cost calculating hypervolume problems age finish iterations within steps still achieves best approximation constant algorithms seedings beneficial test functions however generally performant algorithms age typically gain seedings functions algorithms achieve better approximation faster seeding seems best approximations achieved seeding gap approximation constant achieved without seeding two orders magnitude difference good approximation pareto front basically approximation pareto front tables give numerical comparison assuming fitness evaluations used seeding shown numbers ratios median approximation constant without seeding median approximation constant seeding values indicate seeding beneficial median additionally show statistically significance based test confidence level marks statistically significant improvements marks statistically significant worsenings marks statistically insignificant findings ratios table correspond approximation constants function evaluations left column figures ratios table correspond approximation constants function evaluations right column figures counting statistically significant results functions seedings tables show majority profits seeding algorithms benefit age ibea significant differences depending test function benchmark family profits summing significant results algorithms also worst performance seeding achieved rather difficult wfg functions cornersandcentre seeding achieves algorithms linearcombinations seeding achieves similar analysis assess benefits investigated seeding approaches observe algorithms cornersandcentre seeding yields total bit better linearcombinations seeding yields total order answer question whether statistically significant calculate average rank without seeding runs functions algorithms combined data runs functions algorithms test shows significance confidence level seedings improve upon seeding conclusions seeding result significant reduction computational cost number fitness function evaluations needed observe advantage many common fitness functions even computing initial seeding reduces number fitness function evaluations available moea functions observe dramatic improvement quality needed runtime family practitioners results show worthwhile apply form seeding especially evaluations expensive also investigate different moeas well proven benefit differently seeding observed seeding beneficial experiments could reveal case particular combination seeding algorithm function landscape answer many parts studied mappings benchmark functions create search spaces objective spaces connectedness different local pareto fronts adequacy using seeding procedure much next step towards goal propose investigate seeding combinatorial optimization problems acknowledgements research leading results received funding australian research council arc grant agreement european union seventh framework programme grant agreement sage references agrawal deb simulated binary crossover continuous search space technical report bader deb zitzler faster search using monte carlo sampling multiple criteria decision making sustainable energy transportation systems mcdm vol lecture notes economics mathematical systems springer bringmann friedrich parameterized complexity hypervolume indicator annual conference genetic evolutionary computation conference gecco acm press bringmann friedrich approximating volume unions intersections geometric objects computational geometry theory applications bringmann friedrich approximating least hypervolume contributor general fast practice theoretical computer science bringmann friedrich approximation quality hypervolume indicator artificial intelligence bringmann friedrich neumann wagner evolutionary optimization proc international joint conference artificial intelligence ijcai barcelona spain cheng janiak kovalyov bicriterion single machine scheduling resource dependent processing times siam optimization daskalakis diakonikolas yannakakis good chord algorithm annual symposium discrete algorithms soda deb pratap agrawal meyarivan fast elitist multiobjective genetic algorithm ieee trans evolutionary computation deb thiele laumanns zitzler scalable test problems evolutionary multiobjective optimization evolutionary multiobjective optimization advanced information knowledge processing diakonikolas yannakakis small approximate pareto sets biobjective shortest paths problems siam journal computing durillo nebro alba jmetal framework multiobjective optimization design architecture ieee congress evolutionary computation cec emmerich beume naujoks emo algorithm using hypervolume measure selection criterion international conference evolutionary optimization emo springer evtushenko potapov methods numerical solution multicriterion problem soviet mathematics doklady vol gong jiao multiobjective immune algorithm nondominated selection evolutionary computation grefenstette incorporating problem specific knowledge genetic algorithms genetic algorithms simulated annealing hansen cma evolution strategy comparing review towards new evolutionary computation advances estimation distribution algorithms springer hansen bicriterion path problems multiple criteria decision making theory applications vol lecture notes economics mathematical systems harik goldberg linkage learning probabilistic expression computer methods applied mechanics engineering hatzakis wallace dynamic optimization evolutionary algorithms approach proc annual conference genetic evolutionary computation conference gecco acm press coello coello perez caballero molina seeding initial population evolutionary algorithm using information proc congress evolutionary computation cec ieee press hopper turton empirical investigation heuristic algorithms packing problem european journal operational research huband barone hingston scalable multiobjective test problem toolkit international conference evolutionary optimization emo vol lncs springer ishibuchi tsukamoto sakane nojima evolutionary algorithm hypervolume approximation achievement scalarizing functions annual conference genetic evolutionary computation conference gecco acm press kazimipour qin review population initialization techniques evolutionary algorithms proc congress evolutionary computation cec keedwell khu hybrid genetic algorithm design water distribution networks engineering applications artificial intelligence zhang multiobjective optimization problems complicated pareto sets ieee trans evolutionary computation liaw hybrid genetic algorithm open shop scheduling problem european journal operational research loridan vector minimization problems journal optimization theory applications oman cunningham using case retrieval seed genetic algorithms international journal computational intelligence applications papadimitriou yannakakis approximability optimal access web sources annual symposium foundations computer science focs ieee press papadimitriou yannakakis multiobjective query optimization acm symposium principles database systems pods reuter approximation method efficiency set multiobjective programming problems optimization ruhe fruhwirth bicriteria programs application minimum cost flows computing vassilvitskii yannakakis efficiently computing succinct tradeoff curves theor comput wagner friedrich efficient parent selection evolutionary optimization proc ieee congress evolutionary computation cec ieee wagner neumann fast evolutionary algorithm proc annual conference genetic evolutionary computation conference gecco acm yang zhang hybrid genetic algorithm fitting models electrochemical impedance data journal electroanalytical chemistry zitzler selection multiobjective search international conference parallel problem solving nature ppsn viii vol lncs springer zitzler thiele multiobjective evolutionary algorithms comparative case study strength pareto approach ieee trans evolutionary computation zitzler deb thiele comparison multiobjective evolutionary algorithms empirical results evolutionary computation zitzler laumanns thiele improving strength pareto evolutionary algorithm multiobjective optimization evolutionary methods design optimisation control application industrial problems eurogen figure comparison seeding cornersandcentre left column linearcombinations right column two six dimensions approximation constant pareto front shown function number fitness function evaluations seeded unseeded versions age ibea figures show average repetitions smaller approximation constants indicate better approximation front plots seeded versions shifted number iterations required cornersandcentre seeding iterations linearcombinations seeding iterations circles indicate approximation initial seeding shaded areas illustrate difference seeding seeding specific algorithm plots end prematurely time limit four hours reached figure comparison seeding cornersandcentre left column linearcombinations right column approximation constant pareto front shown function number fitness function evaluations seeded unseeded versions age ibea figures show average repetitions smaller approximation constants indicate better approximation front plots seeded versions shifted number iterations required cornersandcentre seeding iterations linearcombinations seeding iterations circles indicate approximation initial seeding shaded areas illustrate difference seeding seeding specific algorithm plots end prematurely time limit four hours reached function age ibea function age ibea table summary results improvement linearcombinations seeding compare default strategy noseed fitness function evaluations linearcombinations seeding uses fitness function evaluations plus fitness function evaluations table shows ratio median approximation constant noseed divided median approximation constant linearcombinations independent runs values indicate linearcombinations achieves better additive approximation default strategy outcome dividend facilitate qualitative observations show two decimal place marks statistically significant improvements marks statistically significant worsenings marks statistically insignificant findings case moea needed time approximation constant used dashes indicate scenarios even first iteration algorithm completed within allotted function age ibea function age ibea table summary results improvement cornersandcentre seeding compare default strategy noseed fitness function evaluations cornersandcentre seeding uses fitness function evaluations plus fitness function evaluations table shows ratio median approximation constant noseed divided median approximation constant cornersandcentre runs independent values indicate cornersandcentre achieves better additive approximation default strategy outcome dividend facilitate qualitative observations show two decimal place marks statistically significant improvements marks statistically significant worsenings marks statistically insignificant findings case moea needed time approximation constant used dashes indicate scenarios even first iteration algorithm completed within allotted
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feb secure serverless computing using dynamic information flow control kalev tel aviv university cormac flanagan santa cruz sadjad fouladi stanford university leonid ryzhyk vmware research mooly tel aviv university thomas schmitz santa cruz keith winstein stanford university rise serverless computing provides opportunity rethink cloud security present approach securing serverless systems using novel form dynamic information flow control ifc show serverless applications termination channel found existing ifc systems arbitrarily amplified via multiple concurrent requests necessitating stronger guarantee achieve using combination static labeling serverless processes dynamic faceted labeling persistent data describe implementation approach top javascript aws lambda openwhisk serverless platforms present three realistic case studies showing enforce important ifc security properties low overhead introduction may equifax credit reporting agency suffered security breach leaking social security numbers personal information million consumers breach exploited code injection vulnerability apache struts became latest series attacks public private clouds compromising sensitive personal information hundreds million users attacks traced two types faults misconfigurations software vulnerabilities former include issues like incorrect database security attributes choice weak authentication schemes use unpatched software latter include code sql injections file inclusions directory traversals etc simply put enormous trusted computing base tcb modern cloud applications makes intractable enforce information security environments promising avenue smaller tcb lies use information flow control ifc security ifc world information protected global security policy overridden misconfigured application policy explicitly concisely captures constraints information flow system credit card numbers exposed appropriate card associations visa mastercard ifc system enforces policy even buggy malicious applications thus removing application code configuration tcb cloud particular application hijacked code injection attack able bypass enforcement mechanism contrast security models based access control lists capabilities instance compromised program running database administrator privileges easily leak entire database remote attacker despite significant progress ifc remains difficult apply real software dynamic ifc systems incur high runtime overhead static ifc systems shift costs development time work done visiting vmware research usually via use type systems however restrict style programming complicates adoption demonstrate ifc cloud computing feasible implemented low overhead essentially unmodified applications achieve properties leveraging recent developments cloud computing namely rise serverless computing initially popularized amazon aws lambda serverless computing rapidly gaining adoption cloud providers tenants due key benefits elastic scalability ease deployment flexible pricing achieves benefits decoupling application logic resource management serverless model users express applications collections functions triggered response user requests calls functions function written language may request certain runtime environment including specific versions python interpreter libraries however function agnostic environment instantiated physical machine virtual machine container cloud platform manages function placement scheduling automatically spawning new function instances demand requires application state decoupled functions placed shared data store database store allowing function instances access state regardless physical placement cloud argue serverless computing fundamental implications cloud security particular enables practical ifc cloud key observation serverless function constitutes natural unit information flow tracking first serverless function activation handles single request behalf specific user accesses secrets related request second invocation starts clean state get contaminated sensitive data previous invocations conservative assumption secrets obtained function execution propagate outputs track global flow information system monitoring inputs outputs functions comprising based observation develop first ifc system serverless applications called trapeze trapeze encapsulates cloud sandbox unmodified serverless function sandbox intercepts interactions function rest world including functions shared data stores external communication channels redirects security shim shim figure shim tracks information flow enforces global security policy class supported policies along policy enforcement datastore rules defined trapeze dynamic ifc model model addresses weakness existing static dynamic ifc fig trapeze architecture serverless function systems leak information termination channel encapsulated sandbox inputs whereby adversary infer secrets observing termination outputs including invocations program massively parallel nature within outside cloud calls serverless environment amplifies weakness allowing serverless functions reads writes attacker construct information channel shared data stores external communication channels monitored security shim effectively defeating purpose ifc section ifc model eliminates channel enforcing strong security property known tsni handful previous ifc models achieve level security however models either restrictive practical applications introduce significant overhead see section trapeze achieves tsni novel combination static program labeling dynamic labeling data store based faceted store semantics static program labeling restricts sensitivity data serverless function observe ahead time key eliminating termination channel dynamic data labeling crucial securing unmodified applications statically partition data store security compartments faceted store semantics eliminates implicit storage channels present formal proof validated using coq proof assistant model enforces tsni evaluate trapeze three serverless applications online retail store parallel build system image feature extraction service use trapeze secure applications minimal changes application code low runtime overhead thus key contributions work ifc shim architecture serverless computing new ifc model enforces tsni along formal semantics proof correctness experimental evaluation architecture model three serverless applications finally point ifc model limited serverless domain generally speaking applies reactive system decouples computation state algorithmically show tsni enforced systems efficiently examples systems include hadoop apache spark stateless network functions new ifc model serverless discussed serverless function offers convenient unit information tracking enabling practical ifc serverless applications seems natural next step adapting one many existing ifc models serverless environment however take path existing models provide adequate security serverless applications specifically previous ifc models dynamic static enforce security property known tini intuitively tini guarantees attacker deduce secrets stored system outputs however may able deduce part secret fact system stopped producing outputs information channel known termination channel often disregarded low bandwidth typically leaking single bit true serverless systems construct attack serverless application amplifies termination channel spawning many parallel computations leaking one bit example consider serverless system two users benign user bob malicious user eve introduce security labels tag bob eve data respectively labels form lattice labels higher lattice representing secret data bob eve mutually distrusting therefore labels incomparable figure eve launches code injection attack serverless function forcing function execute malicious code figure code designed leak bob secret stored key store eve assume system secured using classical dynamic ifc model upon reading value store function label gets bumped least upper bound function previous label value label note since security shim observe data flow inside function must conservatively taint entire function label function initially runs eve label every line listing annotated current label function reading bob secret function forks instances helper function lines instance leak one bit secret next read secret bumping function dynamic label encode using first keys store store arbitrary value key iff ith bit secret lines helper function also starts initial label short delay line reads key equal function argument store line point function label either bumps corresponding bit secret stays otherwise line attempts send message eve eve handle eve http session succeeds label channel higher equal function compromised fork secret store read secret store write function delay store read eve send diverges label example security lattice secret label top lattice bottom least secret label termination channel classical dynamic ifc model fig security lattice termination channel function current label diverges otherwise eventually attacker end channel receives list bits secret equal implemented attack aws lambda able leak bits per second scaling number threads thus elastically scaling computation serverless architecture also scales termination channel inline theoretical results suggest concurrent system termination channel leak secret time linear size secret therefore aim stronger security guarantee known tsni eliminates termination channel note termination channel example arises function label hence ability send external channel depends labels values function reads store contrast proposed model assigns static security label function activation take advantage fact serverless function always runs behalf specific user assigned corresponding security label complete model presented sections also offers secure way dynamically increase function label without introducing side channel function label determines view data store function observe existence data whose label exceed function label example reading key contains secret function current label store returns result key present store information hiding semantics somewhat tricky maintain multiple functions incomparable labels write store location avoid information leaks situation employing faceted store semantics record contain several values facets different security labels best knowledge trapeze first ifc system combine static program labeling dynamic labeling data using faceting combination eliminates termination storage channels enforces strong security property tsni informal design threat model assumptions assume following entities trusted malicious compromised cloud operator physical hosts network system software hypervisor container manager scheduler serverless runtime shared data stores sandboxing technology assumptions future relaxed help secure enclave technology intel sgx data encryption software verification trust serverless application administrator enforce following invariants application configuration data stores used application configured accessible serverless functions serverless functions system sandboxed finally trust application developer correctly define application information flow policy declassifier functions section rest application untrusted particular assume attacker compromise application code running inside serverless functions including frameworks libraries uses paper focus data confidentiality protecting sensitive data exposed unauthorized users complementary problem enforcing data integrity protecting data unauthorized modification outside scope trapeze although also enforced help ifc techniques finally concerned timing covert channels example channel function running high label communicating function running low label modulating cpu memory usage security lattice start construction ifc model lattice security labels labels represent security classes information flowing system trapeze assign specific semantics labels however practice typically represent users roles system trapeze relies trusted authentication gateway tag external input output channels correct security labels example eve establishes http session system session gets tagged eve label given labeling inputs outputs trapeze applies information flow rules presented enforce information received input channel labeled exposed output channel labeled information flow rules choice information flow rules determines two critical properties ifc system security transparency former characterizes class insecure behaviors system prevents latter characterizes class secure programs system executes unmodified semantics therefore need modified work trapeze trapeze enforces strong security property tsni cost loss transparency argue acceptable serverless systems trapeze assigns runtime security label every serverless function activation label derived event triggered function particular function invoked via http request user obtains user security label alternatively invoked another function inherits caller label function label controls ability send output channel send allowed function label smaller equal channel label trapeze also dynamically labels records data store end security shim intercepts data store operations issued function modifies insert check security labels function creates updates record store record inherits function label see detailed write semantics reading store function observes values whose labels equal label function perspective store behaves contain data function may observe function upgrade label arbitrary higher label using raiselabel operation operation introduce unauthorized information channel decision upgrade depend secrets function previous label secrets simply invisible function upgrade mechanism useful example function running behalf regular user needs update global statistics behalf superuser upgrade operation function label never downgraded current value store semantics trapeze security shim conceals existence data whose security label less equal function tion label maintaining semantics straightforward secret store read writes data store location carry label presence conflicting writes resolution mechanism required secret avoid implicit storage channels conflicting labels store write comparable conflict resolution performed using rule however mechanism function avoids storage channels exists writes incomparable labels following example illustrates problem example implicit storage channel suppose resolve conflicts ignoring writes incomparable label exists store alternative strategies silently overriding existing value failing conflicting write similarly vulnerable figure shows two functions running labels respecfig implicit storage channel via conflicting tively collude leak bob secret eve reads writes secret line however authority send eve directly instead encodes bit secret using record store example lines reconstructs secret attempting write locations reading value back writes locations correspond ignored indicating corresponding bit secret store write store read eve send eliminate unauthorized flows using faceted store semantics record contain several values facets different labels facets created dynamically value new label stored record facet created see section precise semantics read returns recent write visible function thus facets conceal writes label function running label unless example replay example figure faceted store semantics since function labels incomparable respective writes different facets functions observe writes either explicitly indirectly example faceted stores previously introduced ifc research work faceted execution fundamental difference approach read semantics faceted execution read conceptually forks program creating separate branch facet read store program sends external channel branch whose label compatible channel allowed send similar design faceted execution eliminates storage channels however potentially high runtime cost may become impractical system large security lattice contrast trapeze avoids using apriori knowledge function label pruning incompatible facets read time existing faceted execution systems expose termination channel therefore enforce tini whereas trapeze enforces stronger tsni property practical use trapeze faceting exceptional situation trapeze designed run unmodified applications assume conventional store semantics moment multiple facets created store location semantics violated different functions observe different values location trapeze treats situations attempted exploits notifies administrator take recovery actions remove offending function system rollback store previous consistent state meanwhile trapeze guarantees system continues running without exposing sensitive information attacker faceted store semantics emulated security shim top conventional store section implement facets top store yang present design faceted sql database transparency flip side trapeze strong tsni security guarantees protection theoretical loss transparency ability run existing unmodified applications assigning static security label function restrict data visible particular function access values security level even send values anything derived unauthorized channels hand writes data store performed function conservatively labeled function label even carry secrets problems addressed refactoring application particular function gain access secret data via raiselabel operation conversely one avoid tainting data excessively high labels splitting offending function several functions run lower labels however many changes required order adapt existing applications work trapeze create barrier trapeze practical adoption evaluation section indicates practice loss transparency issue serverless applications due common serverless design practice every function accesses values related specific small task therefore likely compatible security labels declassifiers many applications allow limited flow information security lattice example credit reporting agency may make distribution consumers across credit score bands publicly available statistics computed based credit history consumers must therefore labeled least upper bound labels however since aggregate statistics exposes negligible amount information individual consumers safely declassified similar previous ifc models trapeze introduces declassifiers support scenarios declassifier triple security labels serverless function declassifier invoked like serverless function however security label computed using special rules let label calling function declassifier assigned label otherwise design declassifiers violate property therefore formal model proofs section given pure ifc model without declassifiers formal semantics section formalize ifc semantics serverless systems underlying persistent store since computations different security labels might write key store maps key set values facets different label order set sequence according temporal order writes initial store maps key empty sequence label value labeledvalueseq key store thread process processes state value label key labeledvalueseq event outputchannel events operation readcontinuation multisets processes serverless execution state start output nop event read write send fork raiselabel stop value label thread external events operations state transition relation start output run send last run read write run write run fork nop nop nop nop nop label run raiselabel state transition relation refl trans transition relations fig formal semantics state state system consists store store plus multiset currently executing serverless function activations called processes see figure process consists thread plus associated security label label observable events event system include input event start starts new process output event output sends value output channel see figure state transition relation shown figure describes system executes first rule handles incoming event start simply adding multiset processes next five transition rules involve executing particular process next operation operation maximal generality formalize computation language instead assume function run thread operation executes thread returns includes continuation rest thread analogous coinductive definitions used bohannon describe operation corresponding transition rule turn serverless systems spawn new handle incoming event assume incoming event contains new process simplify formal development multiset start output label nop otherwise fig definition projection function send rule checks process permitted output channel label outputchannel label returns security label channel process becomes stuck check fails otherwise generates output event output new process state using continuation returned run read rule reads labeled value sequence store uses projection operation defined figure remove values visible current label passes last entry list recent visible write read continuation note may empty sequence either key never written writes visible current process case last returns passed rule following four generates dummy event nop since externally visible behavior write conventional data store new write would overwrite previous value key contrast faceted store semantics must ensure process unable see new write still read older write hence represent sequence labeled values new write value label remove older writes sequence longer visible namely since process could read also read recent write following function performs garbage collection appends new labeled value write labeledvalueseq value label labeledvalueseq write symbol denotes sequence concatenation fork rule forks new thread continuation original thread threads inherit security label original process raiselabel rule simply raises label current process higher label example permits process read secret data finally allows state perform stuttering nop steps time technical device facilitate proof note rule needed stop operation instead leave stopped processes process multiset simplicity termination sensitive use notation various domains remove information visible observer level see figure example contains values labels visible contains processes labels visible event visible starts process visible outputs channel visible otherwise say nop write denote items appear equivalent observer level proof based projection lemma relates execution full system portion visible level every step corresponding step part vice versa part lemma projection part part proof see appendix based lemma proof termination sensitive theorem single step termination sensitive proof projection part assumption implies projection part therefore transitivity required corollary termination sensitive applied reactive system notion often known proof induction derivation coq formalization semantics proofs available https tsni result states set observable outputs system possible schedules depend inputs visible observer result prevent malicious scheduler leaking secrets prioritizing certain schedules scheduling processes based secrets however mentioned earlier assume scheduler adversarial manner prior work addressed problem assuming scheduler assumption realistic serverless computing implementation evaluation implementation order evaluate proposed security architecture ifc model developed prototype implementation trapeze implementation portable currently runs two popular serverless lambda openwhisk consists three components sandbox security shim authentication service sandbox trapeze sandbox encapsulates application code redirecting inputs outputs security shim figure exact sandboxing technology depends programming language used currently support serverless functions written javascript runtime one common types serverless functions aws ibm cloud functions ibm public openwhisk service encapsulate functions using javascript sandbox many serverless functions rely ability invoke external programs encapsulate external executables using sandbox restricts program activity temporary local directory gets purged every serverless function invocation security shim security shim monitors inputs outputs function enforces ifc rules shim consists multiple adapter modules one supported input output interface three groups adapters data store adapters function call adapters external channel adapters data store adapter implements faceted store semantics top conventional cloud data store trapeze currently supports single type data faceted store implemented top relational database store implements standard dictionary following operations put key value get key del key keys returns keys store store backed relational database table columns key value label table contains entry facet value store used mysql server available aws amazon relational database service security shim passes additional parameter every security label get operation performs sql query returns entries match given key whose label less equal given label del operation deletes facets labels greater equal given label put operation deletes elements del inserts given pair given label addition faceted version also provide conventional insecure store implementation baseline performance evaluation function call adapters support different ways invoke serverless functions making sure callee inherits caller label required ifc model support two invocation mechanisms aws step functions run workflow multiple serverless functions controlled finite automaton amazon kinesis supports asynchronous communication via event streams external channel adapters enable secure communication across cloud boundary supported types channels http sessions email communication via nodemailer module connections external buckets used upload large data objects fit http requests http sessions obtain security labels authentication service see nodemailer adapter uses user database see map email address user security label adapter inherits label user provides login credentials bucket authentication service authentication service responsible associating correct security label every external http session implemented top user database stores credentials email addresses security labels users system entire trapeze framework consists lines javascript code including lines aws openwhisk shim modules lines store lines authentication service evaluation questions evaluation aims answer following questions security trapeze enforce information security serverless applications particular confidentiality requirements applications captured security policy consisting security lattice trusted declassifiers trapeze enforce policy presence buggy malicious code transparency trapeze secure existing serverless applications minimal modifications performance trapeze achieve first two goals low performance overhead case studies answer questions carried three case studies used trapeze add security layer existing serverless applications outline case studies case study hello retail hello retail project serverless team nordstrom goal produce purely serverless web site since architecture competition award serverlessconf austin made several changes hello retail applying trapeze first replaced dynamodb databases hello retail calls trapeze store second replaced calls twilio sms messaging service currently supported trapeze communication third extended hello retail project product purchase subsystem manages online orders credit card payments resulting system consists serverless functions figure shows architecture system system serves several types users store owner manages online catalog processes orders photographers upload product images catalog customers navigate catalog place orders visa credit card authority authorizes card payments behalf customers security lattice figure consists labels matching user categories solid lines diagram show partial order security labels dashed arrows show declassifiers declassifier represented arrow label table summarizes security labels case study following scenario illustrates flow sensitive information hello retail system every step scenario annotated security label data involved step owner owner creates new product description catalog owner sends email one photographers requesting picture product request includes information product description declassified declassifier implements trusted user interface request email photo request photographer registry register photographer api receive photo api get photos step function photos event processor retail event stream aws kinesis create product api photo storage catalog api product release api credit card registry credit card authorization catalog builder product catalog purchase product step function purchase api fig architecture hello retail project circles labeled represent main functions application functions whose names end api directly invoked via client http requests arrows show interactions different components red dashed arrows indicate interactions carry declassified data label owner client clientcc isa description sensitive information managed store owner including product catalog photographers email addresses items catalog visible owner released public via release declassifier labels online purchases specific customer information visible customer store owner since client owner customer credit card information released credit card authority labels external communication channel visa credit card authority product photos uploaded photographer table security classes hello retail case study owner confirmation declassification photographer uploads product image catalog owner ready make product publicly available online catalog declassifies using release declassifier client client orders product catalog order information labeled client visible client well owner since client owner client credit card details labeled clientcc hidden owner clientcc owner visa owner clientcc release photographer authorize client tenant user hello retail feature extraction fig security lattice declassifiers different case studies visa order finalized credit card information sent visa payment authorization external channel labeled isa client response received channel consists one bit information indicating success failure gets declassified isa client authorize declassifier making outcome request visible client owner case study system running parallel software workflows software compilation video processing serverless platforms unit work thunk specifies executable run data dependencies workflow synthesized direct acyclic graph dag thunks recursively executed serverless platform execution engine identifies dependency way using store storage backend consists single serverless function internally runs arbitrary executables invocation function executes exactly one thunk fetching dependencies object store executing thunk storing output back object store use parallel build framework implemented top concrete use case framework extracts workflow dag every thunk corresponds invocation build tool compiler linker project makefile original implementation every authenticated user access sources binaries system use trapeze introduce secure mode mode tenants access source compiled code thunk running behalf tenant taints outputs tenant label tenant may release sources compiled binaries public making available tenants reflected security lattice figure mutually incomparable tenant labels declassifier prior adding mode ported parts written python javascript well modified use store case study image feature extraction serverless application gives users access amazon aws rekognition image analysis service based fetch file store analyse image examples serverless examples collection application consists upload function takes image url fetches image stores store feature extraction function uses aws rekognition extract features image send user use trapeze add security layer example enforcing every user access information extracted images policy expressed simple security lattice figure case study app code modif code declassifiers hello retail feature extraction table size case studies lines code locs locs comprising application lines added modified adapt application work trapeze locs trusted declassifiers security employ trapeze protect sensitive data three case studies original implementation applications either offered protection giving every authenticated user access data system implemented hoc security policies embedded application code example hello retail system design exposes credit card details credit card authority protection relies checks scattered around application code easy get wrong besides bypassed exploit subverts application logic trapeze captures security requirements three case studies security policy consisting security lattice declassifiers shown figures policies simple concise consisting several classes labels declassifiers policies decoupled application logic software architecture instance adding new functions application changing control flow even refactoring database schema affect security policy furthermore trapeze immune malicious compromised application logic simulated code injection attacks case studies replacing original functions malicious functions attempt leak secrets unauthorized users similar examples figures expected simulated attacks failed running application trapeze transparency discussed section trapeze enforces tsni cost reduced transparency may require developer refactor application work trapeze table measures loss transparency case studies reporting size changes application code terms lines code locs required adapt application work trapeze take account changes needed port application use store well compatibility changes fundamental trapeze architecture made unnecessary additional engineering effort none case studies required splitting function multiple functions hello retail required calling raiselabel twice order upgrade label purchase placement function client clientcc saving customer credit card details data store upgrade label payment authorization function client isa order read credit card details store send credit card authority addition case study required minor change due technicality existing code fully compatible sandbox results indicate practice loss transparency issue serverless applications due common software design practices serverless world function assigned single small task accesses values related task therefore likely compatible security labels last column table reports total size declassifiers used case study number characterizes amount trusted application code example function runtime insecure trapeze hello retail update purchase build browse mosh git vim openssh image feature extraction image upload feature extraction table trapeze performance scenario performance measure overhead trapeze running case studies aws lambda serverless platform run case study openwhisk platform ibm cloud formerly ibm bluemix since aws lambda support ptrace use sandbox binary executables section table summarizes runtime overhead trapeze since overhead may depend exact workload constructed several typical workflows case study test hello retail case study two workloads build browse workload simulates construction browsing product catalog update purchase workload simulates updates catalog followed series online purchases test using compile four open source software packages mosh git vim openssh finally consider two scenarios image feature extraction image upload scenario uploads single image data store feature extraction scenario performs feature extraction stored image workload report total number serverless function calls column table function runtime section table reports total runtime functions scenario without protection relative slowdown introduced protection averaged across runs runtimes include additional declassifier calls introduced protected execution accounting declassifiers increased total runtime another build browse workload making measurable impact workloads initial experiments also measured total time execute workload however found due nature varying resource availability serverless environment times varied wildly across different runs provide insights performance trapeze workloads trapeze adds modest overhead function runtime negative overhead mosh workload due noisy ibm cloud environment runtime function varies dramatically across different invocations based load node function scheduled overhead higher hello retail case study update purchase scenario benchmarking revealed bulk overhead due startup time sandbox adds average per function invocation since functions example short runtimes startup time becomes significant contributor total runtime measured two thirds startup time spent loading libraries used application future much overhead eliminated caching preloaded libraries memory finally evaluate storage overhead secure store table compares database sizes kilobytes insecure secure versions store three case studies report results case study insecure trapeze hello retail feature extraction table storage overhead trapeze one workloads case study relative increase database size independent workload secure database requires space stores security label value overhead low examples database stores large objects images source files significant hello retail case study individual values stored database small related work modern technique using security labels dynamically monitor information flow proposed denning lattice model work also defines concept implicit flow information austin describe three techniques monitoring flows namely failure oblivious sensitive upgrade permissive upgrade work exposes fourth choice using faceted values specifically eliminate implicit flows austin flanagan introduce faceted values full enforcement technique rather focusing specifically controlling implicit flows combine faceted value concept concept multiple executions enables precise enforcement cost runtime overhead trapeze enforces strong form tsni existing ifc systems support terminationinsensitive tini previous systems enforce tsni either restrictive costly practical applications multilevel security mls model achieves tsni statically partitioning code data system security compartments model designed primarily systems restrictive applications requires complete existing software smith volpano present security type system enforces tsni imposing harsh restriction loop conditions may depend secret data heintze riecke propose secure typed lambda calculus called slam slam enforces tini sabelfeld sands point version slam lazy evaluation semantics would theoretical result limited practical implications exception haskell none today major programming languages use lazy evaluation indeed stefan implemented haskell library called lio guarantees tsni requiring programmers decompose programs separate threads floating labels analogous processes formalism however lio uses sensitive upgrade rule prevent implicit flows trapeze uses faceted values instead also assume scheduler would inappropriate application serverless computing secure achieves tsni running multiple independent copies program one security class technique introduces cpu memory overhead proportional number security classes acceptable systems security classes devriese piessens consider two classes secret becomes impractical systems potentially millions mutually untrusting users faceted execution potential mitigate drawback bielova rezk proposed theoretical approach extending faceted execution model enforce tsni asbestos applies dynamic ifc granularity process similar trapeze operates granularity serverless function asbestos associates static security label process however label serves upper bound label data process access process effective label changes dynamically enables implicit termination channel best knowledge trapeze first system apply ifc serverless applications several researchers advocate use ifc broader context secure cloud computing however aware practical implementation ideas conclusion advent serverless computing provides opportunity rebuild cloud computing infrastructure based rigorous foundation information flow security present novel promising approach dynamic ifc serverless systems approach combines sandbox security shim monitors operations serverless function invocation static security labels serverless function invocation dynamic faceted labeling data persistent store combination ideas provides strong security guarantee tsni necessary serverless settings avoid termination channel leaks via multiple concurrent requests trapeze implementation approach lightweight requiring new programming languages compilers virtual machines three case studies show trapeze enforce important ifc properties low space time overheads believe trapeze represents promising approach deploying serverless systems rigorous security guarantees help prevent costly information leaks arising buggy application code attacks acknowledgments references airbnb streamalert serverless framework data analysis alerting http amazon aws lambda https amazon aws rekognition https anonymous authors trapeze source code repository link anonymized reviewing apache software foundation apache hadoop https apache software foundation openwhisk https aslan askarov sebastian hunt andrei sabelfeld david sands noninterference leaks bit proc esorics malaga spain thomas austin cormac flanagan efficient information flow analysis proc plas thomas austin cormac flanagan permissive dynamic information flow analysis proc plas thomas austin cormac flanagan multiple facets dynamic information flow proc popl thomas austin tommy schmitz cormac flanagan multiple facets dynamic information flow exceptions acm trans program lang syst article may pages https thomas austin jean yang cormac flanagan armando faceted execution programs proc plas seattle washington usa jean bacon david eyers thomas pasquier jatinder singh ioannis papagiannis peter pietzuch information flow control secure cloud computing ieee transactions network service management andrew baird michael connor patrick brandt running serverless applications enterprise requirements https elliott bell leonard lapadula secure computer systems mathematical foundations technical report mitre nataliia bielova tamara rezk spot difference secure multiple facets european symposium research computer security springer arnab kumar biswas dipak ghosal shishir nagaraja survey timing channels countermeasures acm comput surv march aaron bohannon benjamin pierce vilhelm stephanie weirich steve zdancewic reactive noninterference proceedings acm conference computer communications security acm mark boyd irobot confronts challenges running serverless scale https kuldeep chowhan serverless computing patterns expedia https playstation network breach faq may https cnet magazine computerworld sql injection attacks led heartland hannaford breaches https computerworld authentication oversight led jpmorgan breach https willem groef dominique devriese nick nikiforakis frank piessens flowfox web browser flexible precise information flow control proc ccs dorothy denning lattice model secure information flow comm acm dorothy denning peter denning certification programs secure information flow commun acm july dominique devriese frank piessens noninterference secure proc ieee ssp digital trends latest data breach involves voting records million mexican citizens april https petros efstathopoulos maxwell krohn steve vandebogart cliff frey david ziegler eddie kohler david frans kaashoek robert morris labels event processes asbestos operating system proc sosp https ken ellis reuters replaced websockets amazon cognito sqs marius eriksen server function proc plos project https forbes ebay suffers massive security breach users must change passwords may https forbes hackers broke equifax exploiting patchable vulnerability https sadjad fouladi dan iter shuvo chatterjee christos kozyrakis matei zaharia keith winstein thunk remember make jobs infrastructure review http sadjad fouladi riad wahby brennan shacklett karthikeyan vasuki balasubramaniam william zeng rahul bhalerao anirudh sivaraman george porter keith winstein encoding fast slow video processing using thousands tiny threads proc nsdi boston project source code repository https google google cloud functions https nevin heintze jon riecke slam calculus programming secrecy integrity proc popl san diego california usa tyler hunt zhiting zhu yuanzhong simon peter emmett witchel ryoan distributed sandbox untrusted computation secret data proc osdi savannah usa ibm ibm cloud functions https intel corporation intel software guard extensions programming reference eric jonas shivaram venkataraman ion stoica benjamin recht occupy cloud distributed computing corr http murad kablan azzam alsudais eric keller franck stateless network functions breaking tight coupling state processing proc nsdi boston mckim john announcing winners inaugural serverlessconf architecture competition https microsoft azure functions https andrew myers jflow practical information flow control proc popl andrew myers barbara liskov protecting privacy using decentralized label model tosem national vulnerability database march https nodemailer nodemailer https nordstrom technology hello retail https thomas pasquier jean bacon jatinder singh david eyers access control cloud computing proc sacmat shanghai china pcworld microsoft cloud data breach heralds things come https andrei sabelfeld andrew myers security ieee journal selected areas communications andrei sabelfeld david sands per model secure information flow sequential programs higher order symbol comput march peter sbarski serverless architectures aws examples using aws lambda manning publications shelter island serverless serverless examples https geoffrey smith dennis volpano secure information flow imperative language proc popl san diego california usa deian stefan alejandro russo pablo buiras amit levy john mitchell david mazieres addressing covert termination timing channels concurrent information flow systems acm sigplan notices vol deian stefan alejandro russo john mitchell david flexible dynamic information flow control haskell proc haskell techrepublic massive amazon leaks highlight user blind spots enterprise race cloud july https register rsa explains attackers breached systems april https tom van cutsem mark miller trustworthy proxies virtualizing objects invariants proc ecoop montpellier france https wikipedia anthem medical data breach https wikipedia sony pictures hack https wikipedia yahoo data breaches https wired inside cyberattack shocked government https jean yang travis hance thomas austin armando cormac flanagan stephen chong precise dynamic information flow applications proc pldi santa barbara usa matei zaharia mosharaf chowdhury tathagata das ankur dave justin murphy mccauley michael franklin scott shenker ion stoica resilient distributed datasets abstraction cluster computing proc nsdi san jose stephan arthur zdancewic programming languages information security thesis cornell university zdnet anatomy target data breach missed opportunities lessons learned http http zdnet adultfriendfinder network hack exposes million accounts summary auxiliary semantic details proof details empty sequence concatenation label outputchannel label run thread operation write labeledvalueseq value label labeledvalueseq write means lemma invisibility nop proof omitted lemma projection proof let let proceed cases inversion case let start proceed cases case pick pick start qed case pick pick qed case pick pick qed last case rule let proceed cases case invisibility qed last case resume case analysis case let output run send label pick pick output qed remaining cases omitted lemma projection proof let let proceed cases inversion case let pick qed case pick qed case let output run send label pick pick qed remaining cases omitted
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aug subfiniteness graded linear series huayi chen hideaki ikoma abstract hilbert problem studies finite generation property intersection integral algebra finite type subfield field fractions algebra negative answer due counterexample nagata show subfinite version hilbert problem confirmative answer establish graded analogue result permits show subfiniteness graded linear series depend function field consider finally apply subfiniteness result study geometric arithmetic graded linear series contents introduction weak form hilbert fourteenth problem graded linear series version zariski theorem applications references introduction let integral projective scheme spec cartier divisor graded linear series one refers graded huayi chen partially supported hideaki ikoma supported jsps young scientists partially jsps huayi chen hideaki ikoma graded linear series closely related positivity divisor objects central interest study geometry underlying polarised scheme classically asymptotic behaviour graded linear series type well understood theory hilbert polynomials several results birational algebraic geometry fujita approximation theorem show certain graded linear series even though type still similar asymptotic behaviour generation case recently proposed ideas okounkov method encode asymptotic behaviour dimensions homogeneous components given graded linear series convex body called body euclidean space note graded linear series cartier divisor always graded subalgebra graded algebra type quite natural ask nice birational geometry algebras type namely subalgebras algebra type point view birational geometry convenient consider graded linear series generated extension without specifying polarised model framework graded linear series refer graded polynomial algebra dimensional vector space new construction bodies proposed using ideas arakelov geometry depends choice tower successive extensions extension transcendental transcendence degree construction valid graded linear series type namely contained graded linear series type whose rational functions coincides see one may expect method applies general graded linear series type considering graded linear series however main obstruction strategy priori condition depends extension respect consider graded linear series leads following problem given graded linear series type exist graded linear series type extension contains note problem closely related hilbert fourteenth problem fact given graded linear series contained graded linear series type intersection let field field rational functions variables hilbert fourteenth problem asked whether intersection subfield polynomial algebra finitely generated subfiniteness graded linear series gives graded linear series containing rational functions unfortunately intersection necessarily type shown nagata counterexamples hilbert fourteenth problem note problem actually asks weaker condition generation intersection intersection contained graded linear series type similarly consider following version hilbert fourteenth problem actually positive answer see theorem corollary infra theorem let field integral finite type field fractions let extension contained exists finitely generated containing frac frac method proof consists induction argument respect extension permits reduce problem case extension monogenerated similar method applied graded case subtleties grading structure leads following result gives answer problem graded linear series shows graded linear series absolute condition depend choice extension respect graded linear series considered see theorem corollary infra theorem let field finitely generated field extension let graded linear series subfinite type exists graded linear series finite type recall hilbert fourteenth problem reformulated geometric setting zariski see also survey article note theorem compared following result theorem zariski let field integrally closed finite type frac subextension exist integrally closed finite type ideal fraction field fraction field frac denotes ideal quotient huayi chen hideaki ikoma inspired result establish following projective version zariski theorem deduce alternative proof theorem see corollary infra theorem let finitely generated field extension graded linear series generated homogeneous elements degree assume contains projective spectrum proj normal scheme let subextension exist rational fibration integral normal projective effective divisor following properties rat every sufficiently positive integer iii every dim exists dim moreover transcendence degree replace property every sufficiently positive integer application results establish fujita approximation theorem general graded linear series type see theorem infra upper bound function graded linear series see theorem infra precisely obtain following results theorem let finitely generated field extension graded linear series subfinite type whose dimension nonnegative limit vol lim dimk exists moreover vol equal supremum vol runs set graded linear series contained dimension finally exists function vol dimk also apply results study graded linear series arithmetic setting see theorem infra article organised follows second section prove weaker form hilbert problem result stated subfiniteness graded linear series theorem third section prove graded analogue theorem setting graded linear series fourth section consider problem geometric setting projective analogue zariski result establish theorem finally section develop various applications notation conventions fractions integral domain denoted frac let extension denote transcendence degree let scheme denote set points local ring krull dimension integral scheme denote rat rational functions let projective normal scheme spec weil divisor resp divisor one refers element resp nonnegative say effective denoted nonzero rational function denote principal weil divisor associated namely ordv map rat group homomorphism induces map rat denote divisor rat note graded polynomial algebra rat let extension discrete valuation refer valuation given valuation denote valuation ring maximal ideal residue note extension equal say valuation trivial note case two discrete valuations huayi chen hideaki ikoma said equivalent exists isomorphism let generated extension say discrete valuation divisorial rkz case valuation trivial divisorial condition let subextension let discrete valuation nontrivial restriction discrete valuation ramification index respect unique integer satisfying let integral separated given discrete valuation rat say point centre denotes maximal ideal valuative criterion separation centre exists unique case centre exists denote proper valuative criterion properness every discrete valuation rat centre discrete valuation trivial centre generic point moreover regular point discrete valuation rat whose centre let graded ring denote proj projective quasispectrum graded denote coherent oproj associated see let let particular one sheaf denoted oproj note generated oproj invertible oproj one canonical isomorphisms oproj oproj oproj proj subfiniteness graded linear series let graded ring say essentially integral ideal vanish product two nonzero homogeneous elements positive degree nonzero note essentially integral scheme proj integral see proposition weak form hilbert fourteenth problem let generated integral fractions clearly generated extension let subextension also generated extension see chapitre corollaire consider intersection ask following question could considered weaker form hilbert fourteenth problem exist finitely generated containing frac frac section give answer question definition let say subfinite type type lemma injective homomorphism rings yields dominant morphism spec spec proof let minimal prime ideal since homomorphism rings injective also localised homomorphism hence nonzero particular exists prime ideal equivalently since prime ideal minimal prime ideal one proposition let field subfinite type assume integral domain exists finite type containing also integral domain proof let type lemma one prime ideal since type also proposition thus proved lemma let integral domain field fractions let finite extension generated one element finite type contains exists finite type contains proof let minimal polynomial assume monic let polynomials huayi chen hideaki ikoma let set polynomials claim contained fact suppose element written form euclidean division polynomial written polynomial therefore one lemma let integral domain field fractions let purely transcendental extension transcendence degree finite type contains exists finite type contains proof let transcendental element assume rational function form polynomials one variable let element algebraic closure fact element one written polynomial since transcendental considering variable rational functions specifying value obtain finally applying lemma extension obtain exists type theorem let field subfinite type assume addition integral domain denote field fractions exists finite type proof proposition exists type integral domain containing let fractions generated extension therefore exists sequence extensions extension generated one element extension either generated algebraic element purely transcendental transcendence degree induction obtain exists type theorem thus proved corollary let field integral finite type field fractions let extension subfiniteness graded linear series contained exists finitely generated containing frac frac proof integral type theorem exists type frac clearly one frac frac since frac assertion thus proved graded linear series subfiniteness let generated extension let graded ring polynomials one variable definition graded linear series refer graded dimensional subspace let two graded linear series say contained contains denote let graded linear series generated say finite type contained graded linear series type say subfinite type let graded linear series denote subextension generated elements form nonzero elements exists called field rational functions lemma given graded linear series one every sufficiently positive integer denotes subextension generated elements form proof first note index contains nonzero element fact ghn ghn misuse index since referred base field huayi chen hideaki ikoma changing grading may assume without loss generality generates exist integers nonzero elements vni set lcm observation assume one moreover hypothesis generates zmodule positive integer coprime conclude proof show contains every positive integer since coprime moreover assume positive prx hence contains every integer less remark let graded linear series nonzero element denote graded linear series called twist note twist change rational functions one proposition let graded linear series finite type let integer exist integer family homogeneous elements following conditions fulfilled one integer vector space generated elements form frar natural numbers proof suppose generated claim graded linear series generated let integer since generated obtain subfiniteness graded linear series wdad let element dad since exist integer family elements dad therefore dad concludes claim corresponds number dad finally choose family homogeneous elements forms basis lemma let extensions fields assume extension finitely generated extension finite let graded linear series finite type let graded linear series finite type proof let system generators let basis claim generated fact element written frar huayi chen hideaki ikoma belong writing linear combination obtain lies graded linear series generated lemma thus proved definition let graded linear series assume exists dimension transcendence degree refer readers dimension setting graded linear series cartier divisors line bundles convention dimension theorem let graded linear series assume exists graded linear series finite type contains exists graded linear series finite type proof step reduction case let assertion theorem trivial following assume empty hence subsemigroup let generator subgroup type see example generated van lem changing grading reduce problem case particular exists vector spaces nonzero pick replacing graded linear series generated replacing graded linear series generated procedure change rational functions reduce problem case finally replacing see remark notation nonzero element procedure change fractions see remark reduce problem case moreover replacing graded linear series generated system generators may assume step reduction simple extension case induction explained previous step assume since generated extension assumed contain exist successive extensions extension generated one element subfiniteness graded linear series assume theorem proved case generated one element induction show exists graded linear series type contains fact choose assume chosen graded linear series type let graded linear series generated system generators graded linear series contains without loss generality may assume extension generated one element otherwise replace graded linear series generated generator extension graded linear series type contains rational functions theorem proved simple extension case obtain existence graded linear series type note graded linear series conditions therefore prove theorem prove particular case extension generated one element similarly prove theorem supplementary condition extension algebraic prove particular case extension generated one element algebraic step algebraic extension case step prove theorem assumption extension algebraic explained previous two steps may suppose without loss generality extension generated one element algebraic let minimal polynomial proposition exist integer homogeneous elements generates graded linear series since one moreover since extension generated degree exist polynomials huayi chen hideaki ikoma introduce following polynomials fei note one let graded linear series generated elements graded linear series type remains prove contains clearly let element written form frar consider element fer viewed polynomial written values certain polynomials note one therefore since minimal polynomial euclidean division argument shows written polynomial theorem thus proved particular case algebraic extension step general case step prove theorem general case explained steps may assume extension generated one element transcendental algebraic case already treated step since type exist integer homogeneous elements generate generated exists rational functions subfiniteness graded linear series let element algebraic closure let extension purely transcendental extension generated generated let graded respectively generated let graded linear series generated elements form graded linear series type note therefore let element vbn written form frar belong transcendental obtain implies shows therefore one since already seen let lemma graded linear series type since obtain graded linear series type moreover one extension therefore algebraic extension case theorem proved step obtain existence graded linear series type moreover equality assumption imply graded linear series hence graded linear series theorem thus proved subfinite version zariski theorem preliminaries section collect several basic facts valuations graded rings use show theorem huayi chen hideaki ikoma let generated extension set divisorial valuations discrete valuation either trivial see notation conventions lemma let finitely generated field extension subextension restriction belongs proof let assertion trivially true restriction trivial valuation rest proof assume trivial consequently trivial applying chapitre corollaire one respectively taking sum two inequalities one obtains since belongs one means inequality actually equality hence inequalities equalities particular one lemma let dominant morphism integral separated rat rat discrete valuation centre exists center namely proof since morphism dominant induces injective homomorphism rat rat allows consider recall centre unique point satisfying see notation conventions note subfiniteness graded linear series hence implies since maximal ideal let graded ring let proj homogeneous element let deg deg degree component localisation let dproj spec denote open subscheme proj see notation conventions given set local sections glue global section following lemmas lemma proposition suppose irrelevant ideal finitely generated let finitely generated graded sufficiently positive integer lemma let graded ring proj essentially integral generated finitely many homogeneous elements canonical homomorphism injective element integral proof suppose generated non zerodivisors since essentially integral see notation conventions given one hence covers thus section naturally element intersection taken particular injective given homogeneous element one aei every since generates one obtains moreover induction every implies hence integral see example theorem lemma keep notation lemma suppose noetherian integral domain generated finitely many homogeneous elements huayi chen hideaki ikoma ring exists isomorphic every integrally closed domain isomorphic every proof recall integral domain called ring integral closure fraction generated module note graded rings homogeneous fraction rational functions scheme proj particular homogeneous element belongs homogeneous fraction contained fraction lemma obtain contained integral closure hence module type noetherian hypotheses consider exact sequence oproj ker coker since isomorphic proposition ker hence lemma conclude coker integrally closed argument actually leads since contained integral closure given graded ring positive integer veronese subring divides otherwise lemma proposition natural inclusion induces isomorphism proj proj proj sending moreover open subscheme dproj isomorphic dproj via proj homogeneous element lemma finite type proj projective spec proof chapitre iii proposition lemma exists positive integer natural morphism proj proj isomorphic spec since graded ring generated written quotient suitable polynomial algebra deg proj closed embedding subfiniteness graded linear series proj projective spec sense hartshorne page isomorphisms yield isomorphism homogeneous element hence proj proj isomorphism proj also projective spec proof theorem begin reminder notation hypotheses theorem let generated extension graded linear series see assume generated number homogeneous elements degree moreover assume contains projective spectrum proj normal scheme let subextension purpose construct rational integral normal projective divisor satisfying properties iii predicted theorem set proj note may type hence may proper since assume without loss generality divide rest proof six steps step step give valuation theoretic interpretation required statement let cartier divisor image via lemma one every positive integer therefore every positive integer step step show following claim scheme normal moreover exists equivalent see notation conventions consequence following claim let homogeneous element degree huayi chen hideaki ikoma proof claim inclusion obvious element wdn wdn hence obtain homogeneous element positive degree consider morphism spec spec since normal integrally closed hence claim know krull ring thus particular integrally closed domain given spec one spec see theorem show claim take fact thus contradiction hence know since height step fix every divisible see lemma let graded generated natural inclusion map let proj normalisation integral normal projective rat lemma set see notation conventions let proj proj morphism since normal proj induces since dproj spec subfiniteness graded linear series morphism homogeneous element step step show given one suppose one going deduce contradiction since normal proj dproj homogeneous belonging integral one homogeneous element single nonzero closure equation open subscheme spec dproj hypothesis implies generates unit ideal spec thus exist deg let graded generated proj normalisation proj also integral normal projective rat see step let denote veronese subalgebra see consider natural inclusion homomorphism commutative diagram proj spec proj proj proj dproj spec denote integral set see also lemma let closure open subscheme spec equation invertible since huayi chen hideaki ikoma however hand valuation centre discrete valuation ring thus valuative criterion properness corollaire contained maximal domain mapped therefore hence vanishes leads contradiction step claim step given equivalent see notation conventions claim set finite particular obtain surjective map proof claim inclusion yields morphism nonempty open subscheme theorem generic nonempty open subscheme dim dim proposition thus maps let denote index see set min claim every step finally consider case case regular projective curve spec set canonically bijection moreover surjective map subfiniteness graded linear series hence set min every positive integer rational fibrations associated graded linear series following give alternative proof theorem using projective version zariski result theorem corollary let finitely generated field extension subextension let graded linear series contained graded linear series finite type contained graded linear series finite type proof divide proof three steps step step make several reductions theorem arguments step theorem assume contains claim enlarging necessary assume generated proof claim let homogeneous generators let variables deg every one homogeneous prime ideal contains trdr fact let let homogeneous element degree since morphism spec spec dominant lemma exists homogeneous prime ideal proj set graded linear series injective generated particular assume proj projective scheme invertible sheaf step let normalisation cartier divisor image via choose ample divisor generated huayi chen hideaki ikoma note graded graded linear series type lemma isomorphic proj integral normal applying theorem projective integer rat every ample divisor step let generated let graded linear series generated basis contains number generators every generated theorem comparable existence theorem iitaka line bundles normal projective varieties see example theorem consequence theorem give estimate following type graded linear series type see also corollary theorem infra corollary let finitely generated field extension graded linear series subfinite type let dimension nonnegative exist integral normal projective divisors rational function field kodairaiitaka dimension every sufficiently positive proof existence results arguments corollary thus show existence prescribed properties changing grading may assume generates choose positive integer see lemma let generated set subfiniteness graded linear series lemma let proj every let normalisation let positive integer divisible one ample divisor opb vpn every positive integer see proposition repeating arguments one choose integral normal projective two big cartier divisors two coprime positive integers vpn positive integer moreover one choose ample divisor two coprime positive integers integral resp divisible resp qnd vqn hold every integer since qnd surjective positive integers see example example valid arbitrary characteristics every positive recall arguments lemma theorem fujita appendix let integral normal projective effective cartier divisor dimension section ring finitely generated proof let rat let smooth projective rational function inclusion rational map taking suitable one obtains morphism set min every positive integer hence result reduced classic case curves huayi chen hideaki ikoma remark surface zariski completely cases generated theorem proposition later fujita generalised case dimension one form theorem using iitaka nef big cartier divisor generated semiample see theorem applications section apply criterion theorem study fujita approximation general graded linear series throughout section let generated definition let graded linear series dimension see volume vol lim sup dimk priori invariant takes value see addition graded linear series type volume always positive real number say graded linear series satisfies fujita approximation property vol vol sup finite type dim runs set graded linear series type contained dimension purpose section establish following approximation result theorem graded linear series subfinite type nonnegative dimension satisfies fujita approximation property moreover one vol dimk lim subfiniteness graded linear series proof changing grading may assume without loss generality positive integer let homogeneous fraction note subextension hence generated moreover theorem obtain viewed graded linear series type therefore assertions follow theorem birational consider graded linear series combining results result theorem obtain following upper bound function general graded linear series type theorem let graded linear series kodairaiitaka dimension exists function vol dimk remark result theorem actually provides geometric information graded linear series type let generated transcendental extension let transcendence degree containing extension transcendental transcendence degree let set graded linear series type constructed map set convex bodies following conditions two graded linear series one two graded linear series denotes graded linear series whose homogeneous component space generated graded linear series volume lebesgue measure multiplied allows construct arithmetic analogue bodies general arithmetic graded linear series type using ideas huayi chen hideaki ikoma follows assume number denote set places let absolute value extends either usual absolute value certain absolute value adelic vector bundle spec refer data dimensional vector space family norms exists basis subset satisfying following condition kvr max given adelic vector bundle spec nonzero element arakelov degree deg kskv product formula obtain deg deg moreover arakelov degree det nonzero element det det inf kxr product formula obtain depend choice det let adelic vector bundle rank spec let vectk deg decreasing called minima note number sup rkk coincides minus logarithmic version minima sense roy thunder let sup following let generated extension number let graded linear series type subfiniteness graded linear series equip structure adelic vector bundle spec ksn ksn ksm assume addition lim sup condition implies nonnegative dimension let graded linear series one vnt proposition one proof clearly one prove converse inclusion let integer nonzero elements since exist thus positive integer one therefore implies proposition allows consider birational graded linear series construct body reminded remark concave transform function sending sup condition remark function concave following result generalises theorem case adelically normed graded linear series theorem let finitely generated extension number field graded linear series subfinite type dimension equipped structures adelic vector bundles spec satisfy submultiplicativity condition condition sequence measures rkk rkk huayi chen hideaki ikoma converges weakly boreal probability measure image uniform measure vol concave transform proof graded linear series homogeneous fraction see proposition hence construct decreasing family convex bodies contained described remark moreover two real numbers method obtain desired result references boucksom chen okounkov bodies filtered linear series compositio mathematica bourbaki fascicule xxviii commutative chapitre graduations tions topologies chapitre premiers primaire scientifiques industrielles hermann paris hermann paris fasc xxx commutative chapitres bourbaki masson paris chapitres chen majorations explicites des fonctions mathematische zeitschrift okounkov bodies approach function field arithmetic preprint available https cutkosky asymptotic multiplicities graded families ideals linear series advances mathematics fujita coherent sheaves journal faculty science university tokyo section mathematics approximating zariski decomposition big line bundles kodai mathematical journal grothendieck globale quelques classes morphismes institut des hautes scientifiques publications locale des des morphismes institut des hautes scientifiques publications subfiniteness graded linear series hartshorne algebraic geometry new york graduate texts mathematics kaveh khovanskii bodies semigroups integral points graded algebras intersection theory annals mathematics second series kaveh khovanskii algebraic equations convex bodies perspectives analysis geometry topology progr vol new york lazarsfeld positivity algebraic geometry ergebnisse der mathematik und ihrer grenzgebiete folge series modern surveys mathematics vol berlin classical setting line bundles linear series lazarsfeld convex bodies associated linear series annales scientifiques normale matsumura commutative ring theory second cambridge studies advanced mathematics vol cambridge university press cambridge translated japanese reid mumford hilbert fourteenth problem finite generation subrings rings invariants mathematical developments arising hilbert problems proc sympos pure vol xxviii northern illinois kalb amer math providence nagata addition corrections paper treatise problem hilbert mem coll sci univ kyoto ser math problem hilbert american journal mathematics fourteenth problem hilbert proc internat congress math cambridge univ press new york okounkov inequality multiplicities inventiones mathematicae would multiplicities orbit method geometry physics marseille progr vol boston boston takagi fujita approximation theorem positive characteristics journal mathematics kyoto university zariski hilbert bulletin des sciences zariski theorem high multiples effective divisor algebraic surface annals mathematics second series august huayi chen hideaki ikoma huayi chen paris diderot institut jussieu paris rive gauche sophie germain courrier paris cedex france beijing international center mathematical research peking university beijing china url hideaki ikoma department mathematics kyoto university kyoto japan ikoma
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achievability region regenerating codes multiple erasures jan marwen zorgui zhiying wang center pervasive communications computing cpcc university california irvine usa mzorgui zhiying study problem centralized exact repair multiple failures distributed storage describe constructions achieve new set interior points exact repair constructions build upon layered code construction tian designed exact repair single failure firstly improve upon layered construction general system parameters extend improved construction support repair multiple failures varying number helpers particular prove optimality one point functional repair tradeoff multiple failures parameters finally considering minimum bandwidth cooperative repair mbcr codes centralized repair codes determine explicitly best achievable region obtained among known points including mbcr point ntroduction driven growth applications efficient data storage retrieval become crucial importance several service providers distributed storage systems dss currently widely employed storage dss provide scalable storage high level resiliency face server failures maintain desired level failure tolerance dss utilize replacement mechanism nodes known also repair mechanism allows recover content nodes repair process failed node performed downloading data accessible nodes subset thereof system recovering lost data efficiency dss determined two parameters namely overhead required reliability amount data transferred repair process seminal work proposed new class erasure codes called regenerating codes optimally solve repair bandwidth problem shown one significantly reduce amount bandwidth required repair bandwidth decreases node stores information regenerating codes presented achieve functional repair case replacement nodes required exact copies failed nodes repaired code satisfy reliability constraints however practice often desirable recover exact information failed node called exact repair exact repair codes easier implement maintain thus interest flurry interest designing exact repair regenerating codes moreover growing literature focused understanding fundamental limits exact repair regenerating codes opposed functional regenerating codes recovery many practical scenarios large scale storage systems multiple failures frequent single failure moreover many systems apply lazy repair strategy seeks limit repair cost erasure codes indeed demonstrated jointly repairing multiple failures reduces overall bandwidth compared repairing failure individually distinguish two ways repairing multiple failures cooperative regenerating codes framework replacement node first downloads information nodes helpers replacement nodes exchange information regenerating lost nodes interest work note codes corresponding extreme points cooperative tradeoff developed minimum storage cooperative regenerating mscr codes minimum bandwidth cooperative regeneration mbcr codes centralized regenerating codes upon failure nodes repair carried centralized way contacting helpers available nodes downloading amount information helper content nodes system sufficient reconstruct entire data let size node size entire data code satisfying centralized repair constraints referred code also say code system previous work characterized functional repair tradeoff recovery let normalized functional tradeoff written follows min min inequality gives linear bounds work interested designing centralized exact repair regenerating codes recovering multiple failures tradeoff reduces single point trivially achievable hereafter focus case argued cooperative regenerative codes used construct centralized repair codes total bandwidth case obtained taking account bandwidth obtained helper nodes disregarding communication replacement nodes particular mscr codes achieve performance centralized minimum storage repair msmr codes additionally mbcr codes used centralized repair codes correspond centralized minimum bandwidth codes functional tradeoff points given bcr bcr contributions paper improve upon layered construction presented concerned single node repair construct family regenerating codes capable repairing multiples failures particular system first prove optimality particular constructed point using functional repair tradeoff combining achievable points via construction also mbcr point characterize best achievable region obtained known points remainder paper organized follows description first code construction provided section section iii analyze achievability region system describe second code construction section concluding section notation denote set integers ode onstruction exact repair regenerating codes characterized parameters consider distributed storage system nodes storing amount information data elements distributed across storage nodes node store amount information use denote normalized storage size repair bandwidth respectively system satisfy following two properties reconstruction property connecting nodes sufficient reconstruct entire data repair property upon failure nodes central node assumed contact helpers download amount information exact content failed nodes determined central node called repair bandwidth first describe code construction improvement upon construction based collection subsets called steiner system information first encoded within subset distributed among nodes recall definition steiner systems definition steiner system collection subsets size included subset size appears exactly across subsets steiner systems exist design parameters steiner systems always exist blocks case set family codes describe parameterized construction precoding step consider steiner system generate blocks block indexed set block corresponds information symbols alphabet size encoded using msmr code length dimension alphabet size codeword symbols called repair group comprised moreover assume msmr code possesses optimal repair bandwidth number erasures number helpers total number information symbols code matrix code structure described code matrix size rows indexed integers corresponding different storage nodes columns indexed sets arranged arbitrary chosen order formally define otherwise denotes empty symbol node stores symbols row code matrix checked storage per node given nnr abuse notation terms block repair group used interchangeably requirement alphabet size dictated existence msmr code required property msmr codes known exist example consider steiner system number blocks node number appears times blocks blocks given code matrix given figure let repair nodes simultaneously downloading symbols nodes respectively help repair symbols symbols nodes respectively help repair symbols symbols nodes respectively help repair symbols symbol nodes repair symbol nodes repair symbol nodes repair symbol nodes repair similarly node repair example see repair group tolerates failure nodes therefore code also tolerates failure nodes thus checked construction nodes recover data reconstruction parameter moreover code recover failures therefore possible repair simultaneously failures number helpers flexible satisfies repair bandwidth given propositions two different scenarios proposition using construction possible repair simultaneously set nodes using helpers contribution helper denoted given min proof repair procedure subset missing symbols belonging repair group repaired via msmr repair procedure using available helpers group among chosen helper nodes fixing set helper nodes argue repair feasible indeed let set helpers repair group denote set remaining nodes using follows chosen remaining nodes figure summarizes repair situation given parameters summing repair contributions analyzing limit cases given follows functional tradeoff space sharing construction layered code tradeoff remark seen repair procedure benefit msmr repair property case particular advantages using msmr codes construction maximum distance separable mds codes lower repair bandwidth symmetric repair among helper nodes obviates need expensive procedure duplicating block design adaptability meaning repair strategies multiple erasures help varying number helpers figure shows comparison performance layered code construction system msr repair property clearly helps reduce bandwidth thus repair group enough information across set helpers recover missing components analyze contribution single helper helps simultaneous repair missing symbols repair group size count possible cases repair done help coded symbols among helpers number available coded symbols determines contribution helper dictated msmr repair bandwidth follows corresponding repair group chosen set helpers helper already belongs repair group assumption remaining elements repair group fig repair situation associated given parameters normalized bandwidth per helper total download symbols helper transmits symbols normalized storage per node fig using msr repair property improves upon layered code repair performance technique using msmr codes building blocks outer code constructions used literature instance constructing codes local regeneration fig code matrix system parameters remark argue one use regenerating code corresponding interior point instead msmr code inner code per repair group consider case let code structure given thus code per column length dimension use interior code per repair group let information size per column thus follows repair node download total bandwidth thus obtain achievable point point equally achievable using construction msmr code interior code point optimal tradeoff system optimal point next minimum bandwidth regenerating point proposition considered construction steiner systems study next use general steiner system specific system proposition construction generates code proof consider steiner system let obtain analyze repair bandwidth per helper distinguish two cases case helper node shares block failed nodes design share block either failed nodes thus contributes single symbol bits useful repair missing symbols shared repair group otherwise shares exclusively two blocks failed nodes shared repair group node contributes symbol bits help repair corresponding missing symbol virtue msmr repair property missing symbol repaired helpers case helper number blocks shares number failed nodes given blocks node shares exclusively either failed nodes given therefore contribution helper node repair example illustration proposition similar proposition repair bandwidth identical among helper nodes independent choice failed nodes helpers remark note depend advantage using steiner systems smaller whenever exist induce smaller normalized parameters indeed shown given strictly increasing therefore reduce storage size per node therefore repair bandwidth advantageous use steiner system smallest moreover proposition proposition give iii nalysis achievability system section analyze achievable region system means construction using simplicity steiner system proposition construction generates set achievable points system proof chosen general expression given last equality follows vandermonde identity optimality one achievable point proposition system point achieved optimal interior point proof achieve thus substituting setting obtain therefore point lies functional repair lower bound hence optimal lies first segment bound near msmr point msmr point mbcr point indicated optimal extension property proposition consider system consider optimal point achieved construction proposition one extend system system operating optimal point proposition adding another node system increasing storage per node keeping initial storage content proof let refer parameters old new systems respectively moreover number blocks increased let index new node added new code obtained simply adding another block whose set adding old sets element thus generating another coded symbol corresponding repair group key requirement assume use msmr code accommodate addition extra coded symbols needed done choosing number nodes msmr code large needed may result increase underlying field size old node store extra symbol coming new repair group new node stores newly generated coded symbols old repair groups normalized bandwidth per helper proposition construction gives optimal point system construction also offers following property functional tradeoff space sharing mbcr msmr construction space sharing mbcr space sharing mbcr normalized storage per node fig achievable points system normalized storage per node normalized bandwidth proposition increase fault tolerance system desirable factor acheivability region system subsection seek determine convex hull known achievable points system convex hull denoted smallest convex set example illustrate process extending containing known achievable points obtained convex system system initially repair group combinations among points achieved construction described also mbcr point size code blocks given given objective therefore determine points sufficient describe refer points corner points code matrix given figure presents achievable points system achievable points parameterized denote corresponding point decreases storage adding node system add another block increases abuse notation refer point whose symbols distributed across old point state guiding observations subsequent analysis first one eliminate nodes old blocks become achievable points obtained construction instance point achieves similar bandwidth neighbor point larger storage size points new node stores newly generated coded symbols right also old repair groups new code matrix immediately eliminated outperformed given mbcr point interior point interestingly observe point lies exactly segment joining point mbcr point means point outperformed spacec sharing nonetheless necessary description thus considered corner point following show observations figure property useful systems generalized explicitly determine corner points fault tolerance may deemed insufficient therefore one depending system parameters increase fault tolerance system without sacrificing optimality exact repair tradeoff changing lemma achievable points existing data note also successive application corner points proof seen seen function decreasing fractional function pole convex shown decreases increases monotonically therefore decreasing points interest increases moreover noticing follows points contribute acheivability region points outperformed point terms storage bandwidth lemma implies sufficient consider range define integer show achievable points eliminated considering mbcr point lemma achievability region points points corner points considering mbcr point proof virtue lemma points eliminated consider segment joining points slope segment denoted given slope strictly decreasing means three consecutive points point lies segment joining two extreme points therefore suboptimal analyze achievability region adjoining mbcr point points lemma mbcr point corner point along lemma proof noting concludes result lemma need analyze whether spacesharing mbcr point point may outperform achievable points lemma point outperformed point mbcr point points corner points achievability region proof assumption lemma implies slope segment joining points smaller slope segment lemma slope segment decreasing follows point outperformed across two achievable points including mbcr point therefore determine corner points need successively test increasing values whether point outperformed spacesharing mbcr point let denote smallest outperformed follows lemma following achievability region proposition achievability region given corner points mbcr given proof consider consider mbcr point point compute normalized bandwidth denoted achieved considered intermediate point determine whether using obtain simplification regard function fixed function fixed proof interested analyzing analyze later proof clearly thus sign sign therefore suffices study sign note implies point eliminated thus corner point quadratic function let denote discriminant checked thus exists leading coefficient positive follows one solution say negative solution follows implies set eliminated particular always eliminated let thus outperforms spacesharing follows thus proposition agrees known particular cases eliminated point coincides mbcr point agreement optimal point proposition corner point proposition indeed point lies exactly segment joining mbcr msmr point optimal point proposition corner point proposition characterizes exactly give insight particular point corner point focus analysis sign linear depending sign leading coefficient may exist integer kth kth space sharing enhances achievability region enhance kth point may corner point systems may corner point systems higher reconstruction parameter example follows systems point outperformed spacesharing systems point corner point next proposition addresses cases particular point corner point using similar argument example proposition consider achievable point fixed let pmax kth table specifies scenarios corner point kth kth pmax pmax maximum value satisfying given pmax thus point pmax corner point system pmax point corner point sign sign let solution linear equation simplification proof examine first note point corner point systems indeed follows fixed need determine sign pmax also implies kth equality iff checked point corner point integer necessarily integer follows kth using proposition corollary follows corollary system table summary cases corner point symbol means corner point symbol denotes case also expressed max pmax kth max number corner points given function levels kth pmax final value given pmax example consider setting figure obtain pmax means points corner points number corner points clearly matches observations made figure note rhs integer otherwise odd integer implying mod leads contradiction mod also implies slope ode construction section present another family codes improved upon encapsulates construction special case let denote generator matrix vectorization code every node corresponds set columns different construction allow hence may feed dependent symbols generator matrix let columns corresponding nodes let submatrix consisting maximum amount information stored system upper bounded instance checked generate dependent symbols add another layer inner code construction moreover information symbols assumed finite field appropriately chosen positive integer construction system similarly construction code construction parameterized assume pair let given first information symbols used construct linearized polynomial linearized polynomial evaluated elements obtain viewed vectors linearly independent finally evaluation points fed encoder construction repair repair nodes similar construction contribution helper given note elements construction defined alphabet size evaluation points defined difference resolved viewing vectors applying construction components similarly repair carried linearized polynomial evaluations instance codes proposition shown use codes guarantees reconstruction property regenerating code moreover shows symbols made independent fact rank metric codes may replaced linear codes long reconstruction property satisfied reduce field size furthermore note use codes needed code obtained simply code construction remark construction generalizes construction designed repairing single erasures moreover construction based mds codes rather msmr codes repair scheme based naive repair scheme mds codes finally codes set construction takes arbitrary values remark repair process construction take functional tradeoff space sharing msmr mbcr mbcr point normalized bandwidth per helper columns rank denoted independent choice nodes given min min normalized storage per node fig achievable points using construction system malized storage per node bandwidth coincides construction nornormalized blue curve account dependency introduced codes among intermediate symbols may possible reduce repair bandwidth leveraging dependency varying construction obtain various achievability points construction special case construction corresponding particular constructions coincide parameters simulation shows construction performs better closer msmr point construction performs better closer mbcr point figure plots achievable points construction system various values onclusion studied problem centralized exact repair multiple failures distributed storage first described construction achieves new set interior points particular proved optimality one point functional centralized repair tadeoff moreover considering minimum bandwidth cooperative repair codes centralized repair codes determined explicitly best achievable region obtained among known points system finally described another construction includes first construction special case generates various achievable points general system future work includes investigating outer bounds centralized exact repair problem eferences tian sasidharan aggarwal vaishampayan kumar layered regenerating codes via embedded error correction block designs ieee trans inf theory vol dimakis godfrey wainwright ramchandran network coding distributed storage systems ieee trans inf theory vol shah rashmi kumar ramchandran interference alignment regenerating codes distributed storage necessity code constructions ieee trans inf theory vol suh ramchandran mds code construction using interference alignment ieee trans inf theory vol rawat koyluoglu vishwanath progress msr codes enabling arbitrary number helper nodes information theory applications workshop ita goparaju fazeli vardy minimum storage regenerating codes parameters ieee trans inf theory vol oct cadambe jafar maleki ramchandran suh asymptotic interference alignment optimal repair mds codes distributed storage ieee trans inf theory vol may tamo wang bruck zigzag codes mds array codes optimal rebuilding ieee trans inf theory vol march barg explicit constructions mds array codes optimal repair bandwidth ieee trans inf theory vol april elyasi mohajer determinant coding novel framework regenerating codes ieee trans inf theory vol zorgui wang centralized repair minimum storage regenerating codes ieee international symposium information theory isit june sasidharan senthoor kumar improved outer bound tradeoff regenerating codes ieee international symposium information theory isit ieee duursma outer bounds exact repair codes arxiv preprint sasidharan prakash krishnan vajha senthoor kumar outer bounds bandwidth regenerating codes international journal information coding theory vol duursma shortened regenerating codes arxiv preprint kermarrec scouarnec straub repairing multiple failures coordinated adaptive regenerating codes international symposium network coding netcod ieee shum cooperative regenerating codes ieee trans inf theory vol nov rawat koyluoglu vishwanath centralized repair multiple node failures applications communication efficient secret sharing arxiv preprint zorgui wang centralized repair distributed storage annual allerton conference communication control computing allerton sept cooperative repair regenerating codes distributed storage infocom proceedings ieee ieee wang zhang exact cooperative regenerating codes distributed storage proceedings ieee infocom april kamath prakash lalitha kumar codes local regeneration erasure correction ieee trans inf theory vol rawat silberstein koyluoglu vishwanath optimal locally repairable codes local minimum storage regeneration via codes information theory applications workshop ita ieee tian note rate region regenerating codes arxiv preprint
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feb new topological methods solve equations groups anton klyachko andreas thom abstract show equation associated group word solved hyperlinear group content augmentation lie second term lower central series moreover finite solution found finite extension method proof extends techniques developed gerstenhaber rothaus uses computations homotopy theory cohomology compact lie groups contents introduction main results topology solvability equations acknowledgments references introduction paper solvability equations groups let start briefly recalling analogous situation polynomial equations rational coefficients even though every polynomial root always exists finite field extension solved exists indeed straightforward construct splitting field desired property using machinery commutative algebra side also arguments algebraic topology using notions degree winding number fundamental group used show every polynomial root topological field historically first way provide field extension solved argument essentially goes back gauss work analogous situation one wants solve equations coefficients group algebraic combinatorial approach fails large extend homotopy theoretic approach used obtain positive results see next section definitions precise statements particular consequence able prove conjecture locally residually finite groups first anton klyachko andreas thom want clarify relationship connes embedding problem kervairelaudenbach conjecture observed moreover want extend study cover larger class groups also larger class equations handled methods algebraic topology methods involve detailed study homotopy type simple lie group effect word maps cohomology ring mod coefficients first main result applies prime number group word augmentation lie second step central series equation solved implies second main result says equation augmentation solved hyperlinear group see section details moreover group finite solution found finite extension covers classes singular equations intractable combinatorial methods topological methods developed main results stated explicitly theorems section paper organized follows section collects various preliminaries discusses briefly setup group words equations classes hyperlinear sofic groups section recalls facts cohomology localization theory topological spaces computations homotopy groups spheres section technical part also contains review extension results kishimoto kono section contains proofs main results discusses related results directions main results group words equations denote free group generators group element free product determines word map given evaluation denote natural augmentation sends neutral element call content call group word variables coefficients every group word determines equation variables coefficients obvious way say solved exists overgroup denotes neutral element similarly say solved take denote normal subgroup generated element hhwii clear equation solved natural homomorphism injective similarly equation solved natural homomorphism left inverse study equations groups dates back work bernhard neumann extensive literature equations groups including see also roman kov recent survey topic equations coefficients solvable example equation variable solvable indeed become conjugate overgroup another example hai equation however cases indeed known examples equations solvable equations groups equations whose content trivial call equation singular content trivial otherwise lets put forward following conjecture conjecture let group equation variables coefficients solvable addition finite solution found finite extension case famous conjecture case studied classical work see showed finite every equation one variable solved fact finite extension proof used computations cohomology compact lie groups proof inspired start work work showed fact every equation one variable coefficients unitary group solved already strategy use homotopy theory say associated word map degree map oriented manifolds thus must surjective preimage neutral element provides solution equation key computation degree observe degree depends homotopy class thus since connected change replaced computation degree easy consequence classical computations hopf property solvability group easily seen pass arbitrary cartesian products groups arbitrary quotients groups consequence equations one variable coefficients solved isomorphic subgroup quotient infinite product observation due pestov groups admit embedding called hyperlinear groups see section information class groups see also remark thus result also holds hyperlinear groups particular amenable groups generally sofic groups connes embedding conjecture predicts among things every countable group hyperlinear thus implies conjecture also observed pestov actually studied involved question whether equations form variables solved simultaneously main result case finite generally locally residually finite presentation satisfies second homology integral coefficients vanishes representation associated presentation group obtained glueing bouquet circles according relations amounts certain algebraic condition exponent sum matrix system equations satisfies vanishing condition called jim howie note terminology consistent risk confusion later howie proved result locally indicable groups conjectured hold groups call howie conjecture connes embedding conjecture implies howie conjecture specifically every hyperlinear group satisfies howie conjecture remark equations one variable three occurrences variable solvable result howie however also reduces residually finite anton klyachko andreas thom case uses results similar results proved equations four five occurrences variables remark equations solved combinatorial methods systematic study case started levin conjectured equations one variable coefficients group always solvable conjugate element result direction due first author proved indeed case equations content see moreover group equation whose content proper power neutral element solvable due absence counterexamples conjectured solvability case true even content equation trivial remark existence solutions subtle issue see example hai equation solved finite overgroup even though solutions exist infinite extensions mechanism behind kind examples first discovered higman see statement main results main goal work provide examples equations many variables solvable every hyperlinear group condition equation depends content compared example results gersten conditions depended unreduced word obtained deleting coefficients simplicity concentrate case main result theorem let hyperlinear group equation two variables coefficients solved content lie moreover finite solution found finite extension order prove main result refine study effect word maps cohomology compact lie groups strategy show equations solved sufficiently many specifically prove theorem let prime number let group word equation solved theorem direct consequence work gerstenhaber rothaus however new idea needed show conditions mentioned induced word map surjective denotes special unitary group quotient center strategy replace much simpler homotopic map induced study effect cohomology directly done section necessary preparations section general assumption omitted previous theorem indeed second author showed previous work theorem every every neighborhood exists image lies particular equation solvable equations groups construction proves preceding theorem yields words lie deep derived series contradiction theorem surjectivity word maps without coefficients interesting subject michael larsen conjectured high enough associated word map surjective shown divisibility restrictions words second derived subgroup elkasapy second author similar direction believe high enough divisibility restrictions word map define homotopy class even homotopic map order study words lie deeper lower central series suspect might helpful oberserve induced word map lift simply connected cover lifts even higher connected covers indeed example one show associated word maps lifts complex analogue string group see study related groups topology section collect standard results algebraic topology used proof main theorems classical result samelson says theorem samelson see commutator map particular since sphere map homotopic commutator map must surjective easily implies every group word whose content commutator generators equation solved already consequences best knowledge could proved using combinatorial techniques order treat higher recall aspects algebraic topology methods proof main results follow closely ideas cohomology let positive integer denote special unitary group quotient center projective unitary group denote quotient map coset cohomology rings classical lie groups computed borel example graded ring denote exterior algebra field certain set generators particular degrees product map inversion turn hopf algebra however comultiplication turns trivial situation mainly interested case computation cohomology ring involved also first studied borel later comultiplication cohomology computed work turns lack makes approach work let sumarize situation anton klyachko andreas thom theorem see let odd prime number moreover comultiplication takes form denote kernel natural augmentation start recalling effect various natural word maps cohomology ring lemma let odd prime number consider map given map induced cohomology satisfies mod diagonal embedding proof map arises composition following multiplication map cohomology induces first coproduct followed multiplication cohomology easy verification shows generator multiplied modulo sums products least two generators proves claim also important study effect commutator map cohomology need following result work easy consequence theorem lemma proposition commutator map induces cohomology map sending modulo ideal elements map zero note commutator map induces map also call commutator map first aim show commutator map homotopic map show showing image cohomology class vanish group turns study images fairly straightforward since generators image generators cohomology study last generator considerably complicated rely structure results homotopy type fact final argument rely lemma required result proved directly equations groups lens spaces since natural given scalar multiplication complex number exp considerations study case denote lens space let natural projection space natural one cell dimension see example denote moore space definition space obtained attaching along attaching map defined note letter cause confusion characteristic property unless see chapter example details set denote pinch map collapses hence boundary point note indeed see example fibre bundle embedding sends matrix matrix projection sends matrix first row similarly group admits fibre bundle embedding sends matrix class matrices hexp projection sends matrix first row defined multiplication exp lens space localization prime freely use concept localization topological spaces simply connected abelian fundamental group prime see work background general references see also recent presentation material given topological space abelian fundamental group denote plocalization comes equipped natural map defined certain tower spaces defining properties denotes ring fractions whose denominator divisible continuous map topological spaces abelian fundamental group denote induced map freely use induces isomorphism cohomology coefficients map called homotopy equivalence homotopy equivalence double suspension abelian group denote usual set homotopy classes maps use addition also finite natural maps isomorphisms see example chapter need following computation homotopy groups due bott anton klyachko andreas thom theorem bott generator let consider map given use multiplication group following theorem first proved serre proposition even though without using language localization level topological spaces theorem serre let prime number map homotopy equivalence concentrate case denote composition homotopy inverse projection also following computation homotopy groups odd onto spheres also due serre proposition equal generator generator denotes usual suspension also level maps hence theorem together computation implies refined computation homotopy groups localized prime note covers dimensions whereas bott computation gives information dimension fact used later commutator map induces secondary operation samelson product originally introduced bott already analyzed samelson products maps proved corollary main result theorem element divisible precisely equations groups equal times generator maps induce natural maps note also choose natural map yields generator bott result extends case light computation bott computation samuelson products implies theorem bott let prime number element vanish proof indeed bott result yields image map times generator hence vanish modulo since factors assertion follows samuelson products modulo key understand certain cohomology classes application commutator map work order understand effect commutator map cohomology must study homotopy type restate proposition work lemma exists natural map diagram commutes homotopy using notation introduced section ready state prove extension lemma lemma proof first know proposition exists splitting thus map anton klyachko andreas thom decomposed since map covering get equal multiplication cofiber sequence coming definition induces long exact sequence computations homotopy groups spheres obtain moreover follows group generated map consider map obtain equation sentence equation consider map point going use theorem indeed following identification homotopy classes maps note equation pbi finally implies multiple class conclude finishes proof preceding proof followed closely work claim originality computation equations groups applications cohomology apply computation previous section study effect commutator map cohomology recall denote ideal ring generated corollary let commutator map induced map satisfies mod proof note natural map induces natural homomorphism sends generators zero kernel homomorphism precisely ideal lemma implies composition homotopic since map sends generator cohomology implies claim solvability equations hyperlinear groups related classes groups unitary group equipped natural metric arises normalized frobenius norm vij informally speaking group said hyperlinear multiplication table modelled locally means finite subsets group unitary matrices small mistakes measured normalized frobenius norm precisely definition group called hyperlinear finite subsets exists map anton klyachko andreas thom variations definition equivalent detailed discussion class hyperlinear groups found definition unitary groups metrics replaced symmetric groups sym normalized hamming metrics one obtains definition class sofic groups important class groups introduced gromov weiss order study certain problems ergodic theory since inclusion sym compatible enough metrics every sofic group automatically hyperlinear see details known groups denote set prime numbers need following easy proposition proposition let countable hyperlinear group subgroup quotient proof let enumeration let integer definition hyperlinearity satisfied finite set let corresponding map without loss generality may assume indeed natural diagonal embedding isometric respect normalized frobenius metric replace knk necessary using natural embedding may assume without loss generality image lies similarly replacing suitable number form mnk may assume prime number using dirichlet theorem consider lim unk easy see normal subgroup defines injective homomorphism quotient proves claim remark also true subgroup quotient matter hyperlinear follows results based ideas work see known groups hyperlinear essentially groups known hyperlinear also known sofic proofs main results prove main theorems theorem theorem let first study commutator map follows corollary ideal defined mod denotes subspace use natural tensor product easy see indeed assume equations groups element contributions must minimal possible degree note construction implies product factors form otherwise yield summand unless forbidden proves first step continuous map homotopic commutator map must surjective indeed previous computation shows effect fundamental class cohomology happens map map homotopic surjective attempt extend result words later also allow vary see approach works elements lie first need preparations note free group basis see proposition chapter proposition let satisfies mod independent generally mod side second derived subgroup proof word maps associated word map factored equal msu hence effect cohomology computed explicitly follows results follow directly lemma corollary finishes proof proposition consider heisenberg group sending matrix matrix get sent matrix well known coming back proof theorem anton klyachko andreas thom see succeed strategy mapped central quotient proof theorem result clear may assume assumption implies mapped nontrivially canonical central extension central quotient happens hence main result induced word map fundamental class hence word map must surjective since lift preimage neutral element solves equation proof theorem straightforward consequence theorem proposition claim finite groups follows mal cev theorem stating finitely generated linear groups residually finite related results cases let finish mentioning results beyond second step lower central series far unable exploit mechanisms behind examples order get satisfactory results hyperlinear groups however would also like mention directions possible extensions techniques used paper recall denote commutator iterated commutators defined induction first result goes beyond second step lower central series following result porter theorem porter see map order treat need use sophisticated results algebraic topology related homotopy nilpotence results done rao showing also spin homotopy nilpotent theorem rao map obtain results concerning solvability equations corollary let subgroup let solved group containing proof since every map thus using arguments conclude every content induced word map surjective finally want mention questions appear naturally interface homotopy theory study word maps given topological group seems natural study group words modulo let compact lie group set homotopically trivial define question compute equations groups see partial information particular cases direction following result implied results theorem whitehead see let connected simply connected compact lie group nilpotent dim proof denote degree nilpotency group nil whitehead showed homotopy set group nil dim obtain nil dim dim subgroup generated coordinate projections precisely proves claim let topological group compact lie group call homotopically surjective respect every map homotopy class surjective question let homotopically surjective large acknowledgments work first author supported russian foundation basic research project thanks dresden hospitality visit january thanks institution hospitality second author wants thank tilman bauer interesting comments part paper written visited institute henri trimester random walks asymptotic geometry groups spring research supported erc starting grant leipzig thank unknown referee many useful comments improved exposition references dmitrii baranov anton klyachko efficient adjunction square roots groups sibirsk mat russian russian summary english sib math paul baum william browder cohomology quotients classical groups topology armand borel sur cohomologie des espaces principaux des espaces groupes lie compacts ann math french sur homologie cohomologie des groupes lie compacts connexes amer math french raoul bott space loops lie group michigan math note samelson product classical groups comment math helv aldridge bousfield daniel kan homotopy limits completions localizations lecture notes mathematics vol berlin martin edjvet ayre equations groups algebra colloq nonsingular equations groups comm algebra martin edjvet james howie solution length four equations groups trans amer math soc abdelrhman elkasapy andreas thom method showing surjectivity word maps appear indiana univ math anton klyachko andreas thom anastasia evangelidou solution length five equations groups comm algebra steve gersten reducible diagrams equations groups essays group theory math sci res inst vol springer new york murray gerstenhaber oscar rothaus solution sets equations groups proc nat acad sci misha gromov endomorphisms symbolic algebraic varieties eur math soc jems hiroaki hamanaka daisuke kishimoto akira kono self homotopy groups large nilpotency classes topology appl allen hatcher algebraic topology cambridge university press cambridge graham higman finitely generated infinite simple group london math soc heinz hopf den rang geschlossener liescher gruppen comment math helv james howie pairs systems equations groups reine angew math solution length three equations groups proc edinburgh math soc sergey ivanov anton klyachko solving equations length six groups group theory ioan james emery thomas lie groups proc nat acad sci arye solvability equations groups comm algebra daisuke kishimoto akira kono conjecture topology appl anton klyachko funny property sphere equations groups comm algebra equations groups quasivarieties residual property free group group theory generalize known results equations groups mat zametki russian russian summary english math notes anton klyachko anton trofimov number equation group group theory frank levin solutions equations groups bull amer math soc anatoly malcev isomorphic matrix representations infinite groups rec math mat sbornik peter may kate ponto concise algebraic topology chicago lectures mathematics university chicago press chicago localization completion model categories mamoru mimura goro nishida hirosi toda localization applications math soc japan bernhard neumann adjunction elements groups london math soc nikolay nikolov dan segal generators commutators finite groups abstract quotients compact groups invent math vladimir pestov hyperlinear sofic groups brief guide bull symbolic logic gerald porter homotopical nilpotence proc amer math soc vidhyanath rao spin homotopy nilpotent topology roman kov equations groups groups complex cryptol oskar rothaus group extensions given generators relations ann math hans samelson groups spaces loops comment math helv serre groupes homotopie classes groupes ann math french equations groups trees springer monographs mathematics berlin translated french original john stillwell corrected printing english translation abel stolz andreas thom lattice normal subgroups ultraproducts compact simple groups proc london math soc stephan stolz peter teichner elliptic object topology geometry quantum field theory london math soc lecture note vol cambridge univ press cambridge andreas thom convergent sequences discrete groups canad math bull benjamin weiss sofic groups dynamical systems ser ergodic theory harmonic analysis mumbai george whitehead elements homotopy theory graduate texts mathematics vol springerverlag new york nobuaki yagita homotopy nilpotency simply connected lie groups bull london math soc moscow state university russia address klyachko institut geometrie dresden germany address
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biconnectivity applications dfs using bits sankardeep chakrabortya venkatesh ramana srinivasa rao sattib institute mathematical sciences hbni chennai india seoul national university seoul south korea jul abstract consider space efficient implementations classical applications dfs including problem testing biconnectivity connectivity finding cut vertices cut edges computing chain decomposition given undirected graph vertices edges classical algorithms typically use dfs information vertex building recent implementation dfs due elmasry stacs provide implementations applications dfs algorithms take lgc time small constant central implementation succinct representation dfs tree space efficient partitioning dfs tree connected subtrees maybe independent interest designing space efficient graph algorithms introduction space efficient algorithms becoming increasingly important owing applications presence rapid growth big data proliferation specialized handheld devices embedded systems limited supply memory even mobile devices embedded systems designed large supply memory might useful restrict number write operations example flash memory writing costly operation terms speed also reduces reliability longevity memory keeping constraints mind makes sense consider algorithms modify input use limited amount work space one computational model proposed algorithmic literature study space efficient algorithms memory rom model article focus space efficient implementations fundamental graph algorithms settings without paying much penalty time already rich history designing space efficient algorithms memory model complexity class also known dlogspace class containing decision problems solved deterministic turing machine using logarithmic amount work space computation several important algorithmic results class celebrated reingold method checking undirected graph determine path two given vertices analogue known problem directed graphs respect log space reductions using savitch algorithm problem solved time using bits savitch algorithm space efficient running time superpolynomial among deterministic algorithms running polynomial time directed space efficient algorithm due barnes gave slightly sublinear space using bits algorithm problem running polynomial time know better polynomial time algorithm problem better space bound moreover space used algorithm matches lower bound space solving directed restricted model computation called node naming jumping automata graphs nnjag model introduced especially study directed known sublinear space algorithms problem implemented thus design polynomial time rom algorithm taking space less bits requires significantly new ideas recently use denote logarithm base results announced preliminary form proceedings international symposium algorithms computation isaac lipics volume pages email addresses sankardeep sankardeep chakraborty vraman venkatesh raman ssrao srinivasa rao satti preprint submitted elsevier july improvement space bound special classes graphs like planar hminor free graphs fundamental graph theoretical problems work designing algorithms classical selection sorting problems problems computational geometry among others drawback however graph algorithms using small space sublinear bits running time often polynomial high degree example best knowledge exact running time reingold algorithm undirected connectivity analysed yet know admits large polynomial running time surprising tompa showed directed number bits available natural algorithmic approaches problem require time motivated impossibility results complexity theory inspired practical applications fundamental graph algorithms recently surge interest improving space complexity fundamental graph algorithms without paying much penalty running time reducing working space classical graph algorithms bits little penalty running time generally classical linear time graph algorithms take words equivalently bits space starting paper asano showed one implement dfs using bits improving naive implementation recent series papers presented algorithms basic graph problems namely bfs maximum cardinality search topological sort connected components minimum spanning tree shortest path recognition outerplanar graph chordal graphs among others add small yet growing body algorithm design literature providing algorithms classical graph problems solved using dfs namely problem testing biconnectivity connectivity finding cut vertices cut edges computing chain decomposition among others model computation standard area graph algorithms assume input graph given memory modified algorithm must outputting done separate memory something written memory information read rewritten input read output write addition input output media limited workspace available data workspace manipulated word level standard word ram model machine consists words size bits logical arithmetic bitwise operations involving constant number words take constant amount time count space terms number bits workspace used algorithms historically model called register input model introduced frederickson studying problems related sorting selection assume input graphs represented using adjacency array represented array length entry stores pointer array stores neighbors vertex directed graphs assume input representation adjacency array vertices directed graphs every vertex access two arrays one array array representation become somewhat standard also used recently design various space efficient graph algorithms use denote number vertices number edges respectively input graph throughout paper assume input graph connected graph hence results organization paper asano showed depth first search dfs directed undirected graph performed time bits space elmasry improved time still using bits space build upon results give space efficient implementations several classical applications dfs first warm start simple applications space efficient dfs show following time bits space algorithm compute strongly connected components directed graph section addition also give algorithm output vertices directed acyclic graph topologically sorted order section algorithm find sparse edges spanning biconnected subgraph undirected biconnected graph section using asymptotically time space used dfs using bits time develop fast space efficient algorithms graph problems also applications dfs section develop describe detail space efficient tree covering technique use subsequent sections technique roughly speaking partitions dfs tree connected smaller sized subtrees stored using less space finally solve corresponding graph problem smaller sized subtrees merge solutions across subtrees get overall solution done using less space paying much penalty running time ideas borrowed succinct tree representation literature first application consider section space efficient implementation chain decomposition undirected graph important preprocessing routine algorithm find cut vertices biconnected components cut edges also test among others provide algorithm takes time using bits space improving previous implementations took bits bits space section give improved space efficient algorithms testing whether given undirected graph biconnected biconnected also show one find cut vertices provide space efficient implementation tarjan classical lowpoint algorithm algorithms take time bits space section provide space efficient implementation testing connectivity given undirected graph producing cut edges using time bits space given biconnected graph two distinguished vertices numbering vertices graph gets smallest number gets largest every vertex adjacent lowernumbered vertex finding important preprocessing routine planarity testing algorithm among others section give algorithm determine biconnected graph takes time using bits improves earlier implementations take bits using subroutine section provide improved space effcient implementation two independent spanning tree problem among others direct readers section provide necessary definitions related models several models computation come close model model focus paper comes design graph algorithms single thread common access input tape restricted way streaming model input kept media algorithm tries optimize number passes makes input model elements edges input graph revealed one one extra space allowed algorithm bits observe possible store whole graph dense efficiency algorithm model measured space uses time requires process edge number passes makes stream model one allowed constant number additional variables possible rearrange sometimes even modify input values chan introduced restore model relaxed version memory restricted version model input allowed modified end computation input restored original form motivation example scenarios input original form required application buhrman introduced studied model small amount typically bits clean space provided along additional auxiliary space condition additional space initially arbitrary possibly incompressible state must returned state computation finished input assumed given rom also provided implementations graph algorithms space efficiently preliminaries section list preliminary results graph theoretic definitions used later algorithms develop also discuss briefly high level main technique goes behind almost algorithms paper graph theoretic terminology collect necessary graph theoretic definitions used throughout paper cut vertex undirected graph vertex removed along incident edges graph creates components previously graph connected graph least three vertices biconnected also called graph literature sometimes cut vertex biconnected component maximal biconnected subgraph components attached cut vertices similarly undirected graph bridge cut edge edge removed without removing vertices graph creates components previously graph connected graph least two vertices also called bridgeless sometimes bridge connected component maximal connected subgraph given biconnected graph two distinguished vertices numbering vertices graph gets smallest number gets largest every vertex adjacent vertex numbering vertices called vertices exist biconnected every edge problem given vertices undirected graph natural numbers want find partition sets every every set induces connected graph given graph call set rooted spanning trees independent root vertex every vertex paths spanning trees except endpoints directed graph said strongly connected every pair vertices reachable strongly connected possible decompose strongly connected components maximal set vertices every pair vertices reachable topological sort topological ordering directed acyclic graph linear ordering vertices every directed edge vertex vertex comes ordering let search tree connected undirected directed graph vertex preorder number number vertices visited including preorder traversal similarly postorder number number vertices visited including postorder traversal tree cover space efficient construction implement algorithms bits main idea process nodes dfs tree batches nodes afford store trees size explicitly labels first use algorithm used succinct representations trees partition tree connected subtrees size solve problem dealing smaller subtrees later merge specific order obtain overall solution cases obtain overall solution need generate pairs subtrees explicit node labels process edges specific order describe details tree cover approach section describe algorithms section use following fundamental data structure bitstrings algorithms given bitvector length rank select operations defined follows ranka number occurrences selecta position occurrence following theorem gives efficient structure support operations theorem given bitstring length one construct auxiliary structure support rank select operations time also structure constructed given bitstring time related work dfs recall dfs starts exploring given input graph vertex initially white meaning unexplored becomes gray dfs discovers first time pushed stack colored black finished adjacency list checked completely leaves stack recently elmasry showed following tradeoff result dfs theorem every function computed within resource bound theorem time using bits vertices directed undirected graph visited bits depth first order time lgt particular fixing one obtain dfs implementation runs time using bits build top dfs algorithm provide space efficient implementation various applications dfs directed undirected graphs rest paper simple applications dfs using bits classical applications dfs directed graphs see find strongly connected components directed graph topological sort directed acyclic graph among many others also given undirected biconnected graph dfs used main tool produce sparse spanning biconnected subgraph show topological sort producing sparse spanning biconnected subgraph undirected biconnected graph solved using bits time dfs strongly connected components directed graph obtained using bits time strongly connected components classical two pass algorithm see computing strongly connected components scc given directed graph works follows first step runs dfs reverse graph second pass runs connected component algorithm using dfs processes vertices decreasing order finishing time first pass obtain switching role adjacency arrays present input representation remember vertex ordering first pass due space restriction process batches size reverse order run full dfs obtain store last vertices array ones highest set finishing numbers decreasing order maintain queue size new element finished added queue element earliest finish time end queue deleted pick vertices one one order queue latest finish time start fresh dfs compute connected components output vertices reachable scc output vertices marked bitmap output done vertices restart dfs beginning produce next chunk vertices remembering last vertex produced previous step stop soon hit boundary vertex repeat connected component algorithm chunk vertices continue way clear algorithm produces sccs correctly calling dfs algorithm times total time taken algorithm bits space hence following theorem given directed graph vertices edges represented adjacency array output strongly connected components time bits space topological sort standard algorithm computing topological sort outputs vertices dfs reverse order keep track dfs numbers reversing easy task working space restricted setting bits challenge space keep track dfs order strongly connected components algorithm last section storing outputting vertices batches resulting time algorithm elmasry showed vertices dag output order topological sort within time space bounds dfs plus additional bits also showed perform dfs time bits overall algorithm takes time bits compute topological sorting main idea maintain enough information dfs resume middle apply repeatedly reverse small chunks output produced reverse order one one observe instead storing information restart dfs produce reverse order simply work reverse graph obtained input representation switching role results section announced preliminary form proceedings international computing combinatorics conference cocoon springer lncs volume pages adjacency arrays dfs reverse graph output vertices finished blackened increasing order finishing time see correctness procedure note reverse graph also dag edge dag edge reverse graph become black algorithm performs dfs reverse graph hence placed correct topological sorted order thus following theorem given dag vertices edges black vertices dfs output using space time vertices output topologically sorted order using space time assuming input representation adjacency array graph theorem setting theorem following corollary given dag vertices edges vertices output topologically sorted order using time bits note knew along dfs topological sort take time main contribution theorem shows take space improving result showed topological sort space dfs space bits time adjacency arrays present input topological sort sublinear space note following theorem asano theorem dfs dag performed space bits polynomial time immediately follow theorem topological sort also performed using sublinear bits space one caveat asano algorithm works assuming given dag single source vertex particular determine whether vertex black checking whether reachable source without using gray vertices using sublinear space reachability algorithm algorithm easily extended handle many sources additional log bits simply keep track indices sources dfs explored determine whether vertex black ask reachable earlier source current source without using gray vertices thus following improved theorem theorem dfs dag sources performed using bits polynomial time particular overall space used bits thus theorem theorem obtain following theorem topological sort dag sinks performed using bits polynomial time particular overall space used bits finding sparse biconnected subgraph biconnected graph problem finding spanning subgraph minimum number edges graph known complexity problem decreases drastically want produce sparse spanning subgraph one edges nagamochi ibaraki gave linear time algorithm produces spanning subgraph edges later cheriyan gave another linear time algorithm produced spanning subgraph edges subgraph edges later elmasry gave alternate linear time algorithm producing sparse spanning biconnected subgraph given biconnected graph performing dfs additional bookkeeping follows provide space efficient implementation order start briefly describing elmasry algorithm let dfi denote index integer represents time vertex first discovered vertex performing dfs parent dfs tree let low smallest dfi value among dfi values vertices back edge note quantity different lowpoint value used tarjan classical biconnectivity algorithm basically low captures information regarding deepest back edge going vertex backedges convenience reason become clear following lemma adopt convention low dfi parent edge low deepest backedge note actually tree edge parent backedge algorithm maintains edges dfs tree addition every vertex graph algorithm maintains dfi low values along back edge realizes root dfs tree back edge underlying graph root one child back edge emanating well thus get back edges along tree edges giving subgraph edges elmasry proved resulting graph indeed spanning subgraph algorithm takes time bits space improve space bound albeit slight increase time first proving general lemma following figure part full dfs tree wiggling edges represent tree edges edges arrow heads represent back edges low would come across adjacency array encountering arrays back edge processed back edges since process vertices backedges incident dfs order lemma given undirected graph vertices edges compute report low values deepest back edge going every vertex using bits space time proof aim output deepest back edges every vertex perform dfs always let vertices graph perform dfs usual color array relevant data structures required theorem along one array bits call dbe deepest back edge array initialized zero dbe set algorithm found output deepest back edge emanating vertex whenever white vertex becomes gray visited first time scan adjacency array mark every white neighbor dbe correctness step follows fact visiting vertices dfs order dbe vertex adjacent vertices visited far adjacent deepest back edge emanating hence output edge move next neighbor eventually next step dfs vertices exhausted completes description algorithm see figure illustration see procedure produces deepest back edges every vertex note vertex algorithm reports back edges deepest back edge also tree edges back edge observe convention second case deepest back edge concludes proof lemma performed one dfs produce edges using theorem claimed running time space bounds follow way actually use lemma algorithms finding storing low values vertices state corollary corollary given undirected graph vertices edges set vertices input compute report store low values every vertex dfs tree using bits space time note lemma holds true undirected connected graph follows use lemma give space efficient implementation elmasry algorithm input graph undirected biconnected graph particular show following theorem given undirected biconnected graph vertices edges output edges sparse spanning biconnected subgraph using bits space time proof underlying graph undirected biconnected graph know elmasry algorithm produces sparse spanning subgraph also biconnected order implement given undirected biconnected graph first run algorithm lemma produces reports deepest back edges vertices deepest back edges note actually tree edges convention hence want report multiple time specifically vertex back edge going lemma outputs edge deepest back edge actually tree edge dfs tree order avoid reporting edges perform following scanning adjacency array also check neighbor parent gray report edge parent note back edge one ancestors parent step reports tree edge parent otherwise back edge hence tree edge parent would output dfs exploring outputting deepest back edges parent output edge note test along algorithm lemma using one dfs produce tree edges deepest back edges required elmasry algorithm thus using theorem output edges sparse spanning biconnected subgraph using bits space time tree cover space efficient construction moving handle complex applications dfs undirected graphs namely biconnectivity connectivity etc section discuss common methodology attack problems set machinary section see afterwards use almost similar fashion several problems central algorithms following section decomposition dfs tree use tree covering technique first proposed geary context succinct representation rooted ordered trees high level idea decompose tree subtrees called minitrees decompose yet smaller subtrees called microtrees microtrees tiny enough stored compact table root minitree shared several minitrees represent tree represent connections links subtrees later extended approach produce representation supports several additional operations farzan munro modified tree covering algorithm minitree one node root minitree connected root another minitree simplifies representation tree guarantees minitree exists one node connected root another minitree tree decomposition method farzan munro summarized following theorem theorem parameter rooted ordered tree nodes decomposed minitrees size pairwise disjoint aside minitree roots furthermore aside edges stemming minitree root one edge leaving node minitree child another minitree decomposition performed linear time see figure illustration algorithms apply theorem parameter since number minitrees represent structure minitrees within original tree minitrees connected using bits decomposition algorithm ensures minitree one child minitree minitrees share root structure use property crucially algorithms refer see figure minitree structure tree decomposition shown figure explicitly storing minitrees using pointers requires bits overall one way represent efficiently using bits store using encoding tree store minitrees separately loose ability compute preorder postorder numbers nodes entire tree needed algorithms hence encode entire tree structure using encoding store pointers encoding represent minitrees first encode tree using balanced parenthesis representation summarized following theorem given rooted ordered tree nodes represented sequence balanced parentheses length given preorder postorder number node support subtree size various navigational queries parent child time using additional bits representation support computing child node constant time one using representations produce tree cover representation sufficient need compute next child traverse tree computing subtree sizes subtree figure illustration tree covering technique figure reproduced closed region formed dotted lines represents minitree note minitree one child minitree minitrees share root structure following lemma farzan lemma restated shows minitree split constant number consecutive chunks sequence represent minitree storing pointers set chunks representation together constitute minitree lemma sequence tree bits corresponding form set constant number substrings furthermore substrings concatenated together order form sequence hence one store representation minitrees storing structure stores pointers starting positions chunks corresponding minitree sequence refer representation obtained using tree covering approach representation tree see figure complete example minitree structure along sequence tree figure following lemma shows construct representation dfs tree given graph using additional bits lemma given graph vertices edges algorithm takes time bits perform dfs one create representation dfs tree time using bits proof first construct balanced parenthesis representation dfs tree follows start empty sequence append parentheses perform step dfs algorithm particular whenever dfs visits vertex first time append open parenthesis similarly dfs backtracks append closing parenthesis end dfs algorithm every vertex assigned pair parenthesis length bits need run dfs algorithm construct array hence running time algorithm asymptotically running time dfs algorithm construct auxiliary structures support various navigational operations dfs tree using sequence mentioned theorem takes time space using algorithm use sequence along auxiliary structures navigate dfs tree postorder simulate tree decomposition algorithm farzan munro constructing representation dfs tree reconstruct entire tree pointers intermediate space would bits instead observe tree decomposition algorithm never needs keep temporary components see details addition permanent components component permanent temporary stored storing root component together subtree size since number figure minitree structure tree decomposition shown figure array encodes entire dfs tree using balanced parenthesis representation array demonstrate minitrees split constant number consecutive chunks representation note bottom array actually encoded using bits storing minitrees pointers chunks sequence indicating starting ending positions chunks corresponding minitrees permanent components space required store permanent temporary components point time bounded bits construction algorithm takes time use following lemma description algorithms later sections lemma let graph dfs tree algorithm takes time bits perform dfs using bits one reconstruct minitree given ranges sequence representation along labels corresponding nodes graph time proof first perform dfs construct representation dfs tree construct representation described lemma perform dfs algorithm keeping track preorder number current node step whenever visit new node check preorder number see falls within ranges minitree want reconstruct note mentioned lemma set preorder number nodes belong minitree form constant number ranges since nodes belong constant number chunks sequence within one ranges corresponding minitree constructed add node along label minitree applications dfs using technique section provide bit implementations various algorithmic graph problems use dfs using tree covering technique developed previous section higher level use tree covering technique generate minitrees one one partially solve corresponding graph problem inside minitree finally combining solution across minitrees problems consider include algorithms test biconnectivity connectivity output cut vertices edges find chain decomposition among others test biconnectivity related problems classical algorithm due tarjan computes values defined terms dfstree every vertex checks conditions based values brandes gabow gave considerably simpler algorithms testing biconnectivity computing biconnected components using rules instead call algorithms algorithm due schmidt based chain decomposition graphs determine biconnectivity connected algorithms take time words space roughly approaches compute dfs process dfs tree specific order maintaining auxiliary information nodes start brief description chain decomposition application first providing space efficient implementation chain decomposition schmidt introduced decomposition input graph partitions edge set graph cycles paths called chains used design algorithm find cut vertices biconnected components also test among others follows discuss briefly decomposition algorithm state main result algorithm first performs depth first search let root dfs tree dfs assigns index every vertex namely time vertex discovered first time dfs call dfi imagine back edges directed away tree edges directed towards algorithm decomposes graph set paths cycles called chains follows see figure example first mark vertices unvisited visit every vertex starting increasing order dfi following every back edge originates traverse directed cycle path begins back edge proceeds along tree towards root stops first visited vertex root step mark every encountered vertex visited forms first chain proceed next back edge move towards next vertex increasing dfi order continue process let collection cycles paths notice cardinality set exactly number back edges dfs tree back edge contributes cycle path also initially every vertex unvisited first chain would cycle would end starting vertex using schmidt proved following theorem theorem let chain decomposition connected graph chains partition also denotes minimum degree cycle set first chain decomposition edge bridge contained chain vertex cut vertex first vertex cycle ready describe implementation schmidt chain decomposition algorithm using bits space time using partition dfs tree section following description processing back edge refers step outputting chain directed path cycle containing edge marking encountered vertices visited processing node refers processing back edges node main idea implementation process back edges node preorder schmidt algorithm perform efficiently within space limit bits process nodes chunks size first chunk nodes preorder processed followed next chunk nodes processing back edges chunk process back edges minitrees postorder processing edges minitree processing back edges going different minitree requires edges chunk times minitree thus order process back edges different order process schmidt algorithm argue affect correctness algorithm particular observe following schmidt algorithm correctly produces chain decomposition even process vertices order long process vertex ancestors also processed example level order instead preorder also implies long process back edges coming vertex descendants process back edges going ancestors descendants produce chain decomposition correctly figure illustration chain decomposition input graph dfs traversal resulting along dfis chain decomposition chains paths rest cycles edge bridge contained chain cut vertices process back edge chunk minitree belongs belongs anscestor first output edge traverse path root outputting traversed edges marking nodes visited start another dfs produce minitree containing parent root output path root continue process untill reach vertex marked visited note process terminate since marked ancestor maintain bitvector length keep track marked vertices perform efficiently crucial observation use bounding runtime produce minitree particular pair need produce root marked first time output part chain also generate chunk minitree vertices preorder process edges provide pseudocode see algorithm describing algorithm outputting chain decomposition algorithm chain decomposition let minitrees postorder clg chunks vertices preorder back edges output chain containing edge end end end time taken initial part construct dfs tree decompose minitrees construct auxiliary structures using theorem running time rest algorithm dominated cost processing back edges outlined algorithm process back edges every pair chunk nodes preorder minitree postorder outer loop algorithm generates chunk preorder thus requires signle dfs produce chunks entire execution algorithm inner loop goes minitrees chunk since chunks minitrees prodicing minitree takes time generation pairs takes time particular pair may need generate many minitrees observe happens one back edge every pair since processing first back edge root minitree marked hence chain output afterwards stop root minitree also minitree generated processing pair generated processing pair different since minitree one child minitree thus overall running time dominated generating pairs thus obtain following theorem given undirected graph vertices edges output chain decomposition time using bits testing biconnectivity finding cut vertices algorithm test biconnectivity graph check connected using bits time bfs algorithm checking connectivity gives simple bits algorithm running time another approach use theorem combining criteria mentioned theorem test biconnectivity output cut vertices time using bits show using bits design even faster algorithm running time biconnected also show one find within time space bounds implement classical algorithm tarjan recall algorithm performs dfs computes every vertex value lowpoint recursively defined lowpoint min dfi lowpoint child dfi tarjan proved vertex root cut vertex child lowpoint dfi root dfs tree cut vertex root one child since lowpoint values require bits worst case poses challenge efficiently testing condition biconnectivity bits deal case chain decomposition algorithm compute lowpoint values batches using tree covering algorithm cut vertices encountered process stored separate bitmap show batch processed time using dfs resulting overall runtime computing lowpoint reporting cut vertices first obtain representation dfs tree using decomposition algorithm theorem process minitree postorder minitrees minitree structure process minitree compute lowpoint values nodes minitree except possibly root overall time processing minitree determine vertex cut vertex store information marking corresponding node seperate bit vector minitree reconstructed time using lemma lowpoint value node function lowpoints children however root minitree may children minitress hence root minitree store partial lowpoint value till point used update value subtrees computed lowpoint values possibly minitrees computing lowpoint values minitrees done two step process first step compute store low values node dfi value deepest back edge emanating node belonging minitree using corollary note low values form one component values among find minimum definition lowpoint slight change vertex backedge low nothing min dfi back edge however backedge convention low dfi value parent needs discounted computing lowpoint value easily done also remember dfi value parent every node minitree using bits low values computed stored vertices belonging minitree passed next step computing lowpoint values specifically second step another dfs starting root minitree compute lowpoint values vertices belonging minitree exactly way done classical tarjan algorithm using explicitly stored low values provide code snippet actually shows compute lowpoint values recursively minitree algorithm thus obtain following lemma computing storing lowpoint values nodes minitree performed time using bits algorithm dfs low dfi parent lowpoint dfi else lowpoint min dfi low white dfi dfi dfs lowpoint lowpoint lowpoint lowpoint end end end compute effect roots minitrees lowpoint computation store various bit information minitree roots including lowpoint values rank child subtree process one minitree generate next minitree postorder process similar fashion continue exhaust minitrees store cut vertices bitvector size marking vertex cut vertex reporting end execution algorithm routine task clearly taken bits space total running time run dfs algorithm times overall thus following theorem given undirected graph vertices edges time bits space determine whether connected amount time space compute report cut vertices testing connectivity finding bridges classical algorithm tarjan check connected takes time using words schmidt algorithm based chain decomposition also implemented linear time words purpose section improve space bound bits albeit slightly increased running time use following folklore characterization tree edge parent bridge lowpoint dfi say tree edge bridge vertex descedants dfs tree reach vertex ancestors thus edge removed graph becomes disconnected note since storing lowpoint values requires bits store check criteria mentioned characterization poses challenge efficiently testing condition connectivity bits perform test space efficient manner extend ideas similar ones developed previous section similar biconnectivity algorithm also first construct representation dfs tree using decomposition algorithm theorem process minitree postorder minitrees minitree structure process minitree compute lowpoint values nodes minitree except possibly root overall time processing minitrees come across bridge store separate bitvector end execution algorithm report using lemma know minitree reconstructed time also store root partially computed lowpoint till point done processing minitrees compute lowpoint values vertices belonging minitree using lemma determine lowpoint values vertices belonging minitree generate minitree along node labels easily test whether tree edge bridge using characterization mentioned also need check condition edges connect two minitrees also done within time space bounds store information using bit vector length edge dfs tree bridge thus running another dfs report bridges clearly procedure takes bits space total running time run dfs algorithm times overall hence obtain following theorem given undirected graph vertices edges time bits space determine whether connected connected amount time space compute output bridges vertices undirected graph fundamental tool many graph algorithms planarity testing graph drawing first algorithm vertices biconnected graph due even tarjan simplified ebert tarjan brandes algorithms however preprocess graph using search essentially compute lowpoints turn determine implicit open ear decomposition second traversal required compute actual stordering algorithms take bits space give bits implementation tarjan algorithm first describe two pass classical algorithm tarjan without worrying space requirement algorithm assumes without loss generality exists edge vertices otherwise adds edge starting algorithm moreover algorithm starts dfs vertex edge first edge traversed dfs let parent vertex dfs tree dfi lowpoint usual meaning defined previously first pass depth first search every vertex dfi lowpoint computed stored second pass constructs list initialized vertices numbered order occur obtain addition also sign array bits intialized sign second pass preorder traversal starting root dfs tree works described following pseudocode algorithm algorithm dfs starts edge vertices preorder dfs sign lowpoint insert sign end sign lowpoint insert sign end end clear pseudocode procedure runs linear time using bits space storing elements make space effcient use ideas similar biconnectivity algorithm high level generate lowpoint values first vertices depth first order process due space restriction store list tarjan algorithm instead use sequence dfs tree augment extra information encode final described earlier algorithms algorithm also runs phases phase processes vertices first phase obtain lowpoint values first vertices depth first order run biconnectivity algorithm procedure store explicitly vertices lowpoint values array also execution biconnectivity algorithm sequence generated stored array create two arrays size bits one one correspondence open parentheses sequence use operations defined section map position vertex two arrays corresponding open parenthesis sequence first array called sign storing sign every vertex tarjan algorithm simulate effect list create second array called store relative position every vertex respect parent namely parent comes respectively list algorithm store respectively one crucial observation even though list dynamic relative position vertex change respect position determined time insertion list new vertices may added later follows show use sequence array emulate effect list produce first describe reconstruct list using sequence array main observation use reconstruction node appears nodes child subtrees whose roots marked also nodes child subtrees whose roots marked also nodes subtree appear together consecutively list moreover children marked appear increasing order dfi children marked appear decreasing order dfi thus looking values children node computing subtree sizes determine position among nodes subtree let call child marked similarly marked called let denote subtree rooted vertex dfs tree denotes size let also suppose vertex children children remaining children dfi dfi dfi dfi dfi dfi vertices followed till appear followed finally vertices till appear specifically appears location approach reconstruct list hence output nodes linear time stored memory requires bits perform step bits repeat whole process reconstruction times iteration reproduce sublist ignoring node falls outside range reporting nodes range reconstruction takes time obtain following theorem given undirected biconnected graph vertices edges two distinct vertices output vertices time using bits space applications section show using space efficient implementation theorem immediately obtain similar results applications provide details problem problem given vertices graph natural numbers want find partition sets every every set induces connected graph problem called problem problem even bipartite condition relaxed lovasz proved partition always exists input graph found polynomial time graphs let problem solved following manner let compute note property vertex particular graphs induced always connected subgraph thus applying theorem obtain following theorem given undirected biconnected graph two distinct vertices two natural numbers obtain partition vertex set time using bits space induce connected subgraph problem wada kawaguchi defined following problem call problem actually extension problem defined section problem defined follows input undirected graph vertices edges vertex subset distinct vertices natural numbers output partition vertex set partition vertex set induces connected subgraph note problem extension problem since choosing corresponds original problem wada kawaguchi proved problem always admits solution input graph particular using stordering problem solved following manner suppose inputs let compute note partitioned two sets property know induce connected subgraph moreover using theorem subroutine compute obtain following result theorem given undirected biconnected graph solve problem time using bits space two independent spanning trees recall spanning trees graph independent root vertex every vertex paths spanning trees except endpoints itai rodeh conjectured every graph contains independent spanning trees even though general version conjecture proved yet conjecture shown true also planar graphs particular given graph biconnected generate two independent spanning trees let call following manner choose arbitrary edge say let construct choose every vertex edge choose edge construct choose edge every vertex edge easy prove root independent spanning trees every vertex path root consists vertices except edge whereas along edge consists vertices using theorem compute hard produce thus obtain following theorem given undirected biconnected graph report two independent spanning trees time using bits concluding remarks open problems presented space efficient algorithms number important applications dfs obtaining linear time algorithms maintaining bits space usage interesting challenging open problem one main bottlenecks approach towards finding algorithm dfs also open even though bfs know implementations another challenging open problem remove terms running times algorithms described term running time connectivity algorithm term running time two independent spanning trees algorithm terms seem inherent tree covering approach would interesting find applications tree covering approach space efficient algorithms also plenty applications dfs would interesting study point view space efficiency example planarity one prime example dfs used crucially natural question test planarity given graph using bits references references arora barak computational complexity modern approach cambridge university press asano buchin buchin mulzer rote schulz reprint algorithms simple polygons comput asano izumi kiyomi konagaya ono otachi schweitzer tarui uehara search using bits isaac pages asano kirkpatrick nakagawa watanabe algorithm planar directed graph reachability mfcs lncs pages banerjee chakraborty raman improved space efficient algorithms bfs dfs applications cocoon volume pages springer lncs banerjee chakraborty raman roy saurabh tradeoffs dynamic programming trees bounded treewidth graphs cocoon volume pages springer lncs barba korman langerman sadakane silveira algorithms algorithmica barba korman langerman silveira computing visibility polygon using variables comput barnes buss ruzzo schieber sublinear space polynomial time algorithm directed connectivity siam beame general sequential tradeoff finding unique elements siam brandes eager esa pages chan algorithms computing convex hull simple polygonal line linear time comput buhrman cleve loff speelman computing full memory catalytic space symposium theory computing stoc new york usa may june pages chakraborty pavan tewari vinodchandran yang new upperbounds directed reachability graphs fsttcs pages yeh balanced parentheses strike back acm trans algorithms july chakraborty satti improved linear time algorithms classical graph problems ctw chakraborty raman satti biconnectivity chain decomposition using bits isaac volume lipics pages schloss dagstuhl fuer informatik chakraborty satti algorithms maximum cardinality search stack bfs queue bfs applications cocoon chakraborty raman satti biconnectivity chain decomposition using bits isaac volume lipics pages schloss dagstuhl fuer informatik chan munro raman selection sorting restore model pages cheriyan kao thurimella search sparse certificates improved parallel algorithms connectivity siam cheriyan maheshwari finding nonseparating induced cycles independent spanning trees graphs algorithms clark compact pat trees phd thesis university waterloo canada cook rackoff space lower bounds maze threadability restricted machines siam cormen leiserson rivest stein introduction algorithms mit press curran lee finding four independent trees siam darwish elmasry optimal tradeoff problem esa pages dasgupta papadimitriou vazirani algorithms datta limaye nimbhorkar thierauf wagner planar graph isomorphism ccc pages dyer frieze complexity partitioning graphs connected subgraphs discrete applied mathematics ebert vertices biconnected graphs computing edmonds poon achlioptas tight lower bounds nnjag model siam elberfeld jakoby tantau logspace versions theorems bodlaender courcelle focs pages elberfeld kawarabayashi embedding canonizing graphs bounded genus logspace symposium theory computing stoc new york usa may june pages elberfeld schweitzer canonizing graphs bounded tree width logspace symposium theoretical aspects computer science stacs february france pages elmasry search efficiently identifies two graphs isaac pages elmasry hagerup kammer basic graph algorithms stacs pages even tarjan computing theo comp even tarjan corrigendum computing tcs theor comput farzan ian munro succinct representation dynamic trees theor comput farzan raman rao universal succinct representations trees icalp pages feigenbaum kannan mcgregor suri zhang graph problems model theor comput frederickson upper bounds sorting selection comput syst gabow search strong biconnected components inf process garey johnson computers intractability guide theory freeman geary rahman raman raman simple optimal representation balanced parentheses theor comput geary raman raman succinct ordinal trees queries acm trans algorithms partition conditions graphs combinatorica hagerup kammer succinct choice dictionaries corr munro satti succinct ordinal trees based tree covering acm trans algorithms huck independent trees planar graphs graphs combinatorics itai rodeh approach reliability distributed networks inf kammer kratsch laudahn biconnected components recognition outerplanar graphs mfcs catalytic computation bulletin eatcs lovasz homology theory spanning tress graph acta mathematica hungarica munro tables fsttcs pages munro paterson selection sorting limited storage theor comput munro raman selection memory sorting minimum data movement theor comput munro raman succinct representation balanced parentheses static trees siam munro raman rao space efficient suffix trees algorithms nagamochi ibaraki algorithm finding sparse spanning subgraph graph algorithmica reingold undirected connectivity acm schmidt structure constructions graphs phd thesis free university berlin schmidt simple test inf process schmidt mondshein sequence icalp pages tarjan search linear graph algorithms siam tarjan note finding bridges graph inf process tarjan two streamlined search algorithms fund tompa two familiar transitive closure algorithms admit polynomial time sublinear space implementations siam wada kawaguchi efficient algorithms tripartitioning triconnected graphs graphs pages zehavi itai three journal graph theory
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let dance learning online dance videos daniel castro georgia institute technology steven hickson patsorn sangkloy shickson patsorn sangkloy jan bhavishya mittal sean dai james hays irfan essa sdai hays irfan abstract recent years deep neural network approaches naturally extended video domain simplest case aggregating classifications baseline action recognition majority work area extends imaging domain leading heavy approaches temporal data address issue introduce let dance video dataset growing comprised visually overlapping dance categories require motion classification stress important human motion key distinguisher work given show work visual information sufficient classify categories compare datasets performance using imaging techniques demonstrate inherent difficulty present comparison numerous techniques dataset using three different representations video optical flow pose data order analyze approaches discuss motion parameterization value learning categorize online dance videos lastly release dataset three representations research community use figure row contains frames class represents figure best viewed digitally applied sequences datasets lacking highly dynamic videos goal determine methods best represent motion opposed methods use single properly picked frame identify activity feel approaches devalue necessity video data work introduce video dataset evaluate methods focuses highly dynamic videos requiring motion analysis classification choose domain dance videos large amount dance videos available online diversity dynamics videos provides challenging space problems highly dynamic video analysis enables conduct focused study relevance motion dancing classification broader value motion improving video classification core challenge task attaining adequate representation human motion across clip order highlight trajectory work evaluate current approaches demonstrate value isolating motion properly evaluating approaches dataset introduction video rich medium dynamic information used determine happening scene work consider highly dynamic video video requires parametrization motion extended sequences identify activity performed main challenge highly dynamic video single frame provide sufficient information understand action performed therefore multiple frames leading extended sequence frames need analyzed scene classification one drawbacks current action classification research lack approaches many video classification techniques exist either utilizing single frames late fusion architectures temporal convolutional networks recurrent networks long memory lstm current classification problems often identified single frame present challenging problem wherein class requires use multiple frames adequately classify category specifically propose use optical flow pose estimation motion representations augment traditional video classification approaches comparing approaches enables gain insights inherent encoding motion neural networks difficult understand main contributions analysis baseline approaches video classification general method concurrently learning multiple motion parameterizations video video dataset highly dynamic dance videos contrasted existing video datasets motivate investigation understanding motion parameterization video classification rating temporal data video frame extending convolution kernels size represents temporal extent also point one major challenges using deep learning video classification video datasets comparable quality size image recognition datasets similarly convolutional kernels incorporate spatial domain shown successful action classification security camera depth data recordings wang use similar late fusion approach note without incorporating learned features ensemble method handcrafted features approaches still fail outperform handcrafted approaches combine methods work incorporating preprocessed features optical flow pose detection convolutional kernels order integrate representation motion network architecture another common approach leverage sequential nature long memory lstm specific type recurrent neural network additional gates control flow information lstms process information long term temporal sequences applied video various tasks caption generation learning video representations similarly recurrent convolutional networks lrcns proposed donahue introduce another variation lstm task despite temporal nature approaches less successful encoding motion comparison networks encode spatial temporal domain concurrent architectures effective method classifying motion video still unclear context action recognition many approaches learning features based image context inherent action part commonly used video datasets traditionally generally identified moderately decent accuracy using approaches encode motion parameters specific method encoding motion recently gained traction action recognition use pose detection temporal domain neural networks detecting pose domain provides intrinsic motion subjects scene highlighted earlier initial breakthrough achieved toshev results estimating pose single individual image importance pose demonstrated incorporating pose features cnn action recognition work extended next two years attain jointspecific networks work well partial occluded poses recently implemented detect related work order determine competing approaches examine first present literature review video classification deep networks shown effective classifying localizing segmenting images still unclear properly extend methods video domain common approaches applying proven image classification deep network architectures individual frames video extending convolutional operators convolutions acting time domain preprocessing video images encode motion optical flow running current image architectures processed frames simple way extend neural networks video classification extract features individual frame video technique lead success network learns temporallyinvariant features commonly used baseline approach compare networks incorporate temporal data one common variant late fusion architecture still spatial network stream running parallel alongside temporal network performing classifications based optical flow calculations network architecture significantly outperforms approaches based solely individual frame classification suggesting incorporating temporal component necessary work leverage benefit temporal network incorporating design network architecture karpathy explore direct methods atomic visual actions ava ava dataset contains atomic visual actions movie clips localized within frame work goes beyond simply understanding simple action video clip understanding interaction humans humans objects although somewhat less relevant work demonstrates need understanding motion features human interaction specifically localizing action relevance scene may contain multiple subjects objects figure examples represents different class dataset types ballroom dancing top left waltz top right quickstep bottom left foxtrot bottom right tango let dance dataset main challenge work determining reliable way testing well specific method parameterize motion realized available video datasets tackled known classification problem one could evaluated using extensions available image classification architectures mind developed new dataset prioritizes motion key characteristic classification assembled video dataset containing dynamic visually overlapping dances chose parent category dancing variety measurable features rhythm limb movement represented datasets categories included dataset multiple people within single frame work leverage implementation pose detection demonstrate need motion parameterization classifying highly dynamic video existing datasets handful relevant datasets exist research domain highlight relevant video datasets appropriate work datasets demonstrate growing need understanding type motion features relevant classifying highly dynamic actions explore work ballet flamenco latin square tango dataset contains approximately clips action classes totaling hours data clip length varies largely second seconds depending activity resolution one first datasets tackle human actions video however demonstrate work perframe approaches still perform moderately well dataset illustrating main question seek answer work representation motion classification feature break dancing foxtrot quickstep swing waltz dataset contains videos class video long frames per second videos taken youtube quality includes dancing performances plainclothes practicing examples class seen figure highlight dataset contains four different types ballroom dancing quickstep foxtrot waltz tango seen figure motivation behind picking dances parent category specifically setting dance occurs ballroom satisfies main challenge selecting classes exemplify highly dynamic video note extract two different motion representations input data use community optical flow pose detection attempting detect pose found numerous methods focused pose detection adapted methods multiple individuals given dancing generally group activity see figure kinetics kinetics dataset contains clips action classes minimum videos per class action classes also loosely grouped parent classes break dataset dataset collected curation image classifiers use amazon mechanical turk determine action classes video snippet appropriate class using architecture convolutional neural network image classification vgg classification video achieved based key image frames video sample architecture based caffenet variation alexnet shown figure approach explicitly encode motion determining video classification rather categorizes frame naively selects majority label note although numerous approaches aggregating single class multiple classifications network encode temporal domain late fusion common way adding temporal component deep networks separately performing classification based spatial data single frame temporal data optical flow merging results produces overall classification video shown figure approach computes optical flow two frames time necessarily period entire video frame case considered single instance motion occurred video dancing envision specific move dance figure distribution number people per frame using frames least two people detected dataset dataset two people shot illustrates added complexity task use recent person detector similar approaches seen detecting bounding boxes person scene computed pose individual using positive negative examples methodology seen figure baseline methods figure late fusion architecture color key method incorporates motion optical flow traditional cnn pipeline order better understand need motion parametrization video identified two architectures establish baseline architectures commonly applied video architectures take input per architecture proposed approaches order address challenge categorizing highly dynamic videos implement number methods explicitly encode motion core approaches notion kernels process series video frames classification enables pass short video clips frames approx second network learn overall objective incorporate motion learning pipeline standard approaches assess performance testing approaches evident combining numerous motion parameterizations concurrent deep network architecture would best represent input video approaches extensions successful image classification techniques temporal cnn rgb figure architecture traditional cnn commonly used image recognition stated traditional convolutional neural networks extended video using kernels frames convolved temporal domain cnns tested temporal approaches convolutional network order directly compare potential importance embedding multiple frames learning pipeline addition providing multiple representations original input highlight temporal convolutions computing convolutions input frames although increases complexity model still remains significantly tractable computing convolutions require approximately twice computational power figure pipeline displays skeletal temporal cnn convolution processes initial frames obtain multi person pose estimation input frames obtained performing bounding box person detection processed detecting dancers pose architecture architecture first stack network processes spatial representation input rgb image second stack processes optical flow representation computed frames order accentuate particular motions given dance third stack processes pose visualization explained figure discussed earlier stack essentially encoding number participants detected given dance frame current pose figure demonstration outputs pose detection pipeline top latin dancing positively classified bottom break dancing erroneously classified dancers left leg accurate remaining limbs fail temporal architecture convolve temporal domain focus testing approach discussed embeds spatial temporal information initial convolutional layers propagating information network one main setbacks proposed approach computational time currently takes compute methodologies discuss section temporal architecture utilizes three stacks processes chunks frames time order incorporate temporal component loss network enables learn motion parameters spatial optical flow pose representations visualization pipeline seen figure whose convolutional fully connected layers based standard alexnet architecture temporal cnn skeletal baseline experiments pipeline compute temporal cnn multiperson pose information visualize pipeline figure architecture demonstrates importance leveraging context particular videos dance videos inherently benefit representation given generally multiple people scene use visualization pose able attain comparable results cnn approaches key note method use visual information original frame solely visualized pose information shown figure similar optical flow approach likely method benefits heavily encoding number people frame addition motion implement proposed approaches goal determining approach effective highly dynamic video classification implementation details approach given dataset splits extract individual frames let dance dataset videos resulting frames randomly split per video training testing validation consistent across experiments optical flow pose detection split manner order consistently test approaches figure visualizes workflow temporal cnn uses three convolutional stacks process spatial respective motion components aggregates layers one outputs dance classification frame input merely embedding motion two frames later demonstrate larger frame chunks provide significant improvements approach perform baseline video classification experiment implemented architecture shown figure tensorflow weights network convolutional layers initialized values network ilsvrc dataset final video classification results determined classifying frame video voting determine video overall class initial comparison also tested network optical flow imagery input overall observed significant amounts overfitting original training accuracy hints network learning much appearance specific videos training set class hypothesized using image frames alone results network learning features generalize well dancing categories since way observe motion inherent video testing accuracy peaks iterations network compare results similar framework introduced tested baseline attaining accuracy directly demonstrates possibility solving classification problem carefully selecting right frame versus understanding underlying motion video also ran identical setup using optical flow estimation training optical flow entire dataset used farneback method calculating dense optical flow obtain estimate horizontal vertical components motion incorporate network architecture case saw slightly worse performance approximately testing note overfitting optical flow images subdued given images longer contain background information given number dances occurred similar identical settings background information strong confounding factor original images overall result optical flow performs worse training rgb images given late fusion implemented late fusion architecture shown figure caffe approach follows intuitively previous subsection discuss effects method images optical flow individual stream uses caffenet architecture weights initialized network ilsvrc dataset finetune network training layers end stream concatenated passed final layer outputs respective classifications note architecture method still using single frame input network trained basis chose use caffenet architecture frame initialized ilsvrc weights consistent baseline experiment described previous section allows perform direct comparison approaches determine benefit optical flow dataset approach final video classifications determined classifying individual frame optical flow image pair followed voting determine overall class interesting note total per video classification accuracy method much higher single accuracy although one may compelled argue motion key classification refer back figure figure demonstrates frames throughout dataset also contain tremendously varied number participants see figure optical flow tends visually separate dancers background also explains significant increase algorithm performance addition motion single frame pair comparison comparing accuracy utilizes svm combine streams whereas concatenate final layers convolutional streams fully connected output stated earlier illustrates core issue encountered looking highly dynamic dataset validates motivation introduce lets dance dataset research community figure image dancers performing ballet optical flow estimation see optical flow good job segmenting subjects scene addition encoding motion results discussion order assess temporal architectures compare number approaches explicitly encode motion order determine performance overall become clear need transition traditional cnn approaches conducting video classification evident table methods embed motion significantly outperform traditional methods metrics evaluate approaches necessary order better understand network architecture learning foreground shape representation playing key role classification network results demonstrate improvement independent approach classification accuracy significant increase imaging method optical flow method increase attained combining architecture previous two methods addition single concatenation node fuse data end network demonstrates directly incorporating temporal data network immediately beneficial towards classifying video leveraging network perform full video classification rather classification tested trained network test set videos taking class largest number votes final video label resulted classification accuracy experimentation network architecture saw significant improvement computing unique mean image subtract optical flow increased accuracy final classification rate network performs well classifying ballet waltz tango flamenco foxtrot poor classification accuracy break swing dancing particular interest network performance waltz tango foxtrot occur similar settings network shows capable performing classification within let dance dataset dataset let dance temporal cnn order evaluate approach restructured data chunks needed input convolution network could trained features chunks video network trained method yielded accuracy result particularly impressive demonstrated inherent ability convolution extract motion features explicitly computed major drawback approach complexity convolution inherently takes significant computation unable perform approaches using convolutions due complexity order combat introduce tractable approaches graphics cards current systems utilizes titan pascal graphics cards achieve comparable performance explicitly encoding motion network architecture addition note convolutions limited initial case frames makes difficult encode complex motions last second without sub sampling frames invariably lead loss detail temporal methods invariably suffer limitation given variable inputs convolutional network fully explored table method comparison let dance ucf results obtained results obtained lastly revisit accuracy results activity recognition dataset table illustrates high levels accuracy using standard extensions image classification techniques discuss section important note skeletal temporal cnn order embed human motion data incorporate skeletal images temporal cnn visualize approach frame cnn cnn temporal cnn rgb temporal cnn skeletal cnn temporal cnn conclusion future work testing accuracy work sought understand effect motion classifying videos recent work demonstrated relevance type videos recently seen work conducted demonstrates traditional cnn approaches properly intentionally encode motion methodology fact frequently overlooked testing videos inherently require motion primary motivator work see table convolution methods outperform traditional approaches inherently encoding motion computation prediction similarly methods incorporate optical flow also leverage temporal features significantly improve video classification table comparison numerous approaches testing accuracies dataset pose single image represents pose particular frame attained accuracy note accuracy still performs marginally better approach despite fact utilize spatial rgb representation due computational complexity running concurrent convolutional networks propose stacked convolutional method allows combine multiple streams single graphics card also opens potential future work incorporating optical flow pose data hybrid approaches temporal cnn potential increase algorithm understanding video also developed focused dataset believe research community benefit intentionally selecting highly dynamic actions one specific class tested variety traditional complex methods order begin understand composition dataset baseline performance let dance dataset continue help assess adequate motion parameterization hopefully assist improving learn video data cnn architecture utilizes three data modalities assess singleframe stacked architecture order compare benefits drawbacks shown table approach attains accuracy network performs comparably fusion approach conducted one baselines indicates significant amount information added use skeletal optical flow representations one biggest problems ran throughout research endeavor determining best classes select dataset initially intuition dancing martial arts adequate parent categories quickly saw martial arts represented multiclass problem although dancing exhibits similar overlaps separation much evident performing data collection also alternate different dances partly due availability youtube understanding dances temporal cnn logically extended approach temporal domain stacking image input layers produce chunk approach utilizes input temporal cnn implemented much lower complexity three streams saw method attain best performance methods evaluated accuracy looking successful approaches methods convolution note achieve similar performance classification however two methods equivalent terms computational resources beyond increased workload restrictions inherent appropriately formatting data temporal cnn convolution much training testing time observe even though temporal cnn successful approach may much simpler stacked convolutional network approach available one next steps considered work modifying input data order blur regions video motion considered background would enforce motion parameterization help better understand accomplish improve general video classification algorithms could also explored independently classifying pose estimation significantly challenging seek extend work continuing develop representations intentionally target highly dynamic actions information please visit https references pantofaru sun schmid ross toderici malik sukthankar vijayanarasimhan ricco ava video dataset localized atomic visual actions pishchulin jain andriluka schiele articulated people detection pose estimation reshaping future computer vision pattern recognition cvpr ieee conference pages ieee redmon divvala girshick farhadi look unified object detection arxiv preprint rodriguez ahmed shah action mach maximum average correlation height filter action recognition computer vision pattern recognition cvpr ieee conference pages ieee russakovsky deng krause satheesh huang karpathy khosla bernstein berg imagenet large scale visual recognition challenge international journal computer vision ijcv simonyan zisserman convolutional networks action recognition videos ghahramani welling cortes lawrence weinberger editors advances neural information processing systems pages curran associates simonyan zisserman deep convolutional networks image recognition arxiv preprint soomro zamir shah dataset human actions classes videos wild arxiv preprint srivastava mansimov salakhutdinov unsupervised learning video representations using lstms corr toshev szegedy deeppose human pose estimation via deep neural networks proceedings ieee conference computer vision pattern recognition pages tran bourdev fergus torresani paluri learning spatiotemporal features convolutional networks arxiv preprint venugopalan rohrbach donahue mooney darrell saenko sequence text proceedings ieee international conference computer vision pages wang wang lin wang zuo human activity recognition reconfigurable convolutional neural networks proceedings acm international conference multimedia pages new york usa acm wang qiao tang action recognition descriptors ieee conference computer vision pattern recognition cvpr june abadi tensorflow machine learning heterogeneous systems software available cao simon wei sheikh realtime multiperson pose estimation using part affinity fields june laptev schmid cnn features action recognition proceedings ieee international conference computer vision pages donahue anne hendricks guadarrama rohrbach venugopalan saenko darrell recurrent convolutional networks visual recognition description proceedings ieee conference computer vision pattern recognition pages motion estimation based polynomial expansion image analysis pages springer feichtenhofer pinz zisserman convolutional network fusion video action recognition arxiv preprint gkioxari hariharan girshick malik using detecting people localizing keypoints proceedings ieee conference computer vision pattern recognition pages gkioxari malik finding action tubes proceedings ieee conference computer vision pattern recognition pages yang convolutional neural networks human action recognition pattern analysis machine intelligence ieee transactions jan jia shelhamer donahue karayev long girshick guadarrama darrell caffe convolutional architecture fast feature embedding proceedings acm international conference multimedia pages acm karpathy toderici shetty leung sukthankar video classification convolutional neural networks ieee conference computer vision pattern recognition cvpr june kay carreira simonyan zhang hillier vijayanarasimhan viola green back natsev kinetics human action video dataset arxiv preprint krizhevsky sutskever hinton imagenet classification deep convolutional neural networks pereira burges bottou weinberger editors advances neural information processing systems pages curran associates liu shahroudy wang lstm trust gates human action recognition european conference computer vision pages springer wang xiong wang qiao towards good practices deep convnets arxiv preprint wei ramakrishna kanade sheikh convolutional pose machines arxiv preprint zou zhu deep learning invariant features via simulated fixations video pereira burges bottou weinberger editors advances neural information processing systems pages curran associates
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mar enhancing evolutionary optimization uncertain environments allocating evaluations via bandit algorithms xin qiu risto miikkulainen sentient technologies inc san francisco california sentient technologies inc san francisco california university texas austin austin texas abstract optimization problems uncertain fitness functions common real world present unique challenges evolutionary optimization approaches existing issues include excessively expensive evaluation lack solution reliability incapability maintaining high overall fitness optimization using conversion rate optimization example paper proposes series new techniques addressing issues main innovation augment evolutionary algorithms allocating evaluation budget bandit algorithms experimental results demonstrate bandit algorithms used allocate evaluations efficiently select winning solution reliably increase overall fitness exploration proposed methods generalized optimization problems noisy fitness functions keywords evolutionary computation bandit algorithm uncertain environment introduction many problems fitness evaluations noisy uncertainty makes evolutionary optimization difficult true fitness solution approximated large number samples slowing evaluations making results unreliable concrete example consider conversion rate optimization cro problem cro emerging field web designs optimized increase percentage converted visitors conversion refers desired actions website users making purchase registering new account clicking desired links true conversion rate web design estimated via certain number user recently new technology cro developed applying evolutionary algorithm website design represented genome evolved better ones terms conversion rate evolutionary cro provides considerable advantages traditional multivariant testing exploration evolutionary algorithm covers large design space evolution discovers utilizes effective interactions among variables optimization website design fully automated although impressive improvements human design reported still several open issues evolutionary cro first evaluations candidate designs expensive fair amount traffic may wasted bad designs second weak statistical evidence necessary winner selection among candidate designs reducing reliability optimization outcome third cases target maintain high overall conversion rate optimization process instead identifying single best website design optimization current evolutionary cro technique accommodate demand overcome issues series new mechanisms proposed paper augmenting evolutionary optimization bandit mab algorithms first new framework called developed utilizing mab algorithms traffic allocation fitness evaluation proposed framework aims reducing evaluation cost maintaining optimization performance second enhanced variant designed select winner reliably main idea include addtional verification phase based mab algorithms end evolution process third another variant developed introducing new concept called asynchronous statistics mab algorithms new variant particularly well suited problems overall fitness average fitness evaluations optimization needs maximized empirical studies demonstrate techniques effective simulated cro domain although studies work based cro domain principle ideas easily generalized optimization problems uncertain environments remainder paper organized follows section introduces basic concepts evolutionary cro mab algorithms section explains technical details new approaches discusses underlying rationales section evaluates proposed mechanisms via experiments section provides discussions suggests future research diections background section provides brief description basic concepts existing mab algorithms utilized proposed techniques evolutionary conversion rate optimization metaheuristic inspired natural evolution process individual genome population represents single solution optimization problem individuals evolve crossover mutation survival selection iteratively significant advantage eas make assumption underlying landscape optimization problems thereby leading exceptional ability finding good solutions mathematically intractable problems algorithm algorithm evolutionary cro genome represents web interface design search space web designer space designer specifies elements interface values take instance landing page logo size header image button color content order elements take values evolutionary cro searches good designs space possible combinations values often number millions generation genomes evaluated fixed number user interactions conversion rates evalution used fitnesses genome selection used select parent genomes traditional genetic operations crossover recombination elements two parent genomes mutation randomly changing one element offspring genome performed generate offspring candidates process repeated generation generation termination criterion met usually means reaching fixed number user interactions typical application winning design selected among best candidates estimate future performance campaignmode application winner performance measured overall convergence rate throughout entire experiment require total number arms pull arm observe reward end pull arm max argmaxi log observe reward max max max imax max max max end mab problems due good theoretical guarantees principle behind ucb optimism face uncertainty generally ucb constructs optimistic guess potential reward arm pulls arm highest guess among ucb family algorithms simple yet efficient variant directly applied bernoulli bandits optimistic guess form upper confidence bound derived inequality suppose independent random variables expected values inequality gives exponential upper bound probability value deviates expectation bandit algorithms subsection explains definition bandit problem introduces three representative bandit algorithms used paper bandit problem mab problem slot machine multiple arms given gambler decide arms pull many times pull arm order pull common stochastic mab problem parameterized number arms number rounds fixed unknown reward distributions associated arm arm arm respectively round agent gambler chooses arm set arms pull observes reward sampled reward sample independent past actions observations cro problem special case called bernoulli bandit general stochastic mab problem reward pull either converted cro arm probability success reward algorithm stochastic mab problem must decide arm pull round based outcomes previous pulls classical mab problem goal maximize cumulative sum rewards rounds since agent prior knowledge reward distributions needs explore different arms time exploit seemingly rewarding arms goal aligns application cro clarity statement call type problem classical stochastic mab problem another target stochastic mab problem output recommended arm given number pulls performance mab algorithm evaluated average payoff recommended arm goal aligns application cro called pure exploration problem stochastic mab problem reward pull single arm empirical average reward arm pulls using log represents total number pulls far arms converges zero quickly total number pulls increases upper bound used balance exploration exploitation algorithm shows basic steps algorithm two effects algorithm needs bepmentioned first one arm never pulled corresponding log grow rate faster pulled arms means arm never permanently ruled matter poorly performs effect encourages exploration second one arm pulled many times arms significantly higher empirical mean reward pulled subsequently effect encourages exploitation delicate tradeoff exploration exploitation therefore hallmark thompson sampling except ucb thompson sampling another good alternative mab algorithm classical stochastic mab problem idea assume simple prior distribution parameters reward distribution every arm round play arm according posterior probability best arm effectiveness empirically demonstrated several studies asymptotic optimality theoretically proved bernoulli bandits bernoulli bandits utilizes beta distribution priors family continuous probability distributions interval parameterized two ucb algorithm upper confidence bound ucb algorithm arguably popular approach solving classical algorithm thompson sampling bernoulli bandits algorithm successive rejects algorithm require total number arms end sample beta end pull arm max argmaxi observe reward max max else end end require total number arms total number pulls ensure best arm log log positive shape parameters denoted probability density function pdf beta distribution beta given gamma function mean beta higher lead tighter concentration beta around mean initially assumes arm prior reward distribution beta equivalent uniform distribution round observed successes reward failures reward pulls arm reward distribution arm updated beta algorithm samples updated reward distributions selects next arm pull according sampled reward algorithm describes detailed procedure log original evolutionary cro framework evaluation candidate performed static fashion fixed amount traffic allocated website design fitness candidate measured conversion rate case cro evaluation expensive consuming traffic real website thus overall conversion rate evolution determines real cost optimization process one main drawback original static evaluation large amount traffic may consumed bad candidates thus goal develop new framework allocates traffic dynamically efficient way mab algorithms well suited role specifically candidate web design regarded arm visit website equal pull reward visit single web design assumed follow unknown fixed bernoulli distribution probability getting reward visited user successfully converted probability getting reward visited user converted true conversion rate web design given fixed budget traffic number visits generation bernoulli mab algorithm invoked allocate traffic current candidates fitness candidate equivalent number successful conversions divided total visits numbers counted within current generation based fitnesses standard operations parent selection crossover mutation survival selection conducted generate population next generation algorithm depicts procedure proposed framework namely note goals mab algorithm evaluation phase inherently different mab algorithm cares identifying good arms efficiently whereas aims estimating fitnesses arms spite fact mab algorithm impair optimization performance significantly may even improve shown algorithm elite candidates log times except last two arms pulled times log log therefore total number pulls exceed budget methodology section describes algorithmic details proposed approaches mechanisms basic framework combining evolutionary cro technique mab algorithm presented first two enhanced variants developed tackling different use cases namely best arm identification campaign mode successive rejects algorithm among many existing algorithms solving pure exploration problem successive rejects algorithm stands independent scaling rewards main task algorithm identify best arm arm truly best mean reward fixed number pulls suppose given arms pulls first algorithm divides pulls phases end phase arm lowest empirical mean reward discarded phase arm discarded yet pulled equal number times surviving arm phases recommended best arm details described algorithm result one arm pulled times one arm pulled pull arm rounds end average reward arm pulls end let unique element log algorithm essentially optimal regret difference mean rewards identified best arm true best arm decreases exponentially rate logarithmic factor best possible algorithm algorithm best arm identification mode require population size max maximum number generations number website visits generation percentage elites percentage parents initial population mutation probability archive storing evaluated candidates max perform mab algorithm traffic budget record number conversions number visits within current generation candidate set fitness candidate end create elite pool best percentile candidates current generation create parent pool best percentile candidates current generation initialize offspring pool empty size less perform selection pick parent candidates perform uniform crossover two parents generate offspring perform mutation operation offspring element offspring probability randomly altered offspring add offspring add offspring end end end require control parameters algorithm size elite pool additional traffic best arm identification phase archive storing evaluated candidates initialize elite pool empty max lines algorithm add best percentile candidates current generation elite pool size larger remove worst candidate end create parent pool best percentile candidates current generation initialize offspring pool empty size less lines algorithm end end perform pure exploration mab algorithm traffic budget return identified best candidate tests section show bai mode significantly improve reliability identified best candidate without incurring additional cost one additional modification bai mode removal elite survival mechanism candidate allowed survive one generation candidates next generation totally new purpose modification improve explorative ability framework considering fact evaluations expensive limited number generations less acceptable real cro cases since elite pool bai mode already stored outstanding candidates every generation evolution focus exploring regions search space play important role parent selection survival selection since mab algorithms allocate traffic promising candidates fitnesses actually reliably estimated least promising candidates selection mechanisms relying good candidates therefore enhanced thus proposed framework expected significantly increase overall conversion rate evolution without sacrificing overall optimization performance campaign mode asynchronous bandit algorithm current evolutionary cro technique designed identify good candidate end optimization however scenaria goal cro make overall conversion rate optimization high possible fill need campaign mode based developed introducing new concept existing mab algorithms asynchronous statistics original mab algorithms initialize statistics total reward average reward number pulls etc arms contrast mab algorithms campaign mode run asynchronous manner candidates surviving previous generation preserve statistics use initialize mab algorithm updates usual taking asynchronous example candidate two numbers updated generations candidate fails survive underlying rationale preservation statistics increases survival probability good candidates therefore campaign mode focuses exploitation exploration asynchronous mab algorithms allocate traffic best arm identification mode one important goal cro identify single best design delivered website owner use however normal evolutionary process weak statistical evidence obtained although evidence enough rank candidate designs reproduction enough identify true best candidate solve problem best arm identification bai mode algorithm developed based new framework bai mode applied evolution process concluded mab algorithm pure exploration algorithm performed elite pool collection top candidates generations single winner returned bai phase although additional traffic needed running bai phase cost compensated extracting small portion traffic previous generation empirical algorithm campaign mode asynchronous mab algorithm total elements website needs optimized elements choices respectively basic conversion rate website effect element choice within mean conversion rate possible designs conversion rate best possible design parametric setup similar real world situations simulated visit bernoulli test success probability equal conversion rate web design conducted successful trail corresponds successful conversion givies reward failed trial returns reward require control parameters algorithm excluding initialize total number conversions total number visits candidate max perform asynchronous mab algorithm traffic budget update total number conversions total number visits candidate lines algorithm create parent pool best percentile candidates current generation remove worst percentile candidates initialize offspring pool empty size less lines algorithm offspring initialize total number conversions total number visits offspring add offspring end end end existing elites without reevaluating scratch thus improving overall conversion rate algorithm summarizes structure campaign mode except asynchronous statistics mab algorithms campaign mode differs original two aspects first duplication avoidance mechanism weakened since exploration first priority campaign mode duplications different generations allowed encourage revival underestimated candidates second worst percentile candidates replaced new offspring generated top percentile candidates setting even less portion newly generated offspring limited overall conversion rate stable moreover offspring generated based top candidates overall quality offspring tends better purely random sampling mechanisms crossover mutation operations continue exploration steady pace empirical study section evaluates effectiveness proposed framework mechanisms via simulated experiments conversion rate optimization domain conclusions section supported significance level overall performance evaluation order evaluate performance new framework three representative mab algorithms incorporated empirical comparison three variants original evolutionary cro algorithm conducted original algorithm algorithm except traffic allocation instead varying based mab algorithm candidates evenly share traffic budget convenience original evolutionary cro algorithm named standard method rest paper traffic budget generation fixed maximun number generations set conforming cost limitations cro population size mutation probability different values elite parent percentages tested investigate robustness proposed framework two performance metrics utilized one best conversion rate true conversion rate candidate generation overall conversion rate generation total number conversions one generation divided total number visits generation note overall conversion rate different simply averaging conversion rates candidates traffic allocated candidate may different figure shows results based independent runs figure clear proposed framework significantly increases overall conversion rate evolution without deteriorating optimization performance fact incorporation mab algorithms even improves optimization performance terms best conversion rate regarding influence larger values two parameters lead explorative behaviors cost overall conversion rate early stage cases largest acceptable generation number usually reasonable choice would even less circumstances variants perform best terms overall conversion rate best conversion rate three explanations first mab algorithm allocates traffic promising candidates thereby increasing overall conversion rate evaluation second since top candidates receive traffic mab algorithm reliability best performing candidate enhanced third small quality offspring relies heavily top candidates reliable top candidates tend generate reliable offspring overall quality candidates therefore improved overall conversion rate increased way regarding since average reward simulated cro case low second term log line algorithm plays important role experimental setup cro simulator built simulate interactions users website designs multiple possible choices element website choice increase decrease basic conversion rate website independently interactions elements modeled experiments although important cro play large role traffic allocation effect choice predefined kept fixed cro process experiments section following setup used figure figure shows best conversion rate overall conversion rate generation results averaged independent runs different settings perform significantly better standard method terms measures differences best conversion rate statistically significant generation differences overall conversion rate statistically significant generations best conversion rate generations arm selection encourages evenly allocation traffic thereby leading similar behaviors standard method effectiveness best arm identification mode best conversion rate subsection demonstrates effectiveness bai mode experimental comparison standard method approach average fitness within predefined neighborhood solution space used evaluate candidates experiments neighborhood approach improved considering previous candidates calculating neighbood fitnesses fair comparison neighborhood approach visits per generation bai mode visits generation additional visits bai phase parameters identical algorithms max bai mode algorithm used bai phase neighborhood approach neighborhood size fixed figure compares best conversion rates algorithms independent runs bai mode consistently improves standard method neighborhood approach converges faster early explores efficiently later generation bai mode significantly outperforms even less total traffic neighborhood approach performance gradually improves collection candidates however bai mode still reliable neighborhood approach even later stages based experimental results bai mode allows selecting better winner estimates performance successive rejects thompson sampling standard method successive thompson standard neighborhood approach generation figure best conversion rate generations methods bai phase perform significantly better allow identifying candidate true performance significantly better methods without bai phase neighborhood approach also better methods good bai results averaged independent runs performance differences bai variants variants statistically significant accurately therefore provides important improvements practical applications overall conversion rate generations generalized optimization problems noisy fitness functions main idea utilize mab algorithms allocate evaluation budget although poorly performing candidates receive sufficient traffic accurate estimate true fitnesses optimization performance deteriorate evolution pressure eas comes parent selection survival selection two steps rely primarily good candidates efficient detection good candidates important accurate evaluation bad candidates thus mab algorithms reduce evaluation cost without sacrificing optimization performance bai mode significant improvement settings result reliability critical maintains elite archive collects good candidates throughout optimization process evolution finished pure exploration mab algorithm performed elite archive select final winner process amounts winner selection reliability optimization outcome enhanced additional traffic bai phase extracted previous generations thus extra cost incurred problems overall fitness optimization matters new concepts campaign mode applied asynchronous mab algorithms together high survival probability greedy offspring generation lead high yet stable overall fitness evolution notable mab algorithms without sharing parameters among candidates suitable asynchronization experimental results section fitness works better original fitness candidate pool big enough therefore one promising future direction combine fitness mab algorithms later optimization stages reliability winner candidate improved way another interesting future direction introduce contextual bandit algorithms interrelations among variables explicitly modeled model used mab algorithms allocate evaluation budget efficiently moreover model used crossover mutation operations propagate promising variable combinations often thus increasing overall performance efficiency third direction incorporation asynchronous statistics bai mode initializing elites bai phase statistics main optimization phase may increase reliability best candidate overall conversion rate successive rejects thompson sampling standard method asynchronous asynchronous successive rejects asynchronous thompson sampling generation figure overall conversion rate entire optimization process campaign mode data point generation shows overall conversion rate generation asynchronous versions perform significantly better versions leading better conversion rate entire campaign results averaged independent runs performance differences asynchronous versions original versions statistically significant effectiveness asynchronous mab algorithm campaign mode main difference campaign mode new asynchronous mab algorithm section verifies effectiveness asynchronous statistics mab algorithms via empirical comparison experiments modified run asynchronously compared original versions well standard method parameters used algorithms since campaign mode generally run longer period time max set figure compares results independent runs asynchronous asynchronous perform significantly better original versions asynchronous version better early stages candidates algorithm share parameter total number visits candidates candidate survived long period asynchronous variant lead large significant bias towards less visited candidates introduced traffic allocation line algorithm thereby wasting traffic unreliable candidates candidates share parameters campaign mode works properly asynchronous versions improving overall conversion rate optimization significantly conclusion paper demonstrates mab algorithms used make eas effective uncertain environments first proposed framework makes possible allocate evaluation budget efficiently second reliable winner selection mechanism employed bai mode based pure exploration mab algorithms third campaign mode asynchronous mab algorithms designed achieve high stable overall fitness entire campaign proposed mechanisms shown effective experimental comparisons simulated cro domain ideas conclusions work generalized uncertain optimization problems well discussion future work proposed mechanisms work solves three general issues uncertain optimization problems allocate evaluation budget efficiently make optimization outcome reliable maintain high overall fitness evolution although new approaches demonstrated cro domain references volodymyr mnih csaba audibert empirical bernstein stopping proceedings international conference machine learning icml acm new york usa https herbert robbins aspects sequential design experiments bull amer math soc https khalid salehd ayat shukairy conversion optimization art science converting prospects customers reilly media sebastopol steven scott modern bayesian look bandit appl stoch model bus ind https william thompson likelihood one unknown probability exceeds another view evidence two samples biometrika http richard weber gittins index multiarmed bandits annals applied probability http alekh agarwal daniel hsu satyen kale john langford lihong robert schapire taming monster fast simple algorithm contextual bandits corr shipra agrawal navin goyal analysis thompson sampling bandit problem colt annual conference learning theory june edinburgh scotland http audibert bubeck best arm identification bandits https peter auer paul fischer analysis multiarmed bandit problem machine learning may https bubeck munos gilles stoltz pure exploration multiarmed bandits problems springer berlin heidelberg berlin heidelberg https olivier chapelle lihong empirical evaluation thompson sampling proceedings international conference neural information processing systems nips curran associates usa http herman chernoff measure asymptotic efficiency tests hypothesis based sum observations annals mathematical statistics http carlos domingo ricard osamu watanabe adaptive sampling methods scaling knowledge discovery algorithms data mining knowledge discovery apr https agoston eiben jim smith evolutionary computation evolution things nature http eyal shie mannor yishay mansour action elimination stopping conditions bandit reinforcement learning problems mach learn res http cfm garivier olivier algorithm bounded stochastic bandits beyond granmo solving bernoulli bandit problems using bayesian learning automaton international journal intelligent computing cybernetics https arxiv https wassily hoeffding probability inequalities sums bounded random variables amer statist assoc http yaochu jin branke evolutionary optimization uncertain survey ieee transactions evolutionary computation june https moto kamiura kohei sano optimism face uncertainty supported bandit algorithm biosystems supplement https emilie kaufmann olivier cappe aurelien garivier bayesian upper confidence bounds bandit problems proceedings fifteenth international conference artificial intelligence statistics proceedings machine learning research neil lawrence mark girolami eds vol pmlr palma canary islands http emilie kaufmann nathaniel korda munos thompson sampling asymptotically optimal analysis proceedings international conference algorithmic learning theory alt berlin heidelberg https lai herbert robbins asymptotically efficient adaptive allocation rules advances applied mathematics https maillard remi munos gilles stoltz analysis bandits problems divergences risto miikkulainen neil iscoe aaron shagrin ron cordell sam nazari cory schoolland myles brundage jonathan epstein randy dean gurmeet lamba conversion rate optimization evolutionary computation proceedings genetic evolutionary computation conference gecco acm new york usa https risto miikkulainen neil iscoe aaron shagrin ryan rapp sam nazari patrick mcgrath cory schoolland elyas achkar myles brundage jeremy miller jonathan epstein gurmeet lamba sentient ascend massively multivariate conversion rate optimization proceedings thirtieth innovative applications artificial intelligence conference aaai http miikkulainen risto miikkulainen hormoz shahrzad nigel duffy phil long select winner evolutionary 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lifting deep convolutional pose estimation single image oct denis tome university college london chris russell turing institute university edinburgh crussell lourdes agapito university college london http abstract propose unified formulation problem human pose estimation single raw rgb image reasons jointly joint estimation pose reconstruction improve tasks take integrated approach fuses probabilistic knowledge human pose cnn architecture uses knowledge plausible landmark locations refine search better locations entire process trained extremely efficient obtains results outperforming previous approaches errors introduction estimating full pose human single rgb image one challenging problems computer vision involves tackling two inherently ambiguous tasks first location human joints landmarks must found image problem plagued ambiguities due large variations visual appearance caused different camera viewpoints external self occlusions changes clothing body shape illumination next lifting coordinates landmarks single image still problem space possible poses consistent landmark locations human infinite finding correct pose matches image requires injecting additional information usually form geometric pose priors temporal structural constraints propose new joint approach landmark detection full pose estimation single rgb image takes advantage reasoning jointly estimation landmark locations improve tasks propose novel cnn architecture learns combine image appearance based predictions provided style landmark detectors geometric skeletal information coded novel pretrained model human pose information captured human pose model embedded cnn architecture additional layer lifts landmark coordinates imposing lie space physically plausible poses advantage integrating output proposed landmark location predictors based purely image appearance pose predicted probabilistic model landmark location estimates improved guaranteeing satisfy anatomical constraints encapsulated human pose model way tasks clearly benefit advantage approach training data sources may completely independent deep architecture needs images annotated poses poses human pose model trained independently exclusively mocap data decoupling training data presents huge advantage since augment training sets completely independently instance take advantage extra pose annotations without need ground truth extend training data mocap datasets without need synchronized images contribution work show integrate prelearned human pose model directly within novel cnn architecture illustrated figure joint landmark human pose estimation contrast preexisting methods take pipeline approach takes landmarks given instead show model used part cnn architecture architecture learn use physically plausible reconstructions search better landmark locations method achieves results dataset terms errors related work first describe methods assume joint locations provided input focus solving fusion fusion predicted belief maps fusion loss fusion loss loss final pose fused belief maps projected pose belief maps input image output pose predicted belief maps probabilistic pose model predicted belief maps predicted belief maps projected pose belief maps feature extraction joint prediction projection projected pose belief maps pose stage lifting projection fusion fusion fused belief maps loss stage stage feature extraction joint prediction lifting projection fusion loss feature extraction joint prediction lifting projection probabilistic pose model fusion loss final pose figure multistage deep architecture human pose estimation stage produces output set belief maps location landmarks one per landmark belief maps stage well image used input next stage internally stage learns combine belief maps provided convolutional joint predictors projected pose belief maps proposed probabilistic pose model pose layer responsible lifting landmark coordinates projecting onto space valid poses two belief maps fused single set output proposals landmark locations per stage accuracy landmark locations increases progressively stages loss used stage requires pose annotations overall architecture fully differentiable including new belief maps layers trained using best viewed color lifting problem follow methods learn estimate pose directly images pose known joint positions large body work focused recovering pose people given perfect joint positions input early approaches took advantage anatomical knowledge human skeleton joint angle limits recover pose single image recent methods focused learning prior statistical model human body directly mocap data structure motion approaches nrsfm also recover articulated motion given known correspondences joints every frame monocular video huge advantage unsupervised methods need training data instead learn linear basis poses purely data main drawback need significant camera movement throughout sequence guarantee accurate reconstruction recent work nrsfm applied human pose estimation focused escaping limitations use linear model represent shape variations human body instance defined generative model based assumption complex shape variations decomposed mixture primitive shape variations achieve competitive results representing human pose linear combination sparse set bases pretrained using mocap data also proved popular approach articulated human motion propose convex relaxation jointly estimate coefficients sparse representation camera viewpoint enforce limb length constraints although approaches reconstruct pose single image best results come imposing temporal smoothness reconstructions video sequence recently zhao achieved results training simple neural network recover pose known joint positions although results perfect input data impressive inaccuracies joint estimation modeled performance approach combined joint detectors unknown pose images approaches pose inference directly images fall one two categories models learn regress pose directly image features pipeline approaches pose first estimated typically using discriminatively trained part models joint predictors lifted regression based methods suffer need annotate images ground truth poses technically complex elaborate process pipeline approaches challenge account uncertainty measurements crucial types approaches question incorporate dependencies different body joints leverage useful geometric information inference process many earlier works human pose estimation single image relied discriminatively trained models learn direct mapping image features silhouettes hog sift human poses without passing landmark estimation recent direct approaches make use deep learning approaches train network predict joint locations directly image incorporate model joint dependencies cnn via formalism others impose kinematic constraints embedding ferentiable kinematic model deep learning architecture tekin propose deep regression architecture structured prediction combines traditional cnns supervised learning implicitly encodes dependencies body parts cnns become prevalent joint estimation become increasingly reliable many recent works looked exploit using pipeline approach papers first estimate landmarks later spatial relationships imposed using structured learning graphical models one first propose approach naturally copes noisy detections inherent body part detectors modeling uncertainty propagating shape space satisfying geometric kinematic constraints work also estimates location joints predicting pose using appearance probable pose discovered parts using model another recent example bogo fit detailed statistical body model joint proposals zhou tackles problem pose estimation monocular image sequence integrating temporal information account uncertainties model measurements similar proposed approach zhou method need synchronized training data needs pose annotations train cnn joint regressor separate mocap dataset learn sparse basis unlike approach relies temporal smoothness best performance performs poorly single image finally interpreter network recent approach estimate skeletal structure common objects chairs sofas bears similarities method although approaches share common ground decoupling training data use projection improve predictions network architectures different unlike carry quantitative evaluation human pose estimation network architecture figure illustrates main contribution approach new cnn architecture trained estimate jointly joint locations crucially includes novel layer based probabilistic model human pose responsible lifting poses propagating information skeletal structure convolutional layers way prediction pose benefits information encoded section describes new probabilistic model human pose trained dataset mocap data section describes new components layers cnn architecture finally section describes experimental evaluation dataset obtain results addition show qualitative results images mpii leeds datasets probabilistic model human pose one fundamental challenge creating models human poses lies lack access data sufficient variety characterize space human poses compensate lack data identify eliminate confounding factors rotation ground plane limb length symmetry lead conceptually similar poses unrecognized training data simple preprocessing eliminates factors size variance addressed normalizing data sum squared limb lengths human skeleton one symmetry exploited flipping pose left right aligning human poses training set allowing rotational invariance challenging requires integration data model seek optimal rotations pose rotating poses closely approximated compact gaussian distribution formulate problem optimization set variables given set training poses represented matrix landmark locations number human seek global estimates average pose set orthonormal basis noise variance alongside per sample rotations basis coefficients minimize following estimate arg min tensor analog multiplication vector matrix squared frobenius norm matrix assumed point rotation matrices considered ground plane rotations large number pose samples considered order million training dataset complex samples memory requirements mean possible solve directly joint optimization variables using nonlinear solver ceres instead carefully initialize say set orthonormal basis matrices mean matrix unwrapped vector unit norm orthogonal unwrapped matrices figure visualization training data alignment see section using pca notice poses orientation poses close far poses form another clear cluster alternate performing ppca update updating using ceres minimize error steadily increase size basis target size stops apparent deformations could resolved rotations becoming locked basis early stage empirically leads lower cost solutions initialize use variant algorithm estimate mean pose component altered planar rotations take estimate component mean point direction components interleave components sample concatenate large matrix find rank two approximation calculate replacing adjacent pair rows closest orthonormal matrix rank two take components end result optimization compact lowrank approximation data reconstructed poses appear orientation see figure next section extend model described distribution better capture variations space human poses model human pose although learned gaussian model section directly used estimate see table inspection figure shows data gaussian distributed better described using multiple modal distribution heavily inspired approaches characterize space human poses mixture pca bases related works represent poses interpolation exemplars approaches extremely good modeling tightly distributed poses walking samples testing data likely close poses seen training emphatically case much dataset use evaluation zooming edges figure reveals many isolated paths motions occur never revisited nonetheless precisely regions interested modeling seek coarse representation pose space says something regions low density also characterizes nature pose space represent data mixture probabilistic pca models using clusters trained using using small number clusters important initialize algorithm correctly accidentally initializing multiple clusters single mode lead poor density estimates initialize make use simple heuristic first subsample aligned poses refer compute euclidean distance among pairs seek set samples distance points nearest sample minimized arg min min find using greedy selection holding previous estimate constant iteratively selecting next candidate minimizes cost selection pose samples found using procedure seen rendered poses figure practice stop proposing candidates occur close existing candidates shown samples choose one candidate dominant mode given candidates cluster centers assign aligned point cluster representing nearest candidate run algorithm building mixture probabilistic pca bases new convolutional architecture pose inference pose inference single rgb image makes use multistage deep convolutional architecture trained repeatedly fuses refines poses second module takes final predicted landmarks lifts one last time space final estimate see figure heart architecture novel refinement convolutional pose machine wei reasoned exclusively proposed architecture stage stage stage stage stage stage stage stage stage stage stage stage figure results returned different stages architecture top left evolution skeleton projecting points back space bottom left evolution beliefs landmark left hand stages right skeleton relative mean error per landmark millimeters even incorrect landmark locations model returns physically plausible solution iteratively refined pose estimations landmarks using mixture knowledge image estimates landmark locations previous stage modify architecture generating stage projected pose belief maps fused learned manner standard maps implementation point view done introducing two distinct layers probabilistic pose layer fusion layer see figure figure shows uncertainty belief maps reduced stage architecture accuracy poses increases stage architecture stage sequential architecture consists stages stage consists distinct components see figure predicting use set convolutional pooling layers equivalent used original cpm architecture combine evidence obtained image learned features belief maps obtained previous stage predict updated set belief maps human joint positions lifting output cnnbased belief maps taken input new layer uses new pretrained probabilistic human pose model lift proposed poses projected pose belief maps pose estimated previous layer projected back onto image plane produce new set projected pose belief maps maps encapsulate dependencies body parts fusion layer final layer stage described section learns weights fuse two sets belief maps single estimate passed next stage final lifting belief maps produced output final stage lifted give final estimate pose see figure using algorithm lift poses predicting convolutional pose machines understood updating earlier work ramakrishna use deep convolutional architecture approaches stage landmark algorithm returns dense per pixel belief maps bpt show confident joint center landmark occurs given pixel stages belief maps function information contained image also information computed previous stage case convolutional pose machines work uses architecture summary convolution widths architecture design shown figure details training given predict locations different landmarks captured dataset input output layers stage architecture replaced larger set account greater number landmarks new architecture initialized using weights found cpm model preexisting layers new layers randomly initialized retraining cpms return estimates landmark locations techniques estimation described next section make use locations transform belief maps locations select confident pixel location landmark arg max lifting follow assuming weak perspective model first describe simplest case estimating pose single frame using unimodal gaussian pose model described section model composed mean shape set basis matrices variances compute probable sample results dataset success cases mpii leeds failure cases mpii leeds figure left results dataset identified landmark positions skeleton shown pose taken different actions walking phoning greeting discussion sitting right results images mpii columns leeds datasets last column model trained images diverse contained datasets however often retrieves correct joint positions last row shows example cases method fails either identification landmarks pose refinement figure landmark refinement left predicted landmark positions right improved predictions using projected pose model could give rise projected image arg min orthographic projection matrix known external camera calibration matrix estimated perframe scale although given problem convex unknown rotation matrix problem extremely even known prone sticking local minima using gradient descent local optima often lie far apart pose space poor optima leads significantly worse reconstructions take advantage matrix restricted form allows parameterized terms single angle rather attempting solve optimization problem see consider trivial reparameterization solve let using local methods quantize space possible rotations choice rotation hold fixed solve picking minimum cost solution choice fixed choices rotation terms precomputed finding optimal becomes simple linear least square problem process highly efficient oversampling rotations exhaustively checking locations guarantee solution extremely close global optima found practice using samples refining rotations basis coefficients best found solution using least squares solver obtains reconstruction make use faster option checking locations using best found solution estimate puts close global optima average accuracy finding global optima moreover allows upgrade sparse landmark locations using single gaussian around frames second using python code standard laptop handle models consisting mixture gaussians follow simply solve gaussian independently select probably solution projecting poses onto belief maps projected pose model interleaved throughout architecture see figure goal correct beliefs regarding landmark locations stage fusing extra information physical plausibility given solution previous component estimate physically plausible projected pose directions discussion eating greeting phoning photo posing purchases linkde tekin tekin tekin zhou sanzari single ppca model mixture ppca model sitting sitting smoking waiting walk dog walking walk together average linkde tekin tekin tekin zhou sanzari single ppca model mixture ppca model table comparison pose estimation results approach dataset competitors follow protocol evaluation errors given substantially outperform methods terms average error showing average improvement closest competitor note approaches use video input instead single frame embedded belief map otherwise convolved using gaussian filters fusion belief maps belief maps predicted probabilistic pose model fused belief maps according following equation ftp bpt weight trained part learning set fused belief maps passed next stage used input guide reestimation joint locations instead belief maps used convolutional pose machines objective training following objective cost function minimized stage squared distance generated fusion maps layer ftp belief maps generated gaussian blurring sparse locations landmark training total loss sum layers novel layers implemented extension published code convolutional pose machines inside caffe framework python layers weights updated using stochastic gradient descent momentum details novel gradient updates used lifting estimates pose space given supplementary materials experimental evaluation dataset model trained tested dataset consisting million accurate human poses video mocap dataset female male subjects captured different viewpoints show performing typical activities talking phone walking greeting eating evaluation figure shows predictions improved projected pose model reducing overall mean error per landmark error reduction using full approach estimates comparable magnitude improvement due change architecture moving work zhou architecture reduction pixels pixels see table details evaluation several evaluation protocols followed different authors measure performance pose estimation methods dataset tables show comparisons pose evaluation error yasin rogez mixture ppca model evaluation error bogo mixture ppca model protocol protocol evaluation pixel error zhou trained cpm architecture using refinement table evaluation dataset top two tables compare pose estimation errors competitors protocols bottom table compares pose estimation error competitors approach lifts landmark predictions plausible model projects back image substantially reduces error note use video input knowledge action label estimation previous works take care evaluate using appropriate protocol protocol standard evaluation protocol followed training set consists subjects test set includes subjects original frame rate fps fps evaluation sequences coming cameras trials reported error metric error euclidean distance estimated joints ground truth averaged joints skeletal model table shows comparison approach competing approaches using protocol baseline method using single unimodal probabilistic pca model outperforms almost every method action types exception sanzari still outperforms average across entire dataset mixture model improves offering improvement sanzari closest competitor protocol followed selects subjects training subject testing original video every frame evaluation performed sequences cameras trials error metric reported case pose error equivalent error similarity transformation estimated pose aligned ground truth pose basis using procrustes analysis error averaged joints table shows comparison approach approaches use protocol although model trained using subjects used training protocol one fewer subject still outperforms methods protocol followed selects subjects training testing protocol however evaluation sequences captured frontal camera cam trial original video subsampled error metric used case pose error described protocol error averaged subset joints table shows comparison approach method outperforms bogo almost average even though bogo exploits detailed statistical body model trained thousands body scans captures variation human body shape deformation pose mpii leeds datasets proposed approach trained exclusively dataset used identify landmarks images contained different datasets figure shows qualitative results mpii dataset leeds dataset including failure cases notice probabilistic pose model generates anatomically plausible poses even though landmark estimations correct however shown bottom row even small errors pose lead drastically different poses inaccuracies could mitigated without data annotating additional rgb images training different datasets conclusion presented novel approach human pose estimation single image outperforms previous solutions approach problem iterative refinement proposals help refine improve upon estimates approach shows importance thinking even pose estimation within single image method demonstrating better accuracy approach based upon novel approach upgrading extremely efficient using models tables upgrade stage python code runs approximately frames second realtime approach convolutional pose machines announced integrating systems provide reliable pose estimator natural future direction integrating work simpler approach realtime pose estimation lower power devices acknowledgments work funded secondhands project european union horizon research innovation programme grant agreement chris russell partially supported alan turing institute epsrc grant references agarwal triggs recovering human pose monocular images ieee transactions pattern 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information processing systems pages cho lee complex shape recovery using procrustean normal distribution mixture model international journal computer vision torr lawrence gaussian process latent variable models human pose estimation renals bourlard editors mlmi volume lecture notes computer science pages springer elgammal lee inferring body pose silhouettes using activity manifold learning cvpr fan zheng zhou wang pose locality constrained representation human pose reconstruction european conference computer vision pages springer gotardo martinez computing smooth time trajectories camera deformable shape structure motion occlusion ieee transactions pattern analysis machine intelligence ionescu papava olaru sminchisescu large scale datasets predictive methods human sensing natural environments ieee transactions pattern analysis machine intelligence jain tompson andriluka taylor bregler learning human pose estimation features convolutional networks arxiv preprint jia shelhamer donahue karayev long girshick guadarrama darrell caffe convolutional architecture fast feature embedding arxiv preprint johnson everingham clustered pose nonlinear appearance models human pose estimation proceedings british machine vision conference lee chen determination human body postures single view computer vision graphics image processing lee cho procrustean normal distribution structure motion ieee transactions pattern analysis machine intelligence chan human pose estimation monocular images deep convolutional neural network asian conference computer vision pages springer zhang chan structured learning deep networks human pose estimation proceedings ieee international conference computer vision pages loper mahmood romero black smpl skinned linear model acm transactions graphics tog mori malik recovering human body configurations using shape contexts pami parameswaran chellappa view independent human body pose estimation single perspective image computer vision pattern recognition cvpr 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recognition pages wang flynn wang yuille representing data mixture activated simplices arxiv preprint wang wang lin yuille gao robust estimation human poses single image computer vision pattern recognition cvpr wei ramakrishna kanade sheikh convolutional pose machines arxiv preprint xue lim tian tenenbaum torralba freeman single image interpreter network european conference computer vision pages springer yasin iqbal kruger weber gall approach pose estimation single image ieee conference computer vision pattern recognition cvpr zhao wang martinez simple fast algorithm recover shape landmarks single image arxiv preprint zhou sun zhang liang wei deep kinematic pose regression arxiv preprint zhou zhu leonardos daniilidis sparse representation shape estimation convex relaxation approach arxiv preprint zhou zhu leonardos derpanis daniilidis sparseness meets deepness human pose estimation monocular video arxiv preprint supplementary material computing derivatives lifted model discussed convolution pose machine paper recurrent like architectures problems vanishing gradients effective training require additional loss function defined layer independently drives individual layer return correct predictions regardless information used subsequent layers give derivation gradients emphasized entirely possible train network without using fact similar results obtained using lifting forward pass lifting derivatives rest network additional layers make use custom pythonbased derivatives rather efficient implementation computational reasons might preferable avoid step nonetheless completeness include derivatives two reasons gradients unneeded lifting model use makes best predictions predictions layer closest ground truth constraint naturally enforced objective equation main paper convolutional pose machines architecture suffers problems vanishing gradients overcome wei defined objective layer acted locally strengthen gradients however side effect objective effects happen locally gradients layers little effect learning makes subtle interactions layers less influential forces learning process concentrate simply making accurate predictions layer first give results computing gradients sparse predicted locations see section main paper discussing gradients induced confidence maps sparse locations landmark gradients interests readability neglect use indices indicate stages reader assume variables taken stage similarly dealing mixture gaussians interested computing subgradient reader assume best model already selected forward pass computing gradients using model recall section main body paper mapping initial landmarks projected proposals given arg min discrete set rotations exhaustively minimize number bases owing use discrete rotations mapping piecewise smooth approximation smooth function defined continious induced fixing current state hence remainder section compact notation write matrix size number landmark points number bases formed unwrapping tensor similarly unwrap matrices write also write vector representing unwrapped set landmark positions use vector formed vector followed zeros matrix size formed concatenating matrix values along diagonals zero everywhere else rewrite equation new notation given rewrite equation arg min continuing represent hence truncation mapping belief gradients coordinate transform coordinates predicted landmark induce gaussian belief map change component induces update equivalent difference gaussians component well computational purposes take one pixel induced gradient projected belief map near predicted location induces updating propagated using described equation updating writing set assuming right location given updates dyp update decrease belief bpyp increase anywhere else valid choose sensible update negative step magnitude positive update element magnitude quadrant gaussian width used generate see section main paper direction coordinate references wei ramakrishna kanade sheikh convolutional pose machines arxiv preprint
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neural belief tracker dialogue state tracking nikola diarmuid blaise steve university cambridge apple sjy doseaghdha blaisethom apr abstract one core components modern spoken dialogue systems belief tracker estimates user goal every step dialogue however current approaches difficulty scaling larger complex dialogue domains due dependency either spoken language understanding models require large amounts annotated training data lexicons capturing linguistic variation users language propose novel neural belief tracking nbt framework overcomes problems building recent advances representation learning nbt models reason word vectors learning compose distributed representations user utterances dialogue context evaluation two datasets shows approach surpasses past limitations matching performance models rely semantic lexicons outperforming lexicons provided introduction spoken dialogue systems sds allow users interact computer applications conversation systems help users achieve goals finding restaurants booking flights dialogue state tracking dst component sds serves interpret user input update belief state system internal representation state conversation young probability distribution dialogue states used downstream dialogue manager decide action system user looking cheaper restaurant inform system sure kind user thai food somewhere downtown inform system house serves cheap thai food user inform request address system house regent street figure annotated dialogue states sample dialogue underlined words show rephrasings typically handled using semantic dictionaries perform next system action verbalised natural language generator wen dialogue state tracking challenge dstc series shared tasks provided common evaluation framework accompanied labelled datasets williams framework dialogue system supported domain ontology describes range user intents system process ontology defines collection slots values slot take system must track search constraints expressed users goals informable slots questions users ask search results requests taking account user utterance input via speech recogniser dialogue context system said example figure shows true state user utterance conversation seen example dst models depend identifying mentions ontology items user utterances becomes task confronted lexical variation dynamics context noisy automated speech recognition asr output ood heap affordable budget inexpensive cheaper economic ating igh best highly rated cool chic popular trendy rea entre center downtown central city centre midtown town centre figure example semantic dictionary rephrasings three ontology values restaurant search domain traditional statistical approaches use separate spoken language understanding slu modules address lexical variability within single dialogue turn however training models requires substantial amounts annotation alternatively slu dst coalesced single model achieve superior belief tracking performance shown henderson coupled models typically rely manually constructed semantic dictionaries identify alternative mentions ontology items vary lexically morphologically figure gives example dictionary three pairs approach term delexicalisation clearly scalable larger complex dialogue domains importantly focus english dst research understates considerable challenges morphology poses systems based exact matching morphologically richer languages italian german see paper present two new models collectively called neural belief tracker nbt family proposed models couple slu dst efficiently learning handle variation without requiring resources nbt models move away exact matching instead reason entirely word vectors vectors making user utterance preceding system output first composed intermediate representations representations used decide ontologydefined intents expressed user point conversation best knowledge nbt models first successfully use word vector spaces improve language understanding capability belief tracking models evaluation two datasets show nbt models match performance models make use semantic lexicons nbt models significantly outperform models resources available consequently believe work proposes framework scaling belief tracking models deployment dialogue systems operating sophisticated application domains creation lexicons would infeasible background models probabilistic dialogue state tracking belief tracking introduced components spoken dialogue systems order better handle noisy speech recognition sources uncertainty understanding user goals bohus rudnicky williams young young modern dialogue management policies learn use tracker distribution intents decide whether execute action request clarification user mentioned dstc shared tasks spurred research problem established standard evaluation paradigm williams henderson setting task defined ontology enumerates goals user specify attributes entities user request information many different belief tracking models proposed literature generative thomson young discriminative henderson statistical models systems wang lemon motivate work presented categorise prior research according reliance otherwise separate slu module interpreting user separate slu traditional sds pipelines use spoken language understanding slu decoders detect pairs expressed automatic speech recognition asr output downstream dst model combines information past dialogue context update belief state thomson young wang lemon lee kim perez perez liu sun jang shi dernoncourt liu perez models used raw asr output output potentially one slu decoders williams williams mean models immune drawbacks identified two model categories fact share drawbacks figure architecture nbt model implementation three representation learning subcomponents modified long produce adequate vector representations downstream model components use decide whether current candidate pair expressed user utterance taking account preceding system act dstc challenges systems used output matching systems phoenix wang however robust accurate statistical slu systems available many discriminative approaches spoken dialogue slu train independent binary models decide whether pair expressed user utterance given enough data models learn lexical features good indicators given value capture elements paraphrasing mairesse line work later shifted focus robust handling rich asr output henderson tur slu also treated sequence labelling problem word utterance labelled according role user intent standard labelling models crfs recurrent neural networks used raymond ricardi yao celikyilmaz mesnil peng zhang wang liu lane liu lane approaches adopt complex modelling structure inspired semantic parsing saleh vlachos clark one drawback shared methods resource requirements either need learn independent parameters slot value need manual annotation word level hinders scaling larger realistic application domains joint research belief tracking found advantageous reason slu dst jointly taking asr predictions input generating belief states output henderson sun zilka jurcicek systems used external slu module outperformed systems used external slu features joint models typically rely strategy known delexicalisation whereby slots values mentioned text replaced generic labels dataset transformed manner one extract collection features want food perform belief tracking shared model iterates pairs extracting delexicalised feature vectors making separate binary decision regarding pair delexicalisation introduces hidden dependency rarely discussed identify mentions text toy domains one manually construct semantic dictionaries list potential rephrasings slot values shown use dictionaries essential performance current models though scale rich variety user language general domains primary motivation work presented paper overcome limitations affect previous belief tracking models nbt model efficiently learns able data leveraging semantic information word vectors resolve ambiguity maximising number parameters shared across ontology values flexibility learn domainspecific paraphrasings kinds variation make infeasible rely exact matching delexicalisation robust strategy neural belief tracker neural belief tracker nbt model designed detect pairs make user goal given turn flow dialogue input consists system dialogue acts preceding user input user utterance single candidate pair needs make decision instance model might decide whether goal food talian expressed looking good pizza perform belief tracking nbt model iterates candidate pairs defined ontology decides ones expressed user figure presents flow information model first layer nbt hierarchy performs representation learning given three model inputs producing vector representations user utterance current candidate pair system dialogue acts subsequently learned vector representations interact context modelling semantic decoding submodules obtain intermediate interaction summary vectors used input final module decides whether user expressed intent represented candidate pair representation learning given user utterance system act candidate pair representation learning submodules produce vector representations act input downstream components model representation learning subcomponents make use collections word vectors shown specialising word vectors express semantic similarity rather relatedness essential improving belief tracking performance reason use word vectors wieting throughout work nbt training procedure keeps vectors fixed way test time unseen words semantically related familiar slot values inexpensive cheap recognised purely position original vector space see also means nbt model parameters shared across values given slot even across slots let represent user utterance consisting words uku word associated word vector uku propose two model variants differ method used produce vector representations act constituent ngrams utterance let vin concatenation word vectors starting index vin denotes vector concatenation simpler two models term shown figure model computes cumulative representation vectors summaries unigrams bigrams trigrams user utterance kux vin vectors mapped intermediate representations size wns bsn weight matrices bias terms map cumulative vectors dimensionality denotes sigmoid activation function maintain separate set parameters slot indicated superscript three vectors summed obtain single representation user utterance cumulative representations used model unweighted sums word vectors utterance ideally model learn recognise parts utterance relevant subsequent classification task instance could learn ignore verbs stop words pay attention adjectives nouns likely express slot values figure odel word vectors summed obtain cumulative passed another hidden layer summed obtain utterance representation figure model convolutional filters window sizes applied word vectors given utterance diagram system convolutions followed relu activation function produce summary representations summed obtain utterance representation second model draws inspiration successful applications convolutional neural networks cnns language understanding collobert kalchbrenner kim models typically apply number convolutional filters input sentence followed activation functions following approach model applies different filters lengths figure let fns denote collection filters value word vector dimensionality vin denotes concatenation word vectors starting index let vknu list convolutional filters length run three intermediate representations given fns column intermediate matrices produced single convolutional filter length obtain summary representations pushing representations fied linear unit relu activation function nair hinton time columns matrix get single feature filters applied utterance maxpool relu bsn bsn bias term broadcast across filters finally three summary representations summed obtain final utterance representation vector equation model design better suited longer utterances convolutional filters interact directly subsequences utterance noisy summaries given cumulative semantic decoding nbt diagram figure shows utterance representation candidate slotvalue pair representation directly interact semantic decoding module component decides whether user explicitly expressed intent matching current candidate pair without taking dialogue context account examples matches would want thai food demanding ones pricey restaurant use word vectors comes play model could deal former example would helpless latter case unless human expert provided semantic dictionary listing potential rephrasings value domain ontology let vector space representations candidate pair slot name value given vectors slot summed together nbt model learns map tuple single vector dimensionality utterance representation two representations forced interact order learn similarity metric discriminates interactions utterances pairs either express wcs bsc denotes vector multiplication dot product may seem like intuitive similarity metric would reduce rich set features single scalar multiplication allows downstream network make better use parameters learning interactions sets features context modelling decoder yet suffice extract intents utterances dialogue understand queries belief tracker must aware context flow dialogue leading latest user utterance previous system user utterances important relevant one last system utterance dialogue system could performed among others one following two system acts system request system asks user value specific slot system utterance price range would also tried concatenate pass vector downstream neural network however led weak performance since relatively small datasets suffice network learn model interaction two feature vectors like user answers model must infer reference price range slots area food system confirm system asks user confirm whether specific pair part desired constraints example user responds turkish food yes model must aware system act order correctly update belief state make markovian decision consider last set system acts incorporate context modelling nbt let word vectors arguments system request confirm acts zero vectors none model computes following measures similarity system acts candidate pair utterance representation denotes dot product computed similarity terms act gating mechanisms pass utterance representation system asked current candidate slot pair type interaction particularly useful confirm system act system asks user confirm user likely mention slot values respond affirmatively negatively means model must consider interaction utterance candidate pair slot value pair offered system latter two model consider affirmative negative polarity user utterance making subsequent binary decision binary decision maker intermediate representations passed another hidden layer combined layer maps input vector vector size dim input final binary softmax represents decision given belief state update mechanism spoken dialogue systems belief tracking models operate output automatic speech recognition asr despite improvements speech recognition need make imperfect asr persist dialogue systems used increasingly noisy environments work define simple belief state update mechanism applied asr lists dialogue turn let denote preceding system output let denote list asr hypotheses hti posterior probabilities pti hypothesis hti slot slot value nbt models estimate hti probability expressed given hypothesis predictions hypotheses combined pti hti syst belief state estimate combined cumulative belief state time get updated belief state estimate coefficient determines relative weight previous turns belief state slot set detected values turn given vst informable slots value vst highest probability chosen current goal vst requests slots deemed requested vreq requestable slots serve model user queries require belief tracking across turns experiments datasets two datasets used training evaluation consist user conversations taskoriented dialogue systems designed help users find suitable restaurants around cambridge two corpora share domain ontology contains three informable slots food area price users specify values slots order find restaurants coefficient tuned development set best performance achieved best meet criteria system suggests restaurant users ask values eight requestable slots phone number address two datasets use transcriptions asr hypotheses semantic labels provided dialogue state tracking challenge henderson official transcriptions contain various spelling errors corrected manually cleaned version dataset available training data contains dialogues test set consists dialogues train nbt models transcriptions report belief tracking performance test set asr hypotheses provided original challenge woz wen performed wizard style experiment amazon mechanical turk users assumed role system user dialogue system based ontology users typed instead using speech means performance woz experiments indicative model capacity semantic understanding robustness asr errors whereas dialogues users would quickly adapt system lack language understanding capability woz experimental design gave freedom use sophisticated language expanded original woz dataset wen using data collection procedure yielding total dialogues divided training validation test set dialogues woz dataset available training examples two corpora used create training data two separate experiments dataset iterate train set utterances generating one example slotvalue pairs ontology example consists transcription context list preceding system acts candidate pair binary label example indicates whether utterance context express example candidate pair instance would like irish food would generate positive example candidate pair food rish negative example every pair ontology delexicalised features akin used henderson baseline model supplemented semantic dictionary produced baseline system creators two dictionaries available dictionary contains three rephrasings nonetheless use rephrasings translates substantial gains dst performance see sect believe result supports claim vocabulary used mechanical turkers constrained system inability cope lexical variation asr noise woz dictionary includes rephrasings showing unconstrained language used mechanical turkers setup requires elaborate lexicons evaluation focus two key evaluation metrics introduced henderson goals joint goal accuracy proportion dialogue turns user search goal constraints correctly identified requests similarly proportion dialogue turns user requests information identified correctly models evaluate two nbt model variants train models use adam optimizer kingma crossentropy loss backpropagating nbt subcomponents keeping word vectors fixed order allow model deal unseen words test time model trained separately slot due high class bias constructed examples negative incorporate fixed number positive examples baseline models two datasets compare nbt models baseline system implements wellknown competitive model dataset model henderson model based neural network model recurrent connections turns inside utterances replaces occurrences slot names values generic delexicalised features woz compare nbt models sophisticated belief tracking model presented wen model uses rnn belief state updates cnn feature extraction unlike nbtcnn cnn operates vectors model hyperparameters tuned respective validation sets datasets initial adam learning rate set positive examples included batch size affect performance set experiments gradient clipping used handle exploding gradients dropout srivastava used regularisation dropout rate intermediate representations nbt models implemented tensorflow abadi baseline models map exact matches intents lexiconspecified rephrasings delexicalised ngram features means vectors incorporated directly models results belief tracking performance table shows performance nbt models trained evaluated woz datasets nbt models outperformed baseline models terms joint goal request accuracies goals gains always statistically significant paired moreover statistically significant variation nbt models showing nbt handle semantic relations otherwise explicitly encoded semantic dictionaries nbt performs well across board compare performance two datasets understand strengths improvement baseline greater woz corroborates intuition nbt ability learn linguistic variation vital dataset containing longer sentences richer vocabulary asr errors comparison language subjects dataset less rich compensating asr errors main hurdle given access test set transcriptions nbt models goal accuracy rises dst model model model semantic dictionary eural elief racker eural elief racker goals requests woz goals requests table woz test set accuracies joint goals requests asterisk indicates statistically significant improvement baseline trackers paired indicates future work focus better asr compensation model deployed environments challenging acoustics word vectors info glove xavier goals requests woz goals requests importance word vector spaces nbt models use semantic relations embedded word vectors handle semantic variation produce intermediate representations table shows performance models making use three different word vector collections random word vectors initialised using xavier initialisation glorot bengio distributional glove vectors pennington trained using information large textual corpora semantically specialised vectors wieting obtained injecting semantic similarity constraints paraphrase database ganitkevitch distributional glove vectors order improve semantic content results table show use semantically specialised word vectors leads considerable performance gains vectors significantly outperformed glove xavier vectors goal tracking datasets gains particularly robust noisy data collections vectors consistently outperformed random initialisation gains weaker woz dataset seems large clean enough nbt model learn rephrasings compensate lack semantic content word vectors dataset glove vectors improve randomly initialised ones believe happens distributional models keep related yet antonymous words close together north south expensive inexpensive offsetting useful semantic content embedded vector spaces model showed trends brevity table presents figures table woz test set performance joint goals requests model making use three different word vector collections asterisk indicates statistically significant improvement baseline xavier random word vectors paired conclusion paper proposed novel neural belief tracking nbt framework designed overcome current obstacles deploying dialogue systems dialogue domains nbt models offer known advantages coupling spoken language understanding dialogue state tracking without relying semantic lexicons achieve performance evaluation demonstrated benefits nbt models match performance models make use lexicons vastly outperform available finally shown performance nbt models improves semantic quality underlying word vectors best knowledge first move past intrinsic evaluation show semantic specialisation boosts performance downstream tasks future work intend explore applications nbt dialogue systems well languages english require handling complex morphological variation acknowledgements authors would like thank ivan ulrich paquet cambridge dialogue systems group anonymous acl reviewers constructive feedback helpful discussions references abadi ashish agarwal paul barham eugene brevdo zhifeng chen craig citro greg corrado andy davis jeffrey dean matthieu devin sanjay ghemawat ian goodfellow andrew harp geoffrey irving michael isard yangqing jia rafal jozefowicz lukasz kaiser manjunath kudlur josh levenberg dan rajat monga sherry moore derek murray chris olah mike schuster jonathon shlens benoit steiner ilya sutskever kunal talwar paul tucker vincent vanhoucke vijay vasudevan fernanda oriol vinyals pete warden 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vandyke wen steve young multidomain dialog state tracking using recurrent neural networks proceedings acl vinod nair geoffrey hinton rectified linear units improve restricted boltzmann machines proceedings icml kai sun qizhe xie kai recurrent polynomial network dialogue state tracking dialogue discourse baolin peng kaisheng yao jing wong recurrent neural networks external memory language understanding proceedings national ccf conference natural language processing chinese computing blaise thomson steve young bayesian update dialogue state pomdp framework spoken dialogue systems computer speech language jeffrey pennington richard socher christopher manning glove global vectors word representation proceedings emnlp gokhan tur anoop deoras dilek semantic parsing using word confusion networks conditional random fields proceedings interspeech julien perez spectral decomposition method dialog state tracking via collective matrix factorization dialogue discourse andreas vlachos stephen clark 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capacity scaling law artificial neural networks gerald mario michael krell september sep abstract observed matter ising model ferromagnetism result impossible make artificial neural network sensitive disrupted new information certain threshold reached likewise many input dimensions data points always result overfitting network behave associative memory theoretical derivation backed repeatable empirical evidence shows scaling capacity neural network based two critical points call dimension mackay dimension respectively dimension defines point guaranteed operation memory dimension defines point guaranteed forgetting generalization even high dimensional networks points scale strictly linear number weights exemplified follows binary classifier consisting threshold perceptrons hidden layer bias weighted inputs one binary output perceptron configuration commonly referred multilayer perceptron mlp show classify points input chaotic position see section following behavior observed assuming ideal neural network gating functions handling worst case data derive calculation two critical numbers predicting behavior perceptron networks first derive calculation call lossless memory dimension dimension generalization dimension avoids structured data therefore provides upper bound perfectly fitting training data second derive call mackay dimension limit indicates necessary forgetting lower limit generalization uses network derivations performed embedding ideal network shannon communication model allows interpret two points capacities measured bits validate upper bounds repeatable experiments using different network configurations diverse implementations varying activation functions several learning algorithms bottom line two capacity points scale strictly linear number weights among practical applications result allows network implementations gating functions sigmoid rectified linear units evaluated upper limit independent concrete task dimension bounded dlm introduction dlm samples implies perfect labeling ideal network paper show neural networks like individual perceptrons hopfield networks best understood associative memory one way understand challenges like overfitting forgetting measure capacity memory furthermore closer look error function capacity reveals perceptrons two phase transitions indicated earlier wolfgang kinzel similar ones dimension bounded dlm samples implies chance labeling learned even perfect training ideal network connected perceptron network maximum capacity dimension perceptron scales linearly number weights per perceptron including bias university california berkeley lawrence livermore national lab international computer science institute berkeley authors contributed equally paper sender dataset labeling encoder channel learning dataset method weights identity decoder dataset weights receiver neural dataset network labeling figure shannon communication model applied labeling machine learning dataset consisting sample points ground truth labeling bits sent neural network learning method transfers parametrization network weights dataset kept constant information whole process identity channel forwards everything without loss decoding step network uses weights together dataset try reproduce original labeling using embedding shannon communication model fig capacity measured bits connected methods leading variety claims capabilities limits algorithms see example even though artificial neural networks popular decades article structured follows section understanding processes underlying starts discussion related work usually based solely empirical evidence bounds artificial neural networks particular application domain task see summarize main parts mackay single example perceptron proof section section fact perceptron introduced generalizes result arbitrary since extended works including discussion capacmany variants including limited ities interpreted bits convedescribed perceptron nient consequences validated uses input generates two bounds experimentally using standard output applying linear function inimplementations neural networks reput followed gating function gatsults presented section together ing function typically identity function actual source code reproducibility apthe sign function sigmoid function pendix section finally presents direct pracrectified linear unit relu motitical applications discusses limits vated brain research perceptrons results section concludes paper stacked together networks usually trained future work chain rule backpropagation even though perceptrons utilized long time capacities rarely explored related work yond discussion linear separability versus understanding machine learning opposed xor function general mentions overfitusing black box requires insights ting generalization catastrophic forgetting training testing data available moreover overfitting catastrophic forgetsis space chosen algorithm convergence ting far explained satisfactoand properties optimization rily rithm effect generalization loss catastrophic forgetting describes terms optimization problem formulation effect net first trained one set labels another set labels perceptron quickly looses capability classify one core questions machine learning first set labels recently approach theory focuses complexity introduced overcome catastrophic forpothesis space functions getting avoiding large weight changes eled artificial neural networks weights associated prevition recently become relevant deep ously learned labels interpretation learning seems outperform shallow learning approach valid small alternating deep learning single perceptrons weight changes give better results case linear continuous gating function total number weight updates exceed canated layered fashion techniques like pacity network hence suggest future volutional filters drop early stopping algorithms avoid catastrophic forgetting ularization used tune performance determining capacity avoid overflow dimension measures one measure handles properties given data rademacher complexity understanding properties large neural networks zhang recently performed randomization tests show observed networks memorize data well noise proven evaluating neural networks perfectly learn random labels random data shows dimension analyzed networks size used dataset clear full capacity networks observation also gives good reason smaller size networks outperform larger networks even though lower capacity capacity still large enough memorize labeling data elaborate extension evaluation provided arpit paper indicates lower limit size network different approach using information theory comes group around naftali tishby use information bottleneck principle analyze deep learning layer previous layers treated encoder compresses data better representation decoded labels consecutive layers calculating respective mutual information layer analyze networks behavior training changing amount training data relevant statement work presented immediate consequence dpi data processing inequality information lost one layer recovered higher thereby describing learning capabilities one layer one minimum layers describe overall capabilities network results seem consistent experimental outcomes second higher deeper layers see section however describe learning capabilities neural networks using different information theoretic view namely interpretation neurons memory cells apart speculations section deeper hidden layers able serve error correcting codes previous layers assume ideal network investigate concrete architectures one largest contribution machine learning theory comes vladimir vapnik alexey chervonenkis including dimension dimension well known decades defined largest natural number samples dataset shattered hypothesis space means hypothesis space dimension exists dataset samples binary labeling possibilities exists perfect classifier hypothesis space maps samples perfectly labels due perfect memorizing holds neighbor tight bounds far computed linear classifiers well decision trees definition dimension comes two major drawbacks however first considers potential hypothesis space aspects like optimization algorithm loss regularization function effect choice hypothesis second sufficient provide one example dataset match dimension given complex structure hypothesis space chosen data take advantage structure result shatterability increased increasing structure data aspects matter much simple algorithms major point deep neural networks vapnik suggest determine dimension empirically state conclusion described approach apply neural networks beyond theory far dimension approximated neural networks example abu mostafa argued loosely capacity must bounded number perceptrons recently determined book sigmoid activation function limited amount bits weights loose upper bound dimension set edges consequently number nonzero weights extensions boundaries derived example recurrent neural networks networks piecewise polynomials piecewise linear gating functions another article describes quadratic dimension special case authors use regular grid times points two dimensional space tailor multilayer perceptron directly structure use gates weights memory capacity aware recent questioning approach discussing memory capacity neural networks however occam razor dictates follow path least definition dimension dimension hypothesis space maximum integer dataset cardinality shattered shattered means arbitrary labeling represented hypothesis context learned perceptron maximum holds sumptions perceptrons initially conceived generalizing memory detailed example early works bernard widrow approach also suggested later explained depth sir david mackay fact initial capacity derivations linear separating functions already reported thomas cover also ising model ferromagnetism clearly model used explain memory storage already reported similarities perceptrons also neurons retina mackay first one interpret perceptron encoder shannon communication model chapter article use slightly modified version model depicted fig mackay use shannon model allows measurement memory capacity perceptron bits furthermore allows discussion perceptron capability without taking account number bits used store weights bit doubles also points two distinct transition points error measurement first one discontinuous happens dimension single perceptron offset point dimensionality data point error given perfect training perceptron able generate arbitrary shattering hypothesis space honor describing perceptron shannon communication model allows generalization outlined paper call second point mackay dimension dimension describes largest number samples least possible labelings separated binary classifier dimension single perceptron offset large really required mackay proves sharp drop point performance hence neuron neural network learn much capacity allows catastrophic forgetting occur following section summarize proof rely remainder paper definition general position set points space general position subset size linearly independent lie mackay interprets perceptron encoder shannon communication model compatible interpretation fig input encoder points general position random labeling output encoder weights perceptron decoder receives perfectly learned weights lossless channel question given received set weights knowledge data decoder reconstruct original labels points words perceptron interpreted memory stores labeling points relative data question much information stored training perceptron words ask memory capacity perceptron communication definition advantage mathematical framework information theory applied machine learning also allows predict measure neural network capacity actual unit information bits section functionality perceptron typically explained xor example showing perceptron input variables states model possible output functions xor negation linearly separated single threshold function two variables bias example explanation see section mackay effectively changes computability question labeling question asking given points many possible labelings learned model without error rather computing binary functions variables done mackay uses relationship input dimensionality data number inputs perceptron denoted function indicates number distinct threshold capacity perceptron section summarize proof appearing chapter summary proof interpretation definitions following definitions required tions separating hyperplanes points general position dimensions original function derived calculated recursively iteratively table values function indicating number distinct threshold funct tions points general position sions defined table shows function small turns inexactly samples perforteresting properties mance largely decreases less samples namely largely increases summands middle largest mackay concludes capacity perthis allows derive dimension ceptron therefore error case number possible binary point small follow kinzel physical interlabelings points since pretation understand percept possible labelings input tron error function undergoes two phase transican realized tions first order transition dimenwhen function follows sion continuous one dimension calculation scheme based pascal based interpretation predict gle means bit loss due different phases play role structuring incomplete shattering still highly predictable explaining machine learning algorithms mackay uses error function based therefore throughout paper discuss cumulative distribution standard two points separately sian perform prediction approximate resulting distribution importantly defines second point call capacity perceptron dimension dimension describes network largest number samples typically possible labelings separated binary classifier proofs following definitions useful point large illustrates sharp continuous drop definitions mance point since sum two independent normally distributed random variables remainder article asis normal mean sum sume network weights two means variance sum assumed unit two variances natural see bias counts weight note following section mackay point assumptions architecture linearly additive best case required quick analytic proof definition general position used shown previous section typically used linear gebra general case needed symmetric perceptron uses hyperplane linear sum last summand equal separation see also table neural first etc hence take half networks stricter setting required since neusummands get half values sum ral networks implement arbitrary means means separations perceptron inputs handle half possible labelings samples note definition chaotic position set direct proof actually require large points space chaotic transition sharp sense position subset size possible infer anything positions remaining points needs lossless regardless individual neurons lossy fact tempting call something like entropy perceptron indicates expected number bits labeling perceptron assign set points however empirically easily observed additive leads overestimation capabilities analytically also clear individual hyperplanes intersecting perceptrons connected various ways particularly seems hard assume completely disjoint labeling points perceptrons able implement functions stacked instead proof abstracts concrete network architecture assumes ideal network handling worst case data two critical points described section obtained describing single perceptron memory cell generalization interpret artificial neural network ideal combination memory cells show critical points dlm perceptron network scale linearly note chaotic position implies general position excluding linear inference bear mind slightly distorted grid settings example used general position chaotic position chaotic position equivalent saying inference possible structure data thing machine learner memorize distribution satisfies constraint uniform distribution explained section possible achieve high dimension choice special datasets issue yet learning theory practitioner perspective criticized avoid reported problems consistent embedding shannon communication model therefore propose generalization dimension call lossless memory dimension definition lossless memory dimension dimension dlm maximum integer number dlm theorem capacity scaling law dataset cardinality dlm points chaotic position possible labelings dataset represented function hypothesis space learned even perfect training arbitrary perceptron innote single perceptron dlm puts including potential offset weight cabecause chaotic position implies general pacity either dlm dependtion explained section name ing targeted phase denotes neural corresponding point loss guaranteed network combines respective perceptrons mackay dimension perfectly data points assumed definition mackay dimension chaotic position mackay dimension maximum integer dataset cardinality points chaotic position least possible labelings datasets represented function hypothesis space learned perfect training accuracy proof neuron weights including bias able implement exactly different binary threshold functions sample points maximum number binary labelings points follows perceptron dle able store bits adding lossconsequently higher cardinality less memory cells capacity inimplies less labelings creases capacity linearly amount represented show ideal bits lossy memory cells well fact perceptrons combined ceptron network limit exactly best possible way label disjoint sets sample points correctly two capacity scaling law ceptrons weights maximally extend mackay capacity proof label points transitivity addifrom single perceptron neural networks tion implies dlm additive perit holds many perceptrons generalizing ceptron network words ral network able losslessly memorize dlm dlm case input labels means network whole number sample points neuron exactly responsible fraction labeling dataset words free labels borrow anywhere since monotonically falling increasing assume extreme points dlm furthermore assume capacity asymptotically approaching high values therefore assume phase transitions observed single perceptron generalize network perceptrons perceptrons either lossless lossy combined best possible way proof linear scalability dimension take look ideal network dlm number weights equals number sample points follows neuron exactly responsible fraction labeling dataset thus increased without increasing hard conventional transistor memory would able guarantee assignment labels new points would erase original configuration new label however discussed section hard cutoff assuming perceptron equally contributes labeling incoming points needed doubling fixed result neuron able memorize labeling half points therefore generalizes ideal neural network consequence dlm scale linearly consequences conclusion ideal network need add weights number input points reached guarantee lossless memorization halving number weights results dimension generalizing neural network able losslessly memorize input labels means network whole needs lossless regardless individual neurons lossy note capacities information theory upper bounds measurable dimension given concrete network structure therefore likely lower assumed two lossy memory cells likely complement global learning function used learning function therefore influence empirically measured capacity gating function extreme case using identity gating function mlp capacities limited first perceptron feed forward step get dimensions overall binary separation function still linear matter perceptrons combined following formulas capacity network might come handy practice count bias weights perceptron edge graph number incoming edges number weights identical following holds ceptron weights dlm assuming ideal network points chaotic position practically equation inequality data chaotic position network able exploit redundancies please allow provide following intuition regarding dimension perceptron reduces amount possible labelings half joint reduction best case separate reduction perceptron would result reduction number perceptrons understood taking another look function composed sum two functions one uses weights one uses weights increased new function space composed space functions requires parameters let call difficult space functions uses parameters let call easy could therefore argue network dimension handle easier labelings turn exactly functions modeled dlm twice amount edges know network perceptrons behaves dlm assume hold separation functions however assume ideal network number weights equals previous section perceptron dimension hence unconnected hidden units would capacity dlm connecting hidden units output unit results two observations first output unit perceptron inputs threshold task encoding possible labelings maximum capacity unit add therefore bits input layer hidden layer output layer offset input input output input input offset figure capacity calculation mlp inputs hidden nodes displayed contributions weight joint dimension explained section shannon communication model intuition binary labeling random points bit vector dimension capacity allows weights label points targeted training method words network capacity store labeling points error labeling bits restored output shown previous sections guaranteed dimension case dlm guaranteed capacity therefore bits dimension capacity therefore bits consequence dimensions interpreted using information theoretic measures divergence allows predicting minimally expected error given neural network given input data also worth noting divergence closely related cross entropy turn recently proposed optimization criterion deep learning algorithms second since treat perceptron memory unit care output unit looks restricted space compared input space capacity unit change hence multilayer perceptron mlp hidden units inputs single output node respective offsets get dimension dlm consequently dlm dlm respectively mlp graphical illustration scaling law seen figure scaling law also strongly supported empirical results see section one generalize binary classification follows assume binary classification network dlm means binary labeling implemented one output unit adding second output unit capacity first one must work specific second binary labeling well imagine first output unit away multiclassifier dimension gives every new output node chance implement particular function capacity measurements section describes evaluated dimension empirical means observe theoretical capacities indeed upper limits experimental setup basic principle empirical evaluation obtain samples randomly generated measuring capacities neural network data increase number input points bits follows directly mackay network test network tion perceptron learn possible labelings unit capacity nodes looked implementation consider difficulties mlp class imbalance redundancies higher empirical dimensions due oversampling might achievable mension half possible labelings dimension obviously expect empirical measurements lower theoretical capacities practically neither ideal network perfect training alogrithm exists furthermore higher dimensions able sample hypothesis space could test labelings exhaustively therefore goal create best conditions possible give network highest chance reaching optimal capacity without violating constraints theoretical framework thereby practical workarounds required speedup limitations arise due exponential increase search space mainly used mlp implementation optimizer code provided appendix control randomness ensure consistent results seed randomizers respective index repetition case optimizer fit training data repeat training times data randomly generated sampling normal distribution repeated evaluations different datasets labeling could fitted case dimension labelings could fitted case dimension processing time latter much higher two reasons first larger amount samples analyzed since least labelings evaluated every time second data convergence mlp takes iterations completeness every labeling would tested due symmetry class handling mlp minor speedup achieved testing labelings last sample labeled possible large dimensions testing labelings computationally expensive hence samples tested random selection labelings due approximation results might true values given structure processing effort dimension even worse required limit samples given resources one could imagine better approach multiple random samplings tested median result dimension worst result dimension taken leave future work number tested labels also limits possible dimensions mlp analyzed input dimensions provide reliable results number hidden tuning implementation apart aforementioned implementation tested optimizers like adam sgd well keras library cases net able fit data contrast using hence measured dimensions low could interpreted generalization capability adam sgd optimizer avoiding overfitting note approximates second order derivative makes accurate also computationally expensive prone get stuck local minima also tested different gating functions using identity function network mostly behaved like single perceptron expected tanh logistic function results looked similar relu function needed repetitions processing time expected generation data significant impact results originally tested uniformly sampled data changing sampling normal distribution improved results dramatically empirically measured upper bound came closer theoretical number different tested datasets using distribution minor effect empirical calculation reached limit dimension testing one dataset solely capture randomness training algorithm significant impact empirical results one hidden layer results described setting dimension depicted figure using one hidden neuron behaved always like single perceptron dimension dimension predicted linear relationship well dimensions also observed comparison theoretical empirical dimensions shows similar linear behavior larger dimensions differences get smaller probably due sampling error dimensions important test labelings single misclassification impact whereas dimension effect less severe figure experimental results dimension left dimension right displayed functional dependency top bottom solid lines depict theoretical boundaries whereas respective dotted lines display empirical results black lines display number samples labelings tested anymore random sample makes empirical results less reliable could improved future processing power observed empirical dimension extremely close twice empirical dimension expected theoretical derivations considering aforementioned practical shortcuts clarity result increases confidence validation experiments two hidden layers also performed experiments going deeper one layer expected linear increase capacity network fact case using small obtained results better one sample compared respective dimension one hidden layer architecture cases observed dimension actually far empirical values respective network one hidden layer empirical values dimension come quite close small numbers optimal value increasing number iterations tested interpretation utility datasets also detected three special cases worth pointing perceptron interpreted either unit mlp hidden nodes input models function input variables mensions found dataset done paper memory unit ple samples respectively could tries label sample points binary shattered cases tested digits choice interpretation makes difings sample values exactly one ference example interpreting neuron ple higher therefore binary functional unit one perceptron storage capabilities hidden layer hence able represent different functions output neuron making significant input variables weights task bution resulting learning capabilities able achieve number predicted memory capacity formulation functions grows quickly see also secin paper tion interpreting perceptron ory unit variables able model binary labelings points intuitively given hyperplane already separating space two different regions labeled one bit difference appears almost insignificant already discussed figure however seemingly insignificant difference allows one thing memory stick perfectly stores labelings bits able generalization assume points form two clusters promise hyperplanes used parametrize difference two clusters interpreting perceptrons memory units exclude generalization time allows understand limits better therefore like concur early assumptions understanding neural networks memory indeed beneficial application variation linear separability input data learning function dominate quality prediction network result least three practical applications first overflow capacity even single perceptron serves one definite explanation potentially catastrophic forgetting second experimental methodology including code appendix serves benchmark evaluation neural network implementations using points general position one test learning algorithm network architecture theoretical limit performance efficiency convergence rate third statistically speaking able compare networks simple however important understand bit counting edges comparing capacities derived neural networks total number samples example upper limits optimization binary function variables analysis says ods target functions feature extraction need network structure cessing initialization methods specialized edges mlp means chitectures work still need hidden neurons practical value furthermore explained less edges know proof constructive like dimension chance labeling claude shannon original work learned even higher discussing structure encoder quantifying maximum capabilities additionally main use neural networks conclusion generalization memorization therefore might acceptable lose bits long using information theoretic proof emare right ones pirical evidence show capacities work network architecture agnostic therefore implying deep flat networks capacity interpreted necessarily performance first hidden layer complete exposure input data layers exposure informationreduced output previous layers deeper layers capacity earlier layers assuming amount weights looking smaller input space deep layers free capacity countered results neural networks shortcut hidden layers directly connected input need overall less weights learn labeling still bounded capacity law however also easy show deeper hidden layer serve error correcting code previous layers example output previous hidden layer interpreted repetition code practically speaking concrete architecture dependent perceptrons neural network scale linearly number weights fact guaranteed perfect memorization accuracy neural network learn one bit per weight goal stretched bits per weight without much error error guaranteed large work extension initial work david mackay article first generalize critical points multiple perceptrons derive concrete scaling law show using repeatable experiments see appendix widely used open source framework linear scaling holds theoretical bounds actual upper bounds holds activation function tested including sigmoid relu future work dedicated phase dimension could physically interpreted liquid phase phase clear network make errors lose bits compared input error network complexity number ble configurations use neural networks typically memory abstraction assumed learning qualifying information loss concrete data would therefore allow reason generalization capabilities example insignificant bits noise lost part error network would back hypothesis one learn way avoid overfitting data highly conceivable weight values uniformly distributed especially deeply layered nets would mean capacity individual perceptrons within network overflow well consequence forget previously learned information based learning individual overflow cause interesting effects dynamics overall system gives one possible cause theoretical upper limit always reached investigating methods detect local overflow redirecting input neurons could interesting especially neural networks trained parallelized fashion challenges include understanding deeper hidden layers error correcting codes generalizing results networks complex activation functions radial basis networks finding bounds convolutional networks last least remains provide constructive proofs capabilities depending gating function network architecture unfortunately general finding explicit formula shows certain function implemented network hard even case binary functions would reduce problem boolean satisfiability acknowledgements work performed auspices department energy lawrence livermore national laboratory contract also partially supported lawrence livermore laboratory directed research development grant release number mario michael krell supported federal ministry education research bmbf grant fellowship within fitweltweit program german academic exchange service daad findings conclusions authors necessarily reflect views instead max funders want cordially thank rojas depth discussion chaining function also want thank alfredo metere jerome feldman kannan ramchandran jan hendrik metzen bhiksha raj naftali tishby jaeyoung choi friedrich sommer andrew feit insightful advise barry chen brenda support references information theory complexity neural networks ieee communications magazine november arpit ballas krueger bengio kanwal maharaj fischer courville bengio closer look memorization deep networks jun asian yildiz alpaydin calculating decision trees international symposium computer information sciences pages ieee sep bartlett maiorov meir almost linear dimension bounds piecewise polynomial networks advances neural information processing systems pages bartlett mendelson rademacher gaussian complexities risk bounds structural results journal machine learning research blumer ehrenfeucht haussler warmuth occam razor information processing letters broomhead lowe radial basis functions functional interpolation adaptive networks technical report royal signals radar establishment malvern united kingdom chollet keras https cook complexity theoremproving procedures proceedings third annual acm symposium theory computing pages acm coolidge story binomial theorem american mathematical monthly zhang ren sun delving deep rectifiers surpassing performance imagenet classification ieee international cover geometrical statisticonference computer vision iccv cal properties systems linear inequalpages ieee dec ities applications pattern recognition ieee transactions electronic kinzel phase transitions neural networks philosophical magazine part puters crammer dekel keshet kirkpatrick pascanu rabis singer online nowitz veness desjardins algorithms journal rusu milan quan ramachine learning research malho hassabis clopath kumaran hadsell overcoming catastrophic forgetting neural networks proceedings national academy sciences united states america volume pages national academy sciences mar dekel singer forgetron perceptron budget siam journal computing jan dzugutov aurell koiran sontag neural ani universal relation networks quadratic dimension entropy therjournal computer system sciences modynamical entropy simple liquids feb phys rev aug koiran sontag vapnikchervonenkis dimension recurrent neu feldman dynamic connections ral networks discrete applied mathematneural networks biological cybernetics ics aug dec friedland metere isomorphism lyapunov exponents shannon channel capacity arxiv preprint june https gardner maximum storage capacity neural networks epl europhysics letters krell generalizing decoding optimizing support vector machine classification phd thesis university bremen bremen krell new classifiers based origin separation approach pattern recognition letters feb liu nocedal limited memory bfgs method large scale optimization mathematical programming aug gardner space interactions neural network models journal physics mathematical general mackay information theory inference learning algorithms cambridge university press new york usa http gibbs scientific papers willard gibbs volume longmans green company goodfellow bengio courville mccloskey cohen catasdeep learning mit press trophic interference connectionist networks sequential learning problem harvey liaw mehrapsychology learning motivation bian bounds piecewise linear neural networks proceedings machine learning research morgan deep wide multiple layconference learning theory july ers automatic speech recognition ieee amsterdam netherlands transactions audio speech mar guage processing jan nair hinton rectified linear tishby opening units improve restricted boltzmann mablack box deep neural networks via inchines proceedings internaformation arxiv preprint tional conference international conference machine learning icml pages tishby zaslavsky deep learning information bottleneck principle usa omnipress ieee information theory workshop itw pages april pedregosa varoquaux gramfort michel thirion grisel blon tkacik schneidman berry del prettenhofer weiss dubourg michael bialek ising models vanderplas passos cournapeau networks real neurons arxiv preprint brucher perrot nay machine learning python journal machine learning vapnik nature statistical search feb learning theory springer ratcliff connectionist models vapnik chervonenkis nition memory constraints imposed uniform convergence relative frelearning forgetting functions psychoquencies events probabilities logical review apr theory probability applications jan rojas neural networks systematic introduction https vapnik levin cun measuring learning machine neural computation rojas deepest neural networks sep arxiv july http widrow generalization information storage network adaline neurons pages rosenblatt perceptron bilistic model information storage organization brain psychological zhang bengio hardt recht vinyals understanding deep learnview november ing requires rethinking generalization rumelhart hinton international conference learning repwilliams parallel distributed processresentations iclr ing explorations microstructure cognition vol chapter learning internal representations error propagation pages mit press cambridge usa rumelhart hinton williams neurocomputing foundations research chapter learning representations errors pages mit press cambridge usa theorie der vielfachen understanding machine learning theory algorithms cambridge university press new york usa shannon bell system technical journal mathematical theory communication code print correct rate input dimension numpy random number hidden layers numpy random range dataset size start one sample good results first label fixed zero min different random datasets range numpy random normal distributed data data numpy random normal numpy random range random int bin data converged false repeated runs till converged range mlpclassifier relu lbfgs predict true break short converged break shortcut miss append labelings correct break max print break variable definitions maximum number samples analyzed dimensions analyzed numbers hidden layers maximum number samples random labelings imports import import numpy import random import print correct rate input dimension numpy random number hidden layers numpy random range dataset size start one sample continue shortcut good results first label fixed zero min different datasets range numpy random data numpy random normal numpy random range index random int bin data converged false range mlpclassifier relu lbfgs predict true break short converged labelings correct break short success break short fail append break short success append max print break figure python code used measuring dimension one hidden layer figure python code used measuring dimension one hidden layer variable definitions maximum number samples analyzed dimensions analyzed numbers hidden layers maximum number samples random labelings imports import import numpy import random import
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finite complete rewriting systems finite derivation type automaticity homogeneous monoids may alan cain robert gray malheiro abstract paper investigates class presented monoids homogeneous relations computational perspective properties admitting complete rewriting system derivation type automatic biautomatic investigated class monoids main result shows consistent combination properties negations homogeneous monoid exactly combination properties introduce new concept abstract analogue notion abstract commensurability groups order extend result show statement holds even one restricts attention class nary homogeneous monoids every side every relation length introduce new encoding technique allows extend result partially class multihomogenous monoids introduction numerous interesting algebras arise semigroup algebras field homogeneous semigroup semigroup defined presentation relations examples include algebras yielding solutions equation quadratic algebras skew type see example algebras related young diagrams representation theory algebraic combinatorics plactic chinese algebras see algebras defined permutation relations see examples strong connections structure algebra underlying semigroup motivation studying class comes important semigroups literature admit homogeneous presentations hypoplactic monoid shifted plactic monoid monoids multihomogeneous growth plactic monoid trace monoids author supported fct fellowship later investigador fct fellowship second author partially supported epsrc grant special inverse monoids subgroups structure geometry rewriting systems word problem work partially supported para tecnologia portuguese foundation science technology project centro alan cain robert gray malheiro divisibility monoids queue monoids positive braid monoids investigating semigroup defined homogeneous relations associated semigroup algebra useful first step find good set normal forms canonical representatives generating set elements monoid thus elements algebra see list open problems section importance problem context semigroups defined permutation relations specifically would like set normal forms regular language want able compute effectively normal forms two situations good set normal forms exist monoids admit presentations finite complete rewriting systems see monoids semigroups automatic see properties also implications properties corresponding semigroup algebra indeed semigroup admits finite complete rewriting system semigroup algebra admits finite basis see explanation connection bases complete rewriting systems automaticity semigroup implies algebra automaton algebra sense ufnarovskij see section many examples homogeneous semigroups mentioned shown admit presentations finite complete rewriting systems shown biautomatic see example natural ask extent results generalise arbitrary homogeneous semigroups one ask every homogeneous semigroup admit presentation finite complete rewriting system every semigroup biautomatic within class homogeneous semigroups relationship admitting finite complete rewriting system biautomatic general semigroups properties independent see aim paper make comprehensive investigation questions fact shall consider two different strengths automaticity called automaticity biautomaticity shall also investigate homotopical finiteness property finite derivation type fdt sense squier finiteness property satisfied monoids admit presentations finite complete rewriting systems full definitions concepts given section various degrees homogeneity one impose semigroup presentation shall consider finite presentations homogeneous relations multihomogeneous letter alphabet every relation number occurrences letter equals number occurrences letter homogeneous fixed global constant every relation lengths words multihomogeneous simultaneously homogeneous multihomogeneous homogeneous monoids fcrs fdt biauto auto example see plactic monoid mfcrs auto mfcrs nonauto mfdt biauto mnonfdt biauto fdt mfcrs auto mbiauto fdt mbiauto mfcrs nonauto nonfdt mfcrs biauto auto nonfdt mfcrs nonauto mbiauto example example example example section section section section table summary examples homogeneous monoids exhibiting consistent combinations properties fcrs fdt biauto auto examples combinations properties also exist class homogeneous monoids see section fcrs fdt biauto auto exists see theorem theorem question theorem question question theorem theorem table summary existence examples multihomogeneous monoids consistent combinations properties fcrs fdt biauto auto examples combinations properties also exist class multihomogeneous monoids course restricted class listed class multihomogeneous presentations brevity introduce following terminology four properties interested monoid fcrs admits presentation via finite complete rewriting system respect finite generating set fdt finite derivation type biauto biautomatic auto automatic also use natural negated terms alan cain robert gray malheiro plactic monoid mfdt biauto mfcrs auto mfcrs nonauto mnonfdt biauto figure semilattice showing relationship examples taking free product two examples one obtains new monoid whose properties given taking logical conjunction operation corresponding properties original example monoids corresponds meet operation semilattice interested combinations properties homogeneous monoid since general fcrs implies fdt biauto implies auto combinations possible refer combination properties satisfies restrictions contain property negation consistent first main result shows consistent combination possible within class homogeneous monoids show constructing examples homogeneous monoids consistent combination properties adopt following naming scheme example monoid mab superscript one fcrs fdt nonfdt indicating monoid respectively fcrs thus also fdt fdt fcrs thus also subscript one biauto auto nonauto indicating monoid respectively biauto thus also auto auto biauto thus also section presents fdt fcrs nonfdt fundamental examples mfcrs auto mbiauto mnonauto mbiauto section contains general results behaviour various properties free products monoids use construct remaining examples results summarised table relationship various examples illustrated figure sections introduce new concepts prove new results order study combinations properties occur even restricted classes first introduce investigate notion abstract analogue abstract commensurability groups allows show every consistent combination arise within class homogeneous monoids thus table figure could also describe situation homogeneous monoids develop new encoding technique embeds homogeneous monoids homogeneous monoid multihomogeneous monoid encoding technique allows obtain consistent combinations properties class multihomogeneous multihomogenous monoids specifically allows construct multihomogeneous monoids possile combination properties fcrs biauto auto combination properties fdt biauto auto however allow construct examples separate properties fcrs fdt within class multihomogeneous multihomogeneous monoids table summarises known consistent combinations properties class multihomogeneous monoids using results abstract rees commensurability table also describes situation class multihomogeneous monoids preliminaries subsection derivation graphs homotopy bases finite derivation type complemented alternative formulation concepts terms strict monoidal variations homotopical algebra higher categories however approach using squier complexes one used papers otto wang pride second third authors third author require methods results papers sections information automatic semigroups see assume familiarity basic notions automata regular languages see example transducers rational relations see example although recall key results use frequently background string rewriting systems refer reader words rewriting systems presentations denote empty word alphabet alphabet denote set words generating set monoid every element interpreted either word element words write indicate equal words denote represent element monoid length denoted number symbols denoted denote urev reversal word urev relation denotes smallest monoid congruence generated use standard terminology notation theory string rewriting systems see background reading monoid presentation pair isomorphic quotient case elements called defining relations write set words equal presentation homogeneous respectively multihomogeneous every respectively homogeneous presentation defining relations preserve length multihomogenous presentation defining relations preserve alan cain robert gray malheiro number symbol monoid homogeneous respectively multihomogeneous admits homogeneous respectively multihomogeneous presentation note homogeneous multihomogeneous presentations required finite presentations string rewriting system simply rewriting system pair finite alphabet set pairs usually written known rewriting rules simply rules drawn single reduction relation defined follows exists rewriting rule words xry one obtain substituting word subword rewriting rule reduction relation reflexive transitive closure subscript omitted clear context process replacing subword word rule called reduction application rule iteration process also called reduction word reducible contains subword forms side rewriting rule otherwise called irreducible rewriting system finite finite rewriting system noetherian infinite sequence words noetherian process reduction must eventually terminate irreducible word rewriting system confluent words pair resolves exists word well known noetherian system confluent critical pairs resolve critical pairs obtained considering overlaps sides rewrite rules see details rewriting system confluent noetherian complete monoid admits presentation respect generating set forms finite complete rewriting system monoid fcrs case irreducible elements form set unique normal forms elements monoid thue congruence equivalence relation generated elements monoid presented classes relations coincide let homogeneous monoid let homogeneous presentation without lost generality assume trivial relations form letters since none generators represented decomposed alphabet represents unique minimal generating set generating set must contain minimal generating set two words representing element must length function defined length word representing elsewhere shall write function symbol right easy see homomorphism following definition function called grading homogeneous monoids graded monoids homogeneous monoids derivation graphs homotopy bases finite derivation type associated monoid presentation called squier complex whose vertex set edges corresponding applications relations adjoined instance applications relations see formal definition relations free monoid acts natural way via left right multiplication collection closed paths called homotopy base complex obtained adjoining cells paths generate action free monoid squier complex trivial fundamental groups monoid defined presentation said finite derivation type fdt short corresponding squier complex admits finite homotopy base shown squier property fdt independent choice finite presentation may speak fdt monoids original motivation studying notion squier result says monoid admits presentation finite complete rewriting system monoid must finite derivation type study concepts motivated fact fundamental groups connected components squier complexes called diagram groups turned interesting class groups see recent important work acyclic polygraphs used define homotopical finiteness condition higher categories particular work gives rise definition fdtn extends notion finite derivation type arbitrary dimensions detail monoid presentation associate graph sense serre follows derivation graph infinite graph vertex set edge set consisting collection functions associate edge initial terminal vertices respectively mapping associates edge inverse edge path sequence edges written diagrammatic order path extend mappings paths defining inverse path path path closed path path satisfying two paths composition defined denote set paths vertex include path edges called empty path free monoid acts sides set edges extends naturally action path define alan cain robert gray malheiro figure disjoint derivations paths say parallel write use denote subset pairs parallel paths equivalence relation called homotopy relation contained satisfies following four conditions edges implies implies denotes empty path vertex idea behind condition following suppose word two disjoint occurrences rewriting rules sense let paths give two different ways rewriting word word first apply relation relations applied opposite order see figure want regard two paths essentially achieved condition equivalent paths condition said homotopic disjoint derivations relation also often refereed exchange relation interchange law literature see subset homotopy relation generated smallest respect inclusion homotopy relation containing homotopy relation generated empty set denoted coincides called homotopy base presentation said finite derivation type derivation graph admits finite homotopy base finitely presented monoid said finite derivation type fdt hence theorem finite presentation finite derivation type homogeneous monoids difficult see subset homotopy base set homotopy base thus say set closed paths homotopy base corresponding set homotopy base rational relations references purposes briefly recall basic definitions results regarding rational relations consider relations form set rational relations smallest subset contains empty set singleton sets closed operations union product kleene star note set rational relations also closed kleene plus operation proposition examples regular language relation rational proposition let regular languages rational relation rational relation particular rational relation proof let transducer recognizing let finite automata recognizing respectively adapt simulate inputs first second tapes respectively accept simulated copies accept states adapted transducer recognizes automaticity biautomaticity definition let alphabet let new symbol define mapping mapping alan cain robert gray malheiro definition let finitely generated monoid let finite set generators let regular language every element least one representative define relations pair automatic structure regular language monoid automatic auto admits automatic structure respect finite generating set pair biautomatic structure regular languages monoid biautomatic biauto admits biautomatic structure respect finite generating set note biauto implies auto hoffmann thomas made careful study biautomaticity semigroups distinguish four notions biautomaticity semigroups require least one least one regular notions equivalent groups generally cancellative semigroups theorem distinct semigroups remark biauto clearly implies four notions biautomaticity however shall shortly prove within class homogeneous monoids notions biautomaticity implies biauto see proposition proving regular relation useful strategy prove rational relation relation recognized finite transducer theorem apply following result combination corollary proposition proposition rational relation constant regular shall prove results automaticity biautomaticity class homogeneous monoids unlike situation groups automaticity biautomaticity monoids semigroups dependent choice generating set example however monoids biautomaticity automaticity independent choice semigroup generating sets theorem particular case homogeneous monoids independence choice generating set proposition let homogeneous monoid auto respectively biauto finite generating set language automatic respectively biautomatic structure proof first consider case auto suppose automatic structure notice alphabet alphabet must contain subalphabet representing unique minimal generating set without homogeneous monoids loss generality assume contain alphabet representing minimal generating set let let relation since simply subset obtained taking kleene star finite set elements form definition rational relation let let since composition rational relations rational theorem follows rational relation furthermore thus regular language proposition let kum regular proposition similarly regular hence automatic structure biauto assume biautomatic structure follow reasoning show languages regular proposition let homogeneous monoid let finite generating set let regular language every element least one representative least one least one regular biauto proof suppose regular cases similar proof proposition alphabet must contain unique minimal generating set construct relation proof proposition let let least one least one regular particular rational relations rational relations hence regular proposition since arbitrary proves biautomatic structure alan cain robert gray malheiro despite positive results obtained far note auto imply biauto class homogeneous monoids shall see example fundamental examples fcrs fdt auto homogeneous monoid subsection present homogeneous monoid fcrs thus fdt auto biauto considering reversal semigroup example get homogeneous monoid admits finite complete rewriting system automatic example let mfcrs auto monoid defined presentation consists rewriting rules cxyz cxcz cbca cacb proposition monoid mfcrs auto fcrs proof rewriting system noetherian every rewriting rule either decreases number symbols stays decreases number symbols left symbols see confluent notice overlaps side cbca cacb side rule form cayz cacz however resolve since cacbyz cacbcz cbcayz cbcacz cacbcz therefore confluent proposition monoid mfcrs auto auto proof let language normal forms since finite regular lemma let consider following cases separately must also normal form since side rewriting rule ends hence rational proposition normal form must end side rewriting rule hence cxy cxyb cxcb word cxcb normal form since prefix normal form rewriting rule side cxcb thus cxy cxcb cxy cxy cxcb rational proposition homogeneous monoids normal form must end side rewriting rule either cbc cxy maximal since cbca cacb normal form since normal form side rewriting rule ends cbca maximal cay caya caca word normal form since normal form therefore cay caca cay cay caca union relations rational proposition note also rational hence rational relation moreover lies one relations regular propositions hence mfcrs auto auto suppose aim obtaining contradiction mfcrs auto biauto proposition admits biautomatic structure thus regular language mapping onto mfcrs auto regular contradicts lemma lemma regular language maps onto mfcrs auto regular proof suppose aim obtaining contradiction language exists regular let even number exceeding number states automaton recognizing observe notice represented word similarly represented word furthermore cbn alan cain robert gray malheiro since exceeds number states automaton recognizing apply pumping lemma segment word lies within first letters subword form see hence definition relation normal forms unequal contradicts previous equality thus biauto fcrs fdt homogeneous monoid definition let monoid defined presentation denote rev monoid defined presentation rrev rrev lrev rev called reversal monoid note rev rev fcrs rev defined presenexample let mfcrs nonauto mauto tation rrev presentation defining example rev argue proof since mfcrs nonauto presented proposition rewriting system noetherian overlaps result critical pairs resolve similar way thus mfcrs nonauto also fcrs thus fdt proposition mfcrs nonauto proof suppose aim obtaining contradiction mfcrs nonauto auto let automatic structure mfcrs nonauto fcrs regular since mnonauto homogeneous regular corollary notice rev lrev hence lrev regular language mapping onto mfcrs mfcrs lrev regular since mfcrs nonauto auto nonauto fcrs contradicts lemma mnonauto indeed fdt biauto auto homogeneous monoid following homogeneous monoid introduced katsura kobayashi example showed fdt shall prove biauto thus auto example let let consist rewriting rules abi homogeneous monoids fdt let mfdt biauto mbiauto fdt proposition proposition monoid mfdt biauto example biauto thus auto proof let consist following rewriting rules abi notice every rule consequence indeed using rules consider ordering satisfying lemma order induced noetherian moreover rewriting using rule decreases word respect ordering thus rewriting system noetherian see confluent notice two possible overlaps sides rewriting rules overlap overlap however critical pairs resolve since let language words thus regular prove automatic structure mfdt biauto first show rational relations since mfdt biauto relations equal equality relation hence trivially rational let suppose first may side rewriting rule type rightmost end word case must form since prefix irreducible word irreducible applying rewriting rule yields normal form since alan cain robert gray malheiro side rewrite rule contains except clearly applied one application rewrite rule rightmost end turns normal form word hence lbj lcj lcj hence lbj rational relation proposition suppose reasoning similar previous paragraph shows normal form one application rewrite rule type turns normal form note application rule type might followed one type replaced one type hence ldj hence ldj rational relation proposition suppose rewriting rules apply sequence rule type rewriting bik bik resulting word normal form since way rewriting rule could apply means word would contain contradicts hence bjk bjk bjk finally already normal form hence case case rational relation since mfdt biauto homogeneous furthermore hence regular proposition similar reasoning shows rational relation rewriting normal form consist sequence applications rules type followed possibly one type one rewriting rule required proposition applies show regular hence biautomatic structure mfdt biauto homogeneous monoids biauto auto homogeneous monoid section give example homogeneous monoid thus biauto thus auto example let let rewriting system consisting three rules cab cbb let presentation let mnonfdt biauto monoid presented theorem monoid mnonfdt biauto set unique normal forms set generating set correspondence elements mnonfdt biauto biauto thus auto thus part theorem follow lemma part proved lemma rest subsection devoted proving mnonfdt biauto thus establishing part remark methods use prove example notfdt similar used proof theorem particular use notion critical peaks resolution critical peaks proof refer reader section definitions concepts connection complete rewriting systems fdt let begin fixing notation start adding infinitely many rules form cuab cubb denote set rules notice first precisely rule defined word words cuab cubb represent element monoid mnonfdt biauto since word cuab use relations form pass letter word left right replace cab cbb using relation finally move back right left using relations follows presentations equivalent presentations sense two words equivalent modulo relations equivalent modulo relations particular monoid mnonfdt biauto also defined infinite presentation lemma infinite presentation complete presentation mnonfdt biauto set irreducible words respect complete rewriting system alan cain robert gray malheiro proof fact presentation mnonfdt biauto follows comments made statement lemma considering ordering induced one sees rewriting system noetherian set irreducible words rewriting system set indeed irreducible word contains symbol left symbol otherwise could apply relation moreover word also contains symbol symbols must right rightmost symbol since otherwise could use relation form finally prove confluent suffices consider possible overlaps sides rewriting rules showing critical peaks arising overlaps resolve see section three different ways rewrite rules overlap giving rise three types critical peaks resolved see figure proves confluent thus completes proof lemma lemma monoid mnonfdt biauto biauto proof let prove biautomatic structure mnonfdt biauto previous lemma regular language every element mnonfdt biauto unique representative hence thus rational relation let regardless whether word also lies hence rational relation proposition also lies hand via sequence applications rules hence rational relation proposition cak using sequence applications rules contains least one symbol bak bak bak sequences applications rules sequence applications rules hence cak ubak homogeneous monoids rational relation similar reasoning shows cak ubak thus rational since implies proposition shows images regular hence biautomatic structure mnonfdt biauto let denote derivation graph derivation graph let denote connected components vertex set set words least two occurrences letter likewise let connected component vertex set three infinite families closed paths displayed figure correspond resolutions critical peaks closed path obtain shall call critical circuit let denote critical circuits form denote critical circuits form observe critical circuits since words labelling vertices contain two occurrences letter set critical circuits forms infinite homotopy base see lemma want use infinite homotopy base obtain infinite homotopy base order need take critical circuits transform circuits derivation graph replacing occurrence edge corresponding path mentioned proving equivalent presentations edges realized paths defined alan cain robert gray malheiro cuabc xcuab uab cxuab xcubb ubb cua cxubb cubbc cuacb cub cubcb cucab cucbb cuabvab uabv vab cubbvab cuabvbb ubbv vbb cubbvbb figure resolutions critical peaks derivation graph presentation inductively follows first set rule set path path cuab cubb given commuting using relations applying relation transform ucab ucbb commuting back using relations ending vertex cubb let define mapping set paths set paths let map given defined identity every edge let since forms homotopy base follows infinite homotopy base see corollary observe infinite homotopy base nothing set circuits obtained taking set circuits replacing occurrence edge path defined let denote corresponding set circuits monoid mnonfdt biauto presented finite presentation derivation graph infinite homotopy base lemma discussion definition mnonfdt biauto fdt would finite subset would finite homotopy base sketch since finite homotopy base path homotopic empty path using finitely many paths thus finite subset consisting paths arising way homotopy base homogeneous monoids aim show leads contradiction thus conclude mnonfdt biauto fdt order shall define mapping set paths integral monoid ring define unique map extends mapping nonfdt mnonfdt biauto denotes element mbiauto represented word paths way following basic properties easily verified paths words property follows properties note property implies induces map homotopy classes paths follows shall often omit bars top words images simply write words obvious intended meaning recall set critical circuits form corresponding set circuits let denote free monoid alphabet lemma mnonfdt biauto fdt submodule left generated finitely generated left proof assume mnonfdt biauto fdt therefore fdt since homotopy base derivation graph lemma finite subsets finite homotopy base let arbitrary claim established prove lemma since finite subset lemma since closed path homotopy base write since vertices exactly one relations presentation involve letter follows applying gives claimed complete proof remains compute subset prove submodule finitely generated left recall set critical circuits form alan cain robert gray malheiro set closed paths obtained applying mapping closed paths obtained taking occurrence replacing path equation word equality thus deduce using fact result computing critical circuit map given critical circuit family critical circuit family next lemma completes proof theorem lemma submodule finitely generated left therefore mnonfdt biauto fdt proof suppose aim obtaining contradiction finitely generated left exists finite subset hxizf let maximal length word shall show belongs hxizf suppose either abvj bbvj free monoid clearly contradicts fact conclude finitely generated left follows lemma mnonfdt biauto fdt free products homogeneous monoids section gave four examples homogeneous monoids possess certain combinations properties fcrs fdt biauto auto section use free products construct examples remaining consistent combinations properties note monoids homogeneous presentations free product defined presentation thus also homogeneous first consider interaction free product biauto auto known free product two monoids auto monoids auto theorem theorem would possible extend result biautomaticity general monoids generalization appear literature paper required biautomaticity result homogeneous monoids proofs simpler case homogeneous monoids proposition let homogeneous monoids auto respectively biauto auto respectively biauto proof let homogeneous presentations respectively suppose auto proof biauto similar proposition admits automatic structure since homogeneous word representing identity hence every word representing element must lie thus automatic structure similarly automatic structure hand suppose auto proof biauto similar automatic structures respectively corollary assume without loss generality every element unique representative respectively let note disjoint union languages note language equal every word product words strictly alternating words words since every element represented unique element rational relation let let unique word equal note either another generator equal suppose lies uva unique word equal hand lies form maximal suffix lying case unique word equal unique word equal hence alan cain robert gray malheiro thus rational relation since homogeneous furthermore hence regular proposition thus automatic reasoning essentially proof theorem simplified consider homogeneous semigroups consider interaction free product fcrs within class homogeneous monoids theorem theorem let monoids suppose elements fcrs fcrs proposition let homogeneous monoids fcrs fcrs proof suppose fcrs since homogeneous neither contains elements fcrs theorem converse part follows proposition finally recall following result interaction free product fdt theorem let monoids fdt fdt examples section suffice construct remaining examples figure taking free products regarding figure semilattice examples section correspond elements semilattice decomposition proposition theorem proposition together show taking free product one obtains new monoid whose properties given taking logical conjunction operation corresponding properties given two monoids mab mcd free product mab mcd weaker two properties lie fcrs fdt nonfdt weaker two properties lie biauto auto nonauto thus summarize fdt mfcrs auto mbiauto fdt auto fdt mbiauto mfcrs nonauto fdt nonfdt auto mfcrs biauto auto nonfdt mfcrs nonauto mbiauto theorem consistent combination properties fcrs fdt biauto auto negations exists homogeneous monoid exactly combination properties homogeneous multihomogeneous monoids thus far proved every consistent combination properties fcrs fdt biauto auto negations exists homogeneous monoid exactly properties remainder homogeneous monoids paper show monoids exist restricted class homogenous monoids show monoids consistent combinations properties exist even restricted class multihomogeneous monoids current section describes overall strategy results following two sections develop necessary concepts techniques first introduce investigate theory abstractly semigroups section ultimately proving corollary implies homogeneous monoid listed table obtain homogeneous monoid combination properties thus following analogy theorem homogeneous monoids theorem consistent combination properties fcrs fdt biauto auto negations exists homogeneous monoid exactly combination properties since examples homogeneous monoids consistent combinations properties fcrs fdt biauto auto negations next step extend examples multihomogeneous monoids aim section define investigate embedding homogeneous monoid multihomogeneous monoid stated corollary passing multihomogeneous monoid preserves fdt auto biauto furthermore passing multihomogeneous monoid preserves fcrs unknown whether fcrs preserved passing back original homogeneous monoid equivalently whether nonfcrs preserved passing multihomogeneous monoid applying embedding technique list homogeneous monoids discussed get following results theorem consistent combination properties fcrs biauto auto negations exists multihomogeneous monoid exactly combination properties theorem consistent combination properties fdt biauto auto negations exists multihomogeneous monoid exactly combination properties obtain analogue theorem multihomogeneous monoids would sufficient find multihomogeneous monoid fdt biauto indeed case combining theorems results section noting free product multihomogeneous monoids multihomogeneous would get consistent combination biauto auto negations example fdt multihomogeneous monoid exactly combination properties therefore corollary would get examples multihomogeneous monoids exactly discussed properties joining examples examples theorems would get intended result thus following question question exist multihomogeneous monoid fdt alan cain robert gray malheiro abstractly semigroups introduce new definition inspired notions abstractly commensurable groups rees index semigroups subsemigroup given semigroup finite rees index finite case semigroup said small extension large subsemigroup main interest large subsemigroups small extensions given semigroup share many important properties semigroup see survey definition two semigroups said abstractly reescommensurable finite rees index subsemigroups easy verify abstract equivalence relation semigroups notion naturally extended ideals definition two semigroups said abstract finite rees index ideals idea behind notions abstract semigroups share many important properties fcrs fdt biauto auto proposition fcrs fdt biauto auto preserved abstract known fcrs inherited small extensions theorem large subsemigroups theorem although result small extensions stated context monoids naturally extended semigroups two results imply fcrs preserved abstract rees known fdt inherited small extensions theorem monoids large semigroup ideals theorem recall notion finite derivation type first introduced monoids naturally extended semigroup case section result small extensions easily adapted semigroup case indeed let semigroup consider monoid obtained adding identity fdt derivation graph obtained derivation graph adding extra connected component single vertex corresponding empty word thus fdt small extension semigroup turns monoid small extension monoid therefore theorem monoid fdt large ideal theorem conclude fdt two results small extensions large ideals semigroups show fdt preserved abstract proof homogeneous monoids theorem natural analogue biauto auto biauto inherited small extensions large subsemigroups therefore auto biauto preserved abstract rees preceding result important case multi homogeneous multi homogeneous monoids following result holds proposition every finitely presented multi homogeneous monoid multi homogeneous monoid chosen arbitrarily long greater equal length longest relation proof let finite homogeneous presentation monoid let chosen arbitrarily provided greater equal maximum length relation going construct nary multi homogeneous monoid show abstract finding isomorphic ideals respectively let note ideal finite rees index since complement finite set let urv consider multi homogeneous presentation let monoid defined set ideal moreover finite rees index since contained thue congruence generated define map given thus definition therefore surjective also injective since get therefore routine check homomorphism thus isomorphic finite rees index ideals abstract corollary let set properties preserved abstract exists multi homogeneous monoid satisfying every property exists multi homogeneous monoid satisfying every property note set properties contain negative properties like finitely generated immediate consequence obtain theorem homogeneous multihomogeneous monoids develop embedding technique allows construct multihomogeneous examples technique embeds homogeneous monoid multihomogenous monoid shares several properties alan cain robert gray malheiro study original monoid homogeneous monoid embedded multihomogenous use formulations homogeneous multihomogeneous indicate statements apply general cases let start fixing notation maintained throughout section let finitely generated homogeneous monoid let homogeneous presentation defining recall minimal generating set contains see discussion end subsection define homomorphism denote finitely generated monoid presented denotes set proposition monoid defined presentation multihomogeneous proof presentation homogeneous hence since contains symbols symbols follows hence multihomogeneous presentation homogeneous also relation case get similarly subset code set free generators submonoid generated subsection case set furthermore called prefix code since word proper prefix another word chapter proposition since induces bijection code conclude injective reason called coding morphism proposition monoid presented embeds monoid presented via map words representing elements image precisely words proof consider natural projection onto whose kernel generated since injective kernel composition congruence generated natural projection onto kernel map theorem monomorphism shall turn attention investigating relationship fdt holding fdt holding shall prove results relate squier graphs homogeneous monoids let denote subgraph induced set vertices let extend mapping mapping derivation graph derivation graph mapping edge edge mapping extends mapping paths putting edges lemma mapping following properties maps bijectively every edge every edge every pair words maps bijectively proof follows fact injective homomorphism thus maps bijectively image iii identities follow definition extended map edges fact homomorphism injectivity follows injectivity consider edge suppose without lost generality word decomposes concatenation words words form various since subwords occur start words start lies start word decomposition since words length must also finish end word decomposition hence possibly empty concatenations words thus edge let right inverse injective mapping note elements part lemma extend mapping edges simply setting using fact every edge form unique edge lemma mapping following properties maps bijectively every edge alan cain robert gray malheiro every edge every pair words edge case maps bijectively proof follows immediatly definition lemma edge form image edge since homomorphism definition get required iii suppose edge edge vertex word arguing lemma conclude converse part equivalence follows trivially exists definition get follows definition lemma lemmas regard given encoding graph subgraph identifying lemma let set closed paths set closed paths moreover every closed path proof result follows theorem point strategy starts emerge identification derivation graph allow get finite homotopy base finite homotopy bases lemma let set closed paths set closed paths moreover every closed path proof result follows lemma author uses strict monoidal categories prove analogue theorem indeed context strict monoidal categories notice derivation graph refer theorem prove lemma mapping functor preserves multiplicative structure lemma homotopy relation called satisfying closed path lemma follows closed path next describe remaining connected componnets lemma derivation graph following properties homogeneous monoids vertex maximal factors respect length either vertex unique decomposition form maximal factor factor edge form edge maximal factors corresponding initial terminal vertices proof let maximal factors suppose overlap word without lost generality suppose since sequence words form various subwords occur start words start lies start word decomposition maximality since get since words length must also finish end word maximality get result follows iii definition edge initial terminal vertices words one obtained single application defining relation notice consider maximal factors factors respectively way words initial terminal vertices edge result follows statement lemma path one forms path edges proof lemma iii path consecutive edges form conditions lemma words maximal factors word lemma three possible cases cases corresponds one possible cases iii statement lemma let closed path alan cain robert gray malheiro form closed path proof key observation prove lemma note lemma paths items iii indeed consequence disjoint derivations lemma initial vertex factorized form maximal factor factor notice also lemma edge factorization form edge maximal factors corresponding initial terminal vertices since factor relation going applied hence fixed path also identify set edges relation applied word lemma two consecutive edges get disjoint derivations consequently taking group together edges finding path edges beginning path proceeding way remaining edges get intended result following result viewed generalization lemma proposition monoid fdt monoid fdt proof suppose fdt let finite homotopy base closed paths shall see finite set closed paths homotopy base let closed path lemma closed paths since homotopy base thus lemma turn implies consequently finite homotopy base fdt conversely suppose fdt let finite homotopy base closed paths lemma closed paths homogeneous monoids denote finite set let denote set definition paths satisfy closed path satisfies thus set generates homotopy relation therefore also finite homotopy base note set closed paths conclude proof let closed path since homotopy base lemma get since get required therefore finite homotopy base fdt proposition monoid auto respectively biauto monoid auto respectively biauto proof prove result biauto result auto follows considering multiplication one side suppose biauto proposition biautomatic structure proposition words representing elements image precisely must map onto image map rational relation since thus converse also rational relation let note since rational relation regular follows also regular since must map onto image language maps onto thus since automatic structure relation rational since rational relations follows rational relation since homogeneous regular proposition symmetrical reasoning shows regular hence biautomatic structure suppose biauto proposition biautomatic structure aim construct biautomatic structure let consists words contain subwords equal elements notice particular closed taking subwords note regular future use let note rational relation proposition alan cain robert gray malheiro consider word lemma uniquely factored alternating product words words lies lies except may also note equal word idea build automatic structure language representatives consists alternating products words words language consist words lies formally let note regular since regular homomorphism every element equal word reasoning previous paragraph let factor lies lies except may also consider generator since contain subword since contains subword subwords end symbols words equal precisely words form word equal recally defining relations apply words thus subwords fixed hence rational relation consider generator may end subword thus necessary distinguish three cases case factorization alternating product subwords note since closed taking subwords words equal precisely words form word equal word equal let regular proposition fact since homogeneous contain generators must finite let since regular rational relation proposition rational relation since concatenation rational relations rational relation describes generator case homogeneous monoids case factorization alternating product subwords note last factor words equal precisely words form word equal word equal word lai let lai rational relation describes generator case end suffix form words equal precisely words form word equal let rational relation describes generator case thus rational relation since homogeneous regular proposition similar reasoning shows hence biautomatic structure proposition finite presentation finite presentation defining moreover presentation complete presentation complete thus fcrs fcrs proof using tietze transformations obtain presentation new presentation follows insert generator relation thus obtaining tietze equivalent presentation since homomorphism identifying symbol word performing substitutions words obtain another tietze equivalent presentation defining monoid alan cain robert gray malheiro possible obtain presentation presentation using finitely many tietze transformations obtain presentation presentation using tietze transformations suppose also complete observe relates words alphabet relation set side side thus find indeed since word factor alphabet since word obtained changing factor thus word also contains side relation applied means sides relations applied lefthand sides overlap alternatively apply first relation obtaining word apply relation obtaining word hence theorem rewriting system terminating terminating assumption terminating also terminating since relates words alphabet relations side side also deduce whenever exists therefore relations commute lemma confluent commute also confluent since assumption confluent remains show confluent observe two sides rules overlap since injective get therefore critical pairs thus also confluent conversely suppose complete presentation since contained deduce terminating confluence also holds fact critical pairs resolved particular arising relations hence resolution associated relations involve relations since sides rules belong therefore complete unknown property fcrs preserved true would example satisfying conditions question combining propositions 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madlener automatic monoids versus monoids convergent presentations nipkow rewriting techniques applications lecture notes comput springer squier otto kobayashi condition rewriting systems theoret comput sci harpe topics geometric group theory university chicago press otto kobayashi properties monoids presented convergent systems survey eds advances algorithms languages complexity kluwer academic publishers dordrecht kobayashi otto homotopical homological conditions presented monoids international journal algebra computation lafont new condition monoids presented complete rewriting systems craig squier journal pure applied algebra guiraud malbos categories derivation type theory appl categ otto modular properties monoids nehaniv chrystopher algebraic engineering proceedings international workshop formal languages computer systems kyoto japan march proceedings international conference semigroups algebraic engineering held aizu japan march singapore world url http homogeneous monoids otto properties monoids modular free products certain free products amalgama url http wang finite complete rewriting systems finite derivation type small extensions monoids algebra gray malheiro pride properties inherited monoids groups information computation malheiro finite derivation type large ideals semigroup forum hopcroft ullman introduction automata theory languages computation edition reading berstel transductions languages der angewandten mathematik und mechanik teubner stuttgart baader nipkow term rewriting cambridge university press droste kuich vogler handbook weighted automata edition springer publishing company incorporated guba sapir diagram groups mem amer math soc guiraud malbos normalisation strategies acyclicity advances mathematics serre trees thomas biautomatic semigroups liskiewicz reischuk eds fundamentals computation theory lecture notes computer sciene springer frougny sakarovitch synchronized rational relations words theoret comput sci duncan robertson automatic monoids change generators mathematical proceedings cambridge philosophical society thomas notions automaticity semigroups semigroup forum katsura kobayashi constructing presented monoids complete presentation semigroup forum pride wang rewriting systems conditions associated functions algorithmic problems groups semigroups lincoln trends boston boston jura determining ideals given index presented semigroup demonstratio math cain maltcev elements survey rees index tech wong wong complete rewriting systems large subsemigroups algebra malheiro finite derivation type rees matrix semigroups theoret comput sci thomas automatic semigroups subsemigroups ress index international journal algebra computation howie fundamentals semigroup theory london mathematical society monographs new series clarendon press oxford university press new york kobayashi finite homotopy bases monoids algebra alan cain robert gray malheiro bachmair dershowitz commutation transformation termination siekmann international conference automated deduction vol lecture notes computer science springer berlin heidelberg centro faculdade tecnologia universidade nova lisboa caparica portugal address school mathematics university east anglia norwich united kingdom address departamento centro faculdade tecnologia universidade nova lisboa caparica portugal address ajm
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proposed algorithm minimum vertex cover problem testing gang email abstract paper presents algorithm minimum vertex cover problem problem algorithm computes minimum vertex cover input simple graph tested attached matlab programs stage algorithm applicable yields proved minimum vertex cover tested graphs order tested graphs order stage algorithm applicable tested graphs tested graphs randomly generated graphs random edge density words random probability edge proved stage stage algorithm run time respectively order input graph theoretical proof yet stage applicable graphs stages algorithm proposed general form consistent stages introduction algorithm classified one algorithm design technique strategy maybe suitable suppose need find minimum vertex cover simple graph first part algorithm generate auxiliary simple graph satisfies following four conditions denotes label set set vertex labels different vertices may share label edge however two endpoints different labels component cardinal number thus number called grade component denoted nonempty subset exists maximal clique remark generated steps section graph satisfies conditions remaining part algorithm find maximal clique whose label set proved label set minimum vertex cover graph let neighbor set vertex suppose subgraph satisfies following conditions condition vertex subgraph exists vertex cover condition edge subgraph exists vertex cover component ascending order grade defined condition find maximal subgraph satisfies conditions iterate computation next component nonempty find subgraph contains nonempty subgraph satisfies conditions clique proved claim label set minimum vertex cover graph clique stage algorithm introduced section section stage algorithm proved run time algorithm tested matlab programs found clique tested graphs claim stage algorithm yield minimum vertex cover tested graphs test results detailed section graphs stage algorithm applicable stronger version condition introduced change algorithm reaches stage graphs stage applicable found time stage applicable yields minimum vertex cover tested graphs stage stage except condition replaced stronger version stage actually works tested graphs furthermore exist graphs stage algorithm applicable graphs stages algorithm expressed general form consistent stages stages introduced section steps algorithm firstly graph constructed steps step suppose smallest integer satisfies let define family sets follows definition let first member partition two disjoint sets whose elements consecutive numbers let two sets become members two new members cardinal number larger continue partition process described keep process cardinal number new member also generated reversed way shown attached matlab program example step generate one one component whose label denote component remark grade component defined condition section obvious set components actually set isolated vertices step components generated sequence grades example components constructed construction component components grade larger defined general form follows firstly another definition needed definition suppose member defined definition positive integer let component satisfies call union denote see fig figure examples unions component larger defined follows definition join two unions satisfy following two conditions positive integers let one components denoted corresponds first half containing smaller numbers satisfy conditions generated start construct components grade component generated construction complete call graph claim nonempty subset member satisfies exists maximal clique proof use induction claim obviously holds induction step let subset definition suppose minimal member includes suppose also members let since minimal member includes obvious induction hypothesis exists maximal clique exists maximal clique definition exists maximal clique component minimal member includes member includes includes also component thus exists maximal clique need claim follows understand next step claim subgraph one maximal subgraph satisfies conditions section proof assume two maximal subgraphs satisfy conditions let however obvious also satisfies conditions contradictory assumption step component ascending order grade defined condition section deleting minimal subgraph component get maximal subgraph satisfies conditions suppose grade current component empty iterate computation next component whose grade also otherwise components grade computed nonempty record next step claim follows claim recorded step size maximum clique larger minimum size graph vertex cover proof suppose minimum vertex cover claim exists maximal clique thus satisfies conditions exists component nonempty subgraph satisfying conditions component computed ascending order grade claim maximal subgraph satisfies conditions recorded nonempty step size maximum clique larger step recorded step deleting maximal subset get subgraph contains nonempty subgraph satisfies conditions record next step remark claim tell whether contains nonempty subgraph satisfies conditions finding one maximal subgraph satisfies two conditions recorded step claim follows claim clique label set minimum vertex cover graph proof step contains nonempty subgraph satisfies conditions condition tell exists vertex cover suppose size maximum clique recorded step clique subgraph claim larger minimum size graph vertex cover thus label set minimum vertex cover thus step check whether recorded step clique clique output clique next stage algorithm remark sections explain next stage algorithm claim efficiency algorithm proof two components one component use induction assume claim holds steps algorithm suppose two isomorphic graph satisfy induction hypothesis isomorphic suppose component steps algorithm tell also component thus corollary let component also component need count components neither component suppose kind component definition tell thus definition formula integer means copied times generate different components copied times construct conclusion combine conclusion corollary thus induction succeeds claim proved claim suppose graph order input running time stage algorithm proof step smallest integer satisfies claim log thus asymptotic upper bounds respectively construction runs time step running time computing subgraphs components conditions still also case step therefore stage algorithm takes time testing algorithm algorithm tested two attached programs matlab trial use first file generates saves second file second file generates random graphs random edge density random probability edge testing algorithm edge density program approximately equals ratio complete graph generated normal distribution mean standard deviation testing reasons introduction distribution exist graphs matter labelled unlabeled edge density equals exists amount graphs edge density edge density ordinary personal computer used testing testing parameters results listed follows number tested graphs table testing parameters results standard approximate graphs deviation running time stage edge density applicable minutes hours hours hours hours hours hours hours hours graphs stage applicable remark stage explained section remark running time computer used testing long impossible test large number graphs short period test data reason table recorded edge densities graphs stage applicable recorded edge densities graphs stage applicable table stage applicable tested random graphs step stage yields clique claim step yields label set minimum vertex cover tested graphs stage applicable tested random graphs stage applicable tested random graphs table recorded edge densities tested graphs stage applicable scatter except numbers tested graphs much less numbers unlabeled graphs respective orders however large number tested graphs possibility duplicated graphs algorithm applicable duplicated graphs algorithm applicable reasonable believe applicability ratios tested graphs close applicability ratios unlabeled labelled graphs respective orders stage algorithm graphs stage algorithm applicable stronger version condition introduced follows condition edge subgraph exists vertex cover exists vertex cover change algorithm goes stage graphs stage applicable steps construction apparently need run besides components computed step stage concluded nonempty subgraph satisfying conditions need computed stage condition stronger condition thus stage starts step component nonempty subgraph satisfying conditions following computation stage except condition substituted condition stage algorithm implemented attached matlab program found stage yielded cliques tested graphs stage claims still hold stage stage applicable tested graphs since condition stronger condition tell stage actually works tested graphs definition condition step step require computation thus running time algorithm reaches stage final stage however definition condition looks strange requires expression general way shown following section stages algorithm theoretically proved stage algorithm applicable graphs worthwhile conceive stages algorithm shall general form consistent stages concept hyperedge hypergraph needed hypergraph consists collection vertices collection hyperedges vertex set hyperedges subsets stage algorithm positive integer give following five rules algorithm suppose set hyperedges size let vertex set generated steps hyperedge size larger let subset hyperedge hyperedge size let deleting hyperedges subgraph step subgraph step shall satisfy condition hyperedge subgraph exists vertex cover stage yield clique step start stage step component stage ends difficult show rules consistent stages algorithm claims easily proved hold stage algorithm stage yields minimum vertex cover graph step yields clique step stage hyperedges size vertex sets step step require computation algorithm runs time maximum stage reaches stage conclusions conclusions algorithm summarized follows minimum vertex cover problem graph algorithm runs time maximum stage reaches stage positive integer therefore maximum stage algorithm reaches graph stage algorithm runs time graph respectively stage algorithm applicable yields proved minimum vertex cover tested graphs order larger stage algorithm works tested graphs order larger reasonable believe ratios close real applicability ratios algorithm unlabeled labelled graphs order larger applicability ratio stage tested graphs graphs graphs respectively unlikely applicability ratio stage decreases sharply order graph increases besides stage works tested graphs still stages beyond stage therefore efficient algorithm already applicable practical use summarized performance algorithm extraordinary problems although systematic theoretical explanation yet least important finding valuable research however like many findings conjectures mathematics theoretical explanation may take many years even decades found thus decided make algorithm public explanation improvement become possible tests graphs order necessary valuable however require computer large capacity available author reference levitin introduction design analysis algorithms pearson education boston appendix matlab file first program test algorithm manuspcript shall run running file function order randomly generated graph equals ceil error must positive interger end disp program requires large memory larger size swap file recommended set disp running time ordinary personal computer hours end disp program requires large memory may exceed capacity computer disp running time ordinary personal computer long end error class matrix program shall changed class end disp file please wait larger following saving file may take long time save end step function represents family sets definition algorithm cells arrays represent members family cells size first generated follows end cells larger size generated follows cells rth row represent family member size end end end steps function program component graph expressed adjacency matrix diagonal numbers labels corresponding vertices instead zeros component assigned adjacency matrix however component assigned adjacency matrix component grade larger join unions larger union expressed adjacency matrix diagonal numbers labels corresponding vertices instead zeros matrices components unions first generated follows class used save memory time however shall changed class larger end matrices components unions generated follows disp command show progress running program large min positive integers maximum values size size definition component nonempty larger size continue end isempty represent definition respectively matrix expresses union matrix expresses union ones ones adjacency matrix graph component join expresses component union contains labels used generate component thus generated end else indicates empty next break end end end end end end function function put component union subset explained lines used generate cell thus generated zeros sizeu sizep zeros sizep sizeu command puts component union obvious exists one includes includes ceil includes end end appendix matlab file second program test algorithm manuspcript another file named needs run first function generating random graphs order program finds many tested random graphs yield clique step thus graphs algorithm applicable clique yielded step claim label set label set minimum vertex cover tested graph many random graphs generated tested algorithm ceil error must positive integer end disp file please wait larger following loading file may take long time load file generated disp running time ordinary personal computer compute graph minutes end disp running time ordinary personal computer compute graph long end number tested graphs stage algorithm applicable number tested graphs stage applicable record approximate edge density graph stage applicable record approximate edge density graph stage applicable disp random graph density density randn end density approximately equals ratio complete graph generated normal distribution mean standard deviation density size disp random graph edge continue end bstart stage bstart stage indicates clique disp algorithm applicable random graph else disp algorithm applicable random graph length stage bstart stage indicates clique disp algorithm applicable random graph else disp algorithm applicable random graph length end end disp stage algorithm applicable computed random graphs disp stage algorithm applicable computed random graphs end disp records approximate edge density graph stage applicable disp records approximate edge density graph stage applicable end function density sprandsym density symmetric random sparse matrix approximately density nonzeros however density equals function still generates zeros better performance sprandsym density used works much better although ocassionly still generates zeros logical eye end function bstart stage bstart component graph ascending order grade deleting minimal subgraph get maximal subgraph satisfies conditions stage iterate computation next component nonempty program component graph expressed adjacency matrix diagonal numbers labels corresponding vertices instead zeros component assigned adjacency matrix however component assigned adjacency matrix loop graph components stage never starts component nonempty subgraph component always clique stage isempty indicates nonempty subgraph must satisfies conditions return end end end loop graph components grade larger thus stage stage starts grade component nonempty subgraph satisfying conditions min must positive integers maximum values isempty indicates cell represents nonempty component stage isempty indicates nonempty subgraph satisfies conditions stage command records grade component possible stage start return end end end end end end end function stage recorded step deleting maximal subset get subgraph contains nonempty subgraph satisfies conditions stage record next step vth vertex checked whether deleted whether still nonempty subgraph satisfies conditions stage vth vertex deleted stage isempty indicates still nonempty subgraph satisfies conditions stage command instead next round function would delete edges satisfies obvious ith vertex last remains deletable still ith vertex current else indicates vth vertex shall deleted end end end function function checks whether represents clique size return end size symmetry matrix upper half matrix checked size return end end end end function stage graph defined adjacency matrix function finds maximal subgraph satisfies conditions stage isempty return end stage indicates edges graph expressed satisfy condition stage break end point indicates certain edges graph expressed deleted last round becomes however graph expressed may still edges satisfy condition stage next round necessary end point edges graph expressed satisfy condition stage obvious vertex incident edge condition satisfied thus following part function delete isolated vertices satisfy condition ith vertex checked whether islated vertex whether satisfies condition thus label ith vertex size indicates ith vertex incident edge need check whether label vertex also label vertex cover size follows label xth vertex label ith vertex suppose label vertex cover size note diagonal numbers matrix zeros size indicates label ith vertex label vertex cover size vertex satisfies condition shall deleted else vertex satisfy condition shall deleted end else indicates ith vertex incident edge shall deleted end end end function stage round edges satisfy condition stage input graph expressed deleted however deletion may cause edges longer satisfy condition stage edges may deleted round matrix symmetrical part diagonal checked dia label xth vertex multiplication therefore xth vertex adjacent ith jth vertices stage condition shall satisfied instead condition size indicates kth vertex adjacent ith jth vertices similar explained xth vertex adjacent ith jth kth vertices logical label set indicates vertices satisfy definition condition vertex shall considered element stage end end end point stage end logical label set indicates edge corresponding satisfy condition stage end end end end end function function checks whether label set vertex cover size else end end
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working paper computing testing pareto optimal committees mar haris aziz lang monnot abstract selecting set alternatives based preferences agents important problem committee selection beyond among various criteria put forth desirability committee pareto optimality minimal important requirement asking agents specify preferences exponentially many subsets alternatives practically infeasible assume agent specifies weak order single alternatives preference relation subsets derived using preference extension consider five prominent extensions responsive downward lexicographic upward lexicographic best worst consider corresponding pareto optimality notion study complexity computing verifying pareto optimal outcomes also consider strategic issues four set extensions present pareto optimal strategyproof algorithm even works weak preferences keywords committee selection multiwinner voting pareto optimality algorithms complexity set extensions jel classification introduction pareto optimality central concept economics termed single important tool normative economic analysis moulin outcome aziz csiro unsw sydney australia tel fax lang monnot lamsade paris france lang haris aziz pareto optimal exist another outcome agents like least much least one agent strictly prefers although pareto optimality considered extensively voting social choice settings fair division hedonic games received little attention multiwinner voting outcomes sets alternatives multiwinner voting applies selecting set plans committee hiring team members movie recommendations convenience use terminology committee even results impact far beyond committee elections faliszewski aziz voting setting agents express preferences alternatives single alternative selected pareto optimality context straightforward define achieve verify multiwinner voting difficulty unrealistic assume agents report preferences possible committees since exponential number reason approaches assume report small part preferences extension principle used induce preference possible subsets small input single alternatives preference extensions also widely used social choice settings fair division matching two widely used choices small inputs multiwinner voting rankings linear orders alternatives sets approved alternatives paper make choice generalizes agents report weak orders single alternatives consider five prominent preference extension principles responsive extension set alternatives least preferred set alternatives obtained repeated replacements alternative another alternative least preferred optimistic best respectively pessimistic worst extension orders subsets alternatives according respectively least preferred element downward lexicographic extension lexicographic refinement optimistic extension upward lexicographic extension lexicographic refinement pessimistic worst extension responsive extension roth sotomayor seen ordinal counterpart additivity downward lexicographic extension considered various papers bossert lang klamler best set extension considered number approaches full proportional representation chamberlin courant monroe committee voting settings elkind worst set extension also used klamler skowron captures settings impact bad alternative selection overwhelms benefits good alternatives instance decision crucial issue made one members committee agent ignores one case parent preferences set movies watched child best worst set extensions used coalition formation aziz savani although set extensions implicitly explicitly considered multiwinner voting computational work dealt specific voting rules see related work section instead concentrate pareto optimality consider computation verification pareto optimal committees well computing testing pareto optimal committees computation verification responsive pic prefs downward lexicographic pic set extension upward lexicographic best strict prefs worst pic table complexity computing verifying pareto optimal committees pic coined christos papadimitriou seminar simons institute indicates class problems agents provide input problems admit strategyproof algorithm tence strategyproof algorithm returns pareto optimal outcomes contributions consider pareto optimality respect five aforementioned preference set extensions present various connections pareto optimality notions notions undertake detailed study complexity computing verifying pareto optimal outcomes table summarizes complexity results important message results testing pareto optimality obtaining pareto improvements committees computationally hard even though computing pareto optimal committee easy responsive downward lexicographic extensions give complete characterization complexity testing pareto optimality preferences dichotomous size top equivalence class two unless pareto optimality tested polynomial time size first equivalence classes two best extension show even computing pareto optimal outcome another interesting contrast responsive set extension even preferences dichotomous size top equivalence class two testing pareto optimality contrast extensions worst extension problems computing verifying pareto optimal outcomes admit algorithms also consider requirement strategyproofness top pareto optimality show exist pareto optimal strategyproof algorithms committee voting even weak preferences four five set extensions algorithms considered careful adaptations serial dictatorship committee voting related work first related stream work involves studying specific committee elections rules computational point view generally little focus pareto haris aziz mality focus determining whether committee pareto optimal finding pareto optimal committee sense orthogonal study committee election rules simplest widely used rules electing committee called rules compute score alternative based ranks alternatives best scores elected elkind faliszewski extension principles also used darmann note output rule obviously preferences induced scoring function necessarily respect set extensions klamler klamler compute optimal committees weight constraint single agent therefore optimality equivalent pareto optimality using several preference extensions including worst best downward lexicographic best extension principle used number papers committee elections full proportional representation starting chamberlin courant studied computational point view long series papers procaccia boutilier betzler skowron elkind ismaili rules obviously output pareto optimal committees necessarily extensions set extensions considered paper corresponding analogues extending preferences alternatives preferences lotteries particular set extension corresponds stochastic dominance lottery extension also set extensions considered paper correspond lottery extensions considered works probabilistic social choice brandl aziz cho works based hamming extension agent specifies ideal committee prefers committees less hamming distance ideal committee hamming distance notion used define specific rules minimax approval voting brams selects committee minimizing maximum hamming distance agents although output minimax approval voting always hamming extension good approximations caragiannis note dichotomous preferences hamming extension coincides responsive downward lexicographic extensions therefore computational results responsive set extension dichotomous preferences also hold hamming downward lexicographic extensions second line work concerns understanding classes rules result pareto optimal outcomes works along line bear different type committee elections called voting candidates must declare seat contest kornhauser results existence pareto optimal rules presented sanver kornhauser exactly two candidates per seat designated voting equivalent multiple referenda decision taken series issues computing testing pareto optimal committees setup consider set agents set alternatives preference profile complete transitive relation write denote agent values least much use strict part iff finally denotes indifference relation iff relation results equivalence classes eiki eil eil use equivalence classes represent preference relation agent preference list eiki example denote preferences list agent preferences strict size equivalence class agent preferences dichotomous partitions alternatives two equivalence classes let opwidth maximum size preferred equivalence class opwidth denote max min alternatives maximally minimally preferred respectively thus respectively smallest largest indices eir max min eir let set extensions pareto optimality set extensions set extensions used reasoning preferences agent sets alternatives given preferences single alternatives committee voting responsive extension natural applied various matching settings well roth sotomayor say injection agent weakly prefers define best set extension worst set extension denoted respectively max max side min min downward lexicographic extension agent prefers committee selects alternatives preferred equivalence class case equality one alternatives second preferred equivalence class formally iff smallest eil eil eil eil upward lexicographic extension agent prefers committee selects less alternatives least preferred equivalence class case equality one less alternatives second least preferred equivalence class formally iff largest eil eil remark consider agent preferences let haris aziz idl relations follow definitions efficiency based set extensions set extension define pareto optimality respect committee pareto optimal respect simply exists committee note set extensions coincides standard pareto optimality outcome pareto improvement another agent weakly improves least one agent strictly improves example consider preference profile suppose unique committee unique committee committees committees committees remark consider committee argument follows suppose exists outcome case also hence neither remark always exists committee also pareto improvements harm agent respect relation remark always exists committee also pareto improvements harm agent respect relation figure illustrate relations different efficiency notions later paper present algorithm returns committee hence also make following general observation computing testing pareto optimal committees fig relations five notions efficiency arrow means implies dashed line means always exists committee absence arrow line means sets committees disjoint lemma algorithm compute pareto improvement committee exists algorithm compute eefficient committee set extensions proof start committee recursively apply pareto improvement reach pareto optimal committee best worst extensions pareto improvements one agent improvements since implies let bound number respect paretoimprovement agent strictly improves preferred equivalence class different number alternative outcome increases least one therefore preferred equivalence class improving class pareto improvements similarly number pareto improvements subsequent less preferred equivalence class improves pareto improvement pareto improvements therefore total number paretoimprovements bounded similar argument holds well end section observing set extensions consider set pareto optimal alternatives may pareto dominated consider following example example set consists pareto optimal alternatives pareto dominated set extensions haris aziz responsive set extension trivial way achieve pareto optimality responsive set extension taking decreasing scoring vector consistent ordinal preferences finding total score alternative returning set alternatives maximum scores instance example outcome rule outputs alternatives best borda scores theorem pareto optimal committee responsive set extension committee computed linear time many situations one may already committee one may want find pareto improvement problem testing pareto optimality finding pareto improvement responsive set extension turns much harder task note exists algorithm compute pareto improvement means testing pareto optimality also polynomialtime solvable theorem checking whether committee pareto optimal responsive set extension even dichotomous preferences opwidth strict preferences proof present case opwidth reduction problem vertex cover garey johnson given simple graph minimum vertex cover problem consists finding subset minimum size every edge incident node decision version vertex cover takes input simple graph integer problem deciding exists vertex cover let instance vertex cover one arbitrary edge build following instance pareto optimality edge set agents special agent preferences agent agent reduction clearly done within polynomial time preferences dichotomous check easily committee size pareto optimal exists vertex cover size strict preferences previous reduction replace preferences easy see proof similar using similar reduction hitting set problem also prove theorem concerns parametrized complexity intractability result downey fellows hitting set defined follows given ground set elements collection subsets exist computing testing pareto optimal committees theorem checking whether committee pareto optimal responsive set extension parameter even dichotomous preferences dichotomous preferences present complete characterization complexity according opwidth parameter opwidth pareto improvement committee alternative preferred agent needs kept selected therefore problem checking efficiency easy opwidth theorem problem hard remains case opwidth theorem dichotomous preferences pareto improvement committee respect responsive set extension computed polynomial time opwidth proof consider preference profile dichotomous verifies opwidth let let partition associated first obviously assume case let construction build graph isomorphic iff edge corresponds top two alternatives agent provided one let size optimal vertex cover first claim pareto improvement one follows two conditions satisfied optimal vertex cover containing either least element two elements first show sufficient holds take committee corresponding minimum vertex cover add alternatives add alternatives least one possible holds take committee corresponding minimum vertex cover add alternatives cases obtained committee contains contains least one element contains either two elements element therefore show necessary let pareto improvement containing maximum number alternatives following two properties vertex cover holds since otherwise would hold similar reasons vertex cover adding set size obtain set size haris aziz constitutes pareto improvement necessarily either remains shown checked polynomial time done bipartite indeed construction color sets theorem bipartite graphs problem finding minimum vertex cover equivalent computing maximum matching hence solvable polynomial time check whether optimal vertex cover either holds order check exists transform new bipartite graph add new vertex edge order check let transform new bipartite graph add two new vertices two edges finally test one graphs optimal vertex covers respectively must contain respectively example illustrate algorithm proof theorem let consider dichotomous profile specify top equivalence class agent let construct graph consider four graphs resulting addition new vertex edge results addition two new vertices edges etc two graphs optimal cover size optimal cover optimal cover therefore fig graphs corresponding example note finding algorithm computes pareto improvement committee used decide whether given committee size pareto optimal responsive set extension computing testing pareto optimal committees pareto optimality strategyproofness try achieve strategyproofness simultaneously mechanism strategyproof reporting truthful preferences dominant strategy respect responsive set extension preference profiles note defining strategyproofness way respect extension stronger defining four extensions considered paper nonetheless present positive results respect strategyproofness naive way achieving pareto optimality enumerate list possible winning sets implement serial dictatorship possible outcomes done voting aziz however number possible outcomes exponential responsive preferences result partial order possible winning sets complete transitive order problem solved algorithm viewed computationally efficient serial dictatorship algorithm committee voting serial dictatorship input permutation output last set refined number alternatives yet fixed index permutation agent selects first equivalence classes say agent fixes alternatives eit increment one end pick alternatives add end return theorem exists strategyproof algorithm returns committee pareto optimal responsive set extension proof consider algorithm show stage agent implicitly refines set feasible committees maximal set preferred outcomes set providing additional constraints true base case assume holds note contains alternatives strictly less preferred agents ones respectively fixed moreover agent indifferent alternatives fixes best alternatives value agent requires alternatives selected equivalence class ensured definition haris aziz algorithm follows argument returned set pareto optimal responsive set extension strategyproofness agent turn comes choice fixing alternatives requiring alternatives equivalence class case algorithm already chooses one best possible committees agent note algorithm equivalent serial dictatorship formalized aziz note committee pareto optimal responsive set extension may result serial dictatorship holds even basic voting setting problem serial dictatorship algorithm formalized overly favours agent first permutation one way limit power let choose alternatives note attempt fairer extension serial dictatorship comes expense strategyproofness compromised consider profile preferences preferences permutation outcome agent reports outcome best set extension next consider pareto optimality respect used defining many rules see section theorem unless algorithm compute pareto improvement committee respect even dichotomous preferences opwidth proof show case solve polynomially vertex cover decision problem consider instance vertex cover given simple graph integer assume existence algorithm algo computes pareto improvement committee respect opwidth given profile set alternatives algo returns time polynomial yes pareto optimal respect otherwise returns alternatives pareto dominates prove applying times algo different inputs decide polynomial vertex cover construct following profile set agents agent corresponds edge set alternatives let edge preferences agent preferences last set agents given computing testing pareto optimal committees reduction clearly done within polynomial time set preferences given dichotomous consider following inductive procedure algo pareto optimal respect otherwise return let solution output calls algo polynomial whole procedure polynomial claim vertex cover size iff first prove induction step vertex cover initial step valid vertex cover assume true let prove vertex cover case edge covered assumption covered implies contradiction hence hypothesis deduce strict preference agent equivalently conclusion recursive calls theorem computing committee even dichotomous preferences proof give reduction hitting set let dichotomous preferences exists polynomialtime algorithm compute committee return committee agent gets preferred alternative committee exists committee corresponds hitting set size downward lexicographic set extension point dichotomous preferences responsive set extension coincides downward lexicographic set extension hence get corollary results responsive preferences corollary checking whether committee even dichotomous preferences opwidth note algorithm returns committee reason agent turn refines set possible outcomes preferred subset outcomes committee refined set least preferred respect hence respect committees set possible outcomes theorem exists strategyproof algorithm returns dlefficient committee worst set extension contrast set extensions considered paper pareto optimality respect worst set extension checked polynomial time haris aziz theorem exists algorithm checks whether committee computes pareto improvement possible proof let let eiti least preferred equivalence class eiti want check whether alternatives least agent gets strictly better outcome agents get least preferred outcome check follows let check whether know exists subset strictly preferred least preferred agent reason contains preferred worst alternative agent contains least preferred worst alternative agents means pareto improvement strictly improving possible size winning set less feasible consider strategyproofness together first note algorithm may return outcome however construct suitable strategyproof formalising appropriate serial dictatorship algorithm worst set extension theorem exists strategyproof algorithm returns committee proof consider agents permutation set alternatives initialized reduce set ensuring size least next agent permutation comes deletes maximum number least preferred equivalence classes preferences corresponding alternatives ensuring successive agent permutation gets preferred outcome ensuring agents permutation get least preferred outcome thus algorithm strategyproof pareto optimal respect worst set extension upward lexicographic set extension point dichotomous preferences responsive set extension coincides upward lexicographic set extension hence get corollary results responsive preferences corollary checking whether committee even dichotomous preferences opwidth note algorithm returns committee reason agent turn refines set possible outcomes preferred subset outcomes committee refined set least preferred respect hence respect committees set possible outcomes theorem exists strategyproof algorithm returns committee computing testing pareto optimal committees conclusions considered pareto optimality voting respect number prominent set extensions presented results relations notions well complexity computing verifying pareto optimal outcomes directions future work include considering pareto optimality respect set extensions brandt brill another direction consider compatibility pareto optimality concepts axioms finally remark serial dictatorship algorithm used define multiwinner generalization random serial dictatorship worth investigating raises interesting computational problems acknowledgments extended version ijcai conference paper aziz authors thank felix brandt useful pointers comments also thanks reviewers attendees ijcai comsoc useful comments lang monnot thank anr project references aziz savani hedonic games brandt conitzer endriss lang procaccia editors handbook computational social choice chapter cambridge university press aziz brandt brill computational complexity random serial dictatorship economics letters aziz brandt brill tradeoff economic efficiency strategyproofness randomized social choice proceedings international conference autonomous agents systems aamas pages ifaamas aziz lang monnot computing pareto optimal committees proceedings international joint conference artificial intelligence ijcai pages aziz brandt elkind skowron computational social choice first ten years beyond steffen woeginger editors computer science today volume lecture notes computer science lncs springerverlag forthcoming bossert pattanaik ranking sets objects hammond seidl editors handbook utility theory volume chapter pages kluwer academic publishers kornhauser dictatorship efficient games economic behavior betzler slinko uhlmann computation fully proportional representation jair haris aziz bossert preference extension rules ranking sets alternatives fixed cardinality theory decision brams kilgour sanver minimax procedure electing committees public choice brandl efficiency incentives randomized social choice master thesis technische brandt brill necessary sufficient conditions strategyproofness irresolute social choice functions proceedings conference theoretical aspects rationality knowledge tark pages acm press caragiannis kalaitzis markakis approximation algorithms mechanism design minimax approval voting proceedings aaai conference artificial intelligence aaai pages stable partition problem encyclopedia algorithms pages springer chamberlin courant representative deliberations representative decisions proportional representation borda rule american political science review cho incentive properties ordinal mechanisms games economic behavior pareto efficiency multiple referendum theory decision darmann hard tell condorcet committee mathematical social sciences doi url http downey fellows fundamentals parameterized complexity texts computer science springer elkind ismaili extensions rule proceedings international conference algorithmic decision theory adt pages elkind faliszewski skowron slinko properties multiwinner voting rules proceedings international conference autonomous agents systems aamas pages elkind lang saffidine condorcet winning sets social choice welfare faliszewski skowron slinko talmon multiwinner analogues plurality rule axiomatic algorithmic views proceedings aaai conference artificial intelligence aaai faliszewski skowron slinko talmon multiwinner voting new challenge social choice theory endriss editor trends computational social choice chapter forthcoming garey johnson computers intractability guide theory freeman klamler pferschy ruzika committee selection weight constraints mathematical social sciences computing testing pareto optimal committees lang mengin xia aggregating conditionally lexicographic preferences domains principles practice constraint programming pages boutilier budgeted social choice consensus personalized decision making proceedings international joint conference artificial intelligence ijcai pages aaai press monroe fully proportional representation american political science review moulin fair division collective welfare mit press sanver ensuring referendum voting social choice welfare procaccia rosenschein zohar complexity achieving proportional representation social choice welfare roth sotomayor matching study game theoretic modelling analysis cambridge university press skowron faliszewski slinko achieving fully proportional representation approximability results artif skowron faliszewski lang finding collective set items proportional multirepresentation group recommendation proceedings international joint conference artificial intelligence ijcai pages aaai press
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evolutionary design philosophy theory application tactics college information science engineering ritsumeikan university kusatsu japan faculty engineering kobe university kobe japan abstract although contributed remarkable improvements specific areas attempts develop universal design theory generally characterized failure paper sketches arguments new approach engineering design based semiotics science signs approach combine different design theories product life cycle stages one coherent traceable framework besides bring together designer user understandings notion good product building insight natural sciences complex systems always exhibit meaninginfluential hierarchical dynamics objective laws controlling product development found examination design semiosis process laws applied support evolutionary design products experiment validating theoretical findings outlined concluding remarks given keywords design lifecycle semiotics introduction historically design tasks verifying theories challenging value number domains varied quite objectivistic highly abstract philosophy mathematics rather situated pragmatic economics management essentially subjective loosely structured art sociology surprising present era technocratic civilization design receives significant research attention studied first craft later field engineering promises eventually grow independent discipline developed line traditions classical science effort contribute development discipline academic researchers practitioners although seeing problem fairly different perspectives trying understand justify one another theory useful solving design tasks adherents mathematical approaches habitually assess usefulness theory standpoint logical consistency tractability theories considered good allow generation true theorems comes product design theorems reduced statements following form design right formally determinable quality property see example reference extent reference role theories seen build apparatus terminology axiomatic basis inference methods etc design science main target achieve better control design process however many obstacles preventing broad introduction theories practice fundamental obstacle ontological contradiction notions objectively good right design always subjectively good product contradiction makes development general design theory hardly feasible unless universal notion good product formally clarified proponents different viewpoint claim theories good allow understanding design processes products better obvious analytical flavor theories expected incorporate large amount knowledge assist designer assumed quality product solely depends knowledge possessed designer problem designing good products main problem professional expertise weak point possessing even ideal knowledge neither necessary sufficient condition creation commercially successful product history witnesses individuals little qualification experience well developed creative abilities invented many great products spirit recent advances psychology ethnomethodology anthropology cognitive sciences put forward suggestions highlighting need observationally analytically studying gifted people seem able design devise develop without systematic guidance outside see reference believed research kind would ultimately lead one discovering set thinking techniques methodologies taught designers make designing better products obvious danger however apart usual difficulties application cognitive model many factors affecting designing necessarily missed resultant techniques complexity instability network topology social psychological relationships process product concept creation limited activities designer side generally homogeneous terms well interactions latter could question applicability usefulness obtained techniques well recognized multiple perspectives building design discipline contains part truth none contains entire solution leaving alone marginal unconstructive claims like design learn nature design art never science nonetheless reflect opinions professionals principal question arises whether possible bridge different viewpoints attitudes toward design within one scientific framework would basis strong belief semiotics science signs sign systems signification exceptionally well suited link mathematical social cognitive theories together provide insights design single discipline encourage design community develop new methods tools aid designers lighten routine burden improve quality goodness products justify supposition discuss semiotic approach design study illustrate example following sections paper continues investigation authors started semiotic theory creativity engineering design later gave rise development general semiotic theory evolutionary design semiotics paradigm perhaps significant thing missing traditional design theories failure appreciate products conceived people opposed simply perceived people indeed product comprehend everyday life always artifact physical entity human must adapt product processes complex dynamics regard product concept people activities practice originate objective need creation product subjectively assigning intended meaning later time also subjectively evaluate product value environment actual emergent meaning yet many different views opinions semiotics studies essentially process product among phenomena environment construed sign needs interpreted allow use product throughout life cycle peirce semiotics deals three subjects representamen sign object signified interpretant meaning follows semantically process interpretation representamen main postulate peircean semiotics representamen directly points object sign meaning system interpretance nature system signs sign system therefore representamen sign object sign system necessarily sign systems representamen may signify different object object may signified different representamen etc variety possible combinations limited relations modern biology sociology physics teach richness complexity many natural systems derives strategy organize smaller units larger ones turn arranged still larger ones shall see processes semiosis processes reveal similar hierarchical organization sing systems generally structure subject processes let assume sign system signs level dynamically composed signs level possible combinations signs occur allowed boundary conditions set level signs level constitutive level signs signs level constraining level dynamics semiosis described terms interactions among adjacent levels sign system naturally designer semiosis material process necessarily involves translation rerepresentation information one level another signs level representamina objects phenomena level processes structures level form system interpretance signs physical objects phenomena behavioral dispositions emotions like perceived realized distinctions get representation intermediary level designer cognition respect interpretive laws highest experiential environmentally culturally socially technically economically etc induced level accommodates interpretants assigns meaning representamina design process process introducing sign level meaning level entity materially grounded level entity material relevance level generally sign allowed many possible meanings depending contextual constraints higher interpretive levels people perceive completed product distinctions need interpreted may acquire meaning sign system although human perception relatively uniform consistent natural meaning assigned representamen vary significantly depending subjective dynamics perception level interpretation level well relations adjacent semiotic levels potentially infinite hierarchy interpretative levels signs level turn constitutive objects higher level product perceived conceived ensemble sign system makes language understood general sense limited handling verbal constructions product considered text written language syntax constraining product organization semantics defining meaning product pragmatics reflecting various physiological psychological social associated product design science may seen science evolution language science studies fundamental laws semiosis processes govern product life cycle elements theory industrial semiosis concept industrial semiosis categorizes product processes along three semiotic levels meaning emergence ontogenic level deals life history data future expectations single occurrence product typogenic level holds processes related product type generation phylogenic level embraces processes common past current types occurrences product three levels naturally differ characteristic durational times grouped semiosis processes one moves lowest ontogenic level higher levels objects become larger complicated slower dynamics original interpretation meaning change product concept starts development initially coinciding phylogenesis processes distinct semiotic levels interpretation concept evolved typogenesis works reorganize relationships phylogenesis processes variety objects involved product development increases product types interactions mediate filter buffer levels variety distinctions remains available phylos every object material relevance phylogenic level buffered variations ontogenic level stabilizing mediations typogenic level note three levels product definition mediate global levels product environments dynamics interactions semiotic levels well described terms basic processes variation selection complex system evolution variation stands generation variety simultaneously present distinct entities synchronic variety subsequent distinct states entity diachronic variety variation makes variety increase produces distinctions selection means essence elimination certain distinct entities states reduces number remaining entities states semiotic point view variety product intended operate environment determined devised product structure relations established product parts synchronic variety possible relations product anticipated environment product feasible states potential diachronic variety together aggregate product possible configurations variety defined ontogenic level includes elements description structure environment ontogenesis driven variation goes different configurations product eventually discovers distinction selection every stage product life cycle configurations stable one another constraint configurations imposed resulting selective retention emergence new meaning necessarily new sign typogenic level latter decreases variety specializes ontogenic level distinctions ultimately remain fit environment dynamically stable relation patterns preserved analogously slower timescale typogenesis results emergence new meaning phylogenic level consecutively specializes lower levels thus main semiotic principle product development dynamics processes always seeks decrease number possible relations product environment hence semiosis product life cycle naturally simplified time however natural dynamics augments evolutive potential product concept increasing organizational richness emergence new signs may lead emergence new levels interpretation requires new kind information new descriptive categories must given deal still product formalization among many possible approaches formalization design semiosis chosen algebraic semiotics see reference expressiveness clarity freshness algebraic semiotics deals signs members sign systems defined algebraic theories extra structure semiosis processes specified semiotic morphisms kind mapping algebraic theories instead defining properties sign system reference members algebraic semiotics unlike approaches category theory reference external relationships sign systems sign system represented signs system data set operations called constructors used create signs signs set relations defined system signs set axioms constrain possible signs partially ordered subsort level respectively turn constructors partially ordered priority within level semiotic morphism translation consists partial functions map sorts constructors predicates functions sign system sorts constructors predicates functions sign system retain structure mapping sorts sorts preserves arguments result sorts constructors predicates well subsort ordering change data sorts semiosis laws section put forward bold conjecture dynamics processes described terms basic semiotic components algebraic constructions following form sign system corresponding representation design problem time sign system corresponding representation problem time composition semiotic morphisms specifies interaction variation selection condition information closure requires external elements added current sign system semiotic morphism probability associated number meaningful transformations resultant sign system partial ranking importance ordering constraints every lower ranked constraints violated order higher ranked constraints satisfied morphisms preserve ranking semiotic theory systems postulates scale hierarchy dynamical organization new level emerges new level hierarchy semiotic interpretance emerges development new product always naturally causes emergence new meaning principle emergence directly leads formulation first law semiosis follows semiosis product life cycle represented sequence basic semiotic components least one components well defined sense morphisms isomorphisms least one sequence sense preserve original partial ordering levels present process exists probability distribution possible every component sequence past retrospectively distributions collapses single mapping sequence basic semiotic components degenerated sequence functions future process considered general probabilistic sense terms probability distributions characteristic specific domain social group design approach like seems logical assume successful perhaps sense introduction product market effects introduction settlement corresponding meanings phylogenic semiotic levels let denote number relations product environment formulate second law semiosis follows component represents successful semiosis process morphism natural sense although laws formulated sufficient precision recommended apply alike algebraic semiotics general informal way calling details boundary difficult situations main purpose well laws semiosis guide examination product development usage processes matter design theory even paradigm employed lower applied level evolutionary design evolutionary design relatively new paradigm encompasses recently popular design approaches sustainable design design green design like explicitly recognize social evolutionary nature product development postulate tentative character design solutions making dependent dynamics product following paragraphs give semiotic interpretation paradigm show laws semiosis could applied support evolutionary design semiosis evolutionary design every design based expectations explicitly implicitly determine product intended meaning realized design requirements conceived relation patterns expectations always fit particular environmental conditions often become obsolete product reaches market place universal obvious solution problem increase much possible synchronic variety product contriving appropriate decisions design notorious idea products indeed elaborate structure product concept larger number environmental situations maintain different product configurations fit adapted different situations therefore case dynamic environments design evolution increase synchronic variety making product complex adequately react environmental changes although latter statement contradict lifecycle semiosis laws perhaps true general mean best product must always complex one product maximal synchronic variety due many reasons economical costs technical reliability ecological energy material consumption pollution social ergonomic safety convenience easiness production operation best product simplest possible structure given functionality least possible given environment synchronic variety sense goodness better say adequacy product depends characteristics product environment relation implemented design expectations depends well intended meaning matches meanings emerging phylogenesis design expectations roughly classified two categories functional expectations operation product functional parameters environmental expectations interaction product life cycle distinction dynamics driven violations design expectations dynamics ontogenesis processes relation patterns originally detected interpreted accepted rejected action subject psychological physiological laws instance law differentiated corresponding processes violations functional expectations control product typogenesis violations environmental expectations influence phylogenesis semiosis processes resolving product intended meaning mismatch critical task design life cycle engineering requires development appropriate information technologies tools outline agentbased technology developed detect violations design expectations support way evolutionary design products well assess successfulness life cycle individual product product type product family whole also see reference evolutionary design main idea developed technology evolutionary design support allow evaluation change important design expectations using programmable mobile agents called expectation agents utilize design requirements represented explicitly monitor product functionality usage operational environment expectation agent typically consists static hardware unit including transducers processing preprocessing blocks etc integrated product mobile software part allows agent demonstrate certain level autonomy intelligence proactivity respect product act behalf designer manufacturer agent execute various control procedures transfer registered data via communication line assist product user collect feedback directly user update code data obtained agent analyzed used detect usage environmental patterns shift thereby necessitating optimization synchronic variety product adjusting configuration provide evolutionary support life cycle semiosis processes effective information infrastructure connecting products supplied agents service centers manufacturers designers developed figure gives example agent networking would arranged using instance existing infrastructure web three distinct overlapping layers networking driven phylogenesis processes ontogenic layer expectation agents monitor individual products actual environments agents communicate well design manufacturing maintenance involved parties create product life histories try optimization technical phylogenesis designer cluster environmental cluster communication node environmental cluster product product product product product product product product ontogenesis figure expectation agent networking technologic usage processes associated products product operation expectation agent functionality changed updating agent program code ontogenic layer comprises single products installed dynamic environments essential feature layer order imposed outside agents communication agents evolutionary grouped environmental characteristics specific clusters emergence cluster typogenic layer implies emergence new meaning product type corresponding level semiotic interpretance ideal case number active clusters indicates number product types requisite given environment covered agent network typogenic layer provides systematization design information flows generated agents information data sorted stored depending product type version generation majority actions processes layer defined population products grouped environmental cluster course environmental clusters may naturally grouped still larger units characteristic dynamics different manufacturers upperscale grouping would initiate emergence new meaning product family phylogenic level interpretance phylogenic layer links design production maintenance utilization etc processes required associated realization distinct product technology concept information flows layer relatively stable depend global cultural ecological rather specific technical economic factors first law semiosis regulates main processes phylogenesis makes agent networking really possible figure refrigerator installed expectation agent interactions service center fast slow manufacturer cluster environmental cluster adaptation normal functioning breakdown time figure dynamics interaction figure presents product refrigerator embedded expectation agent part experimental setup built study setup compare behavior virtual products product intended meanings reconstructed virtual objects real products means haptic devices remote sensors see reference gives detailed account experiment analysis empirical data collected setup showed dynamics interactions could serve indicator successful given environment operation product figure depicts characteristic change number interactions product part movements working mode switchings registered expectation agent fluctuations filtered data set block agent although admitted larger scale empirical study required prove efficiency evolutionary design support based expectation agent networking obtained results principally confirmed theoretical conjectures demonstrated technological feasibility approach also see reference discusses case violations design expectations well possible strategies react detected violations light evolutionary design theory conclusions would like conclude paper following remarks first idea semiotic interpretation design process new moreover become almost fashion design community last years many publications subject one principal difference work limited investigation semiotic analysis classification design objects signs representing instead focused processes responsible development objects meanings second see semiotic approach design discriminate true false correct incorrect bad good provide designers new perspective design theories techniques better understand process goes factors subjective objective affecting elaboration semiotic approach believe shed light upon many nonobvious consequences causes application particular design theory technique finally come clean semiotics alone account results insights brought design life cycle theories rather semiotics help merge stages product development together within uniform universal scientific framework latter could seen ultimate goal research acknowledgments authors would like acknowledge financial support japan society promotion science project made research possible greatly indebted goncharenko key contribution experimental part study references suh applications axiomatic design kals van houten eds integration process knowledge design support systems kluwer academic publishers yoshikawa general design theory cad system sata warman eds communication klein knowledge level theory design engineering jacucci olling preiss wozny eds globalization manufacturing digital communication era century innovation agility virtual enterprise kluwer academic publishers hubka eder design science introduction needs scope organization engineering design knowledge london eder wdk engineering design creativity proceedings workshop edc pilsen czech republic heurista ullman herling ambrosio next using problem status determine course action research engineering design latour interobjectivity mind culture activity cavallucci lutz intuitive design method idm new framework design method integration shpitalni proceedings international cirp design seminar haifa israel kryssanov tamaki kitamura understanding design fundamentals synthesis analysis drive creativity resulting emergence artificial intelligence engineering kryssanov goossenaerts goncharenko tamaki semiotic theory evolutionary design concepts illustration kimura proceedings cirp international seminar life cycle engineering tokyo hartshorne weiss eds collected papers charles sanders peirce volumes belknap press harvard university press cambridge salthe development evolution mit press cambridge lemke opening closure semiotics across scales chandler van vijver eds closure emergent organizations dynamics annals new york academy science goossenaerts industrial semiosis founding deployment ubiquitous information infrastructure computers industry heylighen meta systems constraints variation classification natural history metasystem transitions world futures journal general evolution goguen introduction algebraic semiotics applications user interface design nehaniv computation metaphor analogy agents springer lecture notes artificial intelligence goncharenko kryssanov tamaki approach collecting utilizing design information throughout product life cycle fuertes proceedings ieee international conference emerging technologies factory automation etfa upc barcelona spain kryssanov tamaki ueda technology support evolutionary design journal engineering manufacture
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igor zavadskyi taras shevchenko national university kyiv ihorza family fast exact pattern matching algorithms abstract family exact pattern matching algorithms described utilize arrays order process one adjacent text window iteration search cycle approach leads lower average time complexity cost space algorithms family perform well short patterns middle size alphabets case shift window several pattern lengths quite probable main factor algorithm success algorithms outperform algorithm either original version sunday quick search modification wide area pattern length alphabet size plane subareas proposed algorithms fastest among known exact pattern matching algorithms namely perform best alphabet size pattern length parameters typical search natural language text databases key words pattern matching fast search text search introduction pattern matching one fundamental techniques used computer science common pattern matching problem formulated finding exact occurrences given substring larger body text entire presentation use following notation input text pattern searched length input text length pattern alphabet input text pattern size alphabet number different symbols pattern problem systematically studied since beginning seventies number algorithms efficient simplest straight forward search discovered famous algorithm improves time complexity algorithm significantly outperforms average almost known pattern matching algorithms include preprocessing stage preliminary values obtained basing pattern main search cycle text body scanned cases algorithm efficiency strongly depends pattern text lengths also alphabet size since rule dependence main search cycle text length linear time complexity preprocessing stage negligibly small worthwhile compare algorithm efficiency research concerns left area small large area two modifications algorithm namely algorithm bmh sunday quick search considered best decades however number efficient exact pattern matching algorithms invented according fjs tvsbs ebom sbndm fsbndm algorithms mentioned cover three known types patternmatching algorithms fjs tvsbs ebom automata based sbndm fsbndm algorithms utilize operations propose new algorithms almost algorithms type including fjs tvsbs new algorithms exploit idea badcharacter shift originates search compare last character search window last character pattern match shift window long possible bmh algorithm based idea develop generalization bmh algorithm allows performing several badcharacter shifts iteration search cycle bmh algorithm shown fig since analyze algorithms operation level try remove unnecessary subtractions denoting values calculated preprocessing stage bad character shift performed row length equal pos current position search window shift array calculated preprocessing stage ratio small enough symbol likely occur pattern length shift maximum stands maximum length shifts main factor responsible efficiency bmh left area plane interested left upper subarea ratio particularly small case one assume probably character belong pattern characters etc well means search window shifted several window lengths words several adjacent search windows processed iteration search cycle main idea multiwindow search algorithms pos output pos pos fig main search cycle algorithm course least symbols input text must read processed substring length order miss possible pattern occurrence thus least readings input text characters done window length number iterations algorithm like bmh however reduce number operations using arrays arrays occupy rather memory pattern shift arrays search algorithms filling takes preprocessing time nevertheless shown space overheads big comparing memory size modern computers time overheads covered main search cycle resulting significant gain total wide range alphabet size pattern length combinations idea using two text windows searching occurrences pattern new utilized tsw algorithm however windows supposed processed parallel thus tsw algorithm suitable parallel processor structures also idea search array already implemented number algorithms instance algorithm tvsbs ebom however always proposed use two adjacent characters text indices significantly increases probability maximum length shift low check otherwise leads superfluous density checks words even check causes maximum shift high probability need check two adjacent characters shift text window positions case may better perform characters pos could shift text window positions double window algorithm let discuss process two adjacent search windows length could considered one window double length try reduce total number computing operations required process substring length let examine main cycle bmh algorithm shown fig two reads shift table row two iterations replaced one use shift table defined follows leftmost possible position first character pattern assumption shifts defined table divided types shown fig neither belong pattern safely shifted positions forward character belong pattern belongs case safely shifted symbols less namely rightmost occurrence aligned character belongs safely shifted forward less symbols namely rightmost occurrence aligned case pattern matched current position one check coincides pattern proceed forward fig pattern shifts double window algorithm pos pos output pos pos else pos fig double window algorithm main search cycle let calculate number operations bmh double window algorithms required shift text window characters forward case maximum possible shift probable case pattern length small compared alphabet size case rows two iterations bmh algorithm rows one iteration double window algorithm executed note getting element array like equivalent notation requires one addition two readings memory getting element array like equivalent requires two additions one multiplication three readings memory constant calculations shown table number operations double window algorithm twice less compared bmh also noted memory reads generally take longer time operations could performed using processor registers arithmetic operations someone implements discussed algorithms assembler language values variables pos etc stored processor registers arrays implementation memory reads performed two iterations bmh row row iteration memory reads needed one iteration double window algorithm pos pos row thus case efficient assembler implementation double window algorithm main cycle outperforms bmh main cycle even case implementation programming language longest possible shifts performed figure table operational complexity double window algorithms operation window horspool rows rows son rows rows ment memory row row reads row row row row row additions row row row row row cation total one observe even one iteration double window algorithm main cycle requires fewer operations one iteration bmh main cycle case condition met therefore cases shown figures double window algorithm main cycle still executes faster bmh main cycle note case equality holds shift length double window algorithm bmh course advantage main cycle bmh main cycle case lower case case lower case ratio increases balance cases moves close case occurs almost always outperformance main cycle bmh main cycle small case shown figure random text pattern occurs probability regardless value case internal cycle rows executes internal cycle bmh rows one iteration iteration internal cycle requires time one bmh comparison requires readings memory comparison consists additions readings memory result main cycle algorithm essentially faster main cycle bmh algorithm following conditions met alphabet large enough make case frequent ratio small enough make case frequent simulation shows alphabet size condition violation forces algorithm main cycle run slower bmh main cycle random pattern text alphabet size greater could considered large enough violation condition forces algorithm main cycle run approximately speed bmh main cycle however wide range pattern length alphabet size combinations essentially faster bmh range covers range extension let consider possibility processing adjacent text windows one iteration modification algorithm simple array used instead defined follows leftmost possible position beginning pattern assumption figure row changed following way pos thus obtain triple window quadruple window algorithms using notation assignment rewritten course order reduce number multiplications values calculated preprocessing stage values could considered constants however every next dimension adds two additions one multiplication two memory reads calculation routine overhead covered longer shifts value small enough case preprocessing begins play important role since time space complexity grows exponentially depending preprocessing stage discussed next section preprocessing preprocessing stage algorithm values calculated arrays containing elements elements filled values filling array bmh algorithm runs obtaining values time ignorable small realistic values filling array takes almost time following procedure completes task assign value elements replace values rightmost position character pattern replace values rightmost position character pattern first step takes time step time overall time complexity preprocessing stage using special functions copy memory blocks like memcpy library one build implementation faster times conventional method given space complexity algorithms course strongly greater one however memory requirements array even relatively large alphabet containing symbols absolutely admissible computers programs algorithm computational experiment shows efficient values around size array algorithm could efficient maximum size respective array unrolling cycle algorithms based bad character rule bmh could accelerated using unrolling cycle technique consist applying blind shifts without checking end file shift value positive order miss end file text appended fictitious pattern technique could applied multiwindow algorithms following way row replaced endless cycle row check condition break cycle met allows check end file shift array element equal zero speed algorithms computational experiment implement unrolled versions double window number known algorithms language use microsoft visual studio compiler build executables run athlon processor ghz windows platform preprocessing stage algorithms implemented using fast memory fill functions text containing characters randomly built patterns well distribution characters uniform results alphabet size different pattern lengths presented table total running time runs shown results shown since significantly worse results alphabet size algorithm could efficient smaller alphabets short patterns table total running time pattern matching algorithms text seconds bmh conclusions seen triple window algorithm superior known algorithms pattern lengths alphabet size algorithms outperform classical algorithms bmh considered values tvsbs fjs sbndm better short patterns ebom fjs tvsbs sbndm fsbndm pattern length fsbndm algorithm becomes superior technique used algorithm family could applied order accelerate algorithms based comparisons adjacent characters future research direction references knuth morris pratt fast pattern matching strings siam comput boyer moore fast string searching algorithm commun acm horspool practical fast search strings exp sunday fast substring search algorithm commun acm faro lecroq exact online string matching problem review recent results acm computing surveys csur surveys homepage archive volume issue article hudaib suleiman itriq fast pattern matching algorithm two sliding windows tsw comput sci berry ravindran fast string matching algorithm experimental results proceedings prague stringology club workshop
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matrix concentration inequalities aug joel tropp bstract matrix concentration inequalities give bounds deviation random matrix expected value results weak dimensional dependence sometimes always necessary paper identifies one sources dimensional term exploits insight develop sharper matrix concentration inequalities particular analysis delivers two refinements matrix khintchine inequality use information beyond matrix variance reduce eliminate dimensional dependence otivation matrix concentration inequalities provide spectral information random matrix depends smoothly many independent random variables recent years results become dominant tool applied random matrix theory several reasons success approach flexibility matrix concentration applies wide range random matrix models particular obtain bounds spectral norm sum independent random matrices terms properties summands ease use many applications matrix concentration tools require small amount matrix analysis expertise random matrix theory required invoke results power large class examples including independent sums matrix concentration bounds provably close optimal see monograph overview theory comprehensive bibliography matrix concentration inequalities literature suboptimal certain examples weak dependence dimension random matrix removing dimensional term difficult many situations necessary purpose paper identify one sources dimensional factor using insight develop new matrix concentration inequalities qualitatively better current generation results although sacrifice desiderata ultimately hope line research lead general tools applied random matrix theory flexible easy use give sharp results cases atrix hintchine nequality set stage present discuss primordial matrix concentration result matrix khintchine inequality describes behavior special random matrix model called matrix gaussian series result already exhibits key features sophisticated matrix concentration inequalities used derive concentration bounds general models matrix khintchine inequality serves natural starting point deeper investigations matrix gaussian series work focus important class random matrices lot modeling power still supports interesting theory definition matrix gaussian series consider fixed hermitian matrices common dimension let independent family standard normal random variables construct random matrix date march revised april august mathematics subject classification primary secondary key words phrases concentration inequality moment inequality random matrix email jtropp tel tropp refer random matrix form matrix gaussian series hermitian coefficients brevity hermitian matrix gaussian series matrix gaussian series enjoy surprising amount modeling power easy see express random hermitian matrix jointly gaussian entries form generally use matrix gaussian series analyze sum independent random hermitian matrices indeed norm matrices process passing independent sum conditional gaussian series called symmetrization see lem eqn details calculation furthermore techniques gaussian series adapted study independent sums directly without artifice symmetrization note restriction hermitian matrices really limitation also analyze rectangular matrix jointly gaussian entries working hermitian dilation sometimes known wielandt matrix see sec information approach matrix variance many matrix concentration inequalities expressed naturally terms matrix extension variance definition matrix variance let random hermitian matrix matrix variance deterministic matrix var use convention power binds expectation particular consider matrix gaussian series easy verify var see matrix variance clean expression terms coefficients gaussian series easy compute practice matrix khintchine inequality matrix khintchine inequality fundamental fact behavior matrix gaussian series first version result established constants refined papers version adapted sec proposition matrix khintchine consider hermitian matrix gaussian series introduce matrix standard deviation parameter integer symbol denotes schatten lower bound simply jensen inequality section contains short proof upper bound matrix khintchine inequality also yields estimate spectral norm matrix gaussian series type result often useful practice corollary matrix khintchine spectral norm consider hermitian matrix gaussian series dimension introduce matrix standard deviation parameter kvar symbol denotes spectral norm also known operator norm matrix concentration inequalities proof sketch upper bound observe kvar indeed spectral norm bounded schatten apply lyapunov inequality increase order moment one invoke proposition bound trace terms spectral norm finally set simplify constants lower bound note kvar first relation follows optimal inequality last jensen two examples bound shows matrix standard deviation controls expected norm matrix gaussian series factor logarithmic dimension random matrix one may wonder whether lower branch upper branch gives accurate result fact natural examples demonstrate extremes behavior occur integer define diag diag diag diagonal matrix whose entries independent standard normal variables second define goe goe symbol denotes conjugate transposition scaling random matrix goe hermitian part matrix whose entries independent standard normal variables sequence goe called gaussian orthogonal ensemble goe apply matrix khintchine inequality represent matrix hermitian gaussian series diag goe eii written matrix one position zeros elsewhere respectively matrix variances satisfy var diag var goe bound delivers log diag goe relations suppress terms case ratio lower upper bound order log matrix khintchine inequality provide precise information hand examples detailed spectral information available goe diag see sec proof result goe matrix bound diagonal matrix depends familiar calculation expected maximum independent standard normal random variables see norm goe matrix close lower bound provided norm diagonal matrix close upper bound tropp question corollary shows matrix variance controls expected norm matrix gaussian series hand two examples previous section demonstrate need information variance determine norm constant factor therefore must ask parameters allow calculate norm matrix gaussian series precisely matrix variance paper provides first affirmative answer question eyond atrix hintchine nequality section presents new results improve matrix khintchine inequality proposition first motivate type parameters arise try refine result define quantity called matrix alignment parameter describes coefficients matrix gaussian series interact section use alignment parameter state new bound provides uniform improvement matrix khintchine inequality refinements possible consider random matrices highly symmetric distributions introduce class strongly isotropic random matrices section section contains matrix khintchine inequality matrix gaussian series strongly isotropic bound good enough compute norm large goe matrix exactly finally sections discuss extensions related work prospects kind parameters might allow refine proposition result already identity inspiration let work happens var use convention powers bind trace product gaussian variables expectation zero unless indices paired last expression first term comes cases second term comes case matrix variance var emerges new term arises summands indices alternate sense matrix reflects extent coefficient matrices aligned family commutes matrix var term provides new information generally ever coefficients commute quantity expressed terms matrix variance number matrix khintchine inequality proposition gives estimate correct order words commuting coefficients worst possible circumstance previous work matrix concentration implicitly uses model analysis achieve better results need account coefficient matrices interact calculation suggests matrix might contain information need heuristically coefficients fail commute matrix small see idea fruitful need parameter discerning let summarize discussion following observation improve matrix khintchine inequality must quantify extent coefficient matrices commute work builds intuition establish new matrix concentration inequalities matrix alignment parameter section introduce new parameter matrix gaussian series describes much coefficients commute later sections present extensions matrix khintchine inequality rely parameter definition matrix alignment parameter let hermitian matrices dimension matrix alignment parameter sequence quantity max related observation animates theory free probability gives fine description certain large random matrices key fact centered free random variables crossing moments must vanish matrix concentration inequalities maximum takes place triple unitary matrices dimension matrix absolute value defined roughly matrix alignment parameter describes well matrices aligned worst choices coordinates quantity appears mysterious worth paragraphs clarify meaning first let compare alignment parameter matrix standard deviation parameter appears matrix khintchine inequality proposition standard deviation versus alignment let hermitian matrices define standard deviation alignment parameters max proof proposition appears section next let return examples introduction section provide detailed calculations standard deviation alignment parameters diagonal gaussian series diag defined diag diag goe matrix goe defined goe goe matrix alignment parameter tell two examples apart matrix standard deviation remark notation alignment elsewhere abuse notation writing alignment parameter matrix gaussian series even though function coefficient matrices representation series remark unitaries necessary stage may seem capricious include unitary matrices definition fact example section demonstrates alignment parameter would lose value remove unitary matrices hand situations unitary matrices completely arbitrary discussed section matrix khintchine inequality first major result paper improvement matrix khintchine inequality theorem uses information alignment parameter obtain better bounds theorem matrix khintchine consider hermitian matrix gaussian series define matrix standard deviation matrix alignment parameters max maximum takes place triple unitary matrices integer symbol denotes schatten proof theorem appears section also derive bounds spectral norm matrix gaussian series corollary matrix khintchine spectral norm consider hermitian matrix gaussian series dimension define matrix standard deviation matrix alignment parameters max maximum ranges triple unitary matrices log log tropp symbol denotes spectral norm result follows theorem setting potential gain comes reduction power first logarithm matrix khintchine versus matrix khintchine let make comparisons proposition theorem first recall alignment parameter dominated standard deviation parameter proposition therefore bound implies close prediction proposition theorem never significantly worse hand situations theorem gives qualitatively better results particular goe matrix goe bound calculation yield log goe log estimate beats first attempt still falls short correct estimate goe strongly isotropic random matrices seen theorem offers qualitative improvement matrix khintchine inequality proposition nevertheless new result still lacks power determine norm goe matrix correctly obtain satisfactory results specializing attention class random matrices highly symmetric distributions definition strong isotropy let random hermitian matrix say strongly isotropic denotes normalized trace dimension symbol easiest way check random matrix strongly isotropic exploit symmetry properties distribution offer one many possible results direction lem proposition strong isotropy sufficient condition let random hermitian matrix suppose distribution invariant signed permutation every signed permutation strongly isotropic symbol refers equality distribution signed permutation square matrix precisely one nonzero entry row column entry taking values proof suppose signed permutation drawn uniformly random first relation uses invariance signed permutation second relies fact signed permutations unitary averaging fixed matrix signed permutations yields identity times normalized trace matrix proposition applies many types random matrices particular diagonal gaussian matrix diag goe matrix goe strongly isotropic result types distributional symmetry also lead strong isotropy remark group orbits general class matrix gaussian series verify strong isotropy using abstract arguments let unitary representation finite group let fixed hermitian matrix dimension consider random hermitian matrix independent family standard normal variables since acts permutation observation allows perform averaging arguments like one proposition matrix concentration inequalities several ways apply property argue strongly isotropic example suffices also sufficient forms complete tight frame every vector see paper situations condition holds remark spherical designs spherical collection points unit sphere property arbitrary algebraic polynomial variables degree haar measure sphere see paper existence results background references given spherical consider random matrix independent family standard normal variables construction random matrix property variant strong isotropy property sufficient many purposes provided log khintchine inequality strong isotropy second major result paper matrix khintchine inequality valid matrix gaussian series strong isotropy property like theorem result uses alignment parameter control norm random matrix theorem khintchine strong isotropy consider hermitian matrix gaussian series dimension assume strongly isotropic introduce matrix standard deviation matrix alignment parameters max maximum ranges triple unitary matrices integer symbol refers spectral norm schatten proof result appears section also establish lower bound theorem shows moments random matrix controlled standard deviation whenever take schatten essentially spectral norm dimensional factor side negligible therefore implies presence strong isotropy spectral norm matrix gaussian series comparable standard deviation whenever alignment parameter relatively small particular apply result goe matrix goe proposition calculation standard deviation alignment parameters ensures goe observed bound sharp example even take leads good probability bounds via markov inequality furthermore detailed version theorem appearing section precise enough show semicircle law limiting spectral distribution goe hand dependence exponent theorem suboptimal point evident consider diagonal gaussian matrix diag indeed theorem implies bound diag const observed power logarithm tropp discussion paper opens new chapter theory matrix concentration noncommutative moment inequalities main technical contribution demonstrate matrix khintchine inequality proposition last word behavior matrix gaussian series indeed shown matrix variance contain sufficient information determine expected norm matrix gaussian series also identified another quantity matrix alignment parameter allows obtain better bounds every matrix gaussian series furthermore presence extensive distributional information even possible obtain numerically sharp bounds norm certain matrix gaussian series number ways extend ideas results paper alignment consider alignment parameters involving coefficient matrices possible improve term theorem see section additional details matrix series use exchangeable pairs techniques study matrix series form independent family scalar random variables approach potentially quite interesting bernoulli random variables independent sums use conditioning symmetrization apply theorem sum independent random matrices see app example type argument rectangular matrices techniques also give results rectangular random matrices way hermitian dilation sec setting different notion strong isotropy becomes relevant see section elaborated ideas also evidence alignment parameters lead final results matrix concentration related work techniques literature random matrices satisfy three three requirements flexibility ease use power particular many practical applications important able work arbitrary sum independent random matrices chosen study matrix gaussian series simplest instance model may lead insights general problem classical work random matrix theory concerns special classes random matrices books provide overview main lines research field specific subareas random matrix theory address general models monograph gives introduction free probability book chapter describes collection methods banach space geometry monograph covers theory matrix concentration last three works wide scope applicability none provides ultimate description behavior sum independent random matrices one specific strand research would like draw close spirit paper recently bandeira van handel van handel studied behavior real symmetric gaussian matrix whose entries independent centered inhomogeneous variances model random matrix class written indep usual independent family standard normal random variables assume without loss generality situate model context work observe matrix gaussian series significantly general model strongly isotropic model incomparable independententry model see recall strongly isotropic matrices dependent entries time indep diagonal integer need scalar matrix model bandeira van handel established following sharp bound indep indep const maxi log maximum entry maxi plays role formula matrix alignment parameter plays paper paper leans heavily independence assumption clear whether ideas extend general setting matrix concentration inequalities compare result work compute matrix alignment parameter independententry model using difficult extension calculation section effort yields indep maxi see matrix alignment parameter somewhat larger maximum entry maxi thus independent model theorem gives better result classical khintchine inequality proposition somewhat weaker theorem would give result close bound always apply model need strongly isotropic model adequate reach results power scope current generation matrix concentration bounds nevertheless estimate strongly suggests better ways summarizing interactions coefficients hermitian matrix gaussian series alignment parameter one possibility weak variance parameter sup model quantity reduces const maxi idea considering motivated discussion sec well work unfortunately stage clear whether parameters allow obtain simple description behavior gaussian matrix absence burdensome independence isotropy assumptions frontier future work omputation atrix lignment parameters section show compute matrix alignment parameter two random matrices introduction diagonal gaussian matrix goe matrix afterward show example neither theorem theorem hold remove unitary factors matrix alignment parameter diagonal gaussian matrix diagonal gaussian matrix takes form diag eii matrix variance var diag diag follows matrix standard deviation parameters defined satisfy diag diag show matrix alignment parameters defined satisfy diag thus example matrix khintchine inequalities theorem theorem improve matrix khintchine inequality proposition outcome natural given classical result essentially optimal case let evaluate matrix alignment parameter triple unitary matrices form sum eii seii written schur componentwise product transpose operation sum collapses therefore diag max kikp proposition shows diag diag therefore diag diag result follows take limits remark commutativity similar calculation valid whenever family coefficient matrices matrix gaussian series commutes tropp goe matrix goe matrix takes form goe easy calculation shows matrix variance satisfies var goe goe therefore matrix standard deviation parameters defined equal goe goe demonstrate matrix alignment parameters defined satisfy goe large matrix alignment parameters much smaller matrix standard deviation parameters consequence matrix khintchine inequalities deliver substantial gain classical matrix khintchine inequality let compute matrix alignment parameter triple unitary matrices introduce unnormalized sum hard evaluate sum take care first distribute terms line sum two free indices identify four matrix products example first line sum step yields sum remaining indices reach twelve sixteen terms unitary matrices remaining four scaled unitary matrices furthermore trace bounded magnitude worst case applying definition schatten norm triangle inequality unitary invariance find compute goe must reintroduce scaling gives advertised result goe obtain bound simply take limits matrix concentration inequalities unitaries necessary suppose hermitian matrix gaussian series dimension let matrix standard deviation consider alternative alignment parameter quantity suggested discussion section consider general estimate form demonstrate every choice function lower bound const log claim deduce impossible improve classical khintchine inequality using secondorder quantity therefore unitary matrices alignment parameter play critical role argument developed afonso bandeira grateful allowing include introduce pauli spin matrices matrices hermitian unitary furthermore satisfy relations next define calculate indeed positive root quadratic consider gaussian series generated matrices usual independent family standard normal variables series already shown alternative alignment parameter let compute variance standard deviation var kvar expanding random matrix coordinates also find therefore entry centered normal random variable variance obtain counterexample bound fix integer let independent copies gaussian series construct matrix gaussian series spin written direct sum kronecker product matrices diagonal units dimension independent family standard normal variables extending calculations find spin spin meanwhile norm spin bounded absolute value diagonal entries particular log spin max const used fact expected maximum independent standard normal variables proportional log assuming valid sequence estimates obtain const log spin spin therefore function must grow least fast log conclude bound form never improve classical matrix khintchine inequality tropp otation ackground enter body paper let set additional notation state background results first denotes complex linear space matrices complex entries write reallinear subspace consists hermitian matrices symbol represents conjugate transposition write zero matrix identity matrix one position zeros elsewhere dimensions matrices typically determined context hermitian matrix define integer powers usual way iterated multiplication matrix also define complex powers raising eigenvalue power maintaining eigenvectors particular unique square root matrix absolute value defined general matrix rule note positive semidefinite trace normalized trace matrix given use convention power binds trace avoid unnecessary parentheses powers also bind expectation schatten defined arbitrary matrix via rule schatten coincides spectral norm work uses trace powers schatten norms depending one conceptually clearer require inequalities involving trace schatten norms matrices furthermore results drawn chap race oments atrix aussian eries major result paper starting point formula trace moments matrix gaussian series lemma trace moment identity let hermitian matrix gaussian series integer identity easy proof lemma appears next two subsections integration parts foreign study gaussian random matrices example see sec sec exchangeable pairs method establishing matrix concentration also based elementary conceptually challenging analog integration parts lem aside works aware application related techniques prove results matrix concentration preliminaries obtain lemma main auxiliary tool classical integration parts formula function standard normal vector lem form required result derived basic calculus fact gaussian integration parts let vector independent standard normal entries let function whose derivative absolutely integrable respect standard normal measure symbol denotes differentiation respect coordinate also use formula derivative matrix power sec matrix concentration inequalities fact derivative matrix power let differentiable function integer particular symbol refers ordinary matrix multiplication proof lemma let treat random matrix function standard normal vector write distribute sum first factor gaussian integration parts formula fact implies since derivative formula yields completes proof formula hort roof atrix hintchine nequality historically proofs matrix khintchine inequality rather complicated result actually immediate consequence lemma present argument detail appeared literature furthermore approach serves template sophisticated theorems main contributions paper let restate proposition form establish proposition matrix khintchine let hermitian matrix gaussian series define matrix variance standard deviation parameters var trv integer short proof proposition appears next two sections approach parallels exchangeable pairs method used establish matrix khintchine inequality rademacher series cor replace exchangeable pairs conceptually simpler argument based gaussian integration parts reach statement proposition simply rewrite trace terms schatten norm remark noninteger moments proof proposition adapted obtain moment bounds see cor closely related argument preliminaries main idea proof simplify trace moment identity elementary matrix inequality anticipating subsequent arguments state inequality greater generality need right proposition suppose hermitian matrices size let integers satisfy real number range min proof proposition depends numerical fact nonnegative numbers function convex interval achieves minimum therefore min need lift scalar inequality matrices tropp proof without loss generality may change coordinates diagonal eii expanding copies eii eii take absolute values inequality implies remaining trace nonnegative eii components matrix consequence eii reach last identity reversed steps reassemble sum trace proof matrix khintchine inequality may establish proposition let introduce notation quantity interest use integration parts result lemma rewrite trace moment choice apply matrix inequality proposition reach identified matrix variance defined next let identify copy side solve resulting algebraic inequality end invoke inequality trace trv identified quantity second inequality lyapunov since unknown nonnegative solve polynomial inequality reach required result econd rder atrix hintchine nequality section prove theorem matrix khintchine inequality let restate result form establish theorem matrix khintchine let hermitian matrix gaussian series define matrix variance standard deviation parameter var trv define matrix alignment parameter max maximum ranges triple unitary matrices integer proof theorem occupy rest section reach statement introduction rewrite traces terms schatten norms also provide proof proposition section matrix concentration inequalities discussion establish theorem let spend moment discuss proof result theorem based pattern argument matrix khintchine inequality proposition time apply proposition surgically control terms trace moment identity lemma significant new observation use complex interpolation reorganize products matrices arise calculation refine argument several ways first apply complex interpolation care possible define matrix alignment parameter maximum set commuting unitaries given commuting matrices simultaneously diagonalizable improvement might make easier bound matrix alignment parameters second quite clear proof proceed beyond terms example integer obtain results terms quantities max max ordering indices respectively refinement allows reduce order coefficient standard deviation term unfortunately must also compute alignment parameters instead observation shows unproductive press forward approach indeed number orderings indices grows consider longer products awful prospect applications preliminaries proof theorem use two interpolation results reorganize products matrices first one type matrix inequality cor version result specialized setting fact consider finite sequence hermitian matrices dimension let matrix dimension number see lem proof based hadamard theorem prop second result complicated interpolation multilinear function whose arguments powers random matrices proposition multilinear interpolation suppose multilinear function fix nonnegap tive integers let random matrices necessarily independent max max expression random unitary matrix commutes fact proof proposition depends hadamard theorem prop argument standard somewhat involved postpone details appendix overture let commence proof theorem initial steps similar argument leads matrix khintchine inequality proposition introduce notation quantity interest identity follows integration parts result lemma time make summands apply proposition terms remaining values exponent apply proposition reach bound take advantage fact interleaved powers random matrix second term tropp first term treat first term side simply repeat arguments section obtain bound terms quantity trv quantities defined identified copy integration parts continue want break matrix appears second term side perform another gaussian integration parts write invoke fact obtain result follows product rule formula derivative power bound first term side terms standard deviation parameter second term lead matrix alignment parameter finding standard deviation parameter let address first term side first draw sum back trace identify matrix variance defined isolate random matrix apply inequality exponents follow lyapunov inequality thus inequality fact implies trv trv identified combine last three displays arrive identified another copy finding matrix alignment parameter remains study second term side rearranging sums write object apply interpolation result proposition consolidate powers random matrix consider multilinear function since matrix gaussian series moments orders therefore index max max max max matrix concentration inequalities three terms maximum admit bound may well consider third one max max max max max first step definition reach second line use fact commutes cycle trace third line inequality used left unitary invariance matrix absolute value delete next take maximum unitary matrices apply lyapunov inequality draw expectation term involving finally identify quantity note maximum bounded alignment parameter defined similar calculations valid two terms whence since possible choices determine main part argument finished putting pieces together conclude merge bounds obtained solve resulting inequality quantity combine reach clearing factors reach inequality andp nonnegative numbers nonnegative solution quadratic inequality must satisfy follows take square root invoke subadditivity square root twice reach finally simplify numerical constants arrive comparison standard deviation alignment parameters last task section establish proposition states alignment parameter never exceeds standard deviation easiest way obtain result use block matrices inequalities schatten norm fix integer fix triple unitary matrices consider quantity establish proposition suffices show using block matrices converting trace schatten norm write tropp entries block column matrices indexed pairs arranged lexicographic order invoke inequality schatten norms write product two block matrices sum two factors form suffices bound first one indeed identified matrix variance defined applied inequality fact identified invoked unitary invariance schatten norm recognized quantity summary established needed show econd rder atrix hintchine trong sotropy section prove extension theorem gives lower upper bounds trace moments strongly isotropic matrix gaussian series theorem matrix khintchine strong isotropy let hermitian matrix gaussian series assume strong isotropy property define matrix standard deviation parameter matrix alignment parameter max maximum ranges triple unitary matrices integer catp lower bound also requires written catp pth catalan number function normalized trace max proof result appears starting section reach statement theorem introduction rewrite normalized traces terms schatten norms fact states catalan numbers satisfy bound catp gives explicit numerical form upper bound discussion establish theorem let comment proof meaning result important observation estimate extremely accurate least examples particular goe matrix goe defined showed section standard deviation parameter alignment parameter therefore theorem implies goe catp estimate sufficient prove limiting spectral distribution goe semicircle law see sec details derive law trace moments furthermore markov inequality implies norm goe high probability proof theorem lot common arguments leading proposition theorem main innovation use strong isotropy imitate moment identity would hold free probability idea allows remove dependence standard deviation term although may seem proof requires matrix gaussian series analogous techniques based theory exchangeable pairs allow deal types random matrix series observation potential lead universality laws also clear argument could prove related results approximate form strong isotropy matrix concentration inequalities addition possible extend ideas rectangular matrix gaussian series case consider hermitian dilation correct analog strong isotropy observation allows obtain sharp bounds trace moments rectangular gaussian matrices fashion even show limiting spectral density sequence rectangular gaussian matrices distribution provided aspect ratio sequence held constant finally remark similar arguments applied obtain algebraic relations stieltjes transform matrix approach may lead directly limit laws sequences random matrices increasing dimension see sec sec argument species preliminaries aside results collected far proof theorem requires additional ingredients first state basic properties catalan numbers fact catalan numbers pth catalan number defined formula catp particular catp nondecreasing catp catalan numbers satisfy recursion catq next result covariance identity product centered functions gaussian vector thm regarded refinement inequality provides bound variance centered function gaussian vector fact gaussian covariance identity let independent standard normal vectors let functions whose derivatives square integrable respect standard normal measure assume symbol refers differentiation respect coordinate usual statement result involves semigroup given elementary formulation finally need bound solution certain type polynomial inequality estimate related fujiwara inequality sec include proof sketch since could locate precise statement literature proposition polynomial inequalities consider integer fix positive numbers implies proof sketch consider polynomial descartes rule signs implies exactly one positive root say furthermore positive number direct calculation one may verify satisfies means conclude implies tropp normalized trace moments let commence proof theorem first introduce notation normalized trace moments matrix gaussian series clear since symmetric random variable odd trace moments zero remains calculate even trace moments obtain second moment simple argument first identity follows direct calculation using definition matrix gaussian series second identity strong isotropy hypothesis last relation definition take spectral norm see identified standard deviation parameter defined representation moments major challenge compute rest even moments usual first step invoke gaussian integration parts integer lemma implies considering instead makes argument cleaner analyze expression examine index separately subject one treatment fix index first center adding subtracting expectations vanish one zero mean productive think first sum side approximation side second sum perturbation let focus first sum side last display use strong isotropy hypothesis simplify expression last identity follows side note motivation imitate moment identity would hold free sense free probability finally combine last three displays reach observe modified indexing sums step depends facts odd perturbation term next step argument bound perturbation term terms alignment parameter defined use gaussian covariance identity fact end let explain write summand perturbation term covariance let real diagonal matrix diag expanding normalized trace using coordinate indices find apply gaussian covariance identity expectation introduce parameterized family random matrices independent copy matrix concentration inequalities observe distribution although dependent fact fact deliver combining formulas expressing result terms normalized trace find fact expression valid hermitian matrix unitary invariance trace summing last identity reach point alignment parameter starts become visible finding matrix alignment parameter next goal control expression terms alignment parameter use interpolation result proposition bound sum choice indices obtain estimate max max max max max random unitary matrix commutes corresponding random matrix section bound term maximum fashion example consider fourth term max max max first step inequality trace second step inequality expectation last line recall distribution identify finally recognize matrix alignment parameter defined summary shown introduce bound arrive used numerical inequality valid finally sum expression index conclude required bound perturbation term tropp recursion trace moments view shown written indicate expression contains lower bound upper bound normalized trace moment next two sections solve recursion obtain explicit bounds trace moments first obtain upper bound catp result gives inequality afterward assuming establish lower bound catp together estimates yield statement theorem solving recursion upper bound begin proof upper bound first step argument remove lag term recursion using moment comparison fix integer observe increasing matrix first inequality holds second inequality lyapunov introduce estimate recursion obtain polynomial inequality form proposition ensures words using formula apply induction prove catp stated result follows take square root invoke subadditivity let commence induction formula holds noted assuming bound holds integer range verify bound also valid integer range bound implies catq catq exp cat using definition catalan numbers one may verify catq increasing catq case follows inspection latter bound recursion catalan numbers together imply catr take root determine catr used numerical inequality valid combine estimate recursive bound obtain catr matrix concentration inequalities see holds induction may proceed solving recursion lower bound turn proof lower bound assuming use induction show catp result follows take root begin induction recall formula valid suppose valid integer range verify formula lower branch recursion states induction hypothesis yields catq catr catr used fact achieves maximum value one endpoints convexity also applied recursive formula catalan numbers bound implies log therefore second inequality holds catalan numbers nondecreasing combine last three displays arrive catr catr verified formula completes proof ppendix nterpolation esults appendix establish proposition interpolation inequality multilinear function random matrix whose proof appears appendix multivariate complex interpolation interpolation result use body paper consequence general theorem interpolation function several complex variables proposition multivariate complex interpolation let natural number positive number define simplicial prism consider bounded continuous function pair distinct indices assume analytic section property analytic iti itk establish proposition next two sections argument relies principles support standard univariate complex interpolation although seems likely result form already appears literature able locate reference tropp preliminaries proposition depends hadamard theorem prop proposition theorem consider vertical strip complex plane consider bounded continuous function analytic interior sup sup sup see result delivers case proposition proof proposition proof multivariate interpolation result proposition follows induction number arguments let begin base cases function one argument inequality obviously true next consider bivariate function bounded continuous analytic section property fix point define bounded continuous function assumption implies analytic select gives application theorem proposition implies supt sup introducing definition simplifying supt supt sups sups case proposition fix positive integer suppose established inequality functions arguments words assume bounded continuous analytic section property supt iti need extend result functions variables consider bounded continuous function analytic section property fix complex vector define number formula trivial therefore may assume introduce function one may verify inherits boundedness continuity analytic sections therefore induction hypothesis gives supt iti fixed choice index numbers consider function since bivariate case provides iti sup sups isi isk matrix concentration inequalities combine bounds reach supt iti itk since see second product form term first product thus sup step completes induction established proposition interpolation multilinear function random matrices prepared establish interpolation result proposition multilinear function random matrices actually establish somewhat precise version state proposition refined multilinear interpolation suppose multilinear function fix nonnegative integers let random hermitian matrices necessarily independent max unitary matrix commutes establish result next section observe proposition immediately implies proposition interpolation result use body paper indeed recognize large parenthesis side geometric mean bound geometric mean maximum components proof proposition perturbative argument may assume matrix almost surely nonsingular indeed parameter replace modified matrix yei independent family standard normal variables completing argument draw zero obtain inequality original random matrices first step argument perform polar factorization random hermitian matrix unitary almost surely positive definite two factors commute index clarity argument introduce unitary matrices notation perform interpolation matrices next introduce function replacing powers complex variables set simplicial prism defined statement proposition claim function bounded continuous analytic sections properties required apply interpolation result proposition let assume claim holds complete proof relation proposition imply sup fix index product introduce unitary matrix polar factor follows tropp similarly define therefore sup max construction commutes index second line apply jensen inequality relax supremum include unitary matrices commute corresponding replace supremum maximum since unitary group compact function continuous needed show finally must verify claim multilinear function bounded continuous const fix point let applying observation function const const const first estimate follows jensen inequality bound multilinear function second inequality depends unitary invariance spectral norm identity kpkre polar decomposition last bound inequality geometric arithmetic mean since conclude bounded since continuous application dominated convergence theorem shows continuous function well proof analytic sections similar fix vector since multilinear easy check map analytic fixed choice pair distinct indices together morera theorem theorem allow conclude also analytic therefore analytic section property force claim established cknowledgments afonso bandeira responsible argument section ramon van handel offered critical comments parts research completed mathematisches forschungsinstitut oberwolfach mfo instituto nacional pura aplicada impa rio janeiro author gratefully acknowledges support onr award sloan research fellowship gordon betty moore foundation eferences anderson guionnet zeitouni introduction random matrices volume cambridge studies advanced mathematics cambridge university press cambridge bandeira van handel sharp nonasymptotic bounds norm random matrices independent entries available http bhatia matrix analysis volume graduate texts mathematics new york bondarenko radchenko viazovska optimal asymptotic bounds spherical designs ann math bai silverstein spectral analysis large dimensional random matrices springer series statistics springer new york second edition buchholz operator khintchine inequality probability math chen gittens tropp masked sample covariance estimator analysis using matrix concentration inequalities inf inference chen tropp subadditivity matrix concentration random matrices electron matrix concentration inequalities garling inequalities journey linear analysis cambridge university press cambridge kemp math introduction random matrix theory available http oleszkiewicz best constant inequality studia khintchine dans math acad sci paris ledoux talagrand probability banach spaces isoperimetry processes springer berlin marden geometry polynomials second edition mathematical surveys american mathematical society providence mackey jordan chen farrell tropp matrix concentration inequalities via method exchangeable pairs ann nourdin peccati normal approximations malliavin calculus volume cambridge tracts mathematics cambridge university press cambridge stein method universality nica speicher lectures combinatorics free probability volume london mathematical society lecture note series cambridge university press cambridge pisier vector valued completely maps pisier martingale inequalities comm math tao topics random matrix theory volume graduate studies mathematics american mathematical society providence tropp tail bounds sums random matrices found comput tropp introduction matrix concentration inequalities foundations trends machine learning appear available http van handel spectral norm inhomogeneous random matrices available http vershynin introduction analysis random matrices compressed sensing pages cambridge univ press cambridge vale waldron tight frames generated finite nonabelian groups numer algorithms
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deep transform cocktail party source separation via complex convolution deep neural network andrew simpson centre vision speech signal processing university surrey guildford deep neural networks dnn state art many engineering problems yet addressed issue deal complex spectrograms use circular statistics provide convenient probabilistic estimate spectrogram phase complex convolutional dnn typical cocktail party source separation scenario trained convolutional dnn complex spectrograms two source speech signals given complex spectrogram monaural mixture discriminative deep transform used complex convolutional obtain probabilistic estimates magnitude phase components source spectrograms separation results par equivalent based separation approaches index learning supervised learning complex convolution deep transform introduction convolutional deep neural networks dnn capable exploiting geometric assumptions data structure order share network weights convolutional dnn applied sliding window fashion predictions given datapoint may made multiple alternate windowed contexts distribution predictions probabilistic estimate may obtained computer vision problems datapoints representing pixel intensities positive real numbers therefore convolutional dnn used make predictions intensity given pixel parametric statistics may used obtain probabilistic estimate pixel intensity summarizes predictions made various different contexts sliding window containing pixel question audio spectrogram provides intuitive visual counterpart image computer vision problem equivalent application convolutional dnns computer audition less straight forward particular equivalence spectrogram image holds interpretations complex spectrogram limited magnitude component approach ignoring phase component complex spectrogram imply serious limitations classification problems phase may critical audio synthesis phase critical furthermore convolutional dnn applied phase component spectrogram appropriate compute probabilistic estimates phase overlapping windowed predictions using parametric statistics use circular statistics obtain probabilistic estimates phase computed using convolutional dnn illustrate approach context typical cocktail party source separation problem featuring complex spectrograms trained complex convolutional dnn separate speech cocktail party scenario dnn used complex convolutional deep transform trained separate speech two concurrent speakers input layer dnn provided complex spectrogram output layer dnn trained respective complex spectrograms two speakers used trained complex convolutional make probabilsitic predictions new concurrent speech mixtures speakers using objective source separation quality metrics analyzed separation quality results par equivalent probabilsitic source separation techniques offer slightly better sound quality method consider typical simulated cocktail party listening scenario featuring male voice female voice speaking concurrently speakers separately recorded mono reading story two speech signals equalized intensity linearly summed superposed produce competing voice scenario speech signals decimated sample rate khz transformed spectrograms using fourier transform stft window size samples overlap interval sample hanning window provided complex spectrograms frequency bins spectrograms computed first minutes speech signals used training data subsequent seconds data held back later use testing separation model fig probabilistic cocktail party source separation via complex convolutional deep transform upper pair spectrograms plot excerpt original test speech audio single central spectrogram plots linear mixture two speech signals lower pairs spectrograms plot respective source signals separated using complex cdt without output gain adaptation training data mixture component speech spectrograms cut windows time samples windows overlaped intervals samples thus every window training model mixture spectrogram matrix size samples corresponding pair source spectrograms gave approximately training examples testing stage seconds speech spectrogram used overlap intervals sample giving approximately test frames would ultimately applied overlaping convolutional output stage prior windowing complex spectrogram data separated magnitude phase spectrograms respectively spectrogram normalized unit scale magnitude phase data mapped range used dnn size units spectrogram magnitude phase window size unpacked vector length giving two respective vectors size dnn configured input layer vector concatenated mixture magnitude spectrogram samples followed mixture phase spectrogram samples giving total vector length output layer trained synthesize vector featuring sequential concatenation magnitude phase spectrograms respective male female speech component signals meant dnn trained respective components respective concatenated locations output layer vector dnn employed activation function throughout zero bias output layer dnn trained using full iterations stochastic gradient descent sgd iteration sgd featured full sweep training data dropout used training probabilistic testing stage model used signal processing device output layer activations taken synthetic output test data overlap interval sample means test data described speech spectrogram terms sliding window output model sliding window format convolutional model frame input mixture spectrogram passed model produce respective predictions respective magnitude phase spectrograms male female voice respectively sgd trained autoencoder type dnn feature neurons output layer degree invariant persistent activity account activity included output gain adaptation stage mean activation across test frames subtracted individual activations frame time constant adapted output layer predictions accumulated sliding window size moved steps thus separate predictions obtained column output spectrograms speakers respective magnitude phase spectrograms gave output distribution spectrograms contained within matrix indexed using time frequency window index magnitude matrices running average magnitude spectrogram calculated phase spectrogram matrices equivalent circular mean phase angle computed follows phase spectrograms predicted output layer first remapped range range shown convenience transformed matrix unit vectors plane using following elementwise matrix operation subscript indices dropped convenience cos circular sum window size computed matrix operation matrix operation circular mean angle matrix computed taking fourquadrant inverse tangent indices onwards dropped convenience respective estimated magnitude phase spectrograms recombined complex spectrogram following matrix operation exp estimated complex spectrogram subjected inverse stft using procedure separation quality resulting separated audio respect original time domain audio signals measured using toolbox quantified terms ratio sdr ratio sir ratio sar separation quality assessed iteration sgd training order evaluate trajectory performance training measure iii results fig plots spectrograms illustrating stages mixture separation brief excerpt seconds test data model trained iterations top spectrograms plot original male female speech audio single central spectrogram plots linear mixture illustrating large degree overlap feature space next downwards plotted spectrograms complex convolutional probabilistic audio output gain adaptation employed finally bottom plotted spectrograms representing respective complex convolutional probabilistic audio featuring output gain adaptation sets output spectrograms illustrate features closely resemble original signals visual inspection output gain adaptation results less noise somewhat better definition features output spectrograms appear captured bandwidth original signals fricative noise components originals replicated faithfully either case informal listening revealed respective separated output audio good quality noisy fig illustrates case model feature output gain adaptation model tested fig plots mean objective separation quality measures sdr sir sar computed entire test data respect original audio averaged across male female voice signals function training iteration range three functions appear nearing convergence around iterations three functions less monotonic peak separation quality around iterations averaged across two voices sdr sir sar compares well equivalent dnn binary mask approach reported previously identical data conditions achieved slightly worse artefact performance equivalent sir see fig sdr sar minor advantage presumably due ability present approach employ phase directly convolutional dnn binary mask approach phase taken mixture spectrogram present results also compare reasonably well ideal binary mask computed test data using mask computed source spectrograms achieves sdr sir sar present results also much better convolutional probabilistic approach reported previously degree may inclusion phase magnitude present approach provides small advantage terms sampling may broadly equivalent also note similar based convolutional dnn approach reported previously present performance appears superior earlier matrix factorization nmf based approaches featured small scale dnn within nmf pipeline however results methods directly comparable present results insight neural processing auditory system associated perceptual illusions function acknowledgment ajrs supported grant engineering physical sciences research council epsrc references fig complex convolutional separation quality function training iterations mean ratio sir ratio sdr ratio sar computed audio separated using complex convolutional function training iteration measures computed test audio using toolkit averaged across two voices discussion conclusion introduced complex convolutional approach cocktail party source separation using spectrograms probabilistic approach features parametric statistical estimation spectrogram magnitude circular statistical estimation phase convolutional dnn trained two minutes speech two speakers tested seconds new speech speakers separation quality similar binary mask based convolutional dnn aproaches features slightly improved artefact performance although dnn employed layers consider degree abstraction already provided stft inverse stft giving effective depth layers demodulation synthesis surprising approach works well furthermore ability model operate full phase information whilst retaining topographic projection stft appears advantage may also performance present model enhanced use hanning window acts similarly oversampling mitigate aliasing suggested affect dnn learning performance generally circular statistical process probabilistic synthesis described may useful general probabilistic synthesis level complex spectrogram general level model interpreted auditory model featured output gain adaptation appears similar observed early auditory system auditory gain adaptation temporal occurs various timescales tens seconds even minutes principle output gain adaptation present model may interpreted featuring rectangular temporal integration window length seconds entire test data hence present model may interpreted demonstrating neuronal output gain adaptation may useful terms synthesis noise reduction hence findings may provide lecun bengio convolutional networks images speech time series handbook brain theory neural networks lawrence giles tsoi back face recognition convolutional neural network approach neural networks ieee transactions krizhevsky sutskever hinton imagenet classification deep convolutional neural networks advances neural information processing systems simpson ajr deep transform error correction via probabilistic grais sen erdogan deep neural networks single channel source separation acoustics speech signal processing icassp ieee int conf huang kim smaragdis deep learning monaural speech separation acoustics speech signal processing icassp ieee int conf simpson ajr probabilistic source separation convolutional deep neural network mcdermott cocktail party problem curr biol simpson ajr abstract learning via demodulation deep neural network vincent gribonval performance measurement blind audio source separation ieee trans audio speech language processing simpson ajr deep transform cocktail party source separation via probabilistic simpson ajr deep neural network dean harper mcalpine neural population coding sound level adapts stimulus statistics nat neurosci watkins barbour specialized neuronal adaptation preserving input sensitivity nat neurosci rabinowitz willmore schnupp jwh king contrast gain control auditory cortex neuron ulanovsky las nelken processing sounds cortical neurons nat neurosci ulanovsky las farkas nelken multiple time scales adaptation auditory cortical neurons neurosci simpson ajr harper reiss mcalpine selective adaptation oddball sounds human auditory system neurosci
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linkages highly connected directed nov katherine irene paul november abstract study disjoint paths problem kddpp highly strongly connected digraphs integral kddpp even restricted instances input graph connected show integrality condition relaxed allow vertex used two paths problem becomes efficiently solvable highly connected digraphs even part input specifically show absolute constant exists kddpp solvable time connected directed graph function grows rather quickly also show kddpp solvable time connected directed graphs also show deciding feasibility kddpp instances given part input even restricted graphs strong connectivity introduction let positive integer instance directed problem ordered tuple directed graph ordered sets distinct vertices instance integrally feasible exist paths directed path paths pairwise vertex disjoint paths referred integral solution linkage problem disjoint paths problem kddpp takes input instance directed problem problem integrally feasible output integral solution otherwise return problem feasible kddpp notoriously difficult problem shown even restriction fortune hopcroft wyllie attempt make kddpp tractable thomassen asked problem would easier assume graph highly connected define separation directed graph pair exist edge order separation separation trivial graph supported european research council european unions seventh framework programme grant agreement department computer science university rome sapienza rome italy department computer science university rome sapienza rome italy email wollan strongly exist nontrivial separation order let define directed graph integrally every linkage problem integrally feasible thomassen conjectured exists function every connected digraph integrally later answered conjecture negative showing function exists moreover also showed even restricted problem instances graph connected article relax kddpp problem requiring potential solution use vertex twice define directed problem feasible exist paths directed path every vertex contained two distinct paths paths form solution main result article kddpp polynomial time solvable even part input graph sufficiently highly connected define graph every disjoint paths problem feasible theorem integers exists value every strongly graph moreover exists absolute constant given instance kddpp find solution time assumption highly connected theorem omitted usual complexity assumptions theorem determine whether given kddpp instance halfintegrally feasible even assumption connected value theorem grows extremely quickly however fix still efficiently solve kddpp significantly weaker bound connectivity given theorem theorem exists function satisfying following let positive integer given problem connected determine problem feasible output solution time given kddpp even case previous work problem focused various relaxations problem schrijver showed fixed kddpp polynomial time solvable input graph assumed planar later cygan improved result showing kddpp fixed parameter tractable assumption input graph planar recent series articles leading breakthrough showing grid theorem holds directed graphs kawarabayashi kreutzer kawarabayashi showed following relaxation kddpp efficiently resolved fixed showed exists polynomial algorithm given instance kddpp one following find directed paths links every vertex four distinct determine integral solution exists terms hardness results slivkins showed kddpp even restricted acyclic graphs kawarabayashi announced proof slivkins result extended show kddpp also two primary steps proof theorem first show highly connected graph contains large structure use connect appropriate pairs vertices exact structure use bramble depth two bramble set pairwise touching connected strongly connected subgraphs widely studied certificates large directed undirected graphs see sections exact definitions details existence bramble depth two follows immediately kawarabayashi kreutzer proof grid theorem however algorithm given runs polynomial time fixed size bramble show section appropriate assumptions hold proof theorem theorem able find large bramble depth two time graph vertices absolute constant second main step proof theorem show use bramble depth two find desired solution given instance kddpp define linkage set pairwise disjoint paths show section given instance large bramble depth two find smaller along linkage order every element path element distinct subgraph moreover linkage internally disjoint time find linkage distinct subgraphs vertices thus linking appropriate endpoints bramble able find desired solution fact bramble depth two ensures solution find uses vertex twice result given theorem statement proof presented section linking structure bramble depth two instance common technique disjoint path cycle problems undirected graphs see examples main contribution theorem extend technique directed graphs particular simultaneously find linkage linkage made significantly difficult directed case directional nature separations directed graphs fact easy way control separations cross proofs theorems given section construction showing theorem given section directed arborescence directed graph vertex called root property every vertex unique directed path thus every arborescence arises tree selecting root directing edges away root write exists directed path write let directed graph set directed walk first last vertex also contains vertex note every set union strongly connected components let directed graph tree decomposition triple arborescence functions partition sets sets called bags decomposition sets called guards decomposition define incident width smallest integer directed minimum width tree decomposition johnson robertson seymour thomas showed assume fixed positive integers efficiently resolve kddpp restricted directed graphs theorem theorem exists function satisfying following let let problem directed determine integrally feasible output integral solution time simple construction shows result holds efficiently resolve problems halfintegrally graph fixed first define following operation double vertex directed graph create new vertex add edges edges edges edges edges corollary exists function satisfying following let let instance problem directed given input directed width determine problem feasible output solution time proof fix positive integer let instance klinkage problem let directed graph obtained doubling every vertex define problem letting thus feasible integrally feasible moreover integral solution easily converted solution original problem let tree decomposition width observe defined yields tree decomposition width thus theorem determine integrally feasible find solution polynomial time assuming fixed proving claim certificates large directed bramble directed graph set strongly connected subgraphs exists edges links links cover set order bramble minimum size cover bramble number denoted maximum order bramble elements bramble called bags size bramble denoted number bags contains bramble number directed graph gives good approximation seen following theorem formulated theorem exist constants directed graphs holds johnson robertson seymour thomas showed one efficiently either find large bramble directed graph explicitly find directed note result stated algorithmically algorithm follows construction proof additionally looked alternate certificate large namely havens order immediately gives bramble order definitions theorem exist constants directed graphs algorithmically find time either bramble order order moreover find bramble elements long open question johnson robertson seymour thomas whether sufficiently large treewidth directed graph would force presence large directed grid minor let positive integer directed graph defined follows let directed cycles length let vertices labeled odd let directed path vri even let directed path vri directed grid major recent breakthrough kreutzer kawarabayashi confirmed conjecture johnson theorem function given directed graph fixed constant polynomial time obtain either cylindrical grid order butterfly minor directed tree decomposition width purposes use brambles attempting solve kddpp however order ensure paths find use vertex twice require bramble depth two define depth bramble directed graph words bramble depth positive integer vertex contained distinct subgraphs bramble note depth size order least lemma directed contains model bramble size depth two proof let cycles paths vertex labels vij definition cylindrical grid every every let subpath endpoints vli let unique cycle contains vertices let cycles form bramble depth two size desired finding bramble depth two section show given call sufficiently large set vertices directed graph able efficiently find large bramble depth two argument many ways follows diestel proof robertson seymour grid theorem see proof undirected graphs begin collection disjoint linkages show linkages find sublinkage pairwise disjoint need two classic results graph theory namely ramsey menger theorems theorem menger theorem let directed graph subsets maximum number paths equals minimum order separation separates moreover exists algorithm find maximum set paths minimum order separation time theorem ramsey theorem let positive integers every improper two coloring edges undirected clique red blue exists either subgraph every edge colored red subgraph every edge colored blue moreover desired subgraph found time absolute constant first give two preparatory lemmas presenting main result section lemma let digraph vertices let positive integers let let two disjoint sets vertices order let linkage linkage size assume endpoints assume exists permutation endpoints one following holds exist subgraphs forming bramble size depth two exists subset subgraphs pairwise disjoint moreover given input find either time absolute constant proof define auxiliary undirected bipartite graph vertex set edges thus union two perfect matchings component either cycle single edge induced subgraph let note connected component strongly connected subgraph assume case least distinct components theorem exists one following holds subgraphs pairwise intersect subgraphs pairwise disjoint moreover find time first case claim form bramble depth two clearly construction sets form bramble see depth two observe linkages given fact vertex contained must case contained subpath set elements forming disjoint two intersect vertex thus bramble depth two second case every fix contained follows satisfies outcome statement lemma conclude distinct components thus exists component size least let disjoint paths length least assume exists index edges distance least six fix pairwise disjoint subpaths path length five contains exactly three edges distance least five assumption fact starts ends edge corresponding path fact elements pairwise disjoint follows outcome lemma conclude subgraph two edges distance least six corresponding paths intersect follows exists five edge path containing three edges strongly connected subgraph containing ramsey argument applied subgraphs see one desired outcomes holds subset vertices directed graph pair subsets exists directed linkage order lemma let set directed graph vertices let positive integers let pairwise disjoint subsets let linkage order let linkage order one following holds exist holds exists forming bramble size depth two moreover find linkages satisfying outcome bramble time absolute constant proof theorem may assume exist order every element intersects every element definition set exists linkage similarly linkage theorem find linkages time label elements elements let two permutations label elements qxt qyt qxi common endpoint common endpoint similarly qyi common endpoint common endpoint apply lemma linkages qxt may assume get outcome lemma without loss generality may assume subgraphs pairwise disjoint similarly applying lemma qyt subgraphs pairwise disjoint since paths intersects intersects conclude subgraph contains strongly connected subgraph contains since every intersects every forms bramble size depth two required show main result section given sufficiently large set efficiently find large bramble depth two theorem exists function satisfies following let directed graph vertices positive integer let directed path set contains bramble depth two moreover given input find time absolute constant proof given let use notation ftl function iterated times beginning input let fix pairwise disjoint subsets xiin xiout exist subpaths satisfy following xiin xiout traversing directed path vertices xiin occur vertices xiout moreover pick xiin xiout ftt assuming function statement theorem satisfies ftt see xiin xiout exist fix directed linkage xiout jin order ftt theorem find linkages time fix enumeration let define linkages order ftt follows let let pair apply lemma linkages may assume exist disjoint sublinkages size ftt call respectively distinct fix arbitrary subset order ftt fix element linkage construction paths disjoint define strongly connected subgraphs follows fix subgraph along subpath linking endpoints values taken modulo similarly define along subpaths pit linking endpoints along analogous subpaths pit pit think paths laid grid subgraphs natural strongly connected graphs formed following column grid strongly connected graphs formed rows every vertex one subgraphs either path moreover subset contained disjoint contained thus possible two intersect vertex three distinct intersect common vertex theorem exists size either intersect pairwise disjoint observation pairwise intersecting forms bramble size depth two thus may assume distinct similarly exists distinct without loss generality assume set forms bramble size depth two completing proof linking bramble depth two main result section following shows sufficiently large bramble depth two use efficiently resolve given instance kddpp modest assumption connectivity graph theorem exists positive integer connected directed graph contains bramble depth two size every problem instance feasible moreover given bags find solution time begin notation recall doubling vertex directed graph defined section contract set vertices inducing strongly connected subgraph delete create new vertex add edges edges edges edges let depth two bramble directed graph define graph follows first let graph obtained doubling every vertex belonging two bags least one bag vertex denote double let collection subsets obtained replacing vertex belonging bag exactly one bags belongs thus elements pairwise disjoint induces strongly connected subgraph depth bramble let graph obtained contracting element denote set contracted vertices note vertices form bidirected clique observe every double vertex gets contracted vertex write bag corresponding vertices contracted stress bag particular let disjoint subsets vertices directed graph separation separates separation properly separates nonempty positive integer say every separation separating order least let directed graph brambles depth two let say linkage none paths contains internally vertex give quick outline proof proceed let denote approach proving feasability two steps find three sets paths one set paths linking bramble another set linking third linking appropriate ends paths first two sets inside get first two sets paths take advantage high connectivity graph linking inside bramble easy structure allows link pairs vertices like need union three sets paths form solution choose first second sets almost intersect bramble limited way third set paths completely contained underlying idea behind approach finding first two sets paths contract bag bramble doubling vertices two bags try apply menger theorem trying issues arise first want ends paths belong distinct bags concerningly contracting bags bramble may destroy connectivity bramble terminals solve throwing away bounded number bags bramble left highly connected subsection show find first two sets paths lemma modulo finding lemma third set paths lemma show put pieces together prove theorem subsection prove lemma linking inside depth two bramble lemma let connected directed graph bramble depth two size let problem instance find paths pks pkt satisfying following pis directed path vertex pit directed path vertex vertices belong distinct bags say bsk btk respectively every vertex belongs two pks vertex belong two paths say pis pjs similarly every vertex belongs two pkt vertex belong two paths say pit ptj internal vertices pis pit belong one bag distinct pis ptj every vertex belongs two pks pkt moreover given bags find paths pks pkt time prove following lemma intermediate step lemma section lemma let connected directed graph bramble depth two size let problem instance assume disjoint exist brambles kbs kbt also moreover find time first let see lemma implies lemma proof lemma consider brambles given lemma denote vertices belong exactly one bag two bags claim exist paths links kbt suppose menger theorem exists separation order separating kbt consider following separation let kbs kbs kbt kbs kbs kbt intuitively separation viewed graph plus add vertices kbs kbt side easy check separation since every vertex belongs kbs also every vertex belongs two bags every vertex belongs one bag contradicts lemma proves claim choose paths let view paths original graph since kbt vertex except vertex choose exists edge second last vertex let pis path obtained replacing notice pis path paths pks internally disjoint satisfy claim exist paths links kbt moreover vertices distinct suppose menger theorem graph separation order properly separating kbt order properly separates kbt contradicting lemma proves claim may also choose paths viewing paths paths symmetrically obtain paths pkt pit joining paths satisfy let set bsi bti check paths pks pkt satisfy seven assertions lemma statement already established see holds note internally disjoint kbt similarly paths internally disjoint kbt moreover definition every vertex kbt either graphs belongs one bag therefore one bag follows internal vertex pis pit belongs one bag proving see let distinct suppose contradiction vertex belongs pis ptj internal vertex either pis ptj belongs one bag also belongs two bags deduce internal vertex pis ptj since found graph know belongs one bag one bag found graph belongs one bag two bags contradiction proving finally let check suppose contradiction sake vertex belongs three paths must pis pjs pit ptj pis pjs may assume without loss generality path found graph containing must bsi since depth two bsj internal vertex contradicting without loss generality bti internal vertex since belongs two bags implies contradiction remains check indeed find paths time indeed finding brambles takes time using lemma sets paths found time according theorem easily get pks pkt linear time following lemma shows solve linkage problem depth two bramble provided terminals belong distinct bags lemma let directed graph let two ordered vertices suppose bramble depth two belong distinct bags bsk btk respectively exist paths links additionally every vertex two distinct paths finally also holds bsi bti find paths time proof obtain follows definition bramble exist vertices bsi bti either since bsi bti strongly connected exist directed path contained bsi directed path contained bti take concatenation two paths construction belongs bsi bti since bags bsk btk distinct every vertex belongs two distinct bags follows desired collection paths found time overall running time follows deduce theorem lemmas follows proof theorem let pks pkt bsk btk satisfy given lemma satisfying hypothesis lemma let paths guaranteed lemma let pis pit concatenation three paths clearly directed walk linking therefore contains directed path need check paths suppose contradiction sake vertex distinct symmetry consider four cases case lemma bsi bti bsj btj belongs two bags internal vertex contradiction case pis pjs lemma may assume since follows bsi internal vertex pjs bsj well contradiction case pit ptj lemma may assume since follows bti internal vertex ptj bsj well contradiction case pis ptj lemma contradicts contradiction running time bound follows bounds given lemmas finding bramble link section prove lemma need following easy combinatorial lemma lemma let set elements suppose proper subsets exist proof prove lemma induction base case trivial assume lemma holds show must also hold let lemma statement choose element possibly relabelling proper subsets induction exist deduce actually prove lemma one side time applying following lemma symmetric version lemma let integers suppose klinkage problem instance connected directed graph bramble depth two size assume disjoint exists bramble kbs also moreover find time proof lemma assume connected bramble depth two size lemma bramble kbs proof symmetric lemma one show exists bramble kbt moreover running time follows running time bound given lemma proof lemma let let clique contracted vertices set find bramble looking generate sequence brambles contained previous one find one sufficiently highly connected purposes consider algorithm procedure find sequence brambles graphs well separations separating bramble graphs respectively algorithm generating brambles let separation properly separates minimum order end remark separation graph separation obesrve properly separates properly separates observe also disjoint main claim following claim separation order least let first check claim implies lemma using algorithm find separation given claim time take loop algorithm remove less bags bramble follows prove claim need following intermediate claim suppose sake contradiction claim find indices nonempty proof first obtain set indices algorithm gives procedure find algorithm generating set max end max let first check observe since sets correspond distinct sets bags vertex belongs two bags recall similarly since thus let check pair indices satisfies first property claim show holds inductively since base case holds trivially assuming property holds let show holds observe unique element smallest index need show true choice want two indices satisfying second property claim recall follows lemma exist indices since follows nonempty proves claim prove claim proof claim recall assumption fix claim separation separation let let observe also separation meet since strongly connected meets choice meet belongs implies similarly contains contracted vertices thus separation may view separation namely second property claim separation nontrivial strong connectivity must order least deduce rewriting conclude assumption assume let turn attention graph define words obtain replacing vertices belonging bags expanded obtain intermediate graphs corresponding contracted vertices note view separation note consider separation order contradicts minimality proves claim completes proof theorem proofs theorems given theorems easy complete proofs theorems begin theorem proof theorem let function theorem let value necessary size bramble order apply theorem resolve instance kddpp let connected graph vertices let instance kddpp greedily find path note subset vertices thus set theorem find time bramble size least theorem find solution time completing proof theorem proof theorem need two additional results note neither statement algorithmic existence algorithm follows immediately constructive proof lemma let directed graph vertices bramble path intersecting every element given input find path time lemma let directed graph graph vertices bramble order path intersecting every element exists set order given input algorithmically find time proof theorem let instance kddpp let let necessary size bramble order apply theorem resolve instance kddpp let function theorem theorem either find tree decomposition width bramble order given tree decomposition corollary solve time function instead find bramble order apply theorem convert bramble depth two theorem may assume thus time find path intersecting every element lemma lemma time find subset finally applying theorem find find bramble size depth two finally theorem resolve time total algorithm takes time function desired lower bounds section give proof theorem proof theorem give reduction satisfiablity problem boolean formulas reduction variant one given shows integral directed acyclic graphs fix formula variables clauses set choose even set noting construct problem instance strong connectivity construct graph follows see figure example let let variable create directed path containing internally vertex form directed path containing vertex form figure illustration graph formula let set observe strong connectivity least since adjacent vertex consider problem instance clearly polynomial reduction need show satisfiable feasible claim satisfiable feasible fix satisfying assignment consider linkage consisting paths path whose interior concatenation set false set true path contains variable true path contains negation variable false path true path false paths straightforward check vertex used twice linkage proving feasible claim feasible satisfiable suppose solution observe vertex belongs therefore belong path solution terminal thus every path must use either show following assignment satisfies set variable false path used two paths true path used twice paths every path must use path uses used two paths uses used two paths particular must used exactly twice union paths deduce path contains two edges moreover middle vertex path either variable belongs set true negation variable belongs set false proves satisfiable references cygan marx pilipczuk pilipczuk planar directed paths problem tractable foundations computer science focs ieee annual symposium foundations computer science pages oct reinhard diestel tommy jensen konstantin gorbunov carsten thomassen highly connected sets excluded grid theorem journal combinatorial theory series steven fortune john hopcroft james wyllie directed subgraph homeomorphism problem theoretical computer science thor johnson neil robertson paul seymour robin thomas directed journal combinatorial theory series kawarabayashi kreutzer directed grid theorem arxiv november kawarabayashi improved algorithm finding cycles elements international conference integer programming combinatorial optimization pages springer kawarabayashi yusuke kobayashi stephan kreutzer excluded grid theorem digraphs directed disjoint paths problem proceedings annual acm symposium theory computing stoc pages new york usa acm kawarabayashi stephan kreutzer excluded grid theorem digraphs forbidden minors pages kawarabayashi stephan kreutzer directed grid theorem proceedings annual acm symposium theory computing pages acm karl menger zur allgemeinen kurventheorie fundamenta mathematicae bruce reed introducing directed tree width electronic notes discrete mathematics robertson seymour graph minors disjoint paths problem journal combinatorial theory series alexander schrijver finding disjoint paths directed planar graph siam journal computing aleksandrs slivkins parameterized tractability paths directed acyclic graphs siam journal discrete mathematics carsten thomassen graphs european journal combinatorics carsten thomassen highly connected digraphs combinatorica
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aug generators reductions ideals local noetherian ring finite residue field louiza fouli bruce olberding bstract let local noetherian ring residue field much known generating sets reductions ideals infinite case finite less well understood investigate existence lack thereof proper reductions ideal number generators needed reduction case finite field give formula smallest integer every ideal reduction follows local noetherian ring every ideal principal reduction number maximal ideals normalization reduced quotient higher dimensions show positive integer exists ideal reduction dim ideal chosen ntroduction let commutative noetherian ring let ideal reduction subideal equivalently subideal denotes integral closure corresponding ideal northcott rees proved local noetherian ring infinite residue field krull dimension every ideal reduction reduction generated elements result generalizations involving analytic spread underlie many applications theory reductions local algebra example reductions analytic spread instrumental describing asymptotic properties ideal property rees algebra blowup proj spec along subscheme defined however residue field finite may exist ideals reduction applicability reductions case local rings finite residue field limited mathematics subject classification key words phrases reduction integral closure finite field analytic spread first author partially supported grant simons foundation grant fouli olberding article examine extent result northcott rees involving reductions fails case finite residue field prove two main results first devoted rings second rings higher dimension case find optimal choice replacing result northcott rees smallest possible positive integer integer depends size residue field cardinality max rred set maximal ideals normalization rred reduced quotient rred theorem let local noetherian ring finite residue field smallest positive integer every ideal reduction max rred local noetherian ring multiplicity every ideal generated elements theorem thus number theorem local noetherian ring infinite residue field every ideal principal reduction using theorem extend result local rings residue field size see corollary corollary let local noetherian ring residue field every ideal principal reduction max rred particular complete local noetherian domain every ideal principal reduction corollary thus dimension one interesting local noetherian rings finite residue field result northcott rees holds every ideal reduction generated dim elements moreover dimension one even ideals without principal reduction least guaranteed existence bound number elements needed generate reduction moving beyond dimension one use theorem show see theorem unlike case infinite residue field bound exists local noetherian ring finite residue field dimension least theorem let local noetherian ring dimension residue field finite positive integer ideal minimally generated elements proper reduction also ideal chosen reductions ideals notation throughout article denotes total quotient ring ring integral closure denote rred reduced ring nilradical set maximal ideals denoted max reliminaries section develop criterion every ideal local noetherian ring finite residue field reduction criterion proposition used proofs main results next section first lemma routine application properties reductions concerns transfer reductions ring reduced quotient rred lemma let ring let positive integer finitely generated ideal reduction irred reduction proof clear every ideal reduction every ideal rred reduction conversely let ideal suppose ideal jrred reduction irred write thus let krred jrred thus krred reduction irred claim reduction since krred irred proposition krred irred remark nilradical contained every integrally closed ideal conclude thus reduction discussed introduction local noetherian ring infinite residue field every ideal principal reduction removing restriction infinite residue field assert general every height ideal principal reduction proposition let local noetherian ring every height ideal principal reduction proof let height ideal lemma suffices show irred principal reduction may assume without loss generality reduced ring let denote minimal prime ideals since height local may assume fouli olberding minimal prime ideals since reduced prime avoidance exists since prime ideals maximal ideal follows claim since height every element zero divisor thus ideal consists zerodivisors since reduced noetherian ring finite product integral domains see example fact consists zero divisors implies iri iri moreover since power contained thus domain iri xri since conclude proposition fact integral extension implies since also reduction next lemma proposition give criteria every ideal noetherian ring reduction stronger result proposition requires also reduced local light theorem assumption necessary proposition state lemma recall arithmetic rank ara proper ideal noetherian ring least number lemma let noetherian ring let proper ideal let integer ara ideal reduction reduction proof let proper ideal let ideal since noetherian ring ara set nonempty contains maximal element ideal suppose may choose ideal hence assumption ideal maximality forces contradiction fact therefore reduction proposition let reduced local noetherian ring let positive integer following equivalent every ideal reduction reductions ideals survive proof proof use fact every ideal principal ideal seen follows theorem fact reduced local noetherian ring dimension implies finite product noetherian integrally closed domains since also dim dim finite product dedekind domains fact semilocal implies dedekind domains principal ideal domains finite product principal ideal domains ring property every ideal principal ideal let denote maximal ideal first suppose every ideal reduction let let nonzerodivisor ideal since hence contains nonzerodivisor therefore also contains nonzerodivisor assumption reduction let established every ideal principal ideal thus principal ideal necessarily generated nonzerodivisor since contains nonzerodivisor since principal ideals generated nonzerodivisor admit proper reductions obtain therefore since hence survive conversely suppose survive prove every ideal reduction suffices lemma show every ideal reduction since ara proper ideal let let clearly reduction height proposition principal reduction thus remains consider case since every ideal principal ideal since reduced contains nonzerodivisor nonzerodivisor let assumption exists survive since follows tar fouli olberding moreover proof lemma fact tar implies since proves reduction results section prove main results paper proving first theorem deals case indicate theorem introduction follows end section theorem prove theorem introduction theorem let local noetherian ring finite residue field let positive integer every ideal reduction max rred proof lemma every ideal reduction every ideal rred reduction thus suffices prove theorem case reduced ring throughout proof let denote set elements maximal ideal assume since local ring nonzero elements units denote elements cartesian product let prove theorem suppose first max show ideal reduction proposition suffices show every survives assumption max reductions ideals therefore may index set maximal ideals since ideals maximal diagonal map surjective ring homomorphism thus may choose let claim every survives show first show dimension space indeed suppose choice implies since conclude contrary fact choice contradiction shows dimension space particular generated fewer elements let claim survives since generated elements dimension space since generated fewer elements adding many elements needed assume without loss generality generated fewer elements particular nakayama lemma implies generated elements form since space dimension matrix whose entries field rank elementary row operations produce rank matrix reduced row echelon form fouli olberding deleting column yields identity matrix follows observation nakayama lemma let claim first observe choice elements remains show let notice choice therefore proving claim conclude survives shows every survives therefore ideal reduction conversely suppose max prove every ideal reduction suffices proposition show let consider defined claim first contained common maximal ideal suppose maximal ideal suppose way contradiction without loss generality assume since also since contradiction implies next claim since reductions ideals similarly therefore argument similar one preceding paragraph shows since otherwise every contradiction thus last equality follows fact maximal ideal integral extension lies consequently since forces since holds conclude proves contained common maximal ideal next since two distinct set contained maximal ideal follows either one survive least maximal ideals established beginning proof every survives conclude max last inequality given assumption implies impossible since contradiction implies survive particular survive therefore proposition every ideal reduction theorem follows easily theorem setting theorem seek smallest positive integer need smallest positive integer max rred max rred equivalently max rred fouli olberding yields conclusion theorem remark noetherian local domain number maximal ideals number minimal prime ideals corollary moreover reduced local noetherian ring geometrically regular formal fibers number maximal ideals number minimal prime ideals completion theorem thus excellent local noetherian rings bound theorem restated using completion rred rather normalization rred corollary let local noetherian ring residue field every ideal principal reduction max rred proof apply theorem case corollary let local noetherian domain residue field multiplicity every ideal principal reduction proof completion local noetherian domain multiplicity minimal prime ideals follows multiplicity formula given theorem remark max thus corollary consequence corollary remark follows corollary noetherian local domain multiplicity every ideal principal reduction known already reasons local cohenmacaulay ring multiplicity every ideal generated elements theorem since multiplicity every ideal theorem sally vasconcelos prove ring every ideal property every ideal principal reduction reduction number see also lemma corollary let noetherian local domain whose completion two minimal prime ideals every ideal principal reduction proof apply remark corollary reductions ideals corollary every ideal complete local domain principal reduction proof apply corollary next use case theorem show absence bound number generators reductions higher dimensions theorem let local noetherian ring dimension residue field finite positive integer ideal minimally generated elements proper reduction also ideal chosen proof let let denote residue field first assertion theorem suffices show exists ideal generated elements reduction see example proposition since noetherian dimension infinitely many prime ideals dimension one choose positive integer let distinct dimension one prime ideals let reduced local noetherian ring minimal prime ideals direct product integrally closed domains consequently least maximal ideals lemma theorem ideal reduction let ideal generated reduction reduction contrary choice conclude ideal reduction hence minimally generated elements proper reduction remains prove second assertion theorem suppose may choose ideal krull height theorem ideal must exist reduction since reduction would ideal generated elements thus proof complete assume choose ideal since krull height theorem implies fouli olberding arithmetic rank therefore since lemma implies ideal reduction shows ideal reduction conclude minimally generated elements proper reduction remark example example given cohenmacaulay local ring finite residue field maximal ideal fails reduction example abhyankar showed certain canonically defined ideals regular local ring reductions motivates question heinzer shannon question whether every integrally closed ideal regular local ring finite residue field reduction theorem guarantees existence ideals without reductions unclear whether ideals chosen integrally closed general local noetherian ring property every integrally closed ideal reduction yet still possess closed ideals reduction see example xamples section give several examples illustrate ideas section first example show order ideal local noetherian domain principal reduction sufficient every integrally closed ideal principal reduction example integer exists local noetherian domain every integrally closed ideal principal reduction yet ideal minimally generated elements proper reduction let local noetherian domain suffices example since nonzero principal ideal principal reduction suppose choose maximal ideals ring let let denote jacobson radical localization multiplicatively closed set let local noetherian domain maximal ideal normalization see example lemma proposition thus pid max observe reductions ideals thus since pid blow maximal ideal sense lipman ring property localization maximal ideal embedding dimension equal multiplicity therefore theorem every integrally closed ideal principal reduction reduction number however since residue field two elements max lemma theorem imply ideal minimally generated elements reduction proposition ideal proper reduction corollary implies maximal ideal local noetherian domain multiplicity principal reduction corollary guarantees exist local rings multiplicity ideals without principal reduction necessarily local ring residue field field next example appears example shows exists local ring multiplicity whose maximal ideal principal reduction example see example let ring given multiplicity let denote maximal ideal let denote images respectively suppose principal reduction write since let let thus lemma way may assume principal reduction generated element form thus possible generators principal reductions minimal prime ideal however reduction ideal thus principal reductions although ring example domain used produce similar examples domains recall theorem lech theorem complete local noetherian ring maximal ideal completion local noetherian domain ass nonzero integer zerodivisor fouli olberding example local noetherian domain whose maximal ideal principal reduction let example since cohenmacaulay since nonzero integer follows theorem lech exists local noetherian domain completion preceding example maximal ideal principal reduction since maximal ideal extended maximal ideal follows maximal ideal principal reduction moreover since complete intersection multiplicity following example suggested bill heinzer example let one show min every linear form belongs minimal prime hence principal reduction maximal ideal hand number minimal primes thus number maximal ideals rred remark since also corollary remark imply exists ideal principal reduction case ideal following example suggested bernd ulrich example let one show min straightforward show principal reduction number minimal primes even though case principal reduction remark corollary remark must exist different ideal principal reduction one verify ideal moreover another proper reduction principal reduction contained hence minimal reductions minimal generating sets different sizes examples section devoted case next example illustrate failure reductions maximal ideal ring mentioned remark heinzer ratliff rush example given finite field example local ring residue field associated graded ring maximal ideal fails reduction case give simple example phenomenon properties reductions ideals example let ring associated graded ring grm claim reduction indeed example reduction one assume two generators reduction linear forms since residue field linear forms straightforward check ideal generated two linear forms height therefore reduction acknowledgements thank bill heinzer bernd ulrich suggesting examples served original motivation article eferences topics local algebra notre dame mathematical lectures university notre dame press fontana huckaba papick domains monographs textbooks pure applied mathematics marcel dekker new york heinzer ratliff rush reductions ideals local rings finite residue fields proc amer math heinzer shannon abhyankar work dicritical divisors available huckaba commutative rings zero divisors monographs textbooks pure applied mathematics marcel dekker new york katz number minimal prime ideals completion local domain rocky mountain math lech method constructing bad noetherian local rings lecture notes math springer berlin lipman stable ideals arf rings amer math matsumura commutative ring theory cambridge university press northcott rees reductions ideals local rings proc camb phil soc sally numbers generators ideals local rings marcel dekker new sally vasconcelos stable rings pure appl algebra swanson huneke integral closure ideals rings modules london mathematical society lecture note series cambridge university press cambridge epartment athematical ciences exico tate niversity ruces exico usa address lfouli epartment athematical ciences exico tate niversity ruces exico usa address olberdin
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zeitschrift angewandte mathematik und mechanik discretization based hamilton law varying action apr janine roger sina aices graduate school rwth aachen university templergraben aachen germany department engineering science university oxford parks road oxford united kingdom received xxxx revised xxxx accepted xxxx published online xxxx key words hamilton law varying action hermite interpolation nonlinear elastodynamics symplectic integration variational integrators msc develop class time integration methods applicable conservative problems elastodynamics methods based hamilton law varying action action continuous system derive spatially temporally weak form governing equilibrium equations expression first discretized space considering standard finite elements resulting system discretized time approximating displacement piecewise cubic hermite shape functions within time domain thus achieve displacement field velocity field discrete virtual action finally construct class schemes methods examined analytically numerically study linear nonlinear systems well inherently continuous discrete structures numerical examples focus applications provided theory however general valid also problems show favorable candidate denoted converges order four thus especially high accuracy numerical solution required scheme efficient methods lower order exhibits linear simple problems properties similar variational integrators symplecticity remains investigated whether symplecticity holds arbitrary systems numerical results show excellent energy behavior copyright line provided publisher introduction work derive class time integration methods computational analysis deformable solids consider discrete version hamilton law varying action obtain discretization schemes based cubic hermite functions time corresponding author sauer phone fax copyright line provided publisher mergel sauer discretization based hamilton law overview existing methods one common approach numerical analysis elastodynamic problems application semidiscrete procedures spatially temporally continuous system discretized space time separately first mechanical equilibrium equations describing deformation body discretized space means finite element method fem point refer standard literature nonlinear finite elements solids see ref spatially discrete system discretized time using instance finite difference scheme collocation based taylor series expansion discretization schemes type include many methods newmark algorithm method method method besides methods fulfilling equilibrium equations single time steps exist various approaches based weighted residuals consider equilibrium sense back publication zienkiewicz weighted residual approach based cubic hermite interpolation time instance discussed ref see also generalized method proposed modak sotelino addition exists broad literature methods combining finite elements space time general solution schemes constructed forming spatially temporally weak form equations motion discretizing resulting statement means finite elements first approaches accounting finite elements time back ref early publications methods include ref broader literature review time integration methods structural mechanics found ref special class time integration schemes applied mechanical systems formed geometrical integrators geometric integration enables design robust methods provide quantitatively qualitatively accurate results since methods preserve geometric properties flow differential equation able exactly represent main characteristic properties physical process geometric integration methods mainly divided two classes integrators symplectic integrators first class methods fulfills conservation laws energy momentum automatically methods type refer simo tarnow simo gonzalez betsch steinmann leyendecker hesch betsch gautam sauer krenk betsch janz references therein see also generalized method discussed ref second class preserves symplectic form presence symmetries momentum maps additionally shows excellent energy behavior integrators represented class variational integrators conservative systems methods constructed forming discrete version hamilton principle choosing function space suitable numerical quadrature see ref overview dissipative controlled mechanical systems derived discrete version alembert principle within last years variational integrators extended towards constrained stochastic multirate multiscale systems well electric circuits variational integrators combination spatial discretization refer ref references therein besides approaches exists covariant discretization method marsden scheme allows symplecticity space time copyright line provided publisher zamm header provided publisher previously mentioned work solution approximated using piecewise lagrange interpolation mechanical system leads smooth approximation position discontinuities velocity discrete time steps besides approach leok shingel developed variational integrator based piecewise hermite interpolation approach solution discrete eulerlagrange equations also time derivatives approximated sufficient accuracy leads globally smooth approximation solution note ref include combined discretization space time incorporate initial conditions mechanical system explicitly early publications structural dynamics considered hamilton law varying action see argyris scharpf fried bailey simkins borri law regarded generalization hamilton principle accounts initial final velocities considering variations displacement boundaries time domain studies mentioned also include cubic hermite interpolation time displacement based law family methods proposed combines different displacement velocity initial final time addition constructed subsequent application hamilton law time integration method based hermite interpolation variational integrators automatically symplectic due discretized action integral serving generating function clear properties hold integration schemes constructed hamilton law objectives paper derive class integration methods based piecewise cubic hermite interpolation time end consider hamilton law varying action thus directly incorporate additional boundary terms arising variations time integration method using approach first discretize resulting equilibrium equation space time instead deriving additional conditions time derivatives approximated solution done ref consider independent variations position velocities general one could construct variational integrator varying action entire temporal domain deriving variation set discrete equations since cubic hermite approximation however would lead unconditionally unstable numerical method pursue different approach vary action discrete time interval individually leads overdetermined system four equations choosing different combinations equations derive family six different methods one schemes coincides method proposed fact time integration methods variational sense derived virtual action total time domain demonstrate numerically however favorable schemes denoted following shows similar properties like true variational integrators excellent behavior simple harmonic oscillator symplecticity interestingly case variant discussed ref emphasize aim work present construction time integration methods demonstrate important features means various numerical examples analytical investigation copyright line provided publisher mergel sauer discretization based hamilton law including proof order convergence goes beyond scope paper instead addressed future work note like methods based approximations time integration schemes favorable simulation discontinuous changes shock waves mechanical systems instead apply temporally smooth examples include linear nonlinear well intrinsically discrete spatially continuous problems focus applications theory however also valid conservative systems compared formulation method leok shingel based cubic hermite interpolation exhibits higher order convergence addition based desired accuracy may efficient classical methods like newmark algorithm outline remainder paper structured follows section introduces action integral continuous body deforming time hamilton law varying action spatially temporally weak form mechanical equilibrium equation derived section briefly outlines spatial discretization means standard finite elements temporal discretization discussed section providing solution strategy leads class different integration methods methods related approaches literature section study main characteristic properties integration schemes symplecticity convergence behavior favorable scheme applied investigate several numerical examples section section finally concludes paper hamilton law varying action section summarize governing equations describing body undergoing finite motion deformation general theory continuum mechanics reader referred text books consider body deforming within time domain initial configuration body denoted boundary denoted body subjected volumetric loads applied deformations prescribed surface loads applied time deformation body characterized unique mapping material point current position material time derivative corresponds velocity material particle located short also write action continuous system following assume conservation mass hyperelastic material behavior external forces depend deformation start action integral continuous system defined integrand corresponds lagrangian system given assume copyright line provided publisher zamm header provided publisher kinetic energy potential energy due internal strains external forces terms respectively denote initial material density energy density function characterizing material behavior detailed description different material models found ref variation action consider admissible variation deformation dimension euclidean space varying action integral yields variations energy terms given grad tensor denotes cauchy stress derived strain energy density function appearing enforce deformation fixed variations become zero arrive classical hamilton principle see book lanczos instead however leave variations arbitrary case equal following boundary term evaluated term angle brackets corresponds scalar product variation linear momentum body detailed discussion arbitrary variations boundaries found ref see chapter due scalar product depends system initial momentum thus initial velocity explicitly reason expression discussed several early publications studying initial value problems structural dynamics ref overview also found ref following terminology used several papers refer hamilton law varying action interesting comment alternatively equation referred hamilton weak principle hwp see ref copyright line provided publisher mergel sauer discretization based hamilton law origin found paper bailey ref copies hamilton original papers obtained found hamilton furnished called law varying action furnish known hamilton principle evidently latter part century application concepts variational calculus euler lagrange reduced hamilton law hamilton explain later two additional boundary terms caused variations important derivation final integration schemes see section equation finally represents spatially temporally weak form governing equilibrium equations note expression general valid elastodynamic problems nevertheless since focus development analysis new time integration method numerically investigate problems paper detailed study problems may subject future work spatial discretization briefly outline spatial discretization means finite element method regarding nonlinear finite elements solids refer text books ref spatially discretize using nel finite elements element initial position deformation velocity approximated dot indicates derivative respect time vectors xnne xnne contain initial current positions well velocities nne nodes belonging element quantities still continuous respect time array nnne contains nodal shape functions nnne associated using isoparametric concept discretize variations means shape functions write grad suitable definition strain operator inserting relations obtain nel computed nel copyright line provided publisher zamm header provided publisher vectors denote deformation velocity spatial nodes elemental mass matrices force vectors fint fext computed fint fext introducing obtain shorter notation later refer global mass matrix global force vectors fint fext assembled elemental contributions equation finally corresponds spatially discrete version hamilton law varying action temporal discretization discretize spatially discrete virtual action time achieve temporal approximate nodal deformation element cubic hermite shape functions xte xte vectors contain nodal deformations corresponding nodal velocities see appendix definition shape functions vectors write order improve readability also use assembled counterparts accounting spatial nodes denote instead see analogy introduced previous section virtual action single time interval approximations inserted virtual action single time interval yields depends four variables thus reformulated increment following terminology marsden west define discrete momenta copyright line provided publisher mergel sauer discretization based hamilton law introduce two analogous variables appear due hermite discretization since unit momentum time refer discrete four terms computed discrete action given appendix results inserting momenta simplify addition since variations remain arbitrary must fulfill hamilton law varying action applied interval inserting finally obtain spatially temporally discrete version single time interval arbitrary variations note derive first varying continuous action integral discretizing variation space time would however obtain expression first discretized action varied discrete action single time interval solution strategy general variational integrators constructed summing discrete action time intervals taking variation summands would arrive following statement expression equivalent summed entire time domain solving subsequently however results method unconditionally unstable spectral radius larger one observation also discussed ref instead develop class methods arising virtual action individual time interval virtual displacements velocities presumed arbitrary copyright line provided publisher zamm header provided publisher provides nno equations nno number finite element nodes physically first two equations relate discrete momenta linear momenta see fig second two equations arise chosen hermite approach assuming displacement velocity previous time step given need given find interval fig equilibrium equations time interval nno equations determine new state system thus reason set two far arbitrary variations zero approach motivated following new deformation velocity computed remaining two equations finally obtain six methods illustrated fig seems promising approach definition follows enforces matching momenta discrete time steps method seen counterpart one four mixed methods varying displacement velocity copyright line provided publisher mergel sauer discretization based hamilton law scheme corresponds formulation proposed fig illustration six integration schemes interestingly resulting six methods completely different characteristics surprising favorable others section numerically investigate properties scheme terms preservation energy convergence behavior furthermore linear systems analyze stability symplecticity schemes several reasons especially focus first discrete velocities appearing hermite ansatz connected displacements setting linear momenta equal discrete momenta note approach possible explicitly copyright line provided publisher zamm header provided publisher account boundary terms appearing hamilton law varying action second since definition directly follows discrete equations fulfilled automatically chapter hairer interestingly satisfies balance linear momentum generalization conservation momentum averaged time step appendix nevertheless must point six integrators variational becomes apparent especially four mixed methods use one displacement one velocity show however least simple linear problems first two methods similar properties like variational integrators symplecticity addition numerically demonstrate methods show good energypreserving behavior even nonlinear problems multiple degrees freedom future work would interesting compare schemes variational integrators detail investigate symplecticity arbitrary systems would interesting examine whether conserves momentum maps associated symmetries lagrangian implementation general equations nonlinear thus must linearized using newton method provides system linear equations iteratively solved new positions velocities finite element nodes linearization derivatives discrete momenta required see appendix analogy force vectors mass matrix terms computed assembling contributions spatial element denoted possible integrals computed analytically done contributions due kinetic energy due linear elastic internal energy appendix remaining integrals evaluated gaussian quadrature choosing sufficient number quadrature points time integration schemes neither energy total energy system must evaluated explicitly since want investigate numerical examples however discuss quantities appendix relation methods idea applying hamilton law varying action initial value problems structural dynamics goes back first approaches using finite elements space time see ref instead displacement initial final time required hamilton principle publications account initial displacement velocity idea motivated baruch riff combine different either displacement velocity approach results six different methods related several important differences schemes since authors discovered previous work instability solution scheme given propose modified discretization virtual displacements ref approach discretized considering second derivatives shape functions variation copyright line provided publisher mergel sauer discretization based hamilton law displacement thus approximated linear instead cubic function time modification leads different partial derivatives action therefore different integration method even important difference schemes concerns boundaries riff baruch consider boundaries entire time domain derive schemes within time interval approach results six different methods solving equations subsequently contrast possible ref remaining formulations ref equations would solved simultaneously summary one could loosely relate six schemes subsequent application methods riff baruch time interval underlying equations however approximated differently besides references mentioned discuss linear dynamic systems forces depend displacement linearly recently leok shingel proposed variational integrator based hermite finite elements time formulation derived approach addition discrete equations method accounts system equation motion strong form cubic hermite shape functions used schemes velocities computed using expressions inserted temporally discrete action one time interval depends displacements final time integration method leok shingel obtained varying incremental action respect displacements setting total virtual action zero compared six hermite formulations resulting method requires half number unknowns solved within time step rate convergence however lower best schemes see section note combination spatial temporal discretizations discussed ref properties six schemes investigate different properties six formulations first focusing linear problem single degree freedom behavior consider simple harmonic oscillator spring pendulum mass stiffness initial elongation displacement velocity oscillator computed analytically uan cos van sin frequency oscillation given period length duration one oscillation determined following numerical results normalized initial energy system copyright line provided publisher zamm header provided publisher compare hermite schemes implicit newmark algorithm choosing newmark parameters regarding linear systems method unconditionally stable conserve energy see ref besides discussed ref newmark variational addition newmark algorithm consider variational integrator based linear finite elements time see appendix refer method fig shows displacement velocity oscillator three periods coarse time discretization expected six schemes displacement velocity discrete time steps contrast approximates velocity constant along time interval leads discontinuities interval boundaries newmark algorithm displacement velocity evaluated discrete time steps observe oscillation period increases newmark method integrator geradin scheme scheme scheme scheme scheme displacement velocity fig harmonic oscillator displacement velocity three periods oscillation six hermite schemes compared newmark algorithm regarding maximum displacement two mixed methods amplitude oscillation seems increase remarkably fig indicates methods may unstable remaining mixed schemes amplitudes displacement velocity decrease contrast amplitude period oscillation fig shows total energy system oscillation periods observe schemes unstable schemes strongly dissipative agrees results shown fig expected remaining two schemes total energy qualitatively preserved interestingly accurate indicated smaller amplitudes oscillation fig compared linear integrator maximum relative errors smaller one order magnitude even two orders see table copyright line provided publisher mergel sauer discretization based hamilton law integrator geradin scheme scheme scheme scheme scheme total energy integrator scheme scheme zoom fig harmonic oscillator energy behavior periods oscillation six hermite schemes compared table harmonic oscillator maximum error total energy methods shown fig emax stability investigate stability six schemes means harmonic oscillator purpose introduce normalized time step insert following ref six schemes expressed form amplification matrix given appendix terms denote displacement velocity time step fig shows spectral radius schemes table shows maximum permitted time step schemes stable table fig show last two schemes unstable even small time steps contrast schemes seem stable large time steps nevertheless methods numerically dissipative see fig promising methods seem show excellent stability energy preservation symplecticity discussed section linear oscillator preserve energy system well motivates investigate whether methods generally symplectic one way prove symplecticity investigate derivatives phase state previous state linear copyright line provided publisher zamm header provided publisher geradin scheme scheme scheme scheme scheme geradin scheme scheme scheme scheme scheme spectral radius zoom purple line hidden behind blue one fig harmonic oscillator spectral radius six hermite schemes function normalized time step table harmonic oscillator maximum time steps schemes stable momentum given results jacobian according ref mapping symplectic symplectic identity matrix dimension nno harmonic oscillator jacobian reduces case one show determinant equal determinant amplification matrix introduced det det relation condition symplecticity fulfilled determinant equal one det det det copyright line provided publisher mergel sauer discretization based hamilton law means appendix one show fulfill det implies least harmonic oscillator schemes symplectic remains subject investigation whether also true arbitrary systems multiple degrees freedom convergence behavior focus four stable schemes equivalent method order study convergence harmonic oscillator consider maximum errors displacement velocity total energy discrete time steps given emax max uan van fig shows convergence behavior errors stable schemes newmark algorithm addition account results discussed ref cubic hermite interpolation note compared last three methods time integration schemes must account twice number unknowns step nodal displacements nodal velocities considered scaling abscissae introducing factor cdof cdof schemes otherwise cdof already shown fig piecewise approximation velocity discontinuous discrete time steps nevertheless determine discrete velocity computing momentum discrete legendre transformation approach also discussed considering boundary conditions velocity maximum errors velocity energy fig finally obtained expected newmark method conserve energy linear system corresponding error thus lies range machine precision errors displacement velocity however order rate convergence observed agrees discussion ref integrators interpolating displacement linearly time second order schemes also show order since computational effort higher newmark method methods favorable comparison leok shingel prove convergence order three integrator based cubic hermite interpolation theorem ref even better rate convergence achieved favorite candidate errors displacement velocity energy converge order copyright line provided publisher zamm header provided publisher emax max integrator geradin scheme scheme scheme steps steps cdof maximum error displacement cdof maximum error velocity max nsteps cdof maximum error energy fig harmonic oscillator convergence behavior cdof hermite schemes otherwise cdof dashed line labeled nsteps denotes estimated machine precision multiplied number computed steps numerical results elastodynamics shown previous sections symplectic harmonic oscillator possesses highest rate convergence far studied linear problems single degree freedom following section apply spatially continuous problems axial vibration linear elastic bar discuss free axial vibration linear elastic bar appendix spatial discretization either use linear lagrange cubic hermite finite elements see appendix text books ref second type element yields approximation displacement also space copyright line provided publisher mergel sauer discretization based hamilton law consider bar vibrating first lowest characteristic eigenmode test case corresponding displacement velocity energy analytically computed wave equation yields uan cos cos van cos sin amplitude oscillation first natural frequency deformation bar vibrating first mode shown fig six linear elements used spatial discretization expected bar performs sinusoidal oscillations due coarse finite element mesh however structure oscillates frequency slightly higher analytical solution oscillation period discrete system thus smaller analytical solution fig linear elastic bar two oscillations first natural frequency using linear like variational integrators incorporate numerical dissipation damp spurious oscillations thus must carefully adjust size time step spatial element mesh since stability analysis including spatial discretization quite tedious roughly estimate maximum permitted time step either linear hermite elements therefore consider cfl condition problems ccfl max ccfl velocity wave propagation characteristic discretization length linearly interpolated element applies hermite finite element choose take account twice number unknowns thus higher accuracy vary cfl number bar vibrating least oscillations estimated maximum values useful choose appropriate parameters following numerical examples course ensure however stability arbitrary cfl numbers smaller estimates shown stability analysis single degree freedom section methods may also become unstable small ranges parameters becomes apparent fig spectral radius exceeds limit one small range time steps stable larger steps apart cfl condition serve sufficient condition stability linear finite max max elements estimate implies time step fulfill copyright line provided publisher zamm header provided publisher max hermite finite elements obtain max case linear problems newmark method unconditionally stable convergence linear bar reconsider axial vibration discussed previous section following ref introduce displacement velocity discrete including relative errors time steps finite element nodes nno nno corresponds frobenius norm arrays normalized square roots numbers nodes time steps relative errors defined uan van analogy define discrete error energy using fig shows convergence behavior displacement velocity energy spatial discretization either use linear hermite elements consider two fixed cfl numbers refining mesh time step simultaneously linear finite element mesh left column fig three time discretization methods converge order indicates specific problem error caused spatial discretization predominates contrast error due spatial hermite discretization right column fig carries considerably less weight yields significantly higher convergence newmark algorithm note bar discretized linear elements resulting system treated naturally discrete springmass system consisting linear springs system temporally analytical solution cos van cos sin uan cos nno natural frequency determined analyzing eigenmodes discrete system fig show maximum errors displacement velocity arising temporal discretization errors ehu ehv determined inserting analytical solutions given expected orders convergence agree studied single degree freedom section table compares computational cost three methods results shown right column fig ccfl measure computation time required obtain error displacement smaller either see fig although must account twice number unknowns within time step takes due higher order convergence less computation time newmark method note linear example integrals discrete momenta evaluated analytically without numerical quadrature example requiring quadrature also time refer following section copyright line provided publisher mergel sauer discretization based hamilton law integrator scheme integrator scheme linear hermite error displacement spatially linear error displacement spatial hermite linear hermite error velocity spatially linear error velocity spatial hermite linear hermite error energy spatially linear error energy spatial hermite fig linear elastic bar convergence behavior displacement velocity energy refining mesh time step parameter given ccfl dashed line ccfl solid line ccfl copyright line provided publisher zamm header provided publisher ehv ehu linear integrator scheme linear temporal error displacement temporal error velocity fig linear elastic bar convergence temporal errors displacement velocity linear lagrange table linear elastic bar step size computation time tct test cases fig ccfl error displacement smaller tct denotes time one oscillation newmark tct tct tct vibration nonlinear bar numerical examples discussed previous sections cover naturally discrete spatially discretized continuum systems far linear problems internal forces depend displacement linearly investigated therefore consider nonlinear material behavior described appendix bar initially deformed prescribing displacement velocity fig shows deformation bar beginning long period oscillations hermite finite element mesh eight elements chosen better comparison results previous section time step normalized period length first eigenmode since mechanical response system differs linear case however initially sinusoidal oscillations turn set different interfering oscillations fig fig shows behavior nonlinear bar comparing system total energy newmark algorithm like harmonic oscillator section energy oscillates qualitatively preserved comparison methods relative error smaller five orders magnitude copyright line provided publisher mergel sauer discretization based hamilton law deformation deformation fig nonlinear bar deformation bar beginning oscillation long time period using hermite integrator scheme fig nonlinear bar energy behavior approximately periods oscillation using hermite maximum relative error addition investigate convergence behavior nonlinear bar considering hermite finite elements space since case deformation computed analytically compare results fine reference solution using fine mesh small time step like examples shown previous section refine discretizations simultaneously results shown fig accuracy scheme becomes apparent displacement total energy quantities observe significantly higher convergence note caused nonlinear material law require numerical quadrature evaluate time integral internal forces see also appendix demand however sufficient high accuracy fine solution computational cost may still lower newmark algorithm error displacement smaller fig instance requires step size newmark copyright line provided publisher zamm header provided publisher method contrast achieves accuracy already therefore measured computation time significantly lower time required newmark algorithm finally investigate well spatially discrete initial energy preserved time see fig error plotted thus arises temporal discretization comparison fig shows linear oscillator nonlinear bar achieve high order convergence integrator scheme eref eref hermite hermite error displacement reference solution error velocity reference solution ehe eref hermite hermite error total energy reference solution error total energy fig nonlinear bar relative errors respect fine reference solution initial energy spatially discrete system refining mesh time step shown ccfl hermite ref reference solution computed using lref conclusion work derive class time integration methods conservative elastodynamic problems using piecewise hermite interpolation approximate displacement deformable solid functions time velocity body thus entire time domain explicitly account initial velocities system consider generalization hamilton principle referred hamilton law varying action copyright line provided publisher mergel sauer discretization based hamilton law action integral continuous system derive spatially temporally weak form equilibrium equation elastodynamics expression first discretized space using standard galerkin finite element method afterwards spatially discrete system discretized time subdividing temporal domain set smaller time intervals approximating displacement cubic hermite shape functions generally methods belonging class variational integrators constructed varying action integral entire time domain equation one derive discrete equations order develop subsequent time integration method cubic hermite interpolation however resulting variational integrator unconditionally unstable issue discussed also riff baruch therefore first vary action time interval individually derive different methods solve new unknown displacement velocity yields six time integration schemes different properties technically methods variational sense derived virtual action total domain interestingly favorable methods denoted offers similar advantageous properties like variational integrators qualitatively accurate behavior symplecticity simple linear systems first investigate properties schemes considering harmonic oscillator demonstrate numerically shows conditional stability convergence order four afterwards examine linear nonlinear problems accounting either inherently continuous spatially discrete systems results show appropriate spatial discretization provides reasonable computational effort remarkably higher accuracy variational integrators based linear interpolation time scheme also energy system integration remains seen whether method symplectic arbitrary nonlinear systems properties convergence behavior preservation momentum maps presence symmetries addressed analytical study future since scope work construction time integration methods focus elastodynamic problems elaborate study elastodynamic problems either two three dimensions may scope future work addition since numerical results obtained within study spatial hermite finite elements look promising would interesting apply time integrator combination spatially discretizations particular isogeometric approaches contrast lagrangian finite elements isogeometric elements prevent errors require unnecessarily small time steps acknowledgements grateful german research foundation dfg supporting mergel sauer grants thank yuri suris technical university berlin sachin gautam iit guwahati marcus schmidt formerly rwth aachen university helpful comments implementation schemes section contains helpful details implementation time integration schemes copyright line provided publisher zamm header provided publisher shape functions spatial temporal discretizations consider either linear lagrange cubic hermite shape functions usually functions defined master domains denoted discretization time mapping master temporal domain characterized jacobian determinant determinant given six schemes linear lagrange shape functions cubic hermite shape functions schemes discrete lagrangian action integral discrete momenta obtained taking derivatives spatially temporally discrete action lht discrete lagrangian lht determined nel nel fext additionally use relation required total energy system computed implementation integrals pointed section integrals discrete momenta computed analytically purpose split internal force linear nonlinear part fint fnlin linear stiffness matrix case fnlin corresponds linear elasticity represents fully nonlinear case let fnlin fext copyright line provided publisher mergel sauer discretization based hamilton law inserting yields derivatives respect required linearization see section given alternative representation integrating parts obtain copyright line provided publisher zamm header provided publisher thus fulfills since see arrive linear variational integrator section outlines linear variational integrator use comparison section deformation velocity approximated defined analogy introduce interpolation discretized action becomes discretization virtual action follows since nth time step belongs two time intervals obtain integrals within partial derivatives also evaluated gaussian quadrature copyright line provided publisher mergel sauer discretization based hamilton law amplification matrices simple harmonic oscillator amplification matrices six schemes required given equations bar section investigate axial deformation thin bar characterized length density cross section area young modulus study linear nonlinear problems considering different material behavior bar let denote equivalents terms written bold font assume case virtual kinetic energy given reduces linear elastic bar discussed section virtual work given section consider material poisson ratio see ref spatial discretization means finite elements refer ref references argyris scharpf finite elements time space nucl eng des bailey application hamilton law varying action aiaa bailey new look hamilton principle found phys copyright line provided publisher zamm header provided publisher bailey hamilton ritz elastodynamics appl mech bailey method ritz applied equation hamilton comput methods appl mech eng baruch riff hamilton principle hamilton law correct formulations aiaa bauchau joo computational schemes int numer methods eng belytschko liu moran nonlinear finite elements continua structures wiley chichester betsch janz consistent method transient simulations mixed finite elements developed framework geometrically exact shells int numer methods eng doi betsch steinmann conservation properties time method part schemes elastodynamics int numer methods eng borri ghiringhelli lanz mantegazza merlini dynamic response mechanical systems weak hamiltonian formulation comput struct owhadi stochastic variational integrators ima numer anal chadwick continuum mechanics concise theory problems dover publications chung hulbert time integration algorithm structural dynamics improved numerical dissipation method appl mech nonholonomic integrators nonlinearity cottrell hughes bazilevs isogeometric analysis toward integration cad fea wiley cottrell reali bazilevs hughes isogeometric analysis structural vibrations comput methods appl mech eng demoures leyendecker ratiu weinand discrete variational lie group formulation geometrically exact beam dynamics numer math fetecau marsden ortiz west nonsmooth lagrangian mechanics variational collision integrators siam appl dyn syst fried analysis phenomena aiaa fung unconditionally stable accurate hermitian time finite elements int numer methods eng gautam sauer temporal discretization scheme adhesive contact problems int numer methods eng variational method direct integration transient structural response sound vib gonzalez exact energy momentum conserving algorithms general models nonlinear elasticity comput methods appl mech eng betsch steinmann conservation properties time method part higher order energy momentum conserving schemes int numer methods eng hairer lubich wanner geometric numerical integration edition springer berlin heidelberg hamilton general method dynamics study motions free systems attracting repelling points reduced search differentiation one central relation characteristic function phil trans soc london hamilton second essay general method dynamics phil trans soc london hesch betsch mortar method conserving schemes frictionless dynamic contact problems int numer methods eng copyright line provided publisher mergel sauer discretization based hamilton law hesch betsch transient domain decomposition problems mortar constraints conserving integration int numer methods eng hilber hughes taylor improved numerical dissipation time integration algorithms structural dynamics earthq eng struct dyn hodges bless weak hamiltonian finite element method optimal control problems guidance control dyn holzapfel nonlinear solid mechanics continuum approach engineering wiley chichester hughes cottrell bazilevs isogeometric analysis cad finite elements nurbs exact geometry mesh refinement comput methods appl mech eng hughes stability convergence growth decay energy average acceleration method nonlinear structural dynamics comput struct hughes hulbert finite element methods elastodynamics formulations error estimates comput methods appl mech eng hughes reali sangalli duality unified analysis discrete approximations structural dynamics wave propagation comparison finite elements nurbs comput methods appl mech eng hulbert time finite element methods structural dynamics int numer methods eng hulbert hughes finite element methods hyperbolic equations comput methods appl mech eng johnson leyendecker ortiz discontinuous variational time integrators complex multibody collisions int numer methods eng kane marsden ortiz west variational integrators newmark algorithm conservative dissipative mechanical systems int numer methods eng kobilarov marsden sukhatme geometric discretization nonholonomic systems symmetries discrete contin dyn syst ser krenk energy conservation newmark based time integration algorithms comput methods appl mech eng krenk global format based time integration nonlinear dynamics int numer methods eng kuhl crisfield decaying algorithms structural dynamics int numer methods eng kuhl ramm generalized method adaptive shell dynamics comput methods appl mech eng kuhl ramm time integration context energy control locking free finite elements arch comput methods eng lanczos variational principles mechanics edition dover publications new york leimkuhler reich simulating hamiltonian dynamics cambridge university press cambridge leok shingel variational integrators ima numer anal lew marsden ortiz west asynchronous variational integrators arch rational mech anal lew marsden ortiz west overview variational integrators finite element methods beyond edited franca tezduyar masud cimne barcelona copyright line provided publisher zamm header provided publisher lew marsden ortiz west variational time integrators int numer methods eng leyendecker betsch steinmann objective conserving integration constrained dynamics geometrically exact beams comput methods appl mech eng leyendecker marsden ortiz variational integrators constrained dynamical systems angew math mech zamm leyendecker variational approach multirate integration constrained systems multibody dynamics edited samin fisette computational methods applied sciences vol springer netherlands leyendecker marsden ortiz discrete mechanics optimal control constrained systems optim control appl methods marsden patrick shkoller multisymplectic geometry variational integrators nonlinear pdes commun math phys marsden west discrete mechanics variational integrators acta numer modak sotelino generalized method structural dynamics applications adv eng softw newmark method computation structural dynamics asce eng mech div junge marsden discrete mechanics optimal control analysis esaim control optim calc var saake construction analysis higher order galerkin variational integrators adv comput math tao cheng owhadi marsden variational integrators electric circuits comput phys oden general theory finite elements applications int numer methods eng peters izadpanah finite elements domain comput mech reich momentum conserving symplectic integrators physica riff baruch stability time finite elements aiaa riff baruch time finite element discretization hamilton law varying action aiaa simkins unconstrained variational statements initial problems aiaa simkins finite elements initial value problems dynamcs aiaa simo tarnow discrete method conserving algorithms nonlinear elastodynamics angew math phys zamp simo tarnow wong exact conserving algorithms symplectic schemes nonlinear dynamics comput methods appl mech eng suris hamiltonian methods type variational interpretation math model tao owhadi marsden structure preserving multiscale integration stiff odes sdes hamiltonian systems hidden slow dynamics via flow averaging multiscale model simul wolff bucher asynchronous variational integration using continuous assumed gradient elements comput methods appl mech eng wood bossak zienkiewicz alpha modification newmark method int numer methods eng wriggers nonlinear finite element methods springer berlin heidelberg zienkiewicz taylor finite element method solid structural mechanics edition elsevier zienkiewicz new look newmark houbolt time stepping formulas weighted residual approach earthq eng struct dyn copyright line provided publisher
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towards constructive version banaszczyk vector balancing theorem daniel shashwat shachar aleksandar dec centrum wiskunde informatica amsterdam dadush department mathematics computer science eindhoven university technology department computer science engineering university california san diego slovett department computer science university toronto anikolov abstract important theorem banaszczyk random structures algorithms states sequence vectors norm convex body gaussian measure exists signed combination vectors lands inside major open problem devise constructive version banaszczyk vector balancing theorem find efficient algorithm constructs signed combination make progress towards goal along several fronts first contribution show equivalence banaszczyk theorem existence distributions signed combinations case symmetric convex bodies equivalence implies existence universal signing algorithm independent body simply samples subgaussian sign distribution checks see associated combination lands inside body asymmetric convex bodies provide novel recentering procedure allows reduce case body symmetric second main contribution show framework efficiently implemented vectors length log recovering banaszczyk results stronger assumption precisely use random walk techniques produce required signing distributions vectors length log use stochastic gradient ascent method implement recentering procedure asymmetric bodies introduction given family sets universe goal combinatorial discrepancy minimization ispto find discrepancy maximum imbalance made small possible discrepancy theory discrepancy minimization plays major role rich history applications computer science well mathematics refer reader general exposition beautiful question regards discrepancy sparse set systems set systems element appears sets classical theorem beck fiala gives upper bound setting also conjectured bound true would tight improved bound given bukh iterated logarithm function base recently shown ezra lovett bound log holds high probability element assigned sets uniformly random best general bounds sublinear dependence currently depend srinivasan used beck partial coloring method give bound supported supported supported supported nwo veni grant netherlands organisation scientific research nwo project nsf career award sloan fellowship nserc discovery grant log min using techniques convex geometry banaszczyk proved general result vector balancing stated implies log min bound proofs srinivasan banaszczyk bounds provided efficient algorithm construct guaranteed colorings short exhaustive enumeration last years tremendous progress made question matching classical discrepancy bounds algorithmically currently essentially discrepancy bounds proved using partial coloring method including srinivasan made constructive constructive versions banaszczyk result however proven elusive recently recent work first second named authors jointly bansal gave constructive algorithm recovering banaszczyk bound setting well general setting alternate algorithm via multiplicative weight updates also given recently however finding constructive version banaszczyk general vector balancing theorem applications approximating hereditary discrepancy remains open problem theorem stated follows theorem banaszczyk let satisfy kvi pnthen convex body gaussian measure least exists lower bound gaussian measure easily seen tight particular vectors equal must allow gaussian measure small enough clear counterexample hand hard see gaussian measure otherwise exists halfspace containing clearly gaussian measure less banaszczyk theorem gives best known bound notorious conjecture generalization conjecture states thatpfor sequence vectors norm exists constant independent context banaszczyk theorem gives bound log log scaling unit ball gaussian measure banaszczyk theorem together estimates gaussian measure slices ball due barthe guedon mendelson naor give bound log min dimension span reduction see lecture shows bound problem implies log min bound setting results deal case cube banaszczyk theorem also applied cases used give best known bound steinitz conjecture problem input set vectors norm one summing aim find permutation minimise maximum sum prefix vectors rearranged according minimize steinitz conjecture bound always irrespective number vectors using vector balancing theorem banaszczyk proved bound log norm recently banaszczyk theorem applied general symmetric polytopes nikolov talwar approximation algorithm hereditary notion discrepancy hereditary discrepancy defined maximum discrepancy restriction set system subset universe shown effan efficiently computable quantity denoted bounds hereditary discrepancy given set system polylogarithmic factors upper bound used banaszczyk theorem natural polytope associated set system however since known algorithmic version banaszczyk theorem general body known efficiently compute colorings achieve discrepancy upper bounds terms recent work algorithmic bounds setting address general problem banaszczyk proof theorem follows ingenious induction argument folds effect pthe choosing sign body first observation finding point set inside equivalent finding point inducting set immediately possible may longer convex instead banaszczyk shows convex subset gaussian measure least long measure least allows induct base case needs show convex body gaussian measure least must contain origin fact follows easily hyperplane separation theorem indicated extremely elegant banaszczyk proof seen relatively mysterious seem provide tangible insights colorings look like results main contribution help demystify banaszczyk theorem showing equivalent constant factor length vectors existence certain subgaussian coloring distributions using equivalence second main contribution give efficient algorithm recovers banaszczyk theorem log min factor convex bodies improves upon best previous algorithms rothvoss eldan singh recover theorem symmetric convex bodies log min factor major consequence equivalence show sequence short enough vectors exists probability distribution colorings symmetric convex body gaussian measure least random variable lands inside probability least importantly distribution efficiently sampled immediately get universal sampler constructing banaszczyk colorings symmetric convex bodies remark recent work constructs restricted form distributions using random walk techniques show implement approximate version sampler efficiently guarantees conclusion vectors length log min provide details results sections extend results asymmetric convex bodies develop novel recentering procedure corresponding efficient implementation allows reduce asymmetric setting symmetric one reduction slight extension aforementioned sampler yields desired colorings note recentering procedure fact depends target convex body hence algorithms longer universal setting provide details results sections interestingly additionally show procedure extended yield completely different coloring algorithm using sampler achieving log min approximation factor surprisingly coloring outputted procedure essentially deterministic natural analytic description may independent interest continue detailed description results begin terminology reduction given set vectors shall call property hereditary holds subsets vectors note banaszczyk vector balancing bounds restricted set vectors hereditary since bound maximum norm vectors hereditary shall say property def colorings holds linear setting given shift one find coloring distribution colorings satisfies property banaszczyk theorem also extends standard arguments linear setting reducing norm bound factor drop follows example general inequality hereditary linear discrepancy proved lovasz spencer vesztergombi results work fact hold linear setting treating linear setting well known one always reduce case vectors linearly independent setting particular assume given shift linearly independent using standard linear algebraic technique find fractional coloring vectors linearly independent def set fractional coordinates see lecture chapter think reduction coloring linearly independent vectors indexed specifically given define lifting function map takes coloring lifts full coloring also satisfies property find coloring would define well moreover span replace work entirely inside convex bodies gaussian measure least central section gaussian measure least large reduced problem case linearly independent vectors space see section full details shall thus simplicity state results setting vectors linearly independent symmetric convex bodies subgaussian distributions section detail equivalence banaszczyk theorem restricted symmetric convex bodies existence certain subgaussian distributions begin main theorem section note holds general setting banaszczyk result theorem main equivalence let finite set following parameters equivalent universal constant factor independent minimum symmetric convex body gaussian measure least minimum exists random variable supported recall random vector subgaussian parameter unit vector words subgaussian marginals satisfy tail bound gaussian mean standard deviation apply discrepancy set signed combinations vectors context banaszczyk theorem directly implies kvi hence kvi furthermore extends linear setting letting pequivalence pmax mentioned banaszczyk theorem extends setting well existence universal sampler claimed previous section fact proof theorem particular follows directly following lemma lemma let random variable exists absolute constant symmetric convex body gaussian measure least distribution supported simply let denote random variable yields desired universal distribution colorings exactly statement lemma consequence see recover banaszczyk theorem symmetric convex bodies suffices ablep efficiently sample distribution sets type linearly independent norm pnmost rely homogeneity pifn random variable supported proof lemma see section details follows relatively directly convex geometric estimates combined talagrand majorizing measures theorem gives powerful characterization supremum gaussian process unfortunately lemma hold asymmetric convex bodies particular negated first standard basis vector conclusion clearly false matter much scale even though gaussian measure one may perhaps hope conclusion still holds ask either asymmetric setting though know prove note however makes sense support symmetric necessarily hold linear discrepancy setting describe high level idea proof reverse direction namely purpose show existence distribution expressed two player game first player chooses distribution second player tries find direction value game small distribution exists bound value game show appropriate convexification space subgaussianity tests second player associated symmetric convex bodies gaussian measure least use von neumann minimax principle switch first second player deduce value game bounded using definition random walk sampler algorithmic perspective turns subgaussianity natural property context random walk approaches discrepancy minimization results thus seen good justification random walk approaches making banaszczyk theorem constructive high level approaches one runs random walk coordinates fractional coloring coordinates hit either steps walk usually come gaussian increments though necessarily spherical try balance competing goals keeping discrepancy low moving fractional coloring closer since sum small centered gaussian increments subgaussian appropriate parameter natural hope output correctly implemented random walk subgaussian main result setting indeed possible limited extent main caveat walk output subgaussian enough fully recover banaszczyk theorem theorem let vectors norm let expected algorithm outputs random coloring random pnpolynomial time variable log achieve sampler guide random walk using solutions vector program whose feasibility first given nikolov show subgaussianity using martingale concentration bounds interestingly random walk analysis rely phases instead based simple relation walk convergence time subgaussian parameter added bonus also give new simple constructive proof feasibility vector program see section details avoids use sdp solver given results previous section random walk universal sampler constructing following colorings corollary let vectors norm let let symmetric convex body gaussian measure given membership oracle expected polynomial time algorithm outputs coloring log mentioned previously best previous algorithms setting due rothvoss eldan singh find signed combination inside log furthermore algorithms universal heavily depend body note algorithms fact tailored find partial colorings inside symmetric convex body gaussian measure least small enough setting sampler provide guarantees recall prior work random walk based discrepancy minimization random walk approach pioneered bansal used semidefinite program guide walk gave first efficient algorithm matching classic bound spencer combinatorial discrepancy set systems satisfying later lovett meka provided greatly simplified walk removing need semidefinite program recovered full power beck entropy method constructing partial colorings harvey schwartz singh defined another random walk based algorithm unlike previous work similarly algorithm explicitly use phases produce partial colorings random walks depend convex body walk polytope one remains convex body although analysis still applies polyhedral setting directly related paper recent work gives walk viewed randomized variant original proof walk induces distribution colorings coordinate output discrepancy perspective gives sampler finds colorings inside axis parallel box gaussian measure least rotations though universal manner matching banaszczyk result class convex bodies asymmetric convex bodies section explain techniques extend asymmetric setting main difficulty asymmetric setting one hope increase gaussian mass asymmetric convex body simply scaling particular take halfspace origin gaussian measure exactly technical level lack measure increase scaling breaks proof lemma crucial showing subgaussian coloring distributions produce combinations land inside main idea circumvent problem reduce setting mass symmetrically distributed origin particular barycenter induced gaussian measure origin body show constant factor scaling also gaussian mass least yielding direct reduction symmetric setting achieve reduction use novel recentering procedure carefully fix certain coordinates coloring well shift body make mass symmetrically distributed guarantees procedure stated theorem recentering procedure let linearly independent convex body gaussian measure least exists fractional coloring span following holds gaussian measure least gaussian measure barycenter origin convention procedure returns full coloring case since done shall treat conditions satisfied even though high level recentering procedure allows reduce initial vector balancing problem one possibly lower dimension respect convex body smaller gaussian measure particular gaussian measure least interestingly mentioned earlier introduction recentering procedure also extended yield full coloring algorithm explain high level details implementation together extension next subsection explain use fractional coloring theorem get useful reduction recall lifting function defined wepreduce initial vector balancing problem problem finding coloring note construction lift coloring satisfies def guarantee gaussian measure least barycenter origin allows direct reduction symmetric setting namely replace symmetric convex body without losing much gaussian measure formalized following extension lemma directly implies reduction subgaussian sampling section lemma let random variable exists absolute constant convex body gaussian measure least barycenter origin particular exists distribution colorings lemma implies random signed combination lands inside probability least thus asymmetric setting effectively reduced symmetric one claimed crucially recentering procedure theorem implemented probabilistic polynomial time one relaxes barycenter condition exactly small norm see section details furthermore estimate lemma robust perturbations thus constructively recover colorings asymmetric setting still suffice able generate good subgaussian coloring distributions combining sampler theorem together recentering procedure constructively recover banaszczyk theorem general convex bodies log factor theorem weak constructive banaszczyk exists probabilistic polynomial time algorithm input linearly independent set vectors norm log small enough necessarily symmetric convex body gaussian measure least given membership oracle computes coloring high probability far aware theorem gives first algorithm recover banaszczyk result asymmetric convex bodies restriction context note algorithm eldan singh finds relaxed partial colorings fractional coordinates coloring allowed fall outside lying inside convex body gaussian measure least however unclear one could use partial colorings recover result even larger approximation factor recentering procedure section describe details recentering procedure leave thorough description algorithmic implementation however section provide abstract instantiation begin give geometric view vector balancing problem recentering procedure help clarify exposition let linearly independent vectors given target body gaussian measure least wep restate vector balancing problem geometrically finding vertex parallelepiped lying inside choice ensures note condition necessary since otherwise exists halfspace separating gaussian measure least recall linear setting using geometric language banaszczyk theorem implies contains origin kvi need assume convex body gaussian measure least contains vertex thus given target body make situation better replacing shift higher gaussian measure particular given symmetry gaussian measure one would intuitively expect gaussian mass symmetrically distributed around shift increases gaussian measure current language fixing color vector corresponds restricting finding vertex facet lying inside intuitively restricting facet improve situation gaussian measure corresponding slice lower dimension larger make formal note inducting facet dimensional parallelepiped must choose center serve new origin lower dimensional space precisely expressed inducting parallelepiped shifted slice span using dimensional gaussian measure span viewpoint one restate goal recentering procedure finding point smallest facet containing satisfies barycenter origin gaussian measure smaller recall long span gaussian measure least guaranteed geometry mind implement recentering procedure follows compute gaussian mass maximized boundary letting denote facet containing induct slice span interior replace terminate explain achieves desired result firstly maximizer facet standard convex geometric argument reveals gaussian measure span smaller particular smaller thus case recentering procedure fixes color free second case interior variational argument gives barycenter induced gaussian measure must origin namely conclude section explain extend recentering procedure directly produce deterministic coloring satisfying theorem purpose shall assume length log small enough constant begin run recentering procedure returns barycenter origin replace joint scaling large enough constant gaussian mass least point run original recentering procedure following modification every time get situation barycenter origin induct closest facet closest origin precisely situation compute point boundary closest origin letting denote facet containing induct span end return final found vertex notice claimed coloring vertex returned algorithm indeed deterministic reason algorithm works following guarantee original recentering procedure gaussian mass span instead show decreases slowly particular use bound log length vectors show every time induct gaussian mass drops factor generally vectors length small enough drop would order constant since massage gaussian mass least applying modified recentering algorithm indeed allows induct times keeping gaussian mass guarantees final vertex derive bound rate decrease gaussian mass prove new inequality gaussian mass sections convex body near barycenter see theorem may independent interest final remark note unlike subgaussian sampler recentering procedure scale invariant namely jointly scale factor output recentering procedure output original gaussian measure homogeneous scalings thus one must take care appropriately normalize applying recentering procedure achieve desired results give high level overview recentering step implementation first crucial observation context task finding maximizing gaussian measure fact convex program precisely objective function gaussian measure logconcave function feasible region convex hence one hope apply standard convex optimization techniques find desired maximizer turns however one significantly simplify required task noting recentering strategy fact necessarily need exact maximizer even maximizer see note shift larger gaussian measure logconcavity shifts also larger gaussian measure thus find shift larger gaussian measure letting intersection point boundary induct facet containing corresponding slice given essentially ignore constraint treat optimization problem unconstrained last observation allow use following simple gradient ascent strategy precisely simply take steps direction gradient either pass facet gradient becomes small alluded previously gradient exactly equal fixed scaling barycenter current shift induced gaussian measure thus gradient small barycenter close origin good enough purposes last nontrivial technical detail efficiently estimate barycenter note barycenter expectation random point inside purpose simply take average random samples generate samples using rejection sampling using fact gaussian measure large conclusion open problems conclusion shown tight connection existence subgaussian coloring distributions banaszczyk vector balancing theorem furthermore make use connection constructively recover weaker version theorem main open problem leave thus fully recover banaszczyk result explained reduces finding distribution colorings output random signed combination input vectors norm believe approach attractive feasible especially given recent work builds distribution colorings coordinate output random signed combination organization section provide necessary preliminary background material section give proof equivalence banaszczyk vector balancing theorem existence subgaussian coloring distributions section give random walk based coloring algorithm section describe implementation recentering procedure section give algorithmic reduction asymmetric bodies symmetric bodies giving proof theorem section show extend recentering procedure full coloring algorithm section prove main technical estimate gaussian measure slices convex body near barycenter needed algorithm lastly section give constructive proof feasibility vector program acknowledgments would like thank american institute mathematics hosting recent workshop discrepancy theory work done preliminaries basic concepts write log logp logarithm base base respectively pnx eucliean norm let kxk vector define denote unit euclidean ball denote unit sphere denote inner product subsets denote minkowski sum define span smallest linear subspace containing denote boundary use phrase relative span specify computing boundary respect subspace topology span set convex symmetric shall say convex body additionally closed interior note usual terminology convex body also compact bounded state explicitly necessary convex body contains origin interior say need concept gauge function convex bodies bounded symmetric convex bodies functional define standard norm proposition let convex body defining gauge function body kxkk inf following holds finiteness kxkk positive homogeneity triangle inequality ykk kxkk kykk furthermore additionally bounded symmetric norm call norm induced particular additionally satisfies kxkk iff kxkk xkk gaussian subgaussian random variables define standard gaussian random variable density definition subgaussian random variable random variable note canonical example distribution standard gaussian vector valued random variable say one dimensional marginals precisely random variable remark definition follows directly following standard lemma allows deduce subgaussianity upper bounds laplace transform random variable include proof appendix completeness lemma let cosh let random vector assume cosh furthermore standard gaussian cosh gaussian measure define gaussian measure precisely measurable set noting also need lower dimensional gaussian measures restricted linear subspaces thus linear subspace dimension understood gaussian measure within treated whole space convenient also use notation denote treating one dimensional gaussian measure often denote interval simply notational convenience convention define otherwise important concept used throughout paper barycenter induced gaussian measure definition barycenter convex body define barycenter induced gaussian measure note random variable supported probability density extending definition slices linear subspace refer barycenter denote one relative dim gaussian measure treating whole space throughout paper need many inequalities regarding gaussian measure first important inequality inequality states measurable subsets note inequality applies generally logconcave measure measure defined density whose logarithm concave importantly inequality directly implies convex log concave function need following powerful inequality ehrhard provides crucial strengthening gaussian measure theorem ehrhard inequality borel sets power ehrhard inequality allows reduce many inequalities gaussian measure two dimensional ones one use show following standard inequality gaussian measures slices convex body include proof completeness lemma given convex body linear subspace dimension proof clearly suffices prove lemma since gaussian distribution rotation invariant without loss generality let denote slice outside support define defined outside support follows ehrhard inequality concave support hence closed convex body let equivalent showing exists halfspace let distance origin boundary since implies contradicting vector balancing reduction linearly independent case section detail standard vector balancing reduction case vectors linearly independent also cover useful related concepts definitions used throughout paper definition lifting function fractional coloring denote set fractional coordinates define lifting function importantly full colorings thus sends full colorings lifting function useful allows given fractional coloring coordinates set reduce linear vector balancing problem one smaller number coordinates detail following lemma lemma let given fractional coloring following holds proof first part follows computation second part follows since last equivalence part terms reduction lemma says words linear vector balancing problem respect vectors shift set reduces linear discrepancy problem shift set give reduction linearly independent setting lemma let polynomial time algorithm computing fractional coloring vectors linearly independent proof let denote basic feasible solution linear system clearly computed polynomial time note system feasible construction show satisfies required conditions let denote rank matrix since basic must satisfy least least constraints equality particular least bound constraints must tight thus since set fractional coordinates must furthermore vectors must linearly independent since otherwise basic finally needed let apply lemma vector balancing problem subgaussian sampling problem vector balancing problem respect shift first assume convex body gaussian measure least applying lemma get vector balancing reduces one respect shift follows directly lemma part using lifting function let span dim linear independence clearly reduced vector balancing problem looks signed combinations hence may replace note lemma hence reduction reduces problem type addition vectors form basis ambient space subgaussian sampling problem identity lemma sampling random coloring subgaussian clearly reduces sampling random coloring subgaussian since equals furthermore since support support distribution lives test subgaussianity need check marginals thus may assume full space completes needed reductions computational model formalize algorithms interact convex bodies use following computational model interact algorithmically convex body assume presented membership oracle membership oracle input outputs otherwise interestingly since always assume convex bodies gaussian measure least need additional centering known point inside inner contained outer containing ball guarantees runtimes algorithms measured number oracle calls arithmetic operations perform note use simple model real computation assume algorithms perform standard operations real numbers multiplication division addition etc constant time banaszczyk theorem subgaussian distributions section give main equivalences banaszczyk vector balancing theorem existence subgaussian coloring distributions fundamental theorem underlies equivalences known talagrand majorizing measure theorem provides nearly tight characterization supremum gaussian process using chaining techniques state essential consequence theorem sufficient purposes reference see theorem talagrand let convex body random vector standard gaussian kxk absolute constant consequence theorem together geometric estimates proved subsection derive following lemma crucial equivalences reductions lemma reduction subgaussianity let symmetric convex body particular convex body particular proof lemma proof follows immediately combining lemmas theorem note lower bounds probabilities follow directly markov inequality state equivalence need definitions following geometric parameters definition geometric parameters let finite set define least number exists random vector supported define least number symmetric convex body state main equivalence gives quantitative version theorem introduction theorem finite set following holds using language restate banaszczyk vector balancing theorem restricted symmetric convex bodies follows theorem let kvi immediate corollary theorems extended linear setting deduce corollary let kvi furthermore kvi explained introduction equivalence shows existence universal sampler recovering banaszczyk vector balancing theorem symmetric convex bodies constant factor length vectors precisely follows directly lemma part corollary details see proof theorem following theorem need classical minimax principle theorem minimax theorem let compact convex sets let continuous function convex fixed concave fixed min max max min proceed proof theorem proof theorem proof let random variable let symmetric convex body lemma part thus exists since holds needed proof recall definition cosh note cosh convex symmetric cosh cosh define cosh lemma note standard gaussian let set probability distributions goal show exists homogeneity may replace thus assume show existence subgaussian distribution show inf sup proving bound show suffices show existence desired distribution let denote minimizing distribution definition cosh bounds laplace transform lemma needed prove estimate let denote closed convex hull functions precisely closure set functions gwi continuity clearly inf sup inf sup strategy apply minimax theorem hold first need convex compact clear since associated standard simplex construction also convex hence need prove compactness since finite closed subset functions associated natural way closed subset show compactness suffices show set bounded particular suffices show universal constant since every limit convex combinations functions suffices show sup max prove following computation cosh cosh sup sup max sup max sup thus compact needed lastly note function bilinear hence continuous satisfies trivially conditions theorem compactness continuity inf sup min max next minimax theorem min max max min max min sup min gwi take gwi function let task reduces showing since suffices show standard gaussian symmetric convex since convex combination symmetric convex functions follows symmetric convex since markov inequality gwi hence needed analysis recentering procedure give crucial tool reduce asymmetric setting symmetric setting namely recentering procedure corresponding theorem introduction next subsection subsection detail use procedure yield desired reduction proof theorem recentering procedure first recall desired guarantees linearly independent vectors shift convex body rnpof gaussian measure least would like find fractional coloring subspace span following holds shall prove induction note base case reduces statement trivial fractional coloring remember first denotes set fractional coordinates lifting function see definition details let define function pnf log compute maximizer let satisfy let span note first since lemma part assume first interior since maximizer touch boundary kkt conditions must direct computation reveals since touch constraints see hence thus claimed satisfies conditions theorem assume must hence dim next lemma see lemma part thus may apply induction vectors shift convex body recover span get def claim satisfies conditions theorem see note lemma part furthermore since next clearly hence span claim thus follows combining reduction asymmetric symmetric convex bodies explained introduction recentering procedure allows reduce banaszczyk vector balancing theorem convex bodies symmetric case particular task subgaussian sampling give reduction detail let theorem let pnx fractional coloring guaranteed recentering procedure theorem let span shall assume kvi constant chosen later lemma part let guarantees recentering procedure know lemma section standard gaussian kxkc hence markov inequality point using banaszczyk theorem linear setting symmetric bodies loses factor norm bound satisfies homogeneity exists hence reduction symmetric case follows also achieve subgaussian sampler though vectors shorter particular applying corollary exists distribution colorings lemma part applied needed geometric estimates section present required estimates proof lemma following theorem latala oleszkiewicz allow translate bounds gaussian measure bounds gaussian norm expectations theorem let standard gaussian let symmetric convex body let chosen following holds using theorem derive derive bound goods bounds gaussian norm expectations note much weaker elementary estimates given would suffice borell inequality however use stronger theorem achieve better constant lemma let standard gaussian let symmetric convex body kxkk proof let satisfy kxkk note assumption let denote number numerical calculation reveals theorem kxkk thus kxkk kxkk kxkk following lemma shows find large ball centered around origin either gaussian mass large barycenter close origin lemma let convex body following holds particular holds holds particular proof begin part assume sake contradiction exist separator theorem exists unit vector particular strictly contained halfspace thus note clear contradiction assumption furthermore first see hence needed prove part similarly contain ball radius exists halfspace rotational invariance gaussian measure may assume let let clearly see thus get contradiction suffices show function satisfying easy see function maximizing left hand side satisfying must indicator function interval right end point function pushes mass far right possible let denote unique number noting optimizing indicator function direct computation reveals since must using inequalities since assumption yielding desired contradiction furthermore follows direct numerical computation extend bound asymmetric convex bodies barycenter near origin need standard fact gauge function body lipschitz contains large ball recall function lkx lemma let convex body satisfying gauge function proof need show kykk see note kxkk ykk ykk kykk kykk kykk triangle inequality since yielding kxkk kykk inequality follows switching also need following concentration inequality maurey pisier theorem let function standard gaussian inequalities prove main estimate asymmetric convex bodies lemma let convex body standard gaussian following holds kxkk kxkk proof prove part symmetry gaussian measure max kxkk xkk kxkk xkk kxkk needed prove part let first lemma part assumptions thus lemma assume sake contradiction kxkk since kxkk must kxkk kxkk theorem lipschitz proporties kxkk kxkk clear contradiction log random walk log random walk algorithm given algorithm step executed polynomial time either calling sdp solver executing algorithm section feasibility program guaranteed theorem also results matrix step computed cholesky decomposition let first make observations useful throughout analysis notice first random process markovian let row definition algorithm log random walk algorithm input kvi vector output random signs log let log let log let let compute pick uniformly random end set sign return else restart algorithm line end kui equals otherwise get otherwise first analyze convergence algorithm show constant probability random walk fixes coordinates absolute value first prepare lemma lemma let form martingale sequence adapted filtration every denote inf proof define martingale respect defined xmin easily see induction therefore compute min monotone convergence theorem proved next lemma gives convergence analysis random walk lemma probability probability least proof prove first claim induction clearly satisfied assume claim holds prove claim follows inductive hypothesis triangle inequality final inequality follows prove second claim show holds every claim follow union bound let fix arbitrary define let event observe therefore events hold simultaneously observation markov property let holds min otherwise sequence conditioned martingale moreover since get kui sequence conditioned satisfies assumptions lemma lemma since event holds markov inequality bound imply proved prove walk subgaussian need following martingale concentration inequality due freedman theorem let martingale ptadapted filtration let var exp next state main lemma together estimate error due rounding implies subgaussianity lemma random variable proof define notice let fix let need show every first pnobserve bounded need consider finite range indeed lemma probability triangle inequality kyt kvi well therefore rest proof assume observe martingale first prove increments bounded follows boundedness increments coordinates indeed triangle inequality kyt follows next bound variance increments markov property random walk entirely determined denoting description algorithm penultimate equality follows final inequality follows chosen ready apply theorem using notation theorem shown bounds hold probability let first claim indeed theorem calculation imply proves lemma finally state main theorem theorem restatement theorem algorithm runs expected polynomial time outputs random vector random variable log proof let event equivalently algorithm takes returns holds otherwise restarts lemma event occurs probability least constant number restarts expectation since random walk talks steps polynomial input size step also executed polynomial time follows expected running time algorithm polynomial algorithm returns output exactly holds output distributed random vector conditioned let fix vector let random variable let let defined proof lemma let parameter proved subgaussian lemma show conditioned prove observe inequality trivially satisfied since right hand side least range rest proof assume conditional using triangle inequality bound distance sign kvi get conditional implies triangle inequality conditional every therefore conditional every lemma recall every right hand side claimed therefore conditioned since log suffices prove theorem recentering procedure section give algorithmic variant recentering procedure theorem given convex body let barycenter gaussian distribution following lemma shows estimate barycenter close farther origin shifting increases gaussian volume lemma let barycenter point satisfying proof let change variables let random variable drawn gaussian distribution restricted body right hand side equal ehy ehe jensen inequality ekb algorithm recentering algorithm use geometric language section instead pnof vectors shift work directly parallelepiped notice facet corresponds fractional coloring coordinates fixed indeed facet determined subset coloring equals size set equal vertex face dimension equivalent full coloring edges faces dimension linear segments length exactly twice length corresponding vectors say side lengths edge length corresponds requiring maxi kvi given point denote face contains minimal dimension denote subspace span language linear discrepancy problem translated problem finding vertex inside recentering problem also expressed way looking point gaussian measure restricted least close start approximating barycenter close origin already done return far origin moving origin shifting respectively help increasing gaussian volume make move lies outside case start moving towards hit boundary stop induct facet land choosing point boundary stopped new origin show even partial move towards decrease volume moreover ensures origin always stays inside one difficulty efficiently compute barycenter exactly get around use random sampling gaussian distribution restricted estimate barycenter high accuracy return shift body barycenter origin running time polynomial suffices choose inversely polynomial assume access membership oracle convex body algorithm recentering procedure input convex body parallelepiped error probability output see statement theorem return fail set set compute estimate barycenter restricted subspace satisfying probability least return fail otherwise continue return else compute relative set else set end set set end return fail following theorem algorithmic version theorem note guarantees algorithm relatively robust make simpler use within algorithms since may called invalid inputs well output incorrectly small probability theorem let parallelepiped containing origin convex body gaussian measure least given membership oracle let algorithm inputs either returns fail point furthermore input correct probability least returns satisfying gaussian measure least moreover algorithm runs time polynomial proof firstly easy check induction beginning iteration loop prove correctness algorithm must show algorithm returns point satisfying conditions theorem probability least purpose shall condition event barycenter estimates computed line within distance true barycenters denote since run barycenter estimator times union bound occurs probability least defer discussion implement barycenter estimator till end analysis conditioning prove lower bound gaussian mass function number iterations crucial establishing correctness algorithm claim let denote state iterations let denote number iterations time dimension decreases conditioned proof prove claim induction base case beginning first iteration note definition inequality clearly holds since since lemma base case holds thus holds assume bound holds time prove assuming iteration let denote corresponding loop variables denote new values line since iteration since conditioning lemma induction hypothesis note drop dimension going lies boundary relative since minimal face containing lower dimensional examine two cases first case assume relative interior case hence given drop dimension desired bound derived directly equation second case assume interior relative case relative furthermore ehrhard inequality equation get lastly lemma since desired bound follows combining equations prove correctness algorithm conditioned first show conditioned algorithm returns line iteration loop sake contradiction assume instead algorithm returns fail let denote state end loop claim used fact since dimension drop times clear contradiction however since gaussian measure always given assume algorithm returns iteration loop let denote state iteration since return iteration must given barycenter satisfies claim also know since equation correctness algorithm follows runtime note dominated calls barycenter estimator thus long estimator runs poly time desired runtime bound holds remains show estimate barycenter efficiently show appendix theorem failure probability time poly poly needed algorithmic reduction asymmetric symmetric banaszczyk section make algorithmic reduction section asymmetric symmetric case directly imply given algorithm return vertex contained symmetric convex body gaussian volume least half also efficiently find vertex contained asymmetric convex body gaussian measure least half definition symmetric body coloring algorithm shall say symmetric body coloring algorithm given input parallelepiped side lengths nonnegative function symmetric convex body satisfying given membership oracle returns vertex contained probability least let present algorithm uses black box achieves guarantee asymmetric convex bodies constant factor loss length vectors algorithm reducing asymmetric convex bodies symmetric convex bodies input algorithm convex body given membership oracle parallelepiped side lengths output vertex contained call recentering procedure restart line call outputs fail otherwise let output call inside let output vertex return else restart line theorem algorithm correct runs expected polynomial time proof clearly line correctness trivial need argue runs expected polynomial time since runtime recentering procedure algorithm polynomial runs independent need argue line accepts constant probability since recentering procedure outputs correctly probability least may condition correctness output line conditioning guarantees recentering algorithm letting dim thus lemma standard gaussian kxkc hence markov inequality construction side lengths hence also side lengths thus input outputs vertex contained probability least hence check line succeeds constant probability needed directly implies theorem shown proof let parallelepiped containing origin side lengths log let kvi denote vectors input thep random walk sampler algorithm outputs expected polynomial time random supported vertices thus lemma part pick small enough symmetric convex body thus letting denote sampler see satisfies conditions log theorem follows combining algorithm body centric algorithm asymmetric convex bodies section give algorithmic implementation extended recentering procedure returns full colorings matching guarantees theorem interestingly coloring output procedure essentially deterministic randomness effect due random errors incurred estimating barycenters convex body unit vector define shifted slice main technical estimate require section following lower bound gaussian measure shifted slices defer proof estimate section theorem exists universal constants convex body satisfying inequality says barycenter close origin gaussian measure parallel slices fall quickly move away origin recall problem recast finding vertex parallelepiped contained inside convex body parallelepiped thus start calling recentering procedure get barycenter close origin recentering allows rescale constant factor gaussian volume increases replace log log chosen volume rescaling least find point boundary closest origin recurse taking slice abuse notation calling convex body rescaling also facet containing crucial point choose origin space use induction step done maintain induction hypothesis parallelepiped contains origin theorem guarantees lose much gaussian volume lemma given convex body log log proof let standard gaussian lemma kxkk log gives kxkk kxkk kxkk kxkk kxkk kxkk log lemma lemma function theorem kxkk log thus kxkk needed algorithm body centric algorithm general convex bodies input convex body given membership oracle parallelepiped side lengths output vertex contained call recentering procedure parameters restart line call outputs fail otherwise let denote output set repeat call recentering procedure restart line call outputs fail otherwise let denote output set dim compute argmin relative set end dim vertex return else restart line algorithm use parameters min log min theorem give formal analysis algorithm begin explaining compute minimum norm point boundary parallelepiped lemma let parallelepiped side lengths containing origin linearly independent let denote dual basis unique set vectors lying inside span satisfying otherwise let denote element minimum norm set following holds argmin kpk relative furthermore computed polynomial time proof note xivi given easy check show argmin relative since must show relative given vectors unit norm norm equal min assume since assumption must therefore hence needed next must show relative firstly clearly since thus argument choose direct calculation hence satisfies one inequalities see equation equality thus relative note needed show paragraph every element minimal face containing satisfies particular since collinear may claim follows since span previous statement show firstly minimality note thus ksk hvi kvi kvi since side lengths kvi thus claimed prove furthermore let denote matrix whose columns linear independence matrix invertible since see columns note lie construction hence constructed polynomial time since clearly constructed polynomial time dual basis claim proven theorem algorithm correct runs expected polynomial time proof clearly check line correctness trivial need show algorithm terminates expected polynomial time particular suffices show probability run algorithm terminates without restart least purpose show algorithm terminates correctly conditioned event call recentering procedure terminates correctly denote later show event occurs probability least finish proof let denote values line iterations repeat loop easy check induction execution either line variables satisfy establish main invariant loop crucial establishing correctness conditioned claim let denote state successful iterations repeat loop following holds dim conditioned proof prove claim induction state corresponds trivially dim first condition holds conditioned gaussian mass least restricted barycenter norm since lemma thus second condition holds well assume statement holds iterations show holds iteration assuming terminate iteration successfully complete iteration denote state beginning iteration line end iteration first verify induction hypothesis dim construction thus need show dim dim given successfully complete iteration namely call recentering algorithm line return fail may distinguish two cases firstly dim must dim dim since otherwise dim loop would exited previous iteration second dim must entered statement line since dim dim see dim corresponds dimension minimal face containing since boundary relative get dim dim dim needed thus condition holds end iteration claimed show conditioned induction hypothesis recall thus suffices prove note since decrease dimension every iteration argued previous paragraph number iterations loop never exceed thus valid number iterations always particular track change gaussian mass going since recentering procedure line terminates correctly conditioning get dim clearly hence needed dim enter statement line since parallelepiped side length lemma ksk thus lemma needed given ksk applying theorem get since lemma desired estimate follows combining claim see number iterations repeat loop always bounded furthermore conditioned loop successfully terminates satisfying dim since dim implies furthermore equation implies dim hence vertex since get vertex contained needed thus conditioned algorithm returns correctly lower bound analysis note never call recentering procedure times line times line union bound probability one calls fails thus occurs probability least needed estimate gaussian measure slices section prove theorem need following estimate gaussian tails formula lemma gaussian tailbounds let proving theorem first prove similar result special class convex bodies define convex body downwards closed implies notational convenience shall denote first second coordinate vector respectively coordinates shall say slice denote either vertical slice horizontal slice define height maximum point convention let height contain point lemma let downwards closed convex body barycenter satisfying let exists constant height least min proof step reduction wedge first show bodies lemma wedgeshaped see illustration figure namely worst case closed convex bodies form precisely show given body satisfying conditions theorem exists wedge satisfying conditions theorem whose height let satisfy conditions theorem first show contains point line claim gaussian mass choosing small enough clear contradiction see note pushing mass right much possible towards line replace band gaussian mass barycenter right clearly band barycenter right iff hence gaussian mass needed assume height least recall note band corresponding height satisfies conditions theorem thus may assume height note point boundary let denote height since lemma hence thus hence since otherwise would convexity may choose line tangent passing may choose since tangent must since hence given tangent also using conclude show wedge satisfies requirements appropriate choice note conditions already satisfied paragraph let choose note must exist since construction note height remains check bound gaussian mass construction figure triangular region beneath red line given must hence needed remains check purpose note first transform pushing mass right hence since also immediately get derive combining assumption reduction may assume wedge form given equation first take care trivial cases firstly height clearly since get needed assume hence height infinite note always intersects line first part thus desired bound trivially holds may thus assume setting line intersects forms angle figure given normalization note perpendicular distance edge follows maintain parametrization wedge terms using relations sin sin cos recover needed recall let let height want prove already done since note lemma hence may assume goal show min step using barycenter condition derive bound large must given let denote halfspace let simple computation shows sin fact get sin follows sin log sin get following bound log sin log sin gives useful bound since case barycenter even always gaussian measure least half thus max log sin step getting bound construction point lies boundary hence sin cos sin using lemma also since combining two get max max log sin putting max log sin sin max cos cos sin observe giving also thus lower bound holds minimize pmax log sin sin sin min max cos cos first minimize respect make following observations fixed first term inside maximum function second decreasing function log sin first term smaller second term log sin first term greater second term thus two terms must become equal somewhere range log sin log sin particular substituting log sin second term provides lower bound sin log sin log sin sin log sin min cos sin min sin cos expression goes increasing whole interval already done else achieves minimum somewhere let setting derivative zero get sin cos log sin sin log sin satisfies sin log sin two terms get two upper bounds sin sin sin using simplify sin sin sin sin derive two bounds expression one useful small large small bound using small enough sin large bound sin thus min needed prove theorem special case barycenter lies right hyperplane show later reduce theorem case lemma exists universal constants convex body satisfying proof split proof two steps step one reduce problem show suffices prove theorem downwards closed convex body reduction guaranteee barycenter gaussian measure slices parallel correspond natural way slices parallel hyperplane invoke lemma get lower bound height lastly step show implies required lower bound slice measure let note since step reduction case reduce problem one downwards closed convex body specify need specify height boundary define height satisfying ehrhard inequality see fact convex furthermore easy check thus downwards closed convex body may invoke lemma conclude height least suffices give lower bound order derive theorem show step bounding clearly suffices goal show let min split analysis two cases depending whether small big step penultimate inequality holds appropriate choice last inequality uses step use bound penultimate inequality holds appropriate choice last inequality uses come proof theorem theorem restated exist universal constants convex body satisfying proof rotational invariance may assume first standard basis vector possibly replacing may also assume desired lower bound follows directly lemma given may assume deal second case main idea remove portion lying left hyperplane barycenter remaining body lies apply lemma truncated body define let let defined smallest negative number satisfying continuity must exists since left hand side tends positive given note show appropriately chosen constants choice equality see given get show exists constant let note let positive number satisfying pushing mass left towards much possible see next inclusion given satisfies combining equations must thus may set set since lastly using lemma get needed constructive vector section give new proof main result natural sdp problem value proof used duality proof direct immediately yields algorithm compute sdp solution uses basic linear algebraic operations need general sdp solver state main theorem next theorem let vectors euclidean length let exists psd matrix xii matrix whose columns vectors prove theorem make use basic identity inverses block matrices standard use schur complement prove lemma let block matrix matrix matrix matrix matrix assume invertible write block form bij dimensions aij inverse schur complement lemma derive main technical claim used proof theorem lemma let positive definite matrix let columns let bii kvi orthogonal projection matrix onto span proof sufficient prove lemma let matrix columns since positive definite principal minor positive definite well therefore invertible lemma let since symmetric idempotent orthogonal projection matrix moreover rank orthogonal projection matrix onto column span lemma follows proof theorem prove theorem induction base case single vector set clearly vxv proceed inductive step consider first case singular exists vector scale exists apply inductive hypothesis vectors reals get matrix extend matrix padding yij otherwise define xxt easy verify conditions theorem satisfied finally assume invertible let define min arg min max apply inductive hypothesis vectors reals get matrix pad matrix first case define easy verify xii submatrix consisting columns except induction hypothesis since symmetric idempotent orthogonal projection matrix therefore bii kvi lemma maxi therefore max completes proof observe proof theorem easily turned efficient recursive algorithm references milton abramowitz irene stegun editors handbook mathematical functions formulas graphs mathematical tables dover publications new york reprint edition wojciech banaszczyk balancing vectors gaussian measures convex bodies random structures algorithms wojciech banaszczyk series signed vectors rearrangements random struct algorithms url http nikhil bansal constructive algorithms discrepancy minimization ieee annual symposium foundations computer science focs pages ieee computer los alamitos nikhil bansal daniel dadush shashwat garg algorithm conjecture matching banaszczyk bound annual ieee symposium foundations computer science focs new brunswick usa october pages franck barthe olivier shahar 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measures dilatations convex symmetric sets ann avi levy harishchandra ramadas thomas rothvoss deterministic discrepancy minimization via multiplicative weight update method arxiv preprint spencer vesztergombi discrepancy matrices european url http doi shachar lovett raghu meka constructive discrepancy minimization walking edges siam preliminary version focs geometric discrepancy volume algorithms combinatorics berlin illustrated guide john von neumann zur theorie der gesellschaftsspiele mathematische annalen aleksandar nikolov conjecture holds vector colorings arxiv preprint aleksandar nikolov kunal talwar approximating hereditary discrepancy via small width ellipsoids symposium discrete algorithms soda pages paouris concentration mass convex bodies geom funct thomas rothvoss constructive discrepancy minimization convex sets annual ieee symposium foundations computer pages ieee computer los alamitos joel spencer six standard deviations suffice trans amer math joel spencer ten lectures probabilistic method volume siam aravind srinivasan improving discrepancy bound sparse matrices better approximations sparse lattice approximation problems proceedings eighth annual symposium discrete algorithms new orleans pages acm new york michel talagrand generic chaining springer monographs mathematics berlin upper lower bounds stochastic processes appendix estimating barycenter section show efficiently estimate barycenter small accuracy convex body let denote gaussian measure restricted random variable denote covariance cov following lemma shows covariance gaussian random vector shrinks restricted convex body include short proof completeness lemma given convex body let gaussian distribution restricted let random variable distributed according cov proof consider concave follows easy consequence inequality hence hessian negative calculated cov setting completes proof also need use paouris inequality restate slightly theorem random vector mean covariance matrix every absolute constant theorem let convex body given membership oracle algorithm computes barycenter within accuracy probability least time polynomial log proof let barycenter generated constant theorem defining following quantities see estimate barycenter generated averaging random samples distribution difference vector true barycenter thus suffices bound probability large show efficiently generate random samples distribution holds also using lemma yit cov thus cov since distribution hence random vectors easily checked using inequality average random variables also hence random vector using putting using theorem log get log generate random points using rejection sampling generate sequence standard gaussian random variables set first sequence belongs exists set arbitrarily clearly conditional existence furthermore gaussian measure least every union bound probability least distributed according let call event conditional inequality holds algorithm needs generate log gaussian random variables check membership compute average points since operations takes polynomial time polynomial log running time algorithm polynomial proof lemma proof first prove subgaussianity let assumption min cosh cosh min min cosh last inequality follows setting let prove since probabilities always one suffices prove min replacing simplifies showing min see let noting lhs rhs least lhs equal rhs larger thus needed prove furthermore standard gaussian note distributed like hence ehx needed
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modeling formation social conventions populations ismael marti xerxes paul ibec instituto barcelona spain upf universitat pompeu fabra barcelona spain bist barcelona institue science technology barcelona spain icrea catalana recerca estudis barcelona spain abstract order understand formation social conventions need know specific role control learning systems advance direction propose within framework distributed adaptive control dac theory novel reinforcement learning architecture crl account acquisition social conventions populations solving benchmark social problem new crl architecture concrete realization dac theory implements sensorimotor control loop handling agent reactive behaviors reflexes along layer based reinforcement learning maximizes reward apply crl task coordination must achieved order find optimal solution show crl architecture able find optimal solutions discrete continuous time reproduce human experimental data standard metrics efficiency acquiring rewards fairness reward distribution stability convention formation introduction formation social conventions seminal work convention david lewis defines social conventions regularities action emerge solve coordination problems possess two main characteristics conventions arbitrary sense group agents given population continue conform long expect others arbitrary equally acceptable possibilities solve coordination problem understanding conditions conventions formed still open question traditionally studied cooperation games umbrella game theory studies later within evolutionary game theory iconic example game theory prisoner dilemma game proposes situation two arrested criminals decide without communication whether testify defect remain silent cooperate cooperate spend year jail defect spend years one cooperates defects former spend years jail latter freed game theory combination possible actions subsequent rewards represented payoff matrix like one shown figure one key concept game theory nash equilibrium set strategies player interest change unilaterally strategy case classical prisoner dilemma game played nash equilibrium mutual defection defect always render higher payoff regardless player chooses figure payoff matrix prisoner dilemma cell matrix shows reward player gets combination respective choices example player red chooses cooperate player blue chooses defect upper right cell player receives reward player receives reward particularly interesting version game called iterated prisoner dilemma ipd thoroughly studied robert axelrod consists playing several consecutive rounds game one player chooses defect given round player chance retaliate next round defecting back case number rounds random unknown players mutual defection longer nash equilibrium incentive defect countered fear retaliation thus leading emergence cooperation nash equilibrium proven axelrod seminal work ipd shows cooperation emerge repeated interactions pure natural selection following axelrod findings russell hardin postulated social conventions could originated large populations type repeated dyadic interactions due capacity individual generalize similar cases overlapping nature group activities however although classical game theoretic approach remained hegemonic three decades studying humans animals cooperate flaws approach led alternate models one major concerns related ecological validity experiments based ipd arguing conditions experiments conducted hardly ever found nature particularly many studies pointed fact cooperation animals humans usually requires continuous exchange information order emerge feature ipd similar cooperation games lack precisely based turns impose significant delay actions order tackle issue several investigations devised ways modify standard game theory tasks dynamic versions individuals respond actions results point cooperation easily achieved dynamic version task due rapid flow information individuals capacity react one example found hawkins goldstone show interaction helps converge stable strategies game theoretical task compared task modeled also show involved payoffs affect formation social conventions according results suggest coordination problems solved either convention spontaneous coordination solutions depend stake coordination fails illustrate point make comparison two examples coordination problem one hand drive car stakes high fail coordinate outcome could fatal resort convention drive right side road hand try avoid people crowded street fly stakes low risky rely purely reactive behaviors avoidance behavior solve paper propose computational model cognitive agents involved social task called battle exes confront results human data obtained purpose develop reinforcement learning crl cognitive architecture based principles distributed adaptive control theory dacma architecture integrates reactive control loop manage conflicts policy learning algorithm acquire strategies run simulations showing modeled cognitive agents rely policy learning stakes game higher reactive control helps acquire better outcomes terms efficiency fairness provides computational hypothesis explaining key aspects emergence social conventions pure dominance setups provides new experimental predictions tested human coordination tasks reinforcement learning regarding computational modeling tasks already extensive literature study emergence conflict cooperation agent populations tackled especially use reinforcement learning extensive reviews check direction lot focus put recently developing enhanced versions deep network architecture proposed particularly extensions social domain architecture uses reinforcement learning algorithm extracts abstract features raw pixels deep convolutional network along lines researchers modeling type conflicts represented classic tasks ipd ecologically valid environments agent learning based deep instance agents based cognitive model already capable learning play video game pong raw sensory data achieve performance cooperative competitive modes approaches paying attention develop agents achieve good outcomes games complex social dilemmas focusing maintaining cooperation making agent prosocial taking account rewards conditioning behavior solely outcomes computational approaches described able reproduce key aspects formation social behaviors agent populations suited disentangling role reactive control policy learning process even coordination task modeled agents biologically environmentally constrained approaches considering sensorimotor control loops bootstrapping learning higher levels cognitive architecture contrast reinforcement learning crl architecture propose based distributed adaptive control dacma theory learning processes bootstrapped sensorimotor control loops see next section moreover systematically confront results experimental human data collected studying conditions agents able converge towards social conventions purpose use already designed tested task called battle exes explain end following section section describe crl architecture two layers one dealing intrinsic behaviors agent another based reinforcement learning allowing agents acquire rules maximizing reward section iii compare results model existing human data finally conclude study analyzing section main results implications discussing limitations possible extensions current model posing several experimental predictions foundations approach approach relies two main contributions literature describe detail section firstly crl architecture agents follows principles dacma theory learning processes bootstrapped reactive control loops secondly use existing task called battle exes human data available various experimental conditions distributed adaptive control theory dac theory brain mind proposes cognition based four control layers operate different levels abstraction first level soma layer contains whole body agent sensors actuators represents interface agent environment layer also contains physiological needs agent driving force whole system reactive layer physiological needs satisfied internal drives implemented reactive sensorimotor loops maintaining stable homeostasis reactive interactions bootstrap learning instantaneous policies implemented adaptive layer acquiring statespace interaction outside scope paper contextual layer acquires temporally extended policies contribute acquisition abstract cognitive abilities goal selection memory planning representations turn affect behavior lower layers fashion control architecture therefore distributed layers thanks interactions directions well laterally within layer dac makes explicit distinction control one hand reactive layer perceptual behavioral learning hand therefore adequate theoretical framework understanding specific roles reactive control policy learning formation social conventions aim present paper allows identification functions agents need ballistic dynamic conditions battle exes one hand ballistic condition players make decision beginning round cognitive agents need use adaptive layer solving task hand dynamic condition agents need reactive adaptive layer since moving environment sensing acting making abstract discrete decisions section describe detail novel reinforcement learning crl architecture composed reactive adaptive control layer results hybrid model reinforcement learning interacts feedback controller inhibiting specific reactive behaviors benchmark hawkins goldstone conducted study investigate two factors continuity interaction ballistic dynamic stakes interaction high low payoff affected formation social conventions humans social task called battle exes battle exes coordination game similar classic battle sexes proposes following situation couple broke want see coffee break time two coffee shops neighborhood one great coffee high reward one okay coffee low reward great coffee shop come across enjoy break basically want enjoy coffee break coordinate way avoid every day situation modeled within framework game theory payoff relation payoff getting great coffee payoff okay coffee payoff players location regarding stakes interaction payoff matrix manipulated create two different conditions high difference payoffs higher low difference lower payoff matrices figure illustrate two conditions figure payoff matrices original battle exes game numbers indicate reward received player red blue reproduced continuity interaction experiment ballistic dynamic condition ballistic condition classical game theory players choose action beginning every round game without control outcome however dynamic condition players freely change course avatars one reaches reward visual example difference conditions check original videos conditions round ends one players reaches one reward spots represent coffee shops altogether results four conditions total two stakes interaction high low combined two continuity interaction ballistic dynamic experiment pair human players dyads depending payoff condition play high low consecutive rounds together order analyze coordination players dyad use three measures fairness based binmore three levels priority efficiency measures cumulative sum rewards players able earn collectively round divided total amount possible rewards efficiency value means players got maximum amount reward fairness quantifies balance earnings two players fairness value means players earned higher payoff amount times stability measures well strategy maintained time words quantifies predictable outcomes following rounds based previous results using measure surprisal shannon defined negative logarithm probability event results show players dynamic condition achieve greater efficiency fairness counterparts ballistic condition high payoff low payoff setups see figure however key finding dynamic condition players coordinate fly without need strategy payoff low payoff high participants coordinate stable strategies words identified stakes interaction crucial factor formation social conventions interaction happens present paper attempts replicating results described computational model based integrated cognitive architecture order identify specific roles reactive control policy learning emergence social conventions figure human decision making battle exes task analyzed efficiency left fairness center stability right within every plot experimental conditions divided two columns ballistic left dynamic right color high blue low orange note stability measured level surprisal negative logarithm probability event means lower surprisal values imply higher stability adapted methods model crl architecture propose study approach reinforcement learning addition reactive controller approach see crl composed two layers reactive adaptive layer former governs sensorimotor contingencies agent within rounds game whereas latter charge learning across rounds reactive layer reactive layer represents agent sensorimotor control system supposed prewired typically evolutionary processes biological perspective battle exes game considering equip agents two predefined reactive behaviors orienting towards rewards escape agents means even absence learning process agents intrinsically attracted reward spots repulsed intrinsic dynamic bootstrap learning adaptive layer see model layer follow approach inspired valentino braitenberg vehicles simple vehicles consist set sensors actuators motors depending type connections created perform complex behaviors visual depiction two behaviors orienting towards rewards escape agents see figure video figure top view agent body represented large circle agent facing toward top page two thin black rectangles sides represent two wheels controlled speed front agent equipped three types sensors agent sensors sensing proximity agent low reward sensors high reward sensors type agent able sense proximity corresponding entity left right side hence six sensors total representation specific set excitatory inhibitory connections sensors actuators provides agents fundamental reactive behaviors escape agents represented red lines orienting towards rewards low high rewards represented green lines orienting towards rewards behavior made combination crossed excitatory connection direct inhibitory connection reward spot sensors motors plus forward speed constant value set sensor positioned left side robot indicating proximity reward spot either high low reward sensor see figure sensors perceive proximity spot closer reward spots higher sensors activated therefore reward spot detected robot forward speed otherwise activated sensor left right make robot turn direction corresponding reward spot escape agents behavior made opposite combination direct excitatory connection crossed inhibitory connection case agent sensors motors sensor positioned left side robot indicating proximity agent see figure closer reward spots higher sensors activated case well agent detected robot forward speed otherwise activated sensor make robot turn opposite direction agent thus avoiding adaptive layer agent adaptive layer based reinforcement learning algorithm endows agent learning capacities maximizing reward functionally decides agent action beginning round based state previous round policy possible states three high low tie indicate outcome previous round agent agent got high reward previous round state high got low reward state low agents went reward state tie actions three well high low none ballistic condition agents use layer operate two first actions take agent directly respective reward spots high low none action choose randomly round action chosen sampled according actual state observed agent figure representation complete cognitive architecture agent top adaptive layer reinforcement learning control loop composed critic value function actor action policy inhibitor function bottom reactive layer sensorimotor control loop composed three sets sensors corresponding reward agent respectively three functions corresponding orienting towards high reward escape agents behaviors respectively two motors corresponding left right motors action selected passed inhibitor function turn one attraction behaviors depending action selected action high orienting towards low reward reactive behavior inhibited selects low inhibit orienting towards high reward behavior selects none act normally without inhibition dynamic condition agent uses whole architecture adaptive reactive layer working together see figure previous condition agent chooses action beginning round based state previous round policy action signaled inhibit opposite reactive behavior according action selected case chooses action high inhibit orienting towards low reward behavior allowing agent focus high reward conversely chooses action low reactive attraction high reward inhibited cases agent avoidance reactive behavior still operates finally action none selected instead choosing randomly two actions ballistic condition act thus relying completely behaviors play round game adaptive layer implemented reinforcement learning algorithm maximizing accumulated reward rounds action similar one implemented see figure adapted operate discrete state action spaces figure representation algorithm implementing adaptive layer time step critic value function receives state environment current reward input outputs error signal informing agent whether last action performed better worse expected see details signal sent actor action policy back critic actor uses error update policy order converge optimal action state critic uses update value function specifically use temporal difference learning algorithm tdlearning based interaction two main components actor action policy learns mapping states actions define action probability performed state critic value function estimates expected accumulated reward state following policy discount factor reward step critic also estimates actor performed better worse expected comparing observed reward prediction provides learning signal actor optimizing actions performing better resp worse expected reinforced resp diminished learning signal called error error error computed function prediction value function currently observed reward given state discount factor empirically set resp means action performed better resp worse expected error signal sent actor back critic updating current values update actor done two steps first matrix rows indexed discrete actions columns discrete states updated according error learning rate set current action previous state integrates observed errors executing action state initialized kept lower bound used updating probabilities applying laplace law succession number possible actions laplace law succession generalized histogram frequency count assumed value already observed prior actual observation prevents null probabilities data observed returns uniform probability distribution therefore higher probable executed using equations actions performing better expected increase probability chosen next time agent state probability decrease finally critic value function updated following learning rate set experimental setup follow battle exes benchmark experimental design one dimension represents ballistic dynamic versions game whereas dimension composed high low difference payoffs condition played agents paired dyads play together rounds game one high payoff conditions ballistic dynamic rounds one low payoff conditions task developed two versions ballistic dynamic battle exes simulated robotic environment source code available online https representation navigational interface experiment seen figure figure screenshot experimental setup top view blue two cognitive agents initial position start round equipped different sensors described green two reward spots bigger one representing high reward smaller low reward lower payoff white circles delimit tie area rules game follows round game finishes one agents reaches reward spot agents within white circle area happens considered tie get points small spot always gives reward whereas big spot gives depending payoff condition low high respectively see figure reward spots allocated randomly two positions beginning round iii results report main results experiment relation human performance battle exes analyzed using efficiency fairness stability measures report results model plot contrast human data interpret results analyze role layer crl architecture relation data obtained condition efficiency efficiency scores model followed normal distribution four conditions first anova performed showing statistically significant difference groups independent samples showed dynamic high low conditions well significantly efficient ballistic counterparts high low conditions observe statistical tendencies human benchmark data see figure figure efficiency results comparison human data left model right four experimental conditions ballistic high ballistic low dynamic high dynamic low cases dynamic conditions achieved higher efficiency scores ballistic ones fairness fairness scores model anova showed statistically significant difference groups independent samples showed benchmark high ballistic condition significantly less fair two dynamic conditions high low mean fairness score hand statistically significant difference low ballistic condition low dynamic opposed human results low high ballistic conditions model significantly different statistical tendencies observed humans model see figure figure fairness results comparison human data left model right four experimental conditions ballistic high ballistic low dynamic high dynamic low cases dynamic conditions achieved higher fairness scores ballistic counterparts difference ballistic conditions similar well human model data stability regarding stability since results four conditions showed distribution performed showing statistically significant difference groups showed difference ballistic low dynamic low conditions statistically significant dynamic high condition significantly stable ballistic high well statistically significant difference two dynamic conditions two ballistic similar statistical tendencies observed humans model see figure figure comparison stability results human left model right four experimental conditions ballistic high ballistic low dynamic high dynamic low note stability measured level surprisal means lower surprisal values imply higher stability cases difference dynamic high low conditions high condition stable low analysis overall model achieved good fitting benchmark like human experiment observe dynamic version model achieves better results efficiency fairness improvement consistent regardless manipulation payoff difference remarkable increase efficiency dynamic condition due reactive layer key role avoiding conflict agents chosen reward feature ballistic model game achieve results stability model overall less stable benchmark although reflected similar relation conditions difference high low payoff conditions ballistic setup increase stability high dynamic condition compared low dynamic see figure nonetheless results show formation social conventions dominance shown figure examples shown figure illustrate two conventions formed dynamic high condition type equilibria ocurred often rounds three conditions thus explaining higher stability condition conclude besides mentioned differences human benchmark model results consistent human data dynamic interaction helps converge efficient fair stable strategies stakes high figure top panel outcomes two dyads high dynamic condition showing formation left pure dominance right equilibria bar represents outcome round game red bar means got high reward blue bar means player got high reward black bars represent ties middle panel surprisal measure rounds play convention formed surprisal drops outcomes start predictable bottom panel frequency count suprisal values histogram axis represents increasingly higher values surprisal bits longer convention maintained larger initial bars frequency count represents surprisal reliance adaptive layer order identify specific role layer formation social conventions run analysis estimates participation introduction none action function payoff difference based results benchmark model dynamic condition higher payoff differences help achieve higher stability expect increase difference payoffs agents rely adaptive layer testing prediction performed simulation six different conditions varying levels payoff difference high low reward value measure level reliance layer logged number times agent outputted none action action agent relies completely reactive layer solve round figure mean percentage actions high low actions selected agents plotted conditions increasing difference high low payoffs considering possible actions adaptive layer randomly choosing actions observe agent selects action amount times prior learning action policy returns uniform distributions means beginning dyad reliance reactive layer would reliance adaptive layer starting point hypothesis correct expect observe increase reliance adaptive layer payoff difference increases indeed results show seen figure steady increase percentage selection adaptive layer payoff difference augments discussion paper investigated specific role control learning formation social conventions setups proposed based principles dac novel reinforcement learning crl cognitive architecture crl approach reinforcement learning addition reactive controller crl architecture composed module based learning algorithm endows agent learning capacities maximizing reward sensorimotor control loop handling agent reactive behaviors integrated cognitive architecture applied task battle exes coordination two agents achieved shown model able converge pure dominance equilibria moreover demonstrated agent interaction affect formation stable fair effective social coventions compared task modeled results consistent ones hawkins goldstone obtained human subjects addition interpret results context functional cognitive model crl determine specific role reactive adaptive control loops play formation social conventions spontaneous coordination sense assess reactive layer significant role avoiding conflict spontaneous coordination adaptive layer required achieve coordination social conventions addition model supports hypothesis higher payoff differences increase reliance adaptive layer knowledge first model able reproduce human behavioral data social game presented setups furthermore similar experiments take account role sensorimotor control loops solving modeled problems real life scenarios issue consider study constitutes fundamental requirement development fully embodied moreover result work study specific manipulations experimental conditions like increasing differences payoffs presented figure affect outcome task implication control loops solving results allow make predictions later tested human experiments future work several directions continue develop work presented paper one possibility would focus biologically detailed approach could implement reactive layer would also consistent dac theory taking inspiration work done reactive controller model presented also regulates agent internal needs direction could also implement contextual layer like one presented allows agent learning rules maximize delayed reward models allow build causal models world take account context learning optimal action policies extending agent functionalities could allow solving much complicated social coordination problems delayed reward another interesting avenue concerns emergence communication could extend model adding signaling behaviors agents test experimental setups similar seminal games proposed lewis following robot centered approach luc steels addition contextual layer proposed could study emergence complex communicative systems embedding outlined finally work aims help advance towards development functional able survive complex world outlined purpose could extend model study aspects cooperation like wolfpack hunting behavior also competing agent populations ongoing work developing setup simulated cognitive agents presented work compete limited resources massively environment setup also allow test hypothesis proposed role consciousness evolutionary strategy could emerge coevolutionary process triggered cognitive agents acknowledgments research leading results received funding european commission horizon socsmc project european research council cdac project references lewis convention philosophical study john wiley sons von neumann morgenstern game theory economic behavior joh wiley sons new york smith game theory evolution fighting evol smith price logic animal conflict nature smith evolution theory games cambridge university press nash equilibrium points games source proc natl acad sci united states axelrod hamilton evolution cooperation science hardin collective action published resources future johns hopkins university press clements stephens testing models cooperation mutualism prisoner dilemma anim behav riehl frederickson cheating punishment cooperative animal societies philos trans soc london biol sci connor altruism among alternatives prisoner dilemma trends ecol evol gintis bowles boyd fehr explaining altruistic behavior humans evol hum behav cooperation experiments coordination communication versus acting apart together anim behav miller garnier hartnett couzin information social cohesion determine collective decisions animal groups proc natl acad sci taborsky frommen riehl correlated key cooperation philos trans soc london biol sci oprea henwood friedman separating hawks doves evidence continuous time laboratory games econ theory friedman oprea continuous dilemma econ rev oprea charness friedman continuous time communication experiment econ behav organ bigoni casari skrzypacz spagnolo time horizon cooperation continuous time econometrica van doorn riebli taborsky coaction versus reciprocity models cooperation theor biol kephart friedman hotelling revisits lab equilibration continuous discrete time econ sci assoc hawkins conducting multiplayer experiments web behav res methods hawkins goldstone formation social conventions environments plos one busoniu babuska schutter comprehensive survey multiagent reinforcement learning ieee trans syst man cybern appl rev claus boutilier dynamics reinforcement learning cooperative multiagent systems tan reinforcement learning independent cooperative agents proceedings tenth international conference machine learning mnih control deep reinforcement learning nature austerweil littman tenenbaum coordinate cooperate compete abstract goals joint intentions social interaction cogsci leibo zambaldi lanctot marecki graepel reinforcement learning sequential social dilemmas proceedings conference autonomous agents multiagent systems international foundation autonomous agents multiagent systems reinforcement learning model resource appropriation advances neural information processing systems eds guyon curran associates tampuu multiagent cooperation competition deep reinforcement learning plos one lerer peysakhovich maintaining cooperation complex social dilemmas using deep reinforcement learning arxiv peysakhovich lerer prosocial learning agents solve generalized stag hunts better selfish ones arxiv peysakhovich lerer consequentialist conditional cooperation social dilemmas imperfect information arxiv verschure voegtlin douglas environmentally mediated synergy perception behaviour mobile robots nature verschure pennartz pezzulo choice neuronal computational principles philos trans soc london biol sci interactions lowlevel reactive control symbolic rule learning embodied agents verschure synthetic consciousness distributed adaptive control perspective philos trans soc london biol sci fudenberg tirole fudenberg tirole game theory mit press binmore natural justice oxford university press herreros arsiwalla verschure forward model purkinje cell synapses facilitates cerebellar anticipatory control advances neural information processing systems braitenberg vehicles experiments synthetic psychology mit press verschure autonomous development behaviors agent populations computational study joint ieee international conference development learning epigenetic robotics ieee jaynes bretthorst probability theory logic science cambridge university press allostatic control robot behavior regulation comparative study adv complex syst duff sanchez fibla verschure biologically based model integration contingencies rules plans prefrontal cortex based extension distributed adaptive control architecture brain res bull marcos ringwald duff verschure hierarchical accumulation knowledge distributed adaptive control architecture computational robotic models hierarchical organization behavior springer berlin heidelberg steels language games autonomous robots ieee intell syst steels evolving grounded communication robots trends cogn sci verschure two possible driving forces supporting evolution animal communication phys life rev arsiwalla verschure embodied artificial intelligence distributed adaptive control integrated framework weitzenfeld vallesa flores wolf pack multiple robot hunting model ieee latin american robotics symposium ieee muro escobedo spector coppinger canis lupus hunting strategies emerge simple rules computational simulations behav processes arsiwalla herreros verschure consciousness evolutionary strategy springer cham arsiwalla herreros sanchez verschure consciousness control process artif intell res dev arsiwalla herreros verschure morphospace consciousness arxiv
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chinesefoodnet image dataset chinese food recognition oct xin hua zhou liang diao dongyan wang paper introduce new challenging food image dataset called chinesefoodnet aims automatically recognizing pictured chinese dishes existing food image datasets collected food images either recipe pictures selfie dataset images food category dataset consists web recipe menu pictures photos taken real dishes recipe menu well chinesefoodnet contains food photos categories category covering large variations presentations chinese food present efforts build image dataset including food category selection data collection data clean label particular use machine learning methods reduce manual labeling work expensive process share detailed benchmark several deep convolutional neural networks cnns chinesefoodnet propose novel data fusion approach referred tastynet combines prediction results different cnns voting method proposed approach achieves accuracies validation set test set respectively latest dataset public available research achieved https index recognition deep learning chinesefoodnet tastynet fig example images dataset row shows five images one category chinese food top bottom food names sichuan noodles peppery sauce mapo tofu potato silk scrambled egg tomato respectively variations visual appearance images chinese food caused complex background various illumination different angle view different ingredients category etc show challenges visual food recognition image keep original size ntroduction ood plays essential role everyone lives behaviour diet eating impacts everyone health underestimating food intake directly relates diverse psychological implications recent years photographing foods sharing social networks become part daily life consequently several applications developed record daily meal activities personal food log system employed computeraided dietary assessment usage preference experiments calorie measurement nutrition balance estimation one ways input food log automatic recognition dish pictures gives rise research field interest deep convolutional neural networks cnns achieved variety computer vision tasks visual dish recognition task situation quality training datasets always plays important role authors contributed equally work means corresponding author xin chen hua zhou zhu dongyan wang midea emerging technology center san jose usa xin email email hua email dongyan email liang diao midea artificial intelligence research institute shenzhen guangdong china email training deep neural network high performance deep model still extent however best knowledge still exist effective chinese food recognition system matured enough used major reason absence largescale high quality image datasets chinese food dataset includes categories images obviously size dataset sufficient satisfy deep learning training requirements visual dish recognition problem widely considered one challenging computer vision pattern recognition tasks compared types food italian food japanese food difficult recognize images chinese dish following reasons images category appear differently since chinese dish different ingredients different cooking methods images greatly visual different even human vision noise images chinese dishes hard model complex noise variety images chinese food taken various environment complex background example dim light vapour environment strong reflection various utensils chinese dishes color shape ornament etc order give impetus progress visual food classification related computer vision tasks build image dataset chinese dish named chinesefoodnet dataset contains images food categories covering popular chinese food images include web images photos taken real world unconstrained conditions best knowledge chinesefoodnet largest comprehensive dataset visual chinese food recognition images chinesefoodnet shown figure benchmark nine cnns models four deep cnns squeezenet vgg resnet densenet dataset experimental results reveal chinesefoodnet capable learning complex models paper also propose novel data fusion approach voting although simple voting effective way fuse results guided benchmarks try combination different cnns models based results chinesefoodnet take normalization predictive models fusing results voting final result method designated tastynet proposed method achieved accuracy validation set test set respectively compared best results approaches single network structure improvements validation set sets achieved respectively paper takes three major contributions following present image dataset chinesefoodnet chinese food recognition tasks chinesefoodnet made images categories food image users daily life public available research related topics provide benchmark dataset totally nine different models four cnns architectures evaluated presents details methodology used evaluation models public available research propose novel data fusion approach visual food recognition combines predictive results different cnns voting experimental results chinesefoodnet shown approach improves performance compared one deep cnns model shown data fusion alternative way improve accuracy instead increasing numbers layers cnns paper organized follows section briefly reviews public food datasets visual food recognition methods section iii describes procedure building tagging chinesefoodnet dataset section several cnns methods benchmarked chinesefoodnet section details proposed name cnns networks consists letters type cnns following numbers number layers dataset accessed https data fusion approach present results chinesefoofnet paper closes conclusion work future directions section elated ork food dataset scholars developed public food applications dietary assessment computational cooking food recipe retrieval pittsburgh food image dataset pfid collects fast food images dataset images related distinct dishes used near duplicate image retrieval ndir japanese food datasets contain categories respectively datasets twin datasets food categories different images images recipe images additional textual information images selfies chinese food dataset containing total ingredient labels images however aims cooking recipe retrieval ingredient recognition visual dish recognition introducing deep learning techniques classification traditional approaches features applied visual food recognition including pairwise feature distribution ped gabor filters bag visual words bow optimized model textons random forests fisher vector like deep learning applied computer vision tasks cnns models outperformed traditional methods achieve higher higher accuracy deeper deeper cnns however approaches traditional methods deep learning tested image dataset chinese food iii hinese ood arge scale hinese ood mage dataset best knowledge largescale image datasets chinese dish recognition mature enough provided necessary resources datadriven techniques deep learning train complex food recognition models section present procedures build chinesefoodnet labelling image expensive step building dataset paper design develop method accelerate whole process order review fairly discuss data available download paper last access date june fig fifty sample images chinesefoodnet dataset dataset contains chinese food images organized categories images dataset color images resized better presentation category selection data clean label various cooking styles exist chinese food culture sichuan cuisine canton cuisine etc chinese food dataset must cover popular chinese cuisines different styles cooking subsection present efforts meet goal first food categories gathered however dishes missed search engine yet popular searched tomato omelette order cover conduct survey favorite chinese dishes within group combining results survey select categories since chinese cruise categories complex dishes similar visually braised chicken wings cola chicken wings manually merge related categories process categories chinese dish collecting large number food images next step clean data generate proper labels image step first remove images irregular height width large small usually irrelevant images use entropy clean images without content entropy quantitative metric image content calculate value entropy channel value channel small remove image enough useful information following step remove duplicate similar images two steps first calculate deep features last full connection layer alexnet second calculate euclidean distance measure similarity distance threshold consider images similar remove one images clearly categorized specific chinese food name recipe menu images ground truth type image directly extracted however number images limited quality images usually high images shot sufficient light condition good presentation food good angels etc thus type images shows different distributions comparing images captured daily life brings potential impact food recognition tasks real life images usually food photos taken real world conditions images mainly users daily uploads show preferred data distributions food images wild besides type images usually associated metadata metadata viewed description image text format often describes name cooking recipe data collection two resources images web images taken photos web images dataset coming social network chinese food users uploaded chinese food pictures also provided tags labels image also partial images dataset collected group daily life steps number images brought together achieves however images may contain missing labels incorrect labels unclear labels names chinese dish chinesefoodnet also listed https fig show basic architectures four cnns evaluation left right architectures vgg resnet densenet squeezenet respectively information food image procedure metadata utilized filter useful images correct labels particularly manually generate set keywords food class database use set keyword match image metadata images metadata contains keywords certain class selected labeled class noted aforementioned step still number incorrect labels either caused unclear descriptions metadata irrelevant images label validation human labor large number images expensive task terms time costs accelerate label image already labeled samples advance first collect small database food images using platform overlap current dataset class labels shallow cnn model trained food recognition task small database given cnns model classify collected images different classes representing candidate labels specifically top predictions shallow network selected candidate labels one image finally perform manually label validation finalize dataset eliminating wrong labelled images dataset description work category selection data collection data collection cleaning mentioned previous subsections finally chinesefoodnet dataset contains images total size gigabyte images dataset kept original size without processing color total number categories current version dataset image labelled one label split whole dataset training testing validation sets approximately ratio respectively specifically images training validation testing set respectively figure figure show example images dataset enchmark hinese ood dataset section conducted benchmark experiments chinesefoodnet dataset first experimental settings described introduce experimental protocol finally provide experimental results analysis experimental settings experiments conducted using pytorch deep learning framework training phase initial learning rate set momentum set weight decay set set learning rate initial learning rate decayed every epoch number epoch training set training optimization table ecognition rates different deep networks food dataset oth top top accuracy shown validation set test set method validation accuracy accuracy method selected stochastic gradient descent sgd momentum augmentation process applied except resizing mirror training images firstly resized random crop size hoizontal flip probability applied used pretrained models imagenet dataset network food data testing images resized use center crop size feed network experiments conducted centos operation system intel xeon cpu ram nvidia tesla gpus hardware memory experimental protocol dataset split training validation test sets random selection images training set images validation set rest images used testing comprehensive experiments conducted using various popular deep learning network architectures different structures different number layers specifically benchmarked performance squeezenet version layer resnet densenet order fair comparison experiments using input image size procedures implementation details resnet squeezenet illuminated figure fig illustration residual block resnet fire block squeezenet experimental results recognition performance different deep networks shown table top accuracy top accuracy test accuracy accuracy presented table shown best performance validation set achieved accuracies also close best results test set best recognition rate obtained second best results obtained lower deeper cnns models generally achieve better performance results see cnn models obtains significant improvements performance number layers network architecture increased resnet layers recognition rate deeper mode resnet layers achieves improvement validation test sets similar results observed densenet architecture hand deep models wider structure also shows promising performance obtains best results test set worst result validation test sets respectively achieved squeezenet shallow narrow network structure designed fast efficient inference tasty two step data fusion approach methodology shown table accuracy higher higher deeper deeper model would improve furthermore possible way use much deeper cnns models however needs much computation memory resources deeper models easily lead overfitting problem alternative way data confusion approach idea fuse inference results different models shown figure predictions different networks gathered voting approach utilized obtain final fused prediction based results different combinations shown table select combination models achieves best top result voting method average results models algorithm details results analysis different combinations network architectures applied experimental results shown table fig basic scheme data fusion approach first one obtain predictive results different models tastynet use second one combine result one final result voting policy paper use weighted coefficient results first step table xperimental results recognition accuracies different fusion schemes fusion method resnet densenet resnet densenet resnet densenet densenet top accuracy validation test top accuracy validation test algorithm algorithm tastynet input image output number range predictive result predictive result predictive result predictive result predictive result get average result find maximum get output number network validation test set respectively based experimental results select five cnns models components tastynet proposed approach get two conclusions followings applying data fusing approach different deep networks overall performance boosted using single deep network combination different network architectures show benefits improving performance combinations network architectures combination deeper wider networks obtains best results evaluation table conclude overall performance generally increasing different combinations ensemble deep networks fusion results resnet different number layers obtained higher performance top accuracy test set single resnet resnet top accuracy test set also fusion results densenet achieved improvement test set best results achieved single densenet architecture furthermore combination different types cnns networks resnets densenets vgg shown row table improves overall recognition performance best result obtained fusing densenet densenet densenet recognition accuracy validation set test set results higher single onclusion uture ork paper successfully created image dataset chinese dish recognition chinesefoodnet contains images food categories images web images also real world consequence models trained dataset covered food recognition applications also present benchmarks nine cnns models four cnns architectures chinesefoodnet finally propose novel data fusion approach tastynet based experimental results select resnet densenent densenet densenet models voting results model obtain final inference result shown results chinesefoodnet proposed approach shown data fusion effective way obtain better result instead working one type cnns model future work extending number food category applied much applications also investigate new fusion methods fuse different results different models obtain better performance acknowledgment authors would like thank zhi zhang xukai zhang zigang wang xiangping zeng xiaofei dangdang qingqing chang discussions 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submitted annals statistics sep multiclass classification information divergence surrogate risk john khashayar khosravi feng stanford university provide unifying view statistical information measures bayesian hypothesis testing loss functions classification problems elaborating equivalence results objects extending existing results binary outcome spaces general ones consider generalization multiple distributions provide constructive equivalence divergences statistical information sense degroot losses multiclass classification major application results classification problems must infer discriminant function making predictions label datum data representation setting hypothesis testing problem experimental design represented quantizer family possible quantizers setting characterize equivalence loss functions meaning optimizing either two losses yields optimal discriminant quantizer complementing extending earlier results nguyen multiclass case results provide substantial basis standard classification calibration results comparing different losses describe convex losses consistent jointly choosing data representation minimizing weighted probability error multiclass classification problems introduction consider multiclass classification problem decision maker receives pair random variables unobserved wishes assign variable one classes minimize probability misclassification represent decision maker via discriminant function component represents margin score perceived likelihood decision maker assigns class datum goal minimize expected loss partially supported award center research additionally supported stanford graduate fellowship keywords phrases risk surrogate loss function hypothesis test duchi khosravi ruan measures loss margins true label expectation taken jointly loss formulation misclassification probability argmaxy may also consider classical bayesian experiment given random variable drawn according one hypotheses prior probabilities wish choose minimizing expected posterior loss many applications making decisions based raw vector may carry useless information impinging statistical efficiency may need store communicate sample using limited memory bandwidth wish classify person taller shorter centimeters makes little sense track blood type eye color increase number variety measurements collect careful design choices important maintaining statistical power interpretability efficient downstream use mitigating false discovery desire give better representations data led rich body work statistics machine learning engineering highlighting importance careful measurement experimental design data representation strategies nguyen note binary case one thus frequently extends classical formulation include stage maps vector vector number situations suggest approach practical classification scenarios equivalent feature selection reduces dimension increases interpretability second motivation consider decentralized detection problem communication applications engineering remote sensors communicate data limited bandwidth memory case central processor infer distribution communication transformed vector one chooses quantizer family low complexity quantizers fuller abstraction may treat problem bayesian experimental design problem mapping may stochastic chosen family possible experiments observation channels preceding examples incorporation quantizer classification procedure poses complex problem one must simultaneously find data representation discriminant goal paralleling risk multiclass classification information surrogate risk thus becomes joint minimization quantized prespecified family quantizers example error loss functions even discontinuous population empirical minimization intractable thus common replace loss convex surrogate minimize surrogate instead surrogate fisher consistent minimization yields bayes optimal discriminant original loss distribution researchers characterized fisher consistency convex surrogates binary multiclass classification weakness results rely strongly using unrestricted class measurable discriminants thus natural convex losses consistent context major difficulty understand consequences using various surrogate losses requiring restricted quantizer class one approach discovering nuanced properties relationship surrogate bayes risk nguyen tackle binary case considering problem joint selection discriminant function quantizer exhibit precise correspondence binary loss functions similarity two probability distributions developed information theory statistics give general characterization loss equivalence classes divergences interesting consequence results spite positive results fisher consistency binary classification problems essentially losses consistent loss provide extension results multiclass case outline discussion contributions build prior work provide unifying framework relates statistical information measures loss functions generalized notions entropy context classification begin recall generalization applies multiple distributions enumerating analogues positivity properties inequalities discrete approximation available binary case may unfamiliar motivate approach section begin main contributions section establish correspondence loss functions generalized entropy discrete distributions multiway make precise define probability simplex let prior distribution duchi khosravi ruan class label posterior probabilities conditional observation concave degroot defines information associated experiment notion generalized entropy clear gap prior posterior entropy always context value measures uncertainty experimenter appropriate units unknown class label prior belief gap prior posterior uncertainty relate type entropy loss functions recall result loss induces entropy also called pointwise bayes risk via inf argument false drop indicator future defining implicitly show inverse construction providing explicit constructable mapping concave function loss inducing pointwise bayes risk loss convex section also develop natural connections generalized entropies classification calibration explicit generally calibrated section address comparison loss nguyen approach binary case present main results consistency joint selection quantizer data representation discriminant using correspondence concave losses characterize pairs losses equivalent quantizers discriminants minimize quantized risk sense continuous concave inf inf another way understand results providing insight classification calibration bayes act optimal discriminant belong class functions statistician may choose classification problem substantial challenge criticism line work classification calibration surrogate risk consistency results say little restricted families multiclass classification information surrogate risk classifiers context corollary main contribution follows loss calibrated distribution sequence inf implies inf consider collection functions set define class functions measurable translated scenario main corollary assume loss calibrated let denote associated pointwise bayes risk inf implies inf collection mappings set distribution sequence exist corollary reposes connections develop losses uncertainty measures generalized measures statistical information divergence central design communication quantization schemes signal processing characterize divergence measures optimized yield optimal quantizers detectors also provide result showing empirical minimization surrogate risk consistent desired original risk number researchers studied connections divergence measures risk binary experiments point results present indeed blackwell shows quantizer induces distributions larger divergence induced prior probabilities allows tests lower probability error liese vajda give broad treatment using representation difference prior posterior risk binary experiment derive number properties see also paper williamson show arise naturally context classification problems gap prior posterior risk classification work binary case results elucidate characterization fisher consistency quantization binary classification nguyen realize pursue line research draw connections fisher consistency information measures classification surrogate losses divergences duchi khosravi ruan notation let denote vectors respectively vector collection objects define indicator function argument false otherwise argument true otherwise let denote probability simplex tin set let aff denote affine hull set rel int denotes interior relative aff let let epi denote epigraph say convex function closed epi closed set though abuse notation say concave closed epi closed convex function say strictly convex point fenchel conjugate function sup conjugate closed convex chapter measures let denote derivative respect random variables say divergence measures probability distributions significant statistical informationtheoretic applications including optimal testing minimax rates convergence design communication schemes central work introduced ali silvey see overview given distributions defined common set closed convex function satisfying measure dominating denote densities respectively value defined appropriately number classical divergence measures arise taking log yields respectively squared hellinger distance total variation distance central study hypothesis testing classification understanding relationships multiple distributions use following generalization multiple distributions multiclass classification information surrogate risk definition let probability distributions common closed convex function set let satisfying let measure let dpi must specify value integrand case function defined arbitrary rel int dom lim otherwise closed convex function value independent closure perspective function prop time consider perspective implicitly treat closure enumerate properties showing naturally generalize classical binary focus basic properties useful results bayes risk classification hypothesis testing parallel results binary case continuity properties respect discrete approximations satisfy inequalities nemetz original work essentially contains results carefully address infinite values closure strict convexity use definitional building blocks defer proofs supplement first step note definition independent base measure see supp proof generalizing cor lemma expression value divergence depend choice dominating measure moreover inequality strict strictly convex identical given importance quantization come consider discrete approximations divergence countable partition define partitioned duchi khosravi ruan binary case following approximability result generalizing thm possibly infinite integrands quantizers give arbitrarily good approximations see proof proposition closed convex function sup supremum finite partitions binary case satisfy data processing inequalities state processing transforming observation drawn distributions decreases divergence formalize recall markov kernel set probability distribution measurable mapping measurable following general data processing inequality shows holds case well generalizing thm possibly infinite closure include proof appendix proposition let closed convex markov kernel define marginals proposition related relationships risk information quantization develop sections defining quantizer measurable mapping measurable spaces quantized divergence sup ranges finite partitions proposition immediately yields quantization reduces information indicator defines markov kernel yielding corollary let closed convex satisfy quantizer multiclass classification information surrogate risk also see quantizers induces finer partition meaning equality implies type ordering central work multiclass loss induces unique consistency discriminants loss quantization intimately tied preservation relative ordering information related quantized risk risks information measures reviewed basic properties turn detailed look relationships hypothesis tests classification generalized entropies statistical informations relating multiple distributions build correspondence parallels binary experiments classification problems first recapitulate probabilistic model classification bayesian hypothesis testing problems introduction prior probability distributions defined set coordinate drawn according multinomial probabilities conditional draw following degroot refer experiment associated experiment family informations follows let bep posterior distribution given observation dpi dpj given closed concave refer generalized entropy see degroot calls uncertainty function information associated experiment reduction entropy uncertainty prior posterior expectation taken immediate concavity degroot thm shows distributions priors concave section develop equivalence results multiclass classification losses risk entropy measures concretely consider recall risk defined set measurable functions equation introduction loss induces entropy via inf also called pointwise bayes risk section give explicit inverse mapping showing generalized entropy induced least duchi khosravi ruan one convex loss function convex section illustrate consistency properties entropy implies convex loss inducing connect results section loss associated risk exists convex function gap prior bayes best expected loss attainable without observing posterior bayes risk inf inf dfl see inverse direction new given closed convex construct convex losses function associated generalized entropy prior satisfying inf inf generalized entropies losses construct natural bidirectional mapping losses generalized entropies giving examples illustrate loss construction yields closed concave function infimum linear functionals thus generalized entropy uncertainty function gap prior posterior entropy following two examples loss illustrative example loss consider zero one loss lzo max hlzo inf generalized entropy concave nonnegative satisfies hlzo standard basis vector example classification scenarios allow different costs classifying certain classes others example may less costly misclassify benign tumor cancerous opposite cyi case use matrix cyi multiclass classification information surrogate risk cost classifying observation class class assigning instead experiment assume cyy define lcw max cyi max maximal loss indices attaining maxj let column representation cty minl ctl choosing hlcw inf max cyi max min entropy hlcw concave nonnegative satisfies hlcw standard basis vectors example corresponds forward mapping losses entropy straightforward though using convex duality conjugacy arguments show inverse mapping construction new though precursors proper scoring rules predictions probability simplex exist thm characterize proper scoring rules always clear generate convex losses stating proposition recall definition fenchel conjugate supt proposition closed concave losses closed convex satisfy equality proof standard fenchel conjugacy relationships chapter imply inf sup defining write inf inf inf proposition shows associated every concave entropy defined simplex least one set convex loss functions generating duchi khosravi ruan entropy via infimal representation thus mapping loss functions entropies entropies convex losses given loss may construct convex loss lcvx hlcvx mapping entropies loss functions generating losses range satisfy surrogate risk consistency generalized entropies construction loss functions somewhat privileged construction often yields desirable properties convex loss function especially related loss indeed often case convex loss generated fisher consistent make explicit recall following definition definition let classification calibrated loss maxj inf inf max example classification calibrated given matrix cost matrix minj ctj inf inf max tewari bartlett thm zhang thm show importance definition let risk exs classification calibrated respect loss sequence distribution fisher consistency inf implies inf classification calibration respect costweighted risk equivalent surrogate risk consistency loss predominance loss literature use classification calibration without qualification mean classification calibration respect loss show minor restrictions generalized entropy function construction yields classification calibrated losses multiclass classification information surrogate risk definition convex function convex closed say without qualification uniformly convex dom exist norm constant definition holds say uniformly concave uniformly convex definition extension usual notion strong convexity holds essentially quantified notion strict convexity definition following two propositions two propositions whose proofs provide appendix show generalized entropies naturally give rise classification calibrated loss functions provide examples results section come proposition assume closed concave symmetric dom let definition additionally assume strictly concave inf attained uniformly concave classification calibrated even strictly concave give classification calibration results indeed recall example showed maxj proposition let maxj loss defined classification calibrated moreover inf inf divergences risk generalized entropies section show definition precise correspondence generalized entropies losses williamson establish correspondence bayes risk results show important link directly loss begin equation concave generalized entropy loss satisfying inf proposition loss duchi khosravi ruan generality assume correspondence let collection measurable functions posterior bayes risk inf dpi posterior distribution conditional information measure thus gap prior bayes posterior bayes may write inf inf ihl sup dpi dpk dfl dpk definition closed convex function sup supremum affine functions argument closed convex equation shows given loss generalized entropy information measure ihl gap prior posterior distributions identical also give converse result shows every written gap prior posterior risks convex loss function first recall result statistical information based generalized entropy associated except closure operation result known thm closed convex let proposition implicitly use closure perspective def prior expectation taken according multiclass classification information surrogate risk combining propositions infimal representation immediately obtain following corollary explicit construction closed convex loss corollary let closed convex function convex losses defined sup satisfy additionally inf inf expectation conditional binary classification problems nguyen thm provide explicit construction closed convex loss inducing divergence binary case allows complete characterization convex functions appears difficult multiclass case corollary coupled information representation given divergence shows complete equivalence loss functions entropies exists loss function prior conversely loss function prior exists gap examples generalized entropies loss correspondences complement general results illustrate correspondence concave generalized entropies loss construction several examples using propositions guarantee classification calibration example loss example continued use generalized entropy maxj generated loss derive convex loss function gives entropy via representation conjugate max duchi khosravi ruan entries sorted order see proof convex loss named similarity error control hypothesis tests lfw max generates entropy hlfw associated zeroone loss moreover proposition guarantees lfw classification calibrated def appears loss lfw new convex classificationcalibrated loss function rather example later context showing distinct convex losses yield generalized entropy consider multiclass logistic loss loss correspond loss generates shannon entropy information example logistic loss entropy define logistic loss log log entropy associated loss familiar shannon entropy log log inf conjugacy calculation inverse construction loss generate also yields logistic loss multiclass logistic loss calibrated loss immediate negative shannon entropy strongly convex simplex pinsker inequality fact logistic loss shannon entropy dual via proposition yield calibration information measure associated logistic loss mutual information observation label indeed denotes shannon entropy usual shannon mutual information multiclass classification information surrogate risk include one final example show instances many different convex losses yield generalized entropy example hinge losses define pairwise multiclass hinge loss lhin also consider slight extension weighted loss functions address asymmetric losses form example case given set loss matrix lhin cij loss cij yields completely identical set calculations without restriction invariant shifts calculation see completeness shows generalized entropy associated hinge loss loss matrix lhin min hlhin inf losses satisfy number classification calibration guarantees note one essentially due zhang thm completeness include proof appendix observation let bounded belowpconvex function differentiable cyi classification calibrated cost matrix def taking see hinge loss calibrated weighted hinge loss loss taking arbitrary calibrated cost matrix even following quantitative calibration guarantee analogy proposition hin hin lcw strengthening observation prove lemma binary case lem similar quantitative guarantees hold classification calibrated loss hlzo know extends multiclass case duchi khosravi ruan comparison loss functions section demonstrated correspondence loss functions generalized entropies statistical information restricted cases classification calibration correspondences assume decision makers access entire observation often case noted introduction often beneficial data make lower dimensional communicate store efficiently improve statistical behavior thus explore impact quantization concepts motivate consider loss logistic loss loss form convex decreasing observation classification calibrated relates one major criticisms classification calibration bayes classifier minimizer risk functions belong class functions considered classification calibration says little context shed light issue identifying losses consistent calibrated even additional selection quantizer data restriction class possible functions implication introduction model quantization experimental design abstractly treat design experiment choice data representation quantization problem quantizer maps space measurable space loss prior label discriminant consider quantized risk recall given distributions equivalently hypotheses bayesian testing setting collection quantizers criterion choose quantizer allows best attainable risk consider quantized bayes defined infimum risk discriminants inf inf dpiq denotes measure risk measures best attainable risk fixed choice one thus seeks design giving lowest quantized bayes whether computational analytic reasons minimizing loss often intractable loss lzo example multiclass classification information surrogate risk convex discontinuous thus interest understand asymptotic consequences using surrogate loss place desired loss say lzo including setting one incorporates dimension reduction via choice nguyen introduce study problem binary classification giving correspondence loss functions surrogate consistency quantization consequences using surrogate consistency resulting quantization classification procedure multiclass case unclear know using surrogate done without penalty end characterize two loss functions provide equivalent criteria choosing quantizers experimental designs data representations according bayes universal equivalence loss functions recalling arguments section statistical information gap prior posterior risks distributions give quantized version construction analogy results section quantized statistical information ihl inf inf dfl inf convex function defined expression depend quantizer denotes posterior distribution conditional observing extend nguyen notion universal equivalence binary case defining losses equivalent induce ordering quantizers information measure definition loss functions universally equivalent prior denoted distributions quantizers definition evidently equivalent ordering condition inf inf inf inf duchi khosravi ruan distributions quantized bayes definition somewhat stringent losses universally equivalent induce quantizer ordering population distributions quantizer finer losses yield data processing inequality corollary section stronger equivalence notion important nonparametric classification settings underlying distribution weakly constrained neither pair quantizers finer definition representation suggest entropy function associated loss infimal representation associated via construction important equivalence two loss functions indeed case first following result universal equivalence loss functions based associated entropies theorem let bounded losses associated generalized entropies construction universally equivalent respect priors exist ahl also characterize universal equivalence prior theorem let theorem let bounded loss functions associated construction universally equivalent respect prior exist nguyen thm prove theorem binary classification problems using arguments outline proofs apply arbitrary require different set tools section consistency empirical risk minimization major application theorems show certain loss functions loss universally equivalent convex loss functions including variants hinge loss showing associated entropies scalar multiples first application theorems however consider bayes consistency empirical risk minimization selecting multiclass classification information surrogate risk discriminant quantizer analogy thm case receive sample define empirical risk let collection quantizers indexed sample size similarly let collection discriminant functions assume collections satisfy estimation approximation error conditions sup inf inf app additionally let risk funcwhere tional misclassification loss lcw example lcw maxi cyi maxj following result theorem assume conditions approximate empirical minimizers satisfying inf let inf loss classification calibrated universally equivalent loss lcw theorem guarantees estimation approximation conditions empirical risk minimization consistent minimizing quantized bayes risk whenever loss classification calibrated equivalent desired loss proof theorem reposes following risk inequality may independent interest lemma consequence results surrogate risk consistency classification calibration universal equivalence guarantees exhibits power calibration universal equivalence lemma assume universally alent weighted misclassification loss lcw cost matrix duchi khosravi ruan exists continuous concave function inf inf choice cyi cyi may take inf inf lemma shows gap surrogate risk provides guaranteed upper bound true risk case modified hinge losses example gap linear binary case even stronger results possible lemma may take lemma loss universally equivalent relies specific form binary convex loss must take examples familywise loss show fairly losses classification calibrated universally equivalent loss provide proof lemma theorem prove consequence lemma theorem examples universal equivalence section give several examples build theorems showing exist convex losses allow optimal joint design quantizers measurement strategies discriminant functions opening way potentially efficient convex optimization strategies end give two loss functions universally equivalent loss prior distributions also give examples classification calibrated loss functions universally equivalent loss although minimizing without quantization yields classifiers example losses treturn example hin cij case hlhin minl khlcw lcw denotes misclassification error example theorem immediately guarantees weighted hinge loss universally equivalent weighted loss weighted hinge loss lhin also example calibrated misclassification error example losses loss return examt ple let lfw multiclass classification information surrogate risk convex loss example associated entropy hlfw maxj hlzo proposition shows lfw classification calibrated thus lfw loss lzo universally equivalent theorems final example consider logistic loss classification calibrated universally equivalent loss example logistic loss loss llog log shannon entropy log hlzo maxj ahllog theorem shows logistic loss universally equivalent loss spite classification calibration distributions collection quantizers sequence rllog inf rllog rlzo inf rlzo proof theorems remainder main body paper consists major parts arguments theorems divide proof theorems two parts part straightforward substantially complex proof direction give proof theorem theorem identical assume dom dom exist holds definition divergences quantizer adf dpi applying relationship obtain universal equivalence follows immediately turn part proofs theorems roadmap follows first define call order equivalence convex functions related equivalence generalized entropies def two loss functions universally equivalent show associated entropies constructed infimal representation functions generating via expression duchi khosravi ruan order equivalent lemmas provide characterization order equivalent closed convex functions lemma linchpin analysis lemma shows two order equivalent closed convex functions dom dom parameters dom dom proves part theorems yielding desired result present main parts proof body paper deferring technical nuances supplement universal equivalence order equivalence definition equivalent variant stated universally equivalent losses induce ordering quantized information measures divergences next definition captures ordering slightly differently definition let closed convex functions closed convex let arbitrary matrices satisfy columns columns matrices context suggests order equivalence strong connections universal equivalence loss functions associated generalized entropies next two lemmas make explicit lemma losses lower bounded universally equivalent associated entropies construction order equivalent proof let entropy pointwise bayes risk associated noting dom dom inf satisfy let matrices let pmshow expression holds constructing appropriate distributions applying universal equivalence let integer large enough multiclass classification information surrogate risk define vectors let aext ext ext satisfy aext ext let bext denote columns extended matrices let spaces define quantizers define distributions ext ext ext ext ext bjl ail ail bjl ext similarly bjl let uniform prior distribution label note posterior probability aext ext aext similarly ext taking expectation ext ail aext ext aext similarly recalling definitions quantized information sociated aext ext universal equivalence losses immediately implies ext bext iff aext bext noting aext bext rearrange preceding equivalent statements adding aext side obtain satisfy inequality duchi khosravi ruan parallel result possible techniques similar use prove lemma constructing explicit discrete space quantizers defer proof supplementary lemma losses universally equivalent prior def corresponding construction order equivalent characterization order equivalence convex functions lemmas illustrate intrinsic relationship universal equivalence def losses order equivalence def associated generalized entropies therefore natural ask convex functions order equivalent lemma characterizes order equivalence coupled lemmas immediately implies theorems lemma let closed convex functions convex set order equivalent exist proof lemma complex provide partial proof highlighting important parts argument deferring technical details supplement essential idea lemma holds simplices certainly holds cover convex set number overlapping simplices extend result fully supplement demonstrate lemma simplices require definition vectors affinely independent linearly independent set simplex conv affinely independent essential special case lemma following result lemma let conv affinely independent order equivalent exist multiclass classification information surrogate risk proof lemma proceeds series intermediate results provide turn deferring proofs supplement first step argue need prove equivalence results convex functions dense subsets domains lemma prop let closed convex satisfy dense subset first technical lemma prove essentially direct consequence definition order equivalence let satisfy order equivalent thus satisfies iff next lemma shows force equality hold extreme points centroid simplex intuitive free parameters choices lemma let closed convex let affinely independent exist ucent ucent lastly following characterization linearity convex functions convex hulls lemma let convex ucent cent four lemmas prove lemma rotating shifting loss generality assume duchi khosravi ruan functions defined continuous defined convex order equivalent make one reduction let standard basis shorthand vector let ecenter centroid conv lemma guarantees existence ecenter let convex order equivalent satisfy ecenter thus lemma equivalent showing convex order equivalent equal extreme points centroid divide discussion two cases linear case suppose ecenter order equivalence ecenter lemma thus implies linear conv equal vertices hence equal interior nonlinear case convexity ecenter order equivalence lemma implies ecenter conv use shorthand may write fix arbitrary wish show end consider consider gaps due convexity ecenter values relative defining linear functions ecenter ecenter assumption convexity multiclass classification information surrogate risk key order equivalence implies sign sign zero crossing exists prove equality presently find use ecenter ecenter arbitrary dense lemma extends equality expression holds returning sign equivalence may divide ecenter inequalpk ity equivalent ecenter calculation yields ecenter applying lemma immediately yields noting obtain equality defining discussion rather recapitulating contributions point directions believe prove interesting study corollary shows convex losses consistent even restricted families classifiers apply practical case collection discriminants convex subset finitedimensional vector space longstanding problem certainly deserves work another direction bit afield investigate links work objective bayesian approaches reference priors line work one family probability models observation space performing inference chooses prior maximize shannon information tasks minimizing log loss may sensible use notion information entropy corresponding desired loss notions loss equivalence including construction convex losses equivalent losses could provide insight situations duchi khosravi ruan references ali silvey general class coefficients divergence one distribution another journal royal statistical society series bartlett jordan mcauliffe convexity classification risk bounds journal american statistical association benjamini hochberg 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information discrimination ieee transactions information theory poor thomas applications distance measures design generalized quantizers binary decision systems ieee transactions communications pukelsheim optimal design experiments classics applied mathematics siam reid williamson information divergence risk binary experiments journal machine learning research robbins aspects sequential design experiments bulletin american mathematical society schapire freund boosting foundations algorithms mit press steinwart compare different loss functions constructive approximation tewari bartlett consistency multiclass classification methods journal machine learning research tishby pereira bialek information bottleneck method allerton conference communication control computing tsitsiklis decentralized detection advances signal processing vol pages jai press vajda singularity probability measures periodica mathematica hungarica williamson vernet reid composite multiclass losses journal machine learning research appear zhang statistical analysis large margin classification methods journal machine learning research duchi khosravi ruan appendix proofs classification calibration results section prove propositions proving propositions proper state several technical lemmas enumerate continuity properties fenchel conjugates prove useful also collect important definitions related convexity norms use without comment appendix norm recall definition dual norm convex function let denote subgradient set point set rel int dom see chapter technical preliminaries provide background convex functions recall definition uniform convexity uniformly convex closed state related definition smoothness definition function continuous gradient respect norm meaning dom first technical lemma equivalence result uniform convexity lemma let closed convex closed convex set convex rel int inequality holds inequality also holds points multiclass classification information surrogate risk see section proof lemma also natural duality uniform convexity smoothness function fenchel conjugate supu dualities common lemma let closed convex set convex def dom see section proof lemma also two results properties smooth functions whose proofs provide sections respectively lemma let smooth lemma let smooth inf sequence satisfies limn inf proof proposition state two intermediate lemmas proving proposition lemma symmetric closed strictly concave continuously differentiable satisfies proof strictly concave suprema closed concave functions compact sets attained continuously differentiable standard results convex analysis thm let satisfy let assume sake contradiction letting permutation matrix swapping entries vector satisfies thus contradiction assumed optimality must whenever duchi khosravi ruan lemma symmetric inf implies strictly concave proof let satisfy assumed lemma suppose sake contradiction let permutation matrix swaps also symmetric contradiction optimalitypof strictly concave lemma implies minimizing moreover lemma must proof proposition infimum attained lemma gives result otherwise recall inf let let sequence using uniformly concave continuous recall lemma implies recall lemma lim lemma implies thus must case eventually moreover lim inf lim inf thus must would contradict find lim inf sequence tends infimum implies inf loss thus classification calibrated def proof proposition without loss generality assume maxj restricting maxj forces larger multiclass classification information surrogate risk risk maxj present two lemmas based convex duality imply result let vector without ith element let inf lemma otherwise constraints proof introducing lagrange multipliers lagrangian inf otherwise substuting noting gives result thus define matrix columns find strong duality inf sup inf inf sup inf min sup max lemma next lemma immediately implies proposition note hlzo maxj lemma let inf max max proof essentially construct optimal vector supremum definition without loss generality assume maxj result trivial maxj define duchi khosravi ruan definining tlow thigh tlow tlow thigh thigh fact strictly decreasing tlow thigh strictly increasing implies exists unique root tlow thigh define vector min moreover maxl remains show equivalently must show suppose sake contradiction hold know tlow thigh assumption satisfying two constraints two strict inequalities assumption would contradiction choice decreasing thus case giving result proof observation proof essentially trivial modification zhang theorem assume without loss generality let recalling cost matrix let cyi noting without loss generality may assume minl show inf multiclass classification information surrogate risk inf proof complete let sequence satisfying maxj inf first show loss generality assume converges suppose sake contradiction lim supm must lim supm lim supm convex must case remains bounded convergence would contradiction minl thus must lim supm subsequence converging without loss generality assume inf continuity show swapping value value increasing latter slightly always improve value consider three cases noting let additionally sufficently small thus whence let taking clear let using see sufficiently small must point thus biconjugates loss subsection calculate conjugate generalized entropy maxj pointwise bayes risk associated loss demonstrating equality let formulating lagrangian supremum dual variables dual objective sup otherwise duchi khosravi ruan inf strong duality obtains problems linear supremum expression sup inf symmetry without loss generality may assume domain function strictly decreasing thus unique smallest satisfying attaining equality inspection must one makes number terms positive fixing number terms solving gives preceding equality expression follows pointwise infimal risks hinge losses demonstrate equality note cyi inf inf formulating lagrangian introducing dual variable generalized kkt conditions problem given taking subgradients lagrangian optimum must without loss generality assume minj set via min setting min otherwise see minj minj minj kkt conditions satisfied thus optimal yielding expression multiclass classification information surrogate risk classification calibration hinge losses provide quantitative guarantee classification calibration losses let lcw maxi cyi maxj lemma let loss let cyi cyi lcw lcw proof first recall example inf minl defining argmaxj breaking ties arbitrary deterministic order sufficient argue cyi min cty min vector show inequality assume without loss generality exists index minl ctl ctl inequality trivial always minl ctl let suppose maxj without loss generality take inf inf cyj ctl writing lagrangian problem introducing variables inequality maxj equality max set claim optimal indeed set duchi khosravi ruan whose value specify later taking subgradients lagrangian respect diag conv ctk arbitrary scalars satisfying notably choosing small enough clear may take show choose assume without loss generality scaling eliminating extraneous indices see seek setting vector set straightforward take summarizing find preceding choices optimal problem thus min particular min min min ctl min ctl recalling must obtain inequality multiclass classification information surrogate risk proofs technical lemmas smoothness proof lemma let assume inequality holds let rel int stt must exist rel int see lemma chapter similarly tstt multiplying preceding inequalities respectively gives rel int consider case rel int preceding display equivalent uniform convexity condition let rel int rel int define functions closed convex functions thus continuous chapter rel int thus lim lim equivalent uniform convexity condition prove converse assume uniform convexity condition equivalent duchi khosravi ruan let directional derivative direction recalling see taking implies subgradient condition proof lemma first note dom uniformly convex meaning kuk thus dom see proposition strictly convex assumption differentiable theorem moreover closed convex meaning subdifferentiable set whence let must standard results convex analysis corollary use uniform convexity condition lemma see adding equations find inequality dividing side obtain desired result note multiclass classification information surrogate risk proof lemma define using taylor theorem computing final integral gives result proof lemma let suppose sake contradiction subsequence without loss generality take full sequence kgn fix choose later lemma defining kgn kgn particular see contradicts fact inf appendix order equivalence convex functions appendix collect proofs various technical results order equivalent functions definition proof lemma proving lemma give matrix characterization vectors equal sums similar characterization majorization via doubly stochastic matrices temporarily defer proof lemma appendix lemma vectors satisfy exists matrix duchi khosravi ruan returning proof lemma proper let construction hand note dom dom satisfying mini inf given matrices construct distributions quantizers constant use definition loss equivalence show order equivalent mind take positive integer max enlarge matrices aext ext respectively construct matrices let aext adding columns aij bij set ext ext aext bil enlarged matrices ext ext columns belong ext thus satisfy dom dom let spaces define quantizers aext ext lemma guarantees exisl tence matrices zij ext implies matrix satisfies zij define probability distribution zij let distribution defined quantizer design choice distributions prior upon functions implicitly depend ext aext bext multiclass classification information surrogate risk ext zij alj similarly ext loss equivalence recall def obtain aext bext aext bext note aext bext dom moreover ext ext preceding display equivalent order equivalence proof lemma one direction proof easy prove converse using induction result immediate claim loss generality assume indeed let permutation matrices sorted decreasing order construct zet satisfies suppose statement lemma true vectors dimension argue result holds loss generality assume let set additionally set matrix defined let zinner upper define vectors ainner binner ainner binner moreover ainner binner particular inductive hypothesis may choose zinner zinner ainner zinner binner inspection setting zinner duchi khosravi ruan proof lemma discussion section prove lemma sequence steps auxiliary lemmas roadmap follows first show may assume set lemma interior done consider affinely independent subset points whence lemma holds done cover overlapping simplices extend loss generality assume interior lemma let aff affine hull dim dim single point lemma trivial argue lemma holds sets int holds generally thus temporarily assume truth convex int since dim full columnrank matrix interior relative theorem convex set int defining fei order equivalent assumption lemma holds exist heb full column rank exists unique avt mapping linear may take obtain fei whence desired result point forward thus assume int covering sets simplices lemma hand show special case simplices sufficient show general lemma first show simplices essentially cover convex sets lemma let arbitrary points int exist int conv points make affinely independent def exist int conv multiclass classification information surrogate risk proving lemma state technical lemma interior points convex sets lemma lemma let convex set rel int rel int proof lemma take arbitrarily general psfrag replacements fig construction lemma position points affinely independent def define ucent ucent int lemma choose small enough points possible int see figure find ucent ucent rearranging ucent duchi khosravi ruan noting obtain ucent int conv points general position ucent ucent int conv lemma extension single simplex use lemma show following lemma implies lemma lemma addition conditions lemma assume simplices int exist bte exist coupled lemma lemma immediately yields lemma indeed lemma shows simplex conditions lemma holds lemma follows proof define sets int int int int exist divide discussion two cases case first suppose differentiable int case int continuity int must bti result follows taking applying lemma case least one without loss generality say choose pair consider collection sets conv int show know int int int multiclass classification information surrogate risk assume sake contradiction subtracting preceding equations one another multiplying respectively one obtains yields contradiction since assumed either exist int int int thus obtain note int int subtracting equalities find interior immediately implies hence exist sets complete proof showing dense define exist affinely independent set forms dense subset int lemma guarantees existence int thus previous paragraph lemma allows extend equality proofs auxiliary lemmas lemma proof lemma let positive negative parts let rsi qsi defining matrices times times times times times duchi khosravi ruan columns order equivalence implies iff equivalent proof lemma assume conv otherwise result trivial let vectors matrix defined define vector possible full rank choosing obtain algebraic manipulations consider ucent ucent ucent ucent ucent ucent multiclass classification information surrogate risk thus may choose desired equalities hold exists ucent ucent assumption lemma ucent ucent thus setting ucent ucent gives desired result proof lemma without loss generality assume ucent let ucent ucent inequalities must equalities giving result appendix proofs bayes consistency empirical risk minimization duchi khosravi ruan proof lemma definition inf inf inf second two terms note theorem hlcw similarly inf inf inf inf clearly concave remains bound expression end let denote vector losses define function inf sup sup let fenchel biconjugate see proposition corollary well papers proposition defining sup yields desired concave function taking passing note may replace min maxij cij give second result without loss generality may assume vector unique maximal coordinate may otherwise assume deterministic rule breaking ties let argmaxj assumed unique consider sup lcw lcw measurable forward dpi shorthand posterior conditional observing lemma immediately implies sup multiclass classification information surrogate risk either losses cyi notation previous general case hlcw whence may take proof theorem proof theorem almost immediate lemma indeed lemma concave bounded satisfying thus sufficient let minimize show set arbitrarily close minimizing inf consequently expectation bound opt app converges zero desired appendix proofs basic properties section collect proofs characterizations generalized def proof lemma let dominating measures also dominates well dpi dpi dpi dpi dpi dpk dpk dpk duchi khosravi ruan latter two equalities holding surely definition radonnikodym derivative thus obtain dpk dpk dpk dpk dpk dpk dpk dpk dpi definition derivative ppki shows base measure affect integral see positivity may take case jensen inequality implies perspective function convex dpk dpk dpk dpk dpk dpk expectation taken distribution inequality strict strictly convex long dpi meaning exists proof proposition proving proposition first establish general continuity result result direct generalization results thm given let denote restriction measure defined lemma let sequence let lim proof define measure nikodym derivatives vectors via radonf dpkfn martingale adapted filtration standard properties conditional expectation multiclass classification information surrogate risk define measure letting see convex perspective function dpkfn see submartingale gives first result lemma assume limit second statement finite otherwise result trivial using convexity inf final inequality consequence fact closed hence attains infimum particular sequence inf submartingale thus sup inf lim inf integrability guarantee doob second martingale convergence theorem thm yields existence vector lim inf inf inf implies giving lemma give proof proposition proper let base measure let associated densities define increasing sequence partitions sets vectors duchi khosravi ruan let range define lim denotes characteristic function set term previous display denotes generated partition defining limiting operation follows lemma final equality measurability containment proof proposition proving proposition state inequality generalizes classical inequality theorem lemma let convex let nonnegative measurable functions finite measure defined proof recall perspective function defined jointly convex measure defines probability measure jensen inequality implies noting gives result use lemma proposition give proposition proposition implies sup finite partition multiclass classification information surrogate risk consequently loss generality assume finite let dominating measure let dpi letting qpi qpi obtain qpk qpk qpk lemma noting obtain desired result stanford university stanford california jduchi khosravi fengruan
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aug countable tightness free topological groups fucai lin alex ravsky jing zhang abstract given tychonoff space let respectively free topological group free abelian topological group sense markov paper consider two topological properties namely countable tightness provide characterizations countable tightness various special classes spaces furthermore also study countable tightness introduction let respectively free topological group free abelian topological group tychonoff space sense markov every denote subspace consists words reduced length respect free basis subspace defined similarly always use denote therefore statement applies applies one techniques studying topological structure free topological groups clarify relations subspaces well known space discrete therefore space discrete similarly groups locally compact space discrete generally nickolas tkachenko proved one groups almost metrizable space discrete yamada gave characterization metrizable space spaces recently proved stratifiable group countable tightness space separable discrete section refine result giving characterization space countable tightness space implies countable tightness group furthermore since space countable strong pytkeev property countable tightness also discuss topological properties countable strong pytkeev property free topological group ferrando introduced concept frame locally convex spaces concept plays important role study function spaces see know strong pytkeev property general topological groups closely related notion instance topological group strong pytkeev property section shall mathematics subject classification primary secondary key words phrases free topological group free abelian topological group countable tightness countable strong pytkeev property universally uniform first author supported nsfc nos natural science foundation fujian province nos china project abroad fund minnan normal university paper partially written first author visiting school computer mathematical sciences auckland university technology march september wishes thank hospitality host fucai lin alex ravsky jing zhang continuously discuss properties free topological groups motivated following interesting questions question question let submetrizable group question question groups recently gabriyelyan paper leiderman pestov tomita paper given answer questions respectively notations terminology section introduce necessary notations terminology throughout paper topological spaces assumed tychonoff unless otherwise explicitly stated undefined notations terminology refer first let denote sets positive integers rational numbers respectively let topological space closure subspace denoted subspace called bounded every continuous function defined subspace bounded closure every bounded set compact space called space called every compact subset finite space called provided subset closed closed compact subset space particular space called exists family countably many compact subsets subset space closed closed subset space sequentially open sequence converging point eventually space called sequential every sequentially open subset open space countable tightness whenever exists countable set space countable countable family subsets satisfying possible select finite set way sequence convergent point called provided points mutually distinct let family subsets space family called network point open neighborhood exists element family called point whenever sequence converges point arbitrary open neighborhood point exist number element space called countable point call family point whenever sequence converges point arbitrary open neighborhood point element subsequence xni xni furthermore family called whenever compact subset arbitrary open set containing finite subfamily recall regular space countable family called pytkeev network point network every open set set accumulating exists infinite family pytkeev network pytkeev network point space said strong pytkeev property point countable pytkeev network space called regular countable pytkeev network following implications follow directly definitions however none reversed proposition see space strong pytkeev property countable countable tightness free topological groups csf countable strong pytkeev property countable countable tightness sequential space definition topological space stratifiable space open assign sequence open subsets whenever note clearly metrizable space stratifiable consider product natural partial order topological space small base exist subset family open subsets base particular say space countable see proposition given group letter denotes neutral element confusion occurs simply use instead denote neutral element let tychonoff space throughout paper copies let neutral element empty word andn every element call word abelian case word also called form word called reduced contains pair consecutive symbols form follows word reduced different neutral element particular element distinct neutral element uniquely written form xrnn support defined supp given subset put supp supp similar assertions obvious changes commutativity valid every let natural mapping defined bel wel also usel symbol abelian case means natural mapping onto clearly continuous mapping characterization countable tightness free topological groups section mainly discuss countable tightness countable free topological groups first give characterization stratifiable countable tightness equivalently countable tightness show space must belong special class spaces countable following theorem generalizes result theorem let stratifiable following equivalent countable tightness countable tightness space separable discrete fucai lin alex ravsky jing zhang proof since equivalence proved suffices show assume neither separable discrete since stratifiable space diagonal theorem compact subspace metrizable thus sequential since space contains point means set sequentially open exists convergent sequence hence take arbitrary convergent sequence limit point moreover assume space contains uncountable closed discrete subset means extend space countable stratifiable space countable extent cosmic countable network therefore separable obtained contradiction shows exists uncountable discrete closed subset without loss generality may assume let function bijection distinct put put clearly proof proposition order obtain contradiction suffices show countable infinite subset closed let supp set contains point implies theorem since stratifiable space closed follows subgroup generated naturally topologically isomorphic closed furthermore claim compact subset set finite assume contrary compact subset infinite clearly set bounded subset hence subspace supp bounded theorem since space paracompact supp compact however set supp contains infinite many elements since infinite contradiction compactness subspace supp therefore subset closed subset closed hence countable tightness since contradiction obviously following corollary corollary let stratifiable strong pytkeev property separable discrete remark let uncountable discrete space infinite compact metric space theorem hence countable tightness however space separable discrete know whether countable tightness therefore following question question let convergent sequence limit point uncountable discrete space countable tightness furthermore also know answer following question question let space countable tightness countable tightness natural ask whether theorem holds class free abelian topological groups next shall give partial answer question theorem let stratifiable countable tightness set points separable subspace countable tightness free topological groups proof assume contrary set points separable set points space since stratifiable space therefore exists uncountable closed discrete subset point moreover since stratifiable paracompact hence collectionwise normal sequential hence exists family mutually disjoint open subsets moreover take sequence convergent point let since stratifiable closed follows subgroup generated naturally topologically isomorphic free abelian topological group topologically isomorphic implies tightness countable however follows theorem tightness uncountable contradiction however converse theorem hold see theorem moreover proof following result similar proposition thus proof omitted paper theorem stratifiable space separable discrete next shall discuss countable free topological groups contrast theorem shall find situation changes dramatically countable fantightness free topological groups first shall give characterization space countable theorem let space countable space discrete proof consider proof case quite similar since sufficiency obvious shall prove necessity order obtain contradiction assume converse suppose countable space discrete clearly however natural take arbitrary finite subset intersection finite natural therefore closed discrete corollary thus contradiction turns countable imposes strong restrictions space recall subspace space said continuous pseudometric admits continuous extensions theorem let space countable either pseudocompact proof suppose exists infinite compact subset next shall show pseudocompact assume converse exists discrete family open subsetssof space easily verified family also discrete hence closed since set compact intersect finitely many thus without loss generality may assume family discrete since infinite compact set contains point pick put cyn let obviously set closed moreover subgroup generated naturally topologically fucai lin alex ravsky jing zhang isomorphic moreover since follows theorem also hence also next claim closed indeed let follows proof theorem topology determined family compact subsets hence topology determined family therefore shows closed therefore countable moreover easy see natural finite subset topology determined family contradiction corollary let countable either compact discrete remark theorems easy see countable tightness countable convergent sequence limit point countable infinite discrete space however countable remark therefore replace theorem furthermore since countable finally shall discuss strong pytkeev property free topological groups well known space strong pytkeev property countable tightness csf authors showed space strong pytkeev property countable therefore interesting discuss strong pytkeev property free topological groups first shall give theorem proved recall space said closed image metric space theorem let space csf theorem let space strong pytkeev property proof obviously suffices show necessity suppose strong pytkeev property theorem group hence separable follows theorem know whether strong pytkeev property answer positive replace strong pytkeev property strong pytkeev property theorem banakh posed following problem problem let sequential free topological group even know answer following question question let rational number subspace usual topology free topological group countable tightness free topological groups however following result theorem let union countably many proof follows corollary since mapping homeomorphism follows corollary subspacesin hence since union disjoint countably many closing section discuss strong pytkeev property topological spaces well known class regular countably compact spaces property countable tightness equivalent countable corollary moreover compact sequential space countable tightness hence strong pytkeev property hence class regular countably compact spaces property countable tightness equivalent strong pytkeev property moreover well known exists countably compact space natural ask whether class regular countably compact spaces existence implies existence pytkeev network answer also negative see example example exists infinite countably compact space satisfies following conditions space contains infinite compact subset space space strong pytkeev property proof exists infinite countably compact subspace compactification space natural numbers endowed discrete topology every compact subset space finite therefore space however space strong pytkeev property see opposite side recently cai lin proved sequentially compact space metrizable remark proof proposition implies space provided pytkeev network next theorem counterpart result theorem let hausdorff countably compact space pytkeev network space metrizable compact space proof let pytkeev network space first show following claim claim countably compact subset ksof arbitrary open subset exists finite subfamily suppose let inductively choose put since countably compact set cluster point definition pytkeev network follows exists contains infinitely many therefore exists contradicting way chosen claim theorem space metrizable thus compact fucai lin alex ravsky jing zhang free topological groups section shall discuss properties free topological groups motivated questions first recall lemma shall give characterization free topological groups let tgg class topological groups lemma group group group let tgg following equivalent sequential space metrizable contains submetrizable open lemma know sequentiality equivalent class topological groups authors also said would interesting know whether sequentiality equivalent class topological groups countable indeed answer negative see following example example exists topological group countable however sequential proof let compactification infinite discrete space let obviously group theorem follows result countable however well known sequential space since closed free topological group sequential however topological group proof example strong pytkeev property example hence natural pose following question question let topological group strong pytkeev property sequential authors gave affirmative answer question following theorem complements theorem let space either discrete submetrizable proof sufficiency proved suffices show necessity let follows lemma metrizable contains submetrizable open metrizable well known discrete hence may assume lis contains submetrizable open submetrizable open since countable follows finite moreover obvious locally well known locally topological group paracompact problem paracompact since closed thus paracompact space finite since follows theorem countable hence submetrizable therefore easy see submetrizable remark however exists space indeed let convergent sequence limit point uncountable discrete space thus since follows however obvious hence countable tightness free topological groups remark see replace theorem however following theorem add additional assumption space theorem let separable space either countable discrete submetrizable proof adapt proof theorem group instead suffices show submetrizable since separable separable similarly proof theorem see index set theorem countable therefore countable submetrizable next consider topological properties free topological group proof theorem easily obtain following proposition proposition space point countable character therefore following proposition proposition let space point csf proof suffices note compact subset exists see corollary answer following question still unknown question let space theorem let collectionwise normal space containing convergent sequence proof suppose hence exists uncountable closed discrete subset moreover assumption exists convergent sequence limit point without loss generality may assume let since collectionwise normal subspace retract exercises subgroup generated naturally topologically isomorphic however csf result thus however since follows proposition contradiction corollary let stratifiable either discrete separable proof assume discrete since stratifiable paracompact sequential theorem stratifiable space compact cosmic therefore separable recently leiderman pestov tomita showed following two results theorem free abelian topological group uniform space corollary let metrizable space set points subset fucai lin alex ravsky jing zhang metrizable space follows locally compact space set points separable corollary easy see exists space however situation changes much free topological groups let convergent sequence limit point closed discrete space cardinality csf hence particular see however following theorem similar proof proposition obtain following proposition proposition suppose space countable product given uniformizable space finest uniformity compatible topology called fine uniformity universal uniformity tychonoff space said uniform exists uniform structure induces topology particular universal uniformity uniform say universally uniform theorem let universally uniform point proof since universally uniform lit easy see local point proposition see local point local point since open andl closed well known homeomorphic subspace local point since open suffices show suppose universally uniformity one take basis let family base obviously satisfies therefore local however following question still unknown question let space acknowledgements authors wish thank professors salvador boaz tsaban telling information paper moreover authors wish thank professor chuan liu reading parts paper making comments finally hope thank professor shou lin finding gap proof theorem previous version giving key supplement proof references arhangel hurewicz spaces analytic sets spaces functions soviet math arhangel bella countable versus countable tightness comment math univ carolinae arhangel tkachenko topological groups related structures atlantis press world paris arhangel okunev pestov free topological groups metrizable spaces topology banakh topology banakh strong pytkeev property topological spaces http banakh topological uniform spaces http cai lin sequentially compact spaces topology countable tightness free topological groups chis vincenta ferrer salvador boaz tsaban character topological groups via bounded systems kampen duality pcf theory algebra dudley continuity homomorphisms duke math engelking general topology revised completed edition heldermann verlag berlin ferrando pellicer saxon tightness distinguished spaces math anal fletcher lindgren spaces marcel dekker new york generalizations complete metric spaces czech math gabriyelyan leiderman topological groups small base metrizability fund gabriyelyan leiderman strong pytkeev property topological groups topological vector spaces monatsh gabriyelyan topological spaces topological groups certain local countable networks topology gabriyelyan kubzdela lopez pellicer topological properties locally convex spaces weak topology topology gabriyelyan related concepts topology graev free topological groups topology topological algebra translations series vol american mathematical society russian original izvestiya akad nauk sssr ser gruenhage generalized metric spaces kunen vaughan eds handbook topology elsevier science publishers amsterdam gruenhage michael tanaka spaces determined covvers pacific gruenhage tanaka products spaces countable tightness trans amer math guthrie characterization general topology kanatani sasaki nagata new characterizations generalized metric spaces math japonica leiderman pestov tomita topological groups admitting base indentity indexed http lin liu networks free topological groups topology lin liu cao weak countability axioms free topological groups submitted lin tanaka closed maps related results topology markov free topological groups izv akad nauk sssr ser russian amer math soc nickolas tkachenko local compactness free topological groups bull austral math meara paracompactness function spaces topology proc amer math pytkeev maximally decomposable spaces trudy mat inst sipacheva free topological groups spaces subspaces topology continuous images souslin borel sets metrization theorems dokl acad nauk ussr tkachenko spectral decomposition free topological groups usp mat nauk boaz tsaban zdomskyy pytkeev property spaces continuous functions houston yamada characterizations metrizable space every topology yamada tightness free abelian topological groups finite product sequntial fans topology yamada metrizable subspaces free topological groups metrizable spaces topology yamada natural mappings free topological groups metrizable spaces topology fucai lin alex ravsky jing zhang fucai lin school mathematics statistics minnan normal university zhangzhou china address linfucai alex ravsky pidstrygach institute applied problems mechanics mathematics nasu naukova lviv ukraine address oravsky jing zhang school mathematics statistics minnan normal university zhangzhou china address
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approximately optimal motion planning control via probabilistic inference mustafa mukadam cheng xinyan yan byron boots feb environment problem optimal motion planing control fundamental robotics however problem intractable stochastic systems general solution difficult approximate nonlinear performance indices present work provide efficient algorithm pipc probabilistic inference planning control yields approximately optimal policies arbitrary nonlinear performance indices using probabilistic inference gaussian process representation trajectories pipc exploits underlying sparsity problem complexity scales linearly number nonlinear factors demonstrate capabilities algorithm receding horizon setting multiple systems simulation robot view ntroduction fundamental goal robotics efficiently compute trajectories actions drive robot achieve desired behavior seek control policy decision problem maximize performance indices describe example smoothness motion energy consumption likelihood avoiding obstacle hierarchical planning control used solve problem practice idea first generate desired state sequence without considering full system dynamics design robust controller tracking dynamic constraints relaxed becomes possible algorithm plan trajectory satisfies complicated performance indices offering greater flexibility system design samplingbased planning techniques even provide formal guarantees probabilistically complete solutions however recent work started challenge classical viewpoint incorporating dynamic constraints within trajectory planning search solutions improved optimality theoretically elegant approach would address planning control problems within stochastic optimal control framework unfortunately since states actions coupled system dynamics exact solutions become intractable exception simple cases known linearly solvable problems challenges motivated researchers find approximate solutions rather directly approximating original problems hierarchical approaches one simple mustafa mukadam cheng xinyan yan byron boots affiliated institute robotics intelligent machines georgia institute technology atlanta usa cacheng bboots affine systems quadratic instantaneous control cost fully controllable systems fig pipc used holonomic robot blue reach goal red environment dynamic obstacles executed trajectory green current planned horizon black wam arm right arm semitransparent arm goal configuration dotted blue end effector trajectory current planned horizon approach direct policy search uses information find locally optimal policy improve convergence rate differential dynamic programming ddp widely adopted foundation locally optimal algorithms solve local gaussian lqg subproblems iteratively improve suboptimal solutions however systems algorithms would require inefficient sampling construct lqg subproblems even given problem close lqg performance index small set nonlinear factors dynamics small amount nonlinearity compared hierarchical approach algorithms impose strict structural assumption applicable problems measure performance integral instantaneous functions paper propose novel approximately optimal approach motion planning control handle costs expressed arbitrary nonlinear factors exploit problem underlying sparse structure specifically consider problems performance index expressed product quadratic factor instantaneous costs finite number possibly nonlinear factors provide algorithm linear complexity number nonlinear factors moreover show approximately optimal policy computed posterior inference probabilistic graphical model dual performance index convert theoretical results practical algorithm called probabilistic inference planning control pipc recursively updates approximately optimal policy information encountered evaluate approach employ pipc markov decision processes mdps mdps pomdps dynamic environments multiple simulated systems see fig history observations actions time shorthand use boldface denote time trajectory variable denote collection causal stochastic policies formulate motion planning control problem stochastic optimization problem let distribution stochastic policy system dynamics finite set time indices goal find optimal policy maximize performance index max max related work objective function defined expectation product two types factors gaussian process factor nonlinear factor two factors described cover many interesting behaviors often desired planning control problems algorithm contributes growing set research seeks reframe planning control problems probabilistic inference work area formed new class approximately optimal algorithms leverage tools approximate probabilistic inference including expectation propagation expectation maximization common framework based summarizes algorithms well approaches like control contribute field following ways first extend performance index control algorithms incorporate nonlinear factors arbitrary connections time contrast approach methods mentioned generally assume performance indices factor instantaneous terms thus require dense sampling solve problems second provide new approach derive gaussian approximation based laplace approximation gaussian processes third define new class optimal control problems called gleqg generalized solvable transformed dual probabilistic representation particular show gleqg admits solution given posterior inference theoretical result discussed section closes gap duality optimal control inference rest paper structured follows begin section defining objective function joint planning control problems section iii present main results approximately optimal motion planning control section theoretical results summarized online algorithm pipc perform simultaneous planning control partially observable stochastic linear systems dynamic environments validate algorithm present implementation details experimental results section section finally section vii concludes paper roblem otion lanning ontrol begin introducing notation let state action observation continuoustime system time let nonlinear factors define factors form exp model nonlinear couplings frequently used planing problems differentiable nonlinear function defined finite number time indices structure model many performance indices planning example simple nonlinear cost function single time instance penalty based difference initial terminal penalty enforce consistency across landmarks time cost collision factor depends finite number states actions refer corresponding states actions support states support actions gaussian process factors gaussian process factor generalization cost function optimal control literature illustrate consider special case joint factor defined similarly let gpu gaussian process mut let positive definite rgreen function satisfying dirac delta distribution integral length trajectory define gaussian process factor dsds exp mus loosely speaking call probability trau jectory gpu note notation assume measurements taken discrete time time sampling interval constant trajectory time necessarily imply sample path gpu rather use metric intuitively maximization encourages close terms distance weighted solving stochastic optimization problem objective function intractable general implicitly defined however show gpu sum gaussian white noise process linearly transformed process problem tractable also extend classical cost model behaviors realized defining gpu linear stochastic differential equation sde let hidden state derivatives prior set gpu solution dyt dyt hyt system matrices control bias wiener process gaussian white noise process words gaussian process gpu mean covariance functions mut hmyt hkt gpy myt another gaussian process min gts state transition matrix respect derivations please refer therein definitions contain cost exp special case obtained setting general assigns action close even terms derivatives hidden states leads preference toward smooth control signals extension joint factor would also encourage smooth state trajectories smaller derivatives state constructing gaussian process factor sde results one particularly nice property consider joint gaussian process green function sparse see let define gaussian process factor similar double integral broken sum smaller double integrals factorized similar form smaller time interval words treat coordinate exponent written quadratic function tridiagonal hessian matrix please see details sparse property foundation approximation procedure algorithm proposed section iii iii pproximate ptimization nference mixed features planning control domains present two major challenges optimization trajectories nonlinear factors former results infinitedimensional problem often requires dense discretization latter precludes direct use algorithms based bellman equation factors may factorize instantaneous terms work propose new approach inspired approximate probabilistic inference goal derive approximation problem form max local approximation gaussian process approximation call problem gleqg generalizes leqg incorporate exponentials form rest section show gleqg derived using probabilistic interpretation factors show problem solved linear time solution written closedform posterior inference probabilistic interpretation factors begin representing factor probability distribution first introduce additional fictitious observations new variables interpreted events wish robot achieve whose likelihood success reflected proportionally practically help keep track message propagation support later derivations second rewrite gaussian process factor include hidden state joint gaussian process factor introduction joint gaussian process sparse property desired rewrite stochastic optimization new notation let step carried similarly construction let conditional distributions defined system dynamics observation model respectively shown equivalent max define joint distribution likelihoods prior trajectory proceeding clarify notation use simplify writing use denote hoc constructed gaussian process factor use denote probability distribution associated real system always define expectation integral notation denotes expectation probability distributions abuse notation call gaussian processes since results depend rather algebraic form gaussian approximation let derive gleqg approximation notice factorized used markovian property section iiia given conditionally independent random variables therefore reasonably approximated gaussians approximate however notably different topologies distribution trajectories whereas density function finite number random variables therefore approximate need find gaussian process gaussian density gaussian process approximation derive gaussian process approximation result desired conditional gaussian process given first need define system dynamics observation model let assume system governed linear sde also gaussian process gaussian process prior case approximation made therefore case nonlinear systems one approach treat local linear approximation derive solution linearized system alternatively learn conditional distribution data directly gaussian process regression however since main purpose show solution available assume system linear given gaussian density approximation unlike approximation straightforward first may available closed form approximate marginal distribution found previous section given find gaussian approximation via laplace approximation nonlinear factor laplace approximation amounts solving nonlinear optimization problem using sparsity structured gaussian processes defined sdes optimization completed using efficient data structures space constraints omit details please see appendix details summary approximations allow approximate gaussian distribution gaussian process approximation system exact system linear proportional factor moreover shown laplace approximation terms trajectory finding approximately optimal policy system matrices wiener process gaussian noise covariance prior placed similar section shown solution gaussian process since substituting results section approximated optimization problem max one show equivalent problem expressed probabilistic notation however writing problem probabilistically avoid algebraic complications arising attempting solve bellman equation factors requires additional state expansion simplicity reflected optimality condition argmax argmax tmax fig pipc system started currently time support points resolution receding horizon setting represents receding horizon window tmax infinite final time algorithm terminates finite horizon setting tmax denotes dirac delta distribution last equality see solution maximization problem coincides mode posterior distribution result optimal policies time derived forward time performing inference without solving future policies first please see appendix proof call property duality gleqg inference result seems surprising similar ideas traced back duality optimal control estimation optimal value function linear quadratic problem computed backward message propagation without performing maximization compared previous work stronger duality holds gleqg dual inference problem probabilistic graphical model defined random variables section nice property result use exponential performance index enables handle factors naturally without referring hoc derivations based dynamic programming extended states posterior representation policy also found interpreted one step posterior iteration results approximation optimal value function relationship overall stochastic optimization unclear posterior representation reasoned notion predictive policy representation without justification effects whole decision process derive policy based assumption associated distribution approximated gaussian therefore condition approximate policy remains valid easily understood even enforced discussed later section robabilistic otion lanning ontrol section iii show approximated well gaussian distribution stochastic optimization approximately solved posterior inference representation suggests approximately optimal policy updated recursively kalman filtering recurrent policy inference kalman filtering approximately optimal policy viewed belief current action given history fictitious events exploit markovian structure underlying derive recursive algorithm updating belief given belief policy derived marginalization first write conditional distribution defined initialization posterior propagated prediction correction summarized together one step transition given given markovian structure integral depends two adjacent support computed constant time note action actually conditioned action taken notation adopted simplify writing thus view kalman filtering transition dynamics observation process formulation gives flexibility switch policies new observation available provides continuoustime action trajectory interval recurrent policy inference based assumption accurate although assumption necessarily true general practical approximation belief current state concentrated horizon within applies short online motion planning control summarize everything pipc algorithm let compensate local nature recomputing new laplace approximation whenever applying filtering update policy subscript denotes future trajectory see fig leads algorithm receding horizon pipc input horizon start time initial belief output stop criteria hti uti getlaplaceapprox uti environment makeobservation filterpolicy hti uti executepolicy filterstate end end return checksuccess iterative framework solves new approximation knowledge system apply scheme problems finite receding horizon facing dynamic environment pipc updates environmental information new laplace approximation section details receding horizon approach summarized algorithm derived similarly finite horizon case first time step pipc computes laplace approximation current horizon window latest information system environment length preview horizon second pipc recursively updates policy using current observation resolution two steps repeat set criteria met execution fails example robot collision mplementation etails perform experiments receding horizon version pipc four different setups including mdp pomdp scenarios execute receding horizon pipc algorithm ignore policy filtering step instead recursively apply policy given mode found laplace approximation baseline viewed direct generalization include action trajectories laplace approximation implemented using library solves posterior maximization nonlinear optimization defined factor graph algorithm note implementation consider constant time difference two support states actions evaluate algorithms three different systems holonomic robot wam arm state dynamics following defined double version solves new problem iteration laplace approximation available https available https integrator state consisting position velocity fft following define gaussian process factor ggt zero identity matrices holonomic robot wam arm arm positive scalars observation process pomdp modeled state observation additive gaussian noise covariance state dynamics arms assumed feedback linearized real system control would mapped back real torques using inverse dynamics valuation conduct benchmark receding horizon algorithm holonomic robot dynamic environment wam arm right arm static environment see fig case compare algorithms mdp pomdp settings across different number dynamic obstacles nobs case respect success rate time reach goal path length path setting run times unique random generator seed account stochasticity kept across four algorithms fair comparison trial marked successful robot reaches goal within euclidean distance gdist marked failed point robot runs collision runs maximum allotted time tmax robot benchmark simulate holonomic robot radius environment moving obstacles see fig robot sensor returns limited view square centered robot current position video experiments available https cost calculated negative log product factors path table success rate across increasing nobs holonomic robot mdp pomdp path length number obstacles path cost time number obstacles number obstacles fig results successful runs increasing nobs holonomic robot moving obstacles squares start random locations follow stochastic jump process noisy acceleration aobs uniformly sampled within every time step velocities vobs restricted within obstacles confined inside boundary simulation table summarizes success rates fig shows aggregate results successful runs table see mdp pomdp cases algorithms higher success rates algorithms especially difficult problems larger stochasticity system increased complexity environment nobs similar increasing trends also observed difference success rates algorithms majority failed cases arise collision due hitting maximum run time performance pomdp cases slightly worse mdp cases average three metrics time path length path cost fig increase general noise obstacles important plots interpreted alongside success rates since sample size successful trails comparatively sparse harder problems wam benchmark demonstrate scalability pipc higher dimensional systems performing benchmark wam parameters benchmark set follows gdist tmax nip table success rate across increasing wam robot arms mdp pomdp wam robot arms wam robot arms set lab industrial environments respectively openrave task drive robot arm given start goal configuration see fig environments static fully observable times compare algorithms respect increasing table summarizes success rates fig shows aggregate results successful runs similar robot benchmark results show algorithms higher success rate ones three metrics increase noise particular performs even better vii onclusion consider problem motion planning control probabilistic inference propose algorithm pipc parameters benchmark set follows gdist tmax nip wam time path length path cost path cost time path length fig results successful runs increasing wam robot arms solving problem exploit intrinsic sparsity stochastic systems particular pipc address performance indices given arbitrary higherorder nonlinear factors general factor despite pipc solving problem complexity scales linearly number nonlinear factors thus making online simultaneous planning control possible horizon problems acknowledgments authors would like thank jing dong help gtsam interface material based upon work supported nsf crii award nsf nri award eferences bertsekas dynamic programming optimal control athena scientific belmont vol arkin robotics mit press kavraki svestka latombe overmars probabilistic roadmaps path planning configuration spaces robotics automation ieee transactions vol kuffner lavalle efficient approach path planning robotics automation proceedings icra ieee international conference vol ieee byravan boots srinivasa fox functional gradient optimization motion planning robotics automation icra ieee international conference ieee schulman duan lee awwal bradlow pan patil goldberg abbeel motion planning sequential convex optimization convex collision checking international journal robotics research vol mukadam yan boots gaussian process motion planning ieee international conference robotics automation icra may marinho boots dragan byravan gordon srinivasa functional gradient motion planning reproducing kernel hilbert spaces proceedings robotics science systems dong mukadam dellaert boots motion planning probabilistic inference using gaussian processes factor graphs proceedings robotics science systems toussaint newton methods markov constrained motion problems arxiv preprint lavalle kuffner randomized kinodynamic planning international journal robotics research vol tedrake manchester tobenkin roberts lqrtrees feedback motion planning via verification international journal robotics research kappen linear theory control nonlinear stochastic systems physical review letters vol levine koltun guided policy icml deisenroth rasmussen pilco approach policy search proceedings international conference machine learning mayne gradient method determining optimal trajectories systems international journal control vol todorov generalized iterative lqg method feedback control constrained nonlinear stochastic systems american control conference proceedings ieee todorov tassa iterative local dynamic programming ieee symposium adaptive dynamic programming reinforcement learning ieee attias planning probabilistic inference aistats toussaint robot trajectory optimization using approximate inference proceedings annual international conference machine learning acm toussaint storkey probabilistic inference solving discrete continuous state markov decision processes proceedings international conference machine learning acm levine koltun variational policy search via trajectory optimization advances neural information processing systems kappen opper optimal control graphical model inference problem machine learning vol online available http rawlik toussaint vijayakumar stochastic optimal control reinforcement learning approximate inference proceedings robotics science systems viii kumar van schuppen optimal control stochastic systems performance index journal mathematical analysis applications vol rasmussen gaussian processes machine learning sarkka solin hartikainen spatiotemporal learning via bayesian filtering smoothing look gaussian process regression kalman filtering ieee signal processing magazine vol bishop pattern recognition machine learning vol boularias predictive model imitation learning partially observable environments machine learning applications icmla seventh international conference ieee barfoot tong sarkka batch trajectory estimation exactly sparse gaussian process regression proceedings robotics science systems berkeley usa yan indelman boots incremental sparse regression trajectory estimation mapping proceedings international symposium robotics research ppendix laplace approximation factor graphs pipc updates laplace approximation whenever efficiently solving nonlinear problem defined bipartite factor graph obstacle factor fiobs exp eti obs interpoaltion factor fiintp exp eti obs gpinterpolate prior factor figp exp eti recall set support augmented states denotes set factors edges connected example factor graph shown fig trajectory starting length equal sparse set support augmented states uniformly apart connected neighbours via gaussian process factors forming chain note implementation details factor implementation prior factor laplace approximation prior factor placed first hidden state reflecting current belief given past history mdp setting covariance state set diagonal matrix small number indicate high confidence control use original gaussian process factor given together define qprior pomdp setting belief hidden augmented state obtained via kalman filtering heuristically set covariance state mentioned previously gaussian process factors analogous defining gpu define turns define fig corresponds gaussian process factors qgp state transition matrix associated takes system obstacle interpolation factors obstacle avoidance use hinge loss function safety distance compute signed distance field effect defines obstacle factors interpolation factors fig use qobs though abstracted single factor fig two support points multiple nip interpolated factors constructed indexes evenly spaced time apart ensure path safety see details goal factor drive system desired goal configuration example particular position configuration space zero velocity action add goal factor every support point except current state encourages optimizer drive states current horizon window closer goal covariance factor acts weighting term define qgoal monotonically decreases euclidean distance goal prior factor exp prior goal factor figoal exp eti goal fig factor graph example laplace approximation problem showing states white circle used shorthand different kinds factors prior black circle obstacle interpolation white square measurement gray square goal black square update laplace approximation laplace approximation used recursively update policy resolution graph updated construct new nonlinear leastsquare optimization problem done shifting horizon window ahead update factors include environmental changes updated graph prior factor first state given additional kalman filter based hidden states observations pomdp problems treat estimation current state perfect knowledge without uncertainty extra heuristic step compromise makes assumption accurate mean current belief proof prove solution approximate optimization problem max written posterior inference argmax argmax found approximate factor given laplace approximation proof assume length trajectory following first show optimal policy deterministic show corresponds mode posterior distribution optimal policy deterministic write objective function fht dut dht fht therefore equivalently formulated explicitly variational problem density function max fht dut dut deal inequality let write max fht dut dut let lagrangian multiplier turn unconstrained optimization use calculus variations derive solution min max min max fht dut dut suppose optimum let arbitrary continuous function optimality condition given fht dut since arbitrary implies fht given scalar conclude satisfying arg max fht optimal policy mode posterior previous proof know policy corresponds mode fht therefore complete proof need show fht policies optimal first let denote fht policies deterministic define next introduce lemma lemma let max constant independent mean covariance gaussian show induction backward order start last policy write purposefully omit dependency exact value observed performing optimization propagate objective function one step backward given maximization given max max max max max max max max max second equality due policy deterministic third proportionality given last equality given definition therefore backward iteration maintains policy optimization problem dut algebraic form last step since completes proof
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scoring classifying gated jun daniel jiwoong graham taylor school engineering university guelph guelph canada imj gwtaylor abstract perhaps methods representation learning conceptually simple easy train recent theoretical work shed light ability capture manifold structure drawn connections density modeling motivated researchers seek ways scoring furthered use classification gated gaes interesting flexible extension learn transformations among different images pixel covariances within images however much less studied theoretically empirically work apply dynamical systems view gaes deriving scoring function drawing connections restricted boltzmann machines set deep learning benchmarks also demonstrate effectiveness single classification introduction representation learning algorithms machine learning algorithms involve learning features explanatory factors deep learning techniques employ several layers representation learning achieved much recent success machine learning benchmarks competitions however successes achieved purely supervised learning methods relied large amounts labeled data though progress slower likely unsupervised learning important future advances deep learning successful example unsupervised learning conceptually simple easy train via backpropagation various regularized variants model recently proposed well theoretical insights operation practice latent representation learned typically used solve secondary problem often classification common setup train single data classes classifier tasked discriminate among classes however contrasts way probabilistic models typically used past literature common train one model per class use bayes rule classification two challenges classifying using first recently known obtain score data meaning much model likes input second autoencoders even scored scores integrate therefore models need calibrated kamyshanska memisevic recently shown scores computed interpreting dynamical system although scores integrate show one combine unnormalized scores generative classifier learning normalizing constants labeled data paper turn interest towards variant capable learning features data main idea learn relations pixel intensities rather pixel intensities structuring model graph connects hidden units pairs images images different hidden units learn images transform images hidden units encode withinimage pixel covariances learning features yield improved results recognition generative tasks adopt dynamical systems view gated demonstrating scored similarly classical adopt framework conceptually formally developing theory yields insights operation gated addition theory show experiments classification model based gated scoring outperform number representation learning architectures including classical scoring also demonstrate scoring useful structured output task classification gated section review gated gae due space constraints review classical instead direct reader reviews share notation similar classical gae consists encoder decoder standard processes datapoint gae processes inputoutput pairs gae usually trained reconstruct given though also trained symmetrically reconstruct intuitively gae learns relations inputs rather representations inputs example represent sequential frames video intuitively mapping units learn transformations case input copied mapping units learn pixel covariances simplest form gae hidden mapping units given basis expansion however leads parameterization relational features mixed standard features simply adding connections gated least quadratic number inputs thus prohibitively large therefore practice projected onto matrices latent factors respectively number factors must thus model completely parameterized matrices assuming matrix encoder function defined multiplication activation function decoder function defined note parameters usually shared encoder decoder choice whether apply nonlinearity output specific form objective function depend nature inputs example binary categorical assumed inputs simplicity presentation therefore eqs functions use objective also constrain gae symmetric model training reconstruct given given symmetric objective thought analogue modeling joint distribution opposed conditional gated scoring authors showed data could scored interpreting model dynamical system contrast probabilistic views based score matching regularization dynamical systems approach permits scoring models either linear data sigmoid binary data outputs well arbitrary hidden unit activation functions method also agnostic learning procedure used train model meaning suitable various types regularized proposed recently section demonstrate dynamical systems view extended gae vector field representation similar view gae dynamical system vector field defined vector field represents local transformation undergoes result applying reconstruction function repeatedly applying reconstruction function input yields trajectory whose dynamics physics perspective viewed force field point potential force acting point gradient potential energy negative goodness point light gae reconstruction may viewed pushing pairs inputs direction lower energy goal derive energy function call scoring function measures much gae likes given pair inputs normalizing constant order find expression potential energy vector field must able written derivative scalar field check submit integrability criterion open simple connected set continuously differentiable function defines gradient field vector field defined gae indeed satisfies integrability criterion therefore written derivative scalar field derivation given appendix also applies gae symmetric objective function setting input following exact procedure scoring gae mentioned section goal find energy surface express energy specific pair previous section showed criterion satisfied implies write vector field derivative scalar field moreover illustrates vector field conservative field means vector field gradient scalar function case energy function gae hence integrating trajectory gae measure energy along path moreover line integral conservative vector field path independent allows take scalar function ydy ydy auxiliary variable kronecker product moreover decoder terms auxiliary variable get ydy const detailed derivation provided appendix identical activation function know simple compute relationship restricted boltzmann machines section relate gaes scoring function types restricted boltzmann machines factored gated conditional rbm rbm gated factored gated conditional restricted boltzmann machines kamyshanska memisevic showed several hidden activation functions defined gradient fields including sigmoid softmax tanh linear rectified linear function relu modulus squaring activation functions applicable gaes well case sigmoid activation function energy function becomes exp const log exp const note consider conditional gae reconstruct given yields log exp const expression identical constant free energy factored gated conditional restricted boltzmann machine fcrbm gaussian visible units bernoulli hidden units ignored biases simplicity derivation including biases provided appendix restricted boltzmann machines covariance cae introduced specific form symmetrically trained identical inputs tied input weights maintains set relational mapping units model covariance pixels one introduce separate set mapping units connected pairwise one inputs model mean intensity case model becomes mcae theorem consider cae encoder decoder parameters model sigmoid moreover consider covariance rbm visibles hiddens energy function defined energy function cae dynamics equivalent free energy covariance rbm constant log exp const proof given appendix extend analysis mcae using theorem results corollary energy function mcae free energy meancovariance rbm mcrbm visibles bernoullidistributed hiddens equivalent constant energy mcae log exp log exp parameterizes mean mapping units parameterizes covariance mapping units proof proof simple let emc energy mean energy covariance emc energy mcae know theorem equivalent free energy covariance rbm results show equivalent free energy mean classical rbm shown free energy mcrbm equal summing free energies mean rbm covariance rbm classification gated kamyshanska memisevic demonstrated one application ability assign energy scores constructing classifier section explore two different paradigms classification similar work consider usual problem first training using energy functions confidence scores also consider challenging structured output problem specifically case prediction data point may one associated label may correlations among labels classification using gated one approach classification take several models assemble classifier example approach fit several directed graphical models use bayes rule combine process simple models normalized calibrated possible apply similar technique undirected models one must take care calibrate approach proposed train assigns energy data define conditional distribution classes exp exp learned bias class bias terms take role calibrating unnormalized energies note similarly combine energies symmetric gated covariance apply class train covariance classical mean combine sets unnormalized energies follows exp eim eic pmcae exp eim energy comes mean standard autoencoder trained class eic energy comes covariance gated trained class call classifiers covariance scoring caes scoring mcaes respectively training procedure summarized follows train mean individually class mean covariance tied weights encoder decoder covariance gated tied inputs learn calibration terms using maximum likelihood backpropagate gae parameters experimental results followed experimental setup used standard set deep learning benchmarks used stochastic gradient descent optimize parameters training hyperparameters number hiddens number factors corruption level learning rate momentum rate batch sizes chosen based validation set corruption levels selected number hidden factors selected selected learning rate range classification error results shown table first error rates scoring variant methods illustrate across datasets aes outperforms caes mcaes outperforms aes caes models pixel means cae models pixel covariance mcae models mean covariance making naturally expressive observe caes mcaes achieve lower error rates large margin rotated mnist backgrounds final row hand caes mcaes perform poorly mnist random white noise background second row bottom believe phenomenon due inability model covariance dataset mnist random white noise pixels typically uncorrelated rotated mnist backgrounds correlations present consistent classification via optimization label space dominant application deep learning approaches vision assignment images discrete classes object recognition many applications however involve structured outputs output variable highdimensional complex joint distribution structured output prediction may include tasks classification regularities learned output segmentation output input key challenge approaches lies developing models able capture complex high level structure like shape still remaining tractable data svm rbm deep gsm aes caes mcaes rect rectimg convex mnistsmall mnistrot mnistrand mnistrotim rbf table classification error rates deep learning benchmark dataset stands stacked svm rbm results deep gsm results aes though proposed work based deterministic model shown energy scoring function gae equivalent constant conditional rbm model already seen use structured prediction problems gae scoring applied structured output problems type idea let classifier make initial prediction outputs fast manner allow second model case gae clean outputs first model since gaes model relationship input structured output initialize output output model optimize energy function respect outputs input held constant throughout optimization recently proposed compositional high order pattern potentials hybrid conditional random fields crf restricted boltzmann machines rbm provides global shape information prior crf adopting idea learning structured relationships outputs propose alternate approach inputs gae words model covariance autoencoder intuition behind first approach use gae learn relationship input output whereas second method aims learn correlations outputs denote two proposed methods gaexy gaey gaexy corresponds gae trained conditionally whose mapping units directly model relationship input output gaey corresponds gae models correlations output dimensions gaexy defines gaey defines differ terms data vectors consume training test procedures detailed algorithm experimental results consider classification problem classify instances take one label time algorithm structured output prediction gae scoring procedure classification xtrain ytrain train perceptron mlp learn mapping argmin appropriate loss function train gated inputs case gaey set argmin appropriate reconstructive loss test data point xtest initialize output using mlp xtest max iter compute oyt update tolerance rate respect convergence optimization followed experimental set four datasets considered yeast consists biological attributes scene mturk majmin targeted towards tagging music yeast consists biological attributes possible labels scene consists image pixels possible labels mturk majmin consist audio features extracted music possible tags respectively figure visualizes covariance matrix label dimensions dataset see correlations present labels suggests structured approach may improve predictor yeast scene mturk majmin fig covariance matrices datasets yeast scene mturk majmin experiments used loss function compared proposed approaches logistic regression standard mlp two structured crbm training algorithms presented permit fair comparison followed procedure training reporting errors paper cross validated folds training validation test examples randomly separated fold error rate measured averaging errors label dimension method logreg mlp gaesxy gaesy yeast scene mturk majmin table error rate datasets previous work report mean across repeated runs different random weight initializations performance four datasets shown table observed adding small amount gaussian noise input improved performance gaexy however adding noise input much effect suspect adding noise makes gae robust input provided mlp interestingly found performance gaey negatively affected adding noise proposed methods gaesxy gaesy generally outperformed methods except gaesy majmin dataset least datasets clear winner two gaesxy achieved lower error gaesy yeast majmin error rate mturk dataset however gaesy outperforms gaesxy scene dataset overall results show gae scoring may promising means structured output prediction conclusion many theoretical empirical studies however theoretical study gated limited apart gae several intriguing properties classical based ability model relations among pixel intensities rather intensities opens broader set applications paper derive theoretical results gae enable gain insight understanding operation cast gae dynamical system driven vector field order analyze model first part paper following procedure showed gae could scored according energy function perspective demonstrated equivalency gae energy free energy fcrbm gaussian visible units bernoulli hidden units sigmoid hidden activations manner also showed covariance formulated way energy function free energy covariance rbm naturally led connection rbm one interesting observation rbms reported difficult train success training rbms highly dependent training setup attractive alternative even energy function required structured output prediction natural next step representation learning main advantage approach compared popular approaches markov random fields inference extremely fast using optimization scoring function future plan tackling challenging structured output prediction problems references bengio deep generative stochastic networks trainable backprop arxiv preprint boutell luob shen brown learning scene classification pattern recognition cho ilin raiko improved learning restricted boltzmann machines icann droniou sigaud gated autoencoders tied input weights icml elisseeff weston kernel method classification nips guillaume bengio regularized learn data generating distribution iclr kamyshanska memisevic autoencoder scoring icml kamyshanska memisevic potential energy ieee transactions pattern analysis machine intelligence krizhevsky learning multiple layers features tiny images tech department computer science university toronto krizhevsky sutskever hinton imagenet classification deep convolutional neural networks nips larochelle erhan courville bergstra bengio empirical evaluation deep architectures problems many factors variation icml tarlow zemel exploring compositional high order pattern potentials structured output learning cvpr mandel eck bengio learning tags vary within song ismir mandel ellis game collecting music metadata journal new music research memisevic learning image features iccv memisevic zach hinton pollefeys gated softmax classification nips mnih hinton learning detect roads aerial images proceedings european conference computer vision eccv mnih larochelle hinton conditional restricted boltzmann machines structured output prediction uai ranzato hinton modeling pixel means covariances using factorized boltzmann machines cvpr rifai contractive explicit invariance feature extraction icml swersky ranzato buchman freitas marlin autoencoders score matching energy based models icml szegedy liu jia sermanet reed anguelov erhan vanhoucke rabinovich going deeper convolutions arxiv preprint taylor hinton factored conditional restricted boltzmann machines modeling motion style icml vincent connection score matching denoising neural computation vincent larochelle bengio manzagol extracting composing robust features denoising autoencoders icml wang melchior wiskott restricted boltzmann machines modeling natural image statistics tech institut fur neuroinformatik bochum bochum germany gated scoring vector field representation check vector field written derivative scalar field submit integrability criterion open simple connected set continuously differentiable function defines gradient field considering gae note ith component decoder rewritten derivatives respect substituting equation similarly derivatives respect substituting equation yields deriving energy function integrating gae trajectory ydy ydy auxiliary variable kronecker product consider symmetric objective function defined equation also consider vector field system symmetric cases valid mentioned section let well let diag diag block diagonal matrices consequently vector field becomes energy function becomes auxiliary variable moreover note decoder first term equation terms auxiliary variable energy reduces const relation types restricted boltzmann machines gated factored gated conditional restricted boltzmann machines suppose hidden activation function sigmoid moreover define gated consists encoder decoder parameters model note weights tied case energy function gated exp const const log exp consider free energy factored gated conditional restricted boltzmann machine fcrbm energy function fcrbm gaussian visible units bernoulli hidden units defined given conditional probability density assigned fcrbm data point exp exp log exp exp partition function free energy function expanding free energy function get log exp log exp log exp wfhk log exp wfhk log exp note center data subtracting mean dividing standard deviation therefore assume substituting log exp wfhk log exp wfhk log exp wfhk const letting get log exp wkf const hence conditional gated fcrbm equal constant gated restricted boltzmann machines theorem consider covariance encoder decoder parameters model moreover consider covariance restricted boltzmann machine gaussian distribution visibles bernoulli distribution hiddens energy function defined parameters energy function covariance dynamics equivalent free energy covariance restricted boltzmann machine energy function covariance log exp const proof note covariance regular gated setting making factor loading matrices applying general energy equation gae equation covariance get const log exp const consider free energy restricted boltzmann machine mcrbm gaussian distribution visible units bernoulli distribution hidden units log exp log exp log exp log exp center data subtracting mean dividing standard deviation therefore assume substituting log exp letting get log exp const therefore two equations equivalent
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new class tests multinormality garch data based nov empirical moment generating function norbert dolores institute stochastics karlsruhe institute technology karlsruhe germany department statistics operations research university seville seville spain abstract generalize recent class tests univariate normality based empirical moment generating function multivariate setting thus obtaining class affine invariant consistent tests multinormality test statistics suitably weighted provide asymptotic behavior observations well context testing innovation distribution multivariate garch model gaussian study behavior new tests compare criteria alternative existing procedures apply new procedure data set monthly log returns keywords moment generating function test multivariate normality gaussian garch model ams classification numbers introduction evidenced papers arcones batsidis cardoso oliveira ferreira ebner enomoto farrel hanusz henze joenssen vogel kim koizumi mecklin mundfrom pudelko rizzo tenreiro thulin estrada voinov yanada zhou shao ongoing interest problem testing multivariate normality without claiming exhaustive list probably covers publications field since review paper henze recently henze koch provided lacking theory test univariate normality suggested zghoul purpose paper twofold first generalize results henze koch multivariate case thus obtaining class affine invariant consistent tests multivariate normality second contrast paper publications considered independent identically distributed observations also provide asymptotics test statistics context dependence specific let time sequence random column vectors defined common probability space assume distribution absolutely continuous respect lebesgue measure let denote normal distribution mean vector covariance matrix write class normal distributions test multivariate normality test null hypothesis usually test consistent fixed alternative distribution since class closed respect full rank affine transformations genuine test statistic based also affine invariant axn nonsingular see henze critical account affine invariant tests multivariate normality follows let denote sample mean sample covariance matrix respectively means transposition vectors matrices furthermore let scaled residuals provide empirical ization denotes unique symmetric square root notice invertible probability one provided see eaton perlman latter condition tacitly assumed hold follows letting exp denote empirical moment generating function close exp moment generating function standard normal distribution sequel stands euclidean norm unit matrix order statistic proposed paper weighted exp fixed parameter role discussed later notice moment generating function analogue testing multivariate normality see baringhaus henze henze zirkler henze wagner latter statistics originate one replaces empirical characteristic function scaled residuals characteristic function exp standard normal distribution general account weighted see baringhaus principle one could replace general weight function satisfying general conditions special choice however leads test criterion certain extremely appealing features since straightforward calculations yield representation kyn exp kyn exp amenable computational purposes notice condition necessary integral finite later impose restriction prove limit null distribution remark affine invariant since depends mahanalobis rejection large values angles distances rest paper unfolds follows next section shows letting tend infinity yields linear combination two measures multivariate skewness section derive limit null distribution setting section addresses question consistency new tests general alternatives section considers new criterion context multivariate garch models order test normality innovations provides pertaining theory section presents monte carlo study compares new tests competing ones considers real data set financial market article concludes discussions section case section show statistic suitable scaling approaches linear combination two measures multivariate skewness theorem lim kyn kyn multivariate sample skewness sense mardia rohatgi respectively proof let kyn denote multivariate sample kurtosis sense mardia exp result follows tedious straightforward calculations using relations kyn kyn kyn kyn kyn kyn derivation second last expression see proof theorem henze stress although kyn show equations terms cancel derivation final result remark interestingly exhibits limit behavior statistic studied henze based weighted involving empirical characteristic function empirical moment generating function testing multivariate normality based empirical characteristic function see theorem henze first sight theorem seems differ theorem henze koch covers special case careful analysis shows notation paper asymptotic null distribution case section consider case random vectors normal distribution key observation deriving limit distribution fact given notice random element hilbert space equivalence classes measurable functions square integrable respect finite measure borel sets given weight function defined resulting norm denoted notation takes form kwn writing convergence distribution random vectors stochastic processes main result section follows theorem convergence suppose normal distribution centred gaussian random element covariance kernel exp view continuous mapping theorem yields following result corollary null hypothesis remark distribution say positive eigenvalues integral operator associated kernel given theorem exp standard normal random variables succeed obtaining explicit solutions equation however since dsdt see shorack wellner tedious straighforward manipulations integrals yield following result generalizes theorem henze koch theorem proof theorem view affine invariance assume distribution henze authors considered exponentially empirical moment generating function process exp notice notation given kan kmn display propositions henze exp display representation sum yield notice centred random elements since central limit theorem hilbert spaces see bosq shows centered gaussian random element using fact normal distribution relations straightforward algebra shows covariance kernel figuring statement theorem equals consistency next result shows test multivariate normality based consistent general alternatives theorem suppose absolutely continuous distribution exp furthermore let symmetric square root inverse covariance matrix letting mxe exp lim inf mxe almost surely proof affine invariance may assume fix put exp proof theorem henze lim max surely strong law large numbers banach space continuous functions ktk fatou lemma yield lim inf lim inf eet surely since arbitrary assertion follows suppose alternative distribution assumed standardized satisfying conditions theorem since exp least one theorem shows surely since given nominal level sequence critical values based rejects large values converges according theorem test consistent alternative consistent distribution satisfying conditions theorem view reasoning given behavior alternatives difficult problem testing normality garch models section consider multivariate garch mgarch model vector unknown parameters unobservable random errors innovations copies random vector assumed mean zero unit covariance matrix hence conditional variance given explicit expression depends assumed mgarch model see francq detailed description several relevant models interest testing normality innovations stems fact distributional assumption made applications erroneously accepted inferential procedures lead wrong conclusions see spierdijk effect assessment standard risk measures value risk therefore important step analysis garch models check whether data support distributional hypotheses made innovations reason number tests proposed innovation distribution papers klar ghoudi contain extensive review tests well numerical comparisons special case testing univariate normality proposals testing multivariate case rather scarce class garch models proved particularly valuable modeling financial data discussed among others rydberg one stylized features financial data extensive simulation study summary reported section learnt data test normality based exhibits high power distributions reasons section devoted adapt procedure testing normality innovations based data driven equation therefore basis observations wish test null hypothesis law general alternatives notice equivalent hypothesis conditionally law two main differences respect case innovations assumed centered zero unit covariance matrix conditional covariance matrix way depends unknown parameter past observations notice although distribution innovations unobservable context model hence inference distribution innovations based residuals recall observe therefore estimate apart suitable estimator also need specify values write say certain conditions arbitrarily fixed initial values asymptotically irrelevant taking account innovations mean zero unit covvariance matrix work directly residuals without standardizing let mng defined replacing define changed wng wng defined replaced mng order derive asymptotic null distribution wng make assumptions sequel denote generic constants values may vary across text stands true value matrix akj kak denotes use notation euclidean norm vectors estimator satisfies vector measurable functions measurable functions satisfying khj kxj sequence vectors function admits continuous derivatives neighborhood exist sup sup sup next result gives asymptotic null distribution wng theorem convergence wng let strictly stationary process satisfying measurable respect generated assume hold null hypothesis centered gaussian random element covariance kernel cov wng exp theorem continuous mapping theorem following corollary corollary assumptions theorem kwg standard estimation method parameter garch models quasi maximum likelihood estimator qmle defined arg max log comte leiberman bardet wintenberger among others shown certain mild regularity conditions qmle satisfies general mgarch models observed many mgarch parametrizations matrix nevertheless exist partial theoretical results models constant conditional correlation model proposed bollerslev extended jeantheau exception since properties thoroughly studied model decomposes conditional covariance matrix figuring conditional standard deviations conditional correlation matrix according tion matrix diagonal matrix diag denotes hadamard product element element product vector dimension positive elements matrices elements model referred certain weak assumptions qmle parameters model satisfies also hold see francq francq depends equation defining asymptotic null distribution garch model quantities well estimator employed therefore asymptotic null distribution used approximate null distribution following klar estimate null distribution using following parametric bootstrap algorithm calculate residuals test statistic generate vectors distribution let iii calculate residuals approximate null distribution means conditional distribution given data practice approximation step iii carried generating large number bootstrap replications test statistic whose empirical distribution similar steps given function used estimate null distribution proof theorem show one assumes continue hold replaced given data converges law kwg conditional distribution defined theorem therefore bootstrap procedure provides consistent null distribution estimator remark practical application bootstrap null distribution estimator entails parameter estimator residuals must calculated bootstrap resample results procedure following approaches ghoudi tests univariate garch models could use weighted bootstrap null distribution estimator sense burke computational point view provides efficient estimator nevertheless verified consistency weighted bootstrap null distribution estimator requires existence moment generating function true distribution generating innovations rather strong condition specially taking account alternatives interest case next result shows test multivariate normality based consistent general alternatives theorem let strictly stationary process satisfying measurable respect generated assume hold absolutely continuous distribution exp lim inf probability similar comments made theorem case done setting proof theorem proof theorem henze follows wng exp assumption exp exp exp prove result apply theorem billingsley converges law showing positive banach space continuous functions ktk endowed supremum norm positive proof applying central limit theorem martingale differences converge distributions hence prove must show tight aim write exp mean value theorem gives exp exp positive theorem billingsley process tight central limit theorem martingale differences converges law zero mean normal random vector hence product continuous function term tight property holds proof view positive constant holds likewise holds completes proof proof theorem let notice let exp exp exp ktk show result prove result follow using proof case proof let ajk ajk observe exp exp yield large enough inequality gives cng exp exp ktk exp max strong law large numbers banach space continuous functions sup sup sup positive constant ergodic theorem surely positive constant using stationarity finite second moments follows surely hence yields probability concludes proof proof reasoning follows similar steps proof fact proof theorem henze thus omitted monte carlo results section describes summarizes results extensive simulation experiment carried study performance proposed tests moreover consider real data set monthly log returns computations performed using programs written language numerical experiments data upper quantiles null distribution approximated generating samples law table displays critical values convention entry like stands results show large sample sizes required approximate critical values corresponding asymptotic values natural competitor test based test studied henze wagner latter procedure simple compute well affine invariant revealed good power performance regard competitors behaviour test based relation depends whether distribution tried number distributions specifically multivariate laplace distribution finite mixtures normal distributions distribution multivariate khintchine distribution uniform distribution pearson type family distributions observed power proposed test either similar smaller distributions new test outperforms observation appreciated looking table displays empirical power calculated generating samples case significance level following alternatives distribution multivariate student degrees freedom fact also observed zghoul numerically studied test based univariate data simulations tried large number values proposed test well tables display results values giving highest power cases considered said simulations table critical points table percentage rejection nominal level test based next subsection numerical experiments garch data simulations considered bivariate model trivariate model parameters models estimated qmle using package ccgarch language distribution innovations took distribution multivariate normal distribution order study level resulting bootstrap test assess power considered following distributions multivariate distribution coincides normal distribution heavy tails goodman kotz asymmetric exponential power distribution aep whereby univariate aep distribution zhu parameters settings gave useful results practical applications errors garch type models previous subsection also calculated table reports percentages rejections nominal significance level sample size resulting pictures quite similar save space omit results values order reduce computational burden adopted method giacomini works follows rather computing critical points monte carlo sample one resample generated monte carlo sample resampling test statistic computed sample resampling critical values computed empirical distribution determined resampling replications simulations generated monte carlo samples level power looking table conclude actual level proposed bootstrap test close nominal level also true although best knowledge consistency bootstrap null distribution estimator statistic proved univariate case respect power proposed test cases outperforms real data set application illustration consider monthly log returns ibm stock index january december observations data set analyzed example tsay showed cccgarch model provides adequate description data available website http author applied proposed test test testing pvalues obtained generating bootstrap samples values table percentage rejections nominal level aep test based table get leads reject expected looking figure displays scatter plot residuals fitting cccgarch model log returns figure represents histograms marginal residuals probability density function standard normal law superimposed conclusions studied class affine invariant tests multivariate normality setting context testing innovation distribution multivariate garch model gaussian thus generalizing results henze koch two ways test statistics suitably weighted based difference empirical moment generating function scaled residuals data moment generating function standard normal distribution considered moment generating function analogues returns ibm returns figure scatter plot residuals ibm returns returns figure histograms residuals class bhep tests use empirical characteristic function decay weight function figuring test statistic tends infinity test statistic approaches certain linear combination two measures multivariate skewness tests easy implement turn consistent wide range alternatives contrast recently studied statistic henze uses empirical moment generating empirical characteristic function test also feasible larger sample sizes since computational complexity order regarding power new tests outperform distributions acknowledgements partially supported grant spanish ministry economy competitiveness references arcones two tests multivariate normality based characteristic function math methods bardet wintenberger asymptotic normality likelihood estimator multidimensional causal processes ann baringhaus henze consistent test multivariate normality based empirical characteristic function metrika baringhaus ebner henze limit distribution weighted fit statistics fixed alternatives applications ann inst stat batsidis martin pardo zografos necessary power divergence type family tests multivariate normality comm statist simul billingsley convergence probability measures john wiley sons new bollerslev modelling coherence nominal exchange rates multivariate generalized arch model rev econ bosq linear processes function spaces springer new york burke multivariate uniform confidence bands using weighted bootstrap statist probab cardoso oliveira ferreira multivariate extension univariate normality test statist comput comte lieberman asymptotic theory multivariate garch processes multiv consistency tests multivariate normality metrika eaton perlman generalized sample covariance matrices ann ebner asymptotic theory tests multivariate normality cox small multiv enomoto okamoto seo multivariate normality test using srivastava skewness kurtosis sut farrel naczk tests multivariate normality associated simulation studies statist comput francq meintanis tests sphericity multivariate garch models econometrics francq garch models structure statistical inference applications wiley london francq qml estimation class multivariate asymmetric garch models econometric theory ghoudi comparison specification tests garch models computat statist data giacomini politis white method conducting monte carlo experiments involving bootstrap estimators econometric theory goodman kotz multivariate normal distributions multiv hanusz note srivastava hui test multivariate normality multiv hanusz new test multivariate normality based small srivastava graphical methods statist comput henze extreme smoothing testing multivariate normality statist probab henze invariant tests multivariate normality critical review statist papers henze meintanis characterizations multinormality corresponding tests fit including garch models henze koch test normality based empirical moment generating function statist papers henze wagner new approach bhep tests multivariate normality multiv henze zirkler class invariant consistent tests multivariate normality comm statist theory methods jeantheau strong consistency estimators multivariate arch models econometric theory empirical characteristic function process residuals garch models applications test empirical characteristic function tests garch innovation distribution using multipliers stat comput joenssen vogel power study tests multivariate normality implemented stat comput robust test multivariate normality econom kim robustified test multivariate normality econom klar lindner meintanis specification tests error distribution garch models comput statist data koizumi hyodo pavlenko modified tests multivariate normality framework statist mardia measures multivariate skewness kurtosis applications biometrika mecklin mundfrom monte carlo comparison type type error rates tests multivariate normality statist comput rohatgi multivariate skewness kurtosis theory probab appl pudelko new affine invariant consistent test multivariate normality probab math rydberg realistic statistical modelling financial data int stat shorack wellner empirical processes applications statistics wiley series probability mathematical statistics john wiley sons new york spierdijk confidence intervals case heavy tails skewness computat statist data rizzo new test multivariate normality multiv tenreiro affine invariant multiple test procedure assessing multivariate normality comput statist data tenreiro new test multivariate normality combining extreme nonextreme bhep tests commun statist thulin tests multivariate normality based canonical correlations stat meth tsay analysis financial time series wiley hoboken new jersey estrada generalization test multivariate normality commun statist voinov pya makarov voinov new invariant consistent chisquared type tests multivariate normality related comparative simulation study commun statist meth yanada romer richards kurtosis tests multivariate normality monotone incomplete data test zghoul test normality based empirical moment generating function comm statist simul zhou shao powerful test multivariate normality appl zhu properties estimation asymmetric exponential power distribution econometrics
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